International Association of Geodesy Symposia Michael G . Sideris, Series Editor
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International Association of Geodesy Symposia Michael G . Sideris, Series Editor

International Association of Geodesy Symposia Michael G . Sideris, Series Editor

Symposium 101: Global and Regional Geodynamics Symposium 102: Global Positioning System: An Overview Symposium 103: Gravity, Gradiometry, and Gravimetry Symposium 104: Sea SurfaceTopography and the Geoid Symposium 105: Earth Rotation and Coordinate Reference Frames Symposium 106: Determination of the Geoid: Present and Future Symposium 107: Kinematic Systems in Geodesy, Surveying, and Remote Sensing Symposium 108: Application of Geodesy to Engineering Symposium 109: Permanent Satellite Tracking Networks for Geodesy and Geodynamics Symposium 110: From Mars to Greenland: Charting Gravity with Space and Airborne Instruments Symposium 111: Recent Geodetic and Gravimetric Research in Latin America Symposium 112: Geodesy and Physics of the Earth: Geodetic Contributions to Geodynamics Symposium 113: Gravity and Geoid Symposium 114: Geodetic Theory Today Symposium 115: GPS Trends in Precise Terrestrial, Airborne, and Spaceborne Applications Symposium 116: Global Gravity Field and Its Temporal Variations Symposium 117: Gravity, Geoid and Marine Geodesy Symposium 118: Advances in Positioning and Reference Frames Symposium 119: Geodesy on the Move Symposium 120: Towards an Integrated Global Geodetic Observation System (IGGOS) Symposium 121: Geodesy Beyond 2000: The Challenges of the First Decade Symposium 122: IV Hotine-Marussi Symposium on Mathematical Geodesy Symposium 123: Gravity, Geoid and Geodynamics 2000 Symposium 124: Vertical Reference Systems Symposium 125: Vistas for Geodesy in the New Millennium Symposium 126: Satellite Altimetry for Geodesy, Geophysics and Oceanography Symposium 127: V Hotine Marussi Symposium on Mathematical Geodesy Symposium 128: A Window on the Future of Geodesy Symposium 129: Gravity, Geoid and Space Missions Symposium 130: Dynamic Planet - Monitoring and Understanding … Symposium 131: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles Symposium 132: VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy Symposium 133: Observing our Changing Earth Symposium 134: Geodetic Reference Frames Symposium 135: Gravity, Geoid and Earth Observation

Geodesy for Planet Earth Proceedings of the 2009 IAG Symposium, Buenos Aires, Argentina, 31 August - 4 September 2009

Edited by Steve C. Kenyon Maria C. Pacino Urs J. Marti

Editors Steve Kenyon National Geospatial-Intelligence Agency SN L-41 Vogel Rd. 3838 63010-6238 Arnold Montana USA Urs Marti Federal Office of Topography swisstopo Geodesy Department Seftigenstrasse 264 3084 Wabern Switzerland

Maria Cristina Pacino Universidad Nacional de Rosario (UNR) Facultad de Ciencia Exactas Ingenierı´a y Agrimensura (FCEIA) Av. Pellegrini 250 2000 Rosario Argentina

ISBN 978-3-642-20337-4 e-ISBN 978-3-642-20338-1 DOI 10.1007/978-3-642-20338-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011941192 # Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The International Association of Geodesy IAG2009 “Geodesy for Planet Earth” Scientific Assembly was held 31 August to 4 September 2009 in Buenos Aires, Argentina. The theme “Geodesy for Planet Earth” was selected to follow the International Year of Planet Earth 2007–2009 goals of utilizing the knowledge of the world’s geoscientists to improve society for current and future generations. The International Year started in January 2007 and ran thru 2009 which coincided with the IAG2009 Scientific Assembly, one of the largest and most significant meetings of the Geodesy community held every 4 years. The IAG2009 Scientific Assembly was organized into eight Sessions with SubSessions in five of them. Four of the Sessions of IAG2009 were based on the IAG Structure (i.e. one per Commission) and covered Reference Frames, Gravity Field, Earth Rotation and Geodynamics, and Positioning and Applications. Since IAG2009 was taking place in the great Argentine city of Buenos Aires, a Session was devoted to the Geodesy of Latin America. A Session dedicated to the IAG’s Global Geodetic Observing System (GGOS), the primary observing system focused on the multidisciplinary research being done in Geodesy that contributes to important societal issues such as monitoring global climate change and the environment. A Session on the IAG Services was also part of the Assembly detailing the important role they play in providing geodetic data, products, and analysis to the scientific community. A final Session devoted to the organizations ION, FIG, and ISPRS and their significant work in navigation and earth observation that complements the IAG. This volume contains the proceedings of the eight Sessions which are listed below: Session 1: Reference frames implementation for geosciences’ applications: From local to global scales Convenors: Zuheir Altamimi, Claudio Brunini Session 2: Gravity of the Planet Earth Convenors: Yoichi Fukuda, Pieter Visser Session – 2.1: Physics and Geometry of Earth: Focus on satellite altimetry and InSAR Convenors: Cheinway Hwang, Jose´ Luis Vacaflor Session – 2.2: Gravity – An Earth Probing Tool: Focus on CHAMP/GRACE/ GOCE missions, relative/absolute/superconducting gravimetry, and their applications Convenors: Roland Pail, Leonid F. Vitushkin

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vi

Session – 2.3: Modern Height Datum: Focus on definition and realization of GPS-levelling and gravity-base height datum Convenors: Sı´lvio R.C. de Freitas, Dru A. Smith Session – 2.4: Gravity and Geoid Modelling: Focus on global and regional gravity and geoid modelling Convenors: Urs Marti, Yan Ming Wang Session 3: Geodesy and Geodynamics: Global and Regional Scales Convenors: Mike Bevis, Sylvain Bonvalot Session – 3.1: Rotation of the Planet Earth Convenors: Richard Gross, Rodrigo, Abarca del Rio Session – 3.2: Sea level changes and post-glacial rebound Convenors: Juan Fierro, Michael Bevis Session – 3.3: Ocean loading and global water distribution / geophysical fluids Convenors: Tonie van Dam, Richard Gross Session – 3.4: Geodesy, crustal motions and geodynamic processes Convenors: Juan Carlos Baez, Sylvain Bonvalot, Arturo Echalar Session – 3.5: Geodesy and the near-field solid earth response to cryospheric mass changes Convenors: Jim Davis, Gino Casassa Session 4: Positioning and remote sensing of land, ocean and atmosphere Convenors: Sandra Verhagen, Pawel Wielgosz Session – 4.1: Technology and land applications Convenors: Dorota Grejner-Brzezinska, Xiaoli Ding Session – 4.2: Modelling and remote sensing of the atmosphere Convenors: Marcelo Santos, Cathryn Mitchell, Jens Wickert Session – 4.3: Multi-satellite ocean remote sensing Convenors: Shuanggen Jin, Ole Baltazar Andersen Session 5: Geodesy in Latin America Convenors: Denizar Blitzkow, Claudia Tocho Session 6: JOINT ION/FIG/ISPRS session on Navigation and Earth Observation Convenors: Dorota Grejner-Brzezinska, Charles Toth Session – 6.1: Navigation (FIG, ION) Convener: Dorota Grejner-Brzezinska Session – 6.2: Earth Observation (ISPRS)C onvener: Charles Toth Session 7: The Global Geodetic Observing System: Science and Applications Convenors: Richard Gross, Hans-Peter Plag, Luiz Paulo Fortes Session – 7.1: Past Progress and Future Plans Convenors: Hans-Peter Plag, Richard Gross, Luiz Paulo Fortes Session – 7.2: Science and Applications Convenors: Richard Gross, Hans-Peter Plag, Luiz Paulo Fortes Session 8: The IAG International Services and their role for Earth observation Convenors: Ruth Neilan, Rene Forsberg The number and quality of contributions for the eight Sessions clearly demonstrated the important and vital role that Geodesy plays in understanding the earth and its dynamical processes. Satellite, airborne, and terrestrial systems and networks are

Preface

Preface

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continually measuring and analyzing the earth for global change. For this reason the name of these proceedings is “Geodesy for Planet Earth” which reflects the role fourdimensional geodesy plays in understanding the changes to Planet Earth. The 2009 Assembly attracted nearly 500 oral and poster presentations from 370 geodesists from 45 countries and clearly shows the interest and importance of geodesy globally. The approximately 130 papers that are included in these proceedings (about 25% of the total) are intended to cover much of the latest research and projects on-going in the field. These proceedings would not be possible without the tremendous work by the Convenors of each of the Sessions and Sub-Sessions. They devoted a enormous amount of time and energy in organizing the reviews and final acceptance of the papers for their Sessions. We are very grateful to IAG Secretary General Hermann Drewes and IAG President Michael Sideris for all their guidance and help with these proceedings. The Local Organizing Committee in Buenos Aires was invaluable in helping arrange a very memorable Assembly and provided essential support in the development of these proceedings. And lastly, sincere thanks go out to all the participating scientists and graduate students who made the IAG 2009 “Geodesy for Planet Earth” Scientific Assembly and these proceedings a tremendous success. Steve Kenyon Maria Cristina Pacino Urs Marti

.

Contents

Session 1

Reference Frames Implementation for Geoscience’s Applications: From Local to Global Scales Convenors: Z. Altamimi, C. Brunini 1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 C. Brunini, L. Sanchez, H. Drewes, S. Costa, V. Mackern, W. Martı´nez, W. Seemuller, and A. da Silva

2

Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alvaro Santamarı´a-Go´mez, Marie-Noe¨lle Bouin, and Guy Wo¨ppelmann

3

4

5

6

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bruyninx, Z. Altamimi, M. Becker, M. Craymer, L. Combrinck, A. Combrink, J. Dawson, R. Dietrich, R. Fernandes, R. Govind, T. Herring, A. Kenyeres, R. King, C. Kreemer, D. Lavalle´e, J. Legrand, L. Sa´nchez, G. Sella, Z. Shen, A. Santamarı´a-Go´mez, and G. Wo¨ppelmann Enhancement of the EUREF Permanent Network Services and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bruyninx, H. Habrich, W. So¨hne, A. Kenyeres, G. Stangl, and C. Vo¨lksen

11

19

27

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ulla Kallio and Markku Poutanen

35

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008, the ignwd08 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Willis, M.L. Gobinddass, B. Garayt, and H. Fagard

43

7

Towards a Combination of Space-Geodetic Measurements . . . . . . . . . . . A. Pollet, D. Coulot, and N. Capitaine

8

Improving Length and Scale Traceability in Local Geodynamical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Jokela, P. H€akli, M. Poutanen, U. Kallio, and J. Ahola

51

59

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Contents

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10

11

12

How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Drewes Transforming ITRF Coordinates to National ETRS89 Realization in the Presence of Postglacial Rebound: An Evaluation of the Nordic Geodynamical Model in Finland . . . . . . . P. H€akli and H. Koivula Global Terrestrial Reference Frame Realization Within the GGOS-D Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Angermann, H. Drewes, and M. Seitz Comparison of Regional and Global GNSS Positions, Velocities and Residual Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Legrand, N. Bergeot, C. Bruyninx, G. Wo¨ppelmann, A. Santamarı´a-Go´mez, M.-N. Bouin, and Z. Altamimi

67

77

87

95

13

GPS Metrology: Bringing Traceable Scale to a Local Crustal Deformation GPS Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 H. Koivula, P. H€akli, J. Jokela, A. Buga, and R. Putrimas

14

Impact of Albedo Radiation on GPS Satellites . . . . . . . . . . . . . . . . . . . . . . . . 113 C.J. Rodriguez-Solano, U. Hugentobler, and P. Steigenberger

Session 2 Gravity of the Planet Earth Convenors: Y. Fukuda, P. Visser 15

On the Determination of Sea Level Changes by Combining Altimetric, Tide Gauge, Satellite Gravity and Atmospheric Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 G.S. Vergos, I.N. Tziavos, and M.G. Sideris

16

Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite and Submarine Measurements . . . . . . . . . . . . 131 J. Calvao, J. Rodrigues, and P. Wadhams

17

The Impact of Attitude Control on GRACE Accelerometry and Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 U. Meyer, A. J€aggi, and G. Beutler

18

Using Atmospheric Uncertainties for GRACE De-aliasing: First Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 L. Zenner, T. Gruber, G. Beutler, A. J€aggi, F. Flechtner, T. Schmidt, J. Wickert, E. Fagiolini, G. Schwarz, and T. Trautmann

19

Challenges in Deriving Trends from GRACE . . . . . . . . . . . . . . . . . . . . . . . . . 153 A. Eicker, T. Mayer-Guerr, and E. Kurtenbach

20

AIUB-GRACE02S: Status of GRACE Gravity Field Recovery Using the Celestial Mechanics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A. J€aggi, G. Beutler, U. Meyer, L. Prange, R. Dach, and L. Mervart

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21

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.C. Gunter, T. Wittwer, W. Stolk, R. Klees, and P. Ditmar

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination of LEO Orbit Based on GNSS Observations . . . . . . . . . 179 A. Shabanloui and K.H. Ilk

23

Pure Geometrical Precise Orbit Determination of a LEO Based on GNSS Carrier Phase Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A. Shabanloui and K.H. Ilk

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field: Boundary Problems and a Target Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 P. Holota and O. Nesvadba

25

Moho Estimation Using GOCE Data: A Numerical Simulation . . . . . 205 M. Reguzzoni and D. Sampietro

26

CHAMP, GRACE, GOCE Instruments and Beyond . . . . . . . . . . . . . . . . . 215 P. Touboul, B. Foulon, B. Christophe, and J.P. Marque

27

The Future of the Satellite Gravimetry After the GOCE Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 P. Silvestrin, M. Aguirre, L. Massotti, B. Leone, S. Cesare, M. Kern, and R. Haagmans

28

Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 R. Mayrhofer and R. Pail

29

Local and Regional Comparisons of Gravity and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C. Jekeli, O. Huang, and T.L. Abt

30

Combination of Local Gravimetry and Magnetic Data to Locate Subsurface Anomalies Using a Matched Filter . . . . . . . . . . . . 247 T. Abt, O. Huang, and C. Jekeli

31

On the Use of UAVs for Strapdown Airborne Gravimetry . . . . . . . . . . 255 Richard Deurloo, Luisa Bastos, and Machiel Bos

32

Updating the Precise Gravity Network at the BIPM . . . . . . . . . . . . . . . . . 263 Z. Jiang, E.F. Arias, L. Tisserand, K.U. Kessler-Schulz, H.R. Schulz, V. Palinkas, C. Rothleitner, O. Francis, and M. Becker

33

Precise Gravimetric Surveys with the Field Absolute Gravimeter A-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 R. Falk, Ja. M€ uller, N. Lux, H. Wilmes, and H. Wziontek

34

Reconstruction of a Torsion Balance and the Results of the Test Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 L. Vo¨lgyesi and Z. Ultmann

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Contents

35

The Superconducting Gravimeter as a Field Instrument Applied to Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 C.R. Wilson, H. Wu, L. Longuevergne, B. Scanlon, and J. Sharp

36

Local Hydrological Information in Gravity Time Series: Application and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 M. Naujoks, S. Eisner, C. Kroner, A. Weise, P. Krause, and T. Jahr

37

Signals of Mass Redistribution at the South African Gravimeter Site SAGOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 C. Kroner, S. Werth, H. Pflug, A. G€ untner, B. Creutzfeldt, M. Thomas, H. Dobslaw, P. Fourie, and P.H. Charles

38

Gravity System and Network in Estonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 To˜nis Oja

39

Evaluation of EGM2008 Within Geopotential Space from GPS, Tide Gauges and Altimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 N. Dayoub, P. Moore, N.T. Penna, and S.J. Edwards

40

Fixed Gravimetric BVP for the Vertical Datum Problem . . . . . . . . . . . . 333 ˇ underlı´k, Z. Fasˇkova´, and K. Mikula R. C

41

Realization of the World Height System in New Zealand: Preliminary Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 R. Tenzer, V. Vatrt, and M. Amos

42

Comparisons of Global Geopotential Models with Terrestrial Gravity Field Data Over Santiago del Estero Region, NW: Argentine . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 L. Galva´n, C. Infante, E. Laurı´a, and R. Ramos

43

Intermap’s Airborne Inertial Gravimetry System . . . . . . . . . . . . . . . . . . . . 357 Ming Wei

44

Galathea-3: A Global Marine Gravity Profile . . . . . . . . . . . . . . . . . . . . . . . . . 365 G. Strykowski, K.S. Cordua, R. Forsberg, A.V. Olesen, and O.B. Andersen

45

Dependency of Resolvable Gravitational Spatial Resolution on Space-Borne Observation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 P.N.A.M. Visser, E.J.O. Schrama, N. Sneeuw, and M. Weigelt

46

A Comparison of Different Integral-Equation-Based Approaches for Local Gravity Field Modelling: Case Study for the Canadian Rocky Mountains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 R. Tenzer, I. Prutkin, and R. Klees

47

Global Topographically Corrected and Topo-Density Contrast Stripped Gravity Field from EGM08 and CRUST 2.0 . . . . . . . . . . . . . . . 389 R. Tenzer, Hamayun, and P. Vajda

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48

Local Gravity Field Modelling in Rugged Terrain Using Spherical Radial Basis Functions: Case Study for the Canadian Rocky Mountains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 R. Tenzer, R. Klees, and T. Wittwer

49

A Sensitivity Analysis in Spectral Gravity Field Modeling Using Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Vassilios D. Andritsanos and Ilias N. Tziavos

50

Investigation of Topographic Reductions for Marine Geoid Determination in the Presence of an Ultra-High Resolution Reference Geopotential Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 C. Tocho, G.S. Vergos, and M.G. Sideris

51

Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Robert Kingdon, Petr Vanı´cˇek, and Marcelo Santos

52

Evaluation of Gravity and Altimetry Data in Australian Coastal Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 S.J. Claessens

53

Development and User Testing of a Python Interface to the GRAVSOFT Gravity Field Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 443 J. Nielsen, C.C. Tscherning, T.R.N. Jansson, and R. Forsberg

54

Progress and Prospects of the Antarctic Geoid Project (Commission Project 2.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Mirko Scheinert

55

Regional Geoid Improvement over the Antarctic Peninsula Utilizing Airborne Gravity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 J. Schwabe, M. Scheinert, R. Dietrich, F. Ferraccioli, and T. Jordan

56

Auvergne Dataset: Testing Several Geoid Computation Methods . . . 465 P. Valty, H. Duquenne, and I. Panet

57

In Pursuit of a cm-Accurate Local Geoid Model for Ohio . . . . . . . . . . . 473 K.R. Edwards, Dorota Grejner-Brzezinska, and Dru Smith

58

Adjustment of Collocated GPS, Geoid and Orthometric Height Observations in Greece. Geoid or Orthometric Height Improvement? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 I.N. Tziavos, G.S. Vergos, V.N. Grigoriadis, and V.D. Andritsanos

Session 3 Geodesy and Geodynamics: Global and Regional Scales Convenors: M. Bevis, S. Bonvalot 59

Regional Geophysical Excitation Functions of Polar Motion over Land Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 J. Nastula and D.A. Salstein

60

Geophysical Excitation of the Chandler Wobble Revisited . . . . . . . . . . 499 Aleksander Brzezin´ski, Henryk Dobslaw, Robert Dill, and Maik Thomas

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On the Origin of the Bi-Decadal and the Semi-Secular Oscillations in the Length of the Day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 S. Duhau and C. de Jager

62

Future Improvements in EOP Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 W. Kosek

63

Determination of Nutation Coefficients from Lunar Laser Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 L. Biskupek, J. M€ uller, and F. Hofmann

64

A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 W. Keller, M. Kuhn, and W.E. Featherstone

65

Stabilization of Satellite Derived Gravity Field Coefficients by Earth Orientation Parameters and Excitation Functions . . . . . . . . . 537 Andrea Heiker, Hansjo¨rg Kutterer, and J€ urgen M€uller

66

The Statistical Characteristics of Altimetric Sea Level Anomaly Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 T. Niedzielski and W. Kosek

67

Testing Past Sea Level Reconstruction Methodology (1958–2006) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 J. Viarre and R. Abarca-del-Rı´o

68

Precise Determination of Relative Mean Sea Level Trends at Tide Gauges in the Adriatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 M. Repanic´ and T. Basˇic´

69

Quantile Analysis of Relative Sea-Level at the Hornbæk and Gedser Tide Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 S.M. Barbosa and K.S. Madsen

70

Assessment of the FES2004 Derived OTL Model in the West of France and Preliminary Results About Impacts of Tropospheric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 F. Fund, L. Morel, and A. Mocquet

71

Gravimetric Time Series Recording at the Argentine Antarctic Stations Belgrano II and San Martı´n for the Improvement of Ocean Tide Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 Mirko Scheinert, Andre´s F. Zakrajsek, Lutz Eberlein, Reinhard Dietrich, Sergio A. Marenssi, and Marta E. Ghidella

72

Mass-Change Acceleration in Antarctica from GRACE Monthly Gravity Field Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Lo´ra´nt Fo¨ldva´ry

73

Mass Variations in the Siberian Permafrost Region from GRACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 Holger Steffen, J€ urgen M€ uller, and Nadja Peterseim

Contents

xv

74

Seasonal Variability of Land Water Storage in South America Using GRACE Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Claudia Tocho, Luis Guarracino, Leonardo Monachesi, Andre´s Cesanelli, and Pablo Antico

75

Water Storage Changes from GRACE Data in the La Plata Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 A. Pereira, S. Miranda, M.C. Pacino, and R. Forsberg

76

Second and Third Order Ionospheric Effects on GNSS Positioning: A Case Study in Brazil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 H.A. Marques, J.F.G. Monico, G.P.S. Rosa, M.L. Chuerubim, and Ma´rcio Aquino

77

Advanced Techniques for Discontinuity Detection in GNSS Coordinate Time-Series. An Italian Case Study . . . . . . . . . . . . 627 A. Borghi, L. Cannizzaro, and A. Vitti

78

Traditional and Alternative Network Adjustment Approach for the TAMDEF GPS in Antarctica . . . . . . . . . . . . . . . . . . . . . . . 635 G. Esteban Va´zquez, Dorota A. Grejner-Brzezinska, and Burkhard Schaffrin

79

Impact of Loading Phenomena on Velocity Field Computation from GPS Campaigns: Application to ResPyr GPS Campaign in the Pyrenees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 J. Nicolas, F. Perosanz, A. Rigo, G. Bliguet, L. Morel, and F. Fund

80

Comparison of the Coordinates Solutions Between the Absolute and the Relative Phase Center Variation Models in the Dense Regional GPS Network in Japan . . . . . . . . . . . . . . . 651 S. Shimada

81

The 2009 Horizontal Velocity Field for South America and the Caribbean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 H. Drewes and O. Heidbach

82

New Estimates of Present-Day Crustal/Land Motions in the British Isles Based on the BIGF Network . . . . . . . . . . . . . . . . . . . . . . 665 D.N. Hansen, F.N. Teferle, R.M. Bingley, and S.D.P. Williams

83

GURN (GNSS Upper Rhine Graben Network): Research Goals and First Results of a Transnational Geo-scientific Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 M. Mayer, A. Kno¨pfler, B. Heck, F. Masson, P. Ulrich, and G. Ferhat

84

Determination of Horizontal and Vertical Movements of the Adriatic Microplate on the Basis of GPS Measurements . . . . . . 683 M. Marjanovic´, Zˇ. Bacˇic´, and T. Basˇic´

85

Determination of Tectonic Movements in the Swiss Alps Using GNSS and Levelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 E. Brockmann, D. Ineichen, U. Marti, S. Schaer, A. Schlatter, and A. Villiger

xvi

Contents

86

A Compilation of a Preliminary Map of Vertical Deformations in New Zealand from Continuous GPS Data . . . . . . . . . . . . . . . . . . . . . . . . . . 697 R. Tenzer, M. Stevenson, and P. Denys

87

Detection of Vertical Temporal Behaviour of IGS Stations in Canada Using Least Squares Spectral Analysis . . . . . . . . . . . . . . . . . . . . 705 James Mtamakaya, Marcelo C. Santos, and Michael Craymer

Session 4 Positioning and Remote Sensing of Land, Ocean and Atmosphere Convenors: S. Verhagen, P. Wielgosz 88

Positioning and Applications for Planet Earth . . . . . . . . . . . . . . . . . . . . . . . . 713 S. Verhagen, G. Retscher, M.C. Santos, X.L. Ding, Y. Gao, and S.G. Jin

89

Report of Sub-commission 4.2 “Applications of Geodesy in Engineering” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 G. Retscher, A. Reiterer, and G. Mentes

90

A Fixed-s Digital Representation of a Random Scalar Field . . . . . . . . 725 K. Becek

91

The Impact of Adding SBAS Data on GPS Data Processing in Southeast of Brazil: Preliminary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 W.C. Machado, F. Albarici, E.S. Fonseca Junior, J.F.G. Monico, and W.G.C. Polezel

92

First Results of Relative Field Calibration of a GPS Antenna at BCAL/UFPR (Baseline Calibration Station for GNSS Antennas at UFPR/Brazil) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 S.C.M. Huinca, C.P. Krueger, M. Mayer, A. Kno¨pfler, and B. Heck

93

Medium-Distance GPS Ambiguity Resolution with Controlled Failure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 D. Odijk, S. Verhagen, and P.J.G. Teunissen

94

Toward a SIRGAS Service for Mapping the Ionosphere’s Electron Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 ´ ngel, C. Brunini, F. Azpilicueta, M. Gende, A. Arago´n-A M. Herna´ndez-Pajares, J.M. Juan, and J. Sanz

95

Assisted Code Point Positioning at Sub-meter Accuracy Level with Ionospheric Corrections Estimated in a Local GNSS Permanent Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 M. Crespi, A. Mazzoni, and C. Brunini

96

Semi-annual Anomaly and Annual Asymmetry on TOPEX TEC During a Full Solar Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 F. Azpilicueta, C. Brunini, and S.M. Radicella

97

Numerical Simulation and Prediction of Atmospheric Aerosol Extinction Using Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 775 J. Shin, S. Lim, C. Rizos, and K. Zhang

Contents

xvii

98

Impact of Atmospheric Delay Reduction Using KARAT on GPS/PPP Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 Ryuichi Ichikawa, Thomas Hobiger, Yasuhiro Koyama, and Tetsuro Kondo

99

Modelling Tropospheric Zenith Delays Using Regression Models Based on Surface Meteorology Data . . . . . . . . . . . . . . . . . . . . . . . . . . 789 Tama´s Tuchband and Szabolcs Ro´zsa

100

Calibration of Wet Tropospheric Delays in GPS Observation Using Raman Lidar Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 P. Bosser, C. Thom, O. Bock, J. Pelon, and P. Willis

101

Generation of Slant Tropospheric Delay Time Series Based on Turbulence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 M. Vennebusch and S. Scho¨n

102

Fitting of NWM Ray-Traced Slant Factors to Closed-Form Tropospheric Mapping Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 Landon Urquhart, Marcelo Santos, and Felipe Nievinski

103

Estimation of Integrated Water Vapour from GPS Observations Using Local Models in Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 Sz. Ro´zsa

104

GNSS Remote Sensing in the Atmosphere, Oceans, Land and Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Shuanggen Jin

105

Mean Sea Surface Model of the Caspian Sea Based on TOPEX/Poseidon and Jason-1 Satellite Altimetry Data . . . . . . . . . 833 S.A. Lebedev

Session 5 Geodesy in Latin America Convenors: D. Blitzkow, C. Tocho 106

Combination of the Weekly Solutions Delivered by the SIRGAS Processing Centres for the SIRGAS-CON Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845 L. Sa´nchez, W. Seem€uller, and M. Seitz

107

Report on the SIRGAS-CON Combined Solution, by IBGE Analysis Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853 S.M.A. Costa, A.L. Silva, and J.A. Vaz

108

Processing Evaluation of SIRGAS-CON Network by IBGE Analysis Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 S.M.A. Costa, A.L. Silva, and J.A. Vaz

109

ProGriD: The Transformation Package for the Adoption of SIRGAS2000 in Brazil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 Marcos F. Santos, Marcelo C. Santos, Leonardo C. Oliveira, Sonia A. Costa, Joa˜o B. Azevedo, and Maurı´cio Galo

xviii

Contents

110

The New Multi-year Position and Velocity Solution SIR09P01 of the IGS Regional Network Associate Analysis Centre (IGS RNAAC SIR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 W. Seem€ uller, M. Seitz, L. Sa´nchez, and H. Drewes

111

Analysis of the Crust Displacement in Amazon Basin . . . . . . . . . . . . . . . 885 G.N. Guimara˜es, D. Blitzkow, A.C.O.C. de Matos, F.G.V. Almeida, and A.C.B. Barbosa

112

The Progress of the Geoid Model for South America Under GRACE and EGM2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 D. Blitzkow, A.C.O.C. de Matos, J.D. Fairhead, M.C. Pacino, M.C.B. Lobianco, and I.O. Campos

113

Combining High Resolution Global Geopotential and Terrain Models to Increase National and Regional Geoid Determinations, Maracaibo Lake and Venezuelan Andes Case Study . . . . . . . . . . . . . . . . 901 E. Wildermann, G. Royero, L. Bacaicoa, V. Cioce, G. Acun˜a, H. Codallo, J. Leo´n, M. Barrios, and M. Hoyer

114

Evaluation of a Few Interpolation Techniques of Gravity Values in the Border Region of Brazil and Argentina . . . . . . . . . . . . . . . 909 R.A.D. Pereira, S.R.C. De Freitas, V.G. Ferreira, P.L. Faggion, D.P. dos Santos, R.T. Luz, A.R. Tierra Criollo, and D. Del Cogliano

115

RBMC in Real Time via NTRIP and Its Benefits in RTK and DGPS Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 S.M.A. Costa, M.A. de Almeida Lima, N.J. de Moura Jr, M.A. Abreu, A.L. de Silva, L.P. Souto Fortes, and A.M. Ramos

Session 6 Joint ION/FIG/ISPRS Session on Navigation and Earth Observation Convenors: D.A. Grejner-Brzezinska, C.K. Toth 116

Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations and Partial Integer Decorrelation in the Presence of Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 P. Henkel

117

Real Time Satellite Clocks in Precise Point Positioning . . . . . . . . . . . . . 935 R.J.P. van Bree, S. Verhagen, and A. Hauschild

118

Improving the GNSS Attitude Ambiguity Success Rate with the Multivariate Constrained LAMBDA Method . . . . . . . . . . . . . . 941 G. Giorgi, P.J.G. Teunissen, S. Verhagen, and P.J. Buist

119

An Intelligent Personal Navigator Integrating GNSS, RFID and INS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949 G. Retscher

120

Integration of Image-Based and Artificial Intelligence Algorithms: A Novel Approach to Personal Navigation . . . . . . . . . . . . . 957 Dorota A. Grejner-Brzezinska, Charles K. Toth, J. Nikki Markiel, Shahram Moafipoor, and Krystyna Czarnecka

Contents

xix

121

Modernization and New Services of the Brazilian Active Control Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 L.P.S. Fortes, S.M.A. Costa, M.A. Abreu, A.L. Silva, N.J.M Ju´nior, K. Barbosa, E. Gomes, J.G. Monico, M.C. Santos, and P. Te´treault

122

magicSBAS: A South-American SBAS Experiment with NTRIP Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973 I. Alcantarilla, J. Caro, A. Cezo´n, J. Ostolaza, and F. Azpilicueta

Session 7 The Global Geodetic Observing System: Science and Applications Convenors: R. Gross, H.-P. Plas, L.P. Forles 123

Scientific Rationale and Development of the Global Geodetic Observing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 G. Beutler and R. Rummel

124

GGOS Bureau for Standards and Conventions: Integrated Standards and Conventions for Geodesy . . . . . . . . . . . . . . . . 995 U. Hugentobler, T. Gruber, P. Steigenberger, D. Angermann, J. Bouman, M. Gerstl, and B. Richter

125

VLBI2010: Next Generation VLBI System for Geodesy and Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 W.T. Petrachenko, A.E. Niell, B.E. Corey, D. Behrend, H. Schuh, and J. Wresnik

126

The New Vienna VLBI Software VieVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 J. Bo¨hm, S. Bo¨hm, T. Nilsson, A. Pany, L. Plank, H. Spicakova, K. Teke, and H. Schuh

127

Estimating Horizontal Tropospheric Gradients in DORIS Data Processing: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 P. Willis, Y.E. Bar-Sever, and O. Bock

Session 8

The IAG International Services and their Role for Earth Observation Convenors: R. Neilan, R. Forsberg 128

The BIPM: International References for Earth Sciences . . . . . . . . . . 1023 E.F. Arias

129

Development of the GLONASS Ultra-Rapid Orbit Determination at Geodetic Observatory Pecny´ . . . . . . . . . . . . . . . . . . . . . 1029 J. Dousa

130

AGrav: An International Database for Absolute Gravity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 H. Wziontek, H. Wilmes, and S. Bonvalot

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043

.

Session 1 Reference Frames Implementation for Geoscience’s Applications: From Local to Global Scales Convenors: Z. Altamimi, C. Brunini

.

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame C. Brunini, L. Sanchez, H. Drewes, S. Costa, V. Mackern, W. Martı´nez, W. Seemuller, and A. da Silva

Abstract

The SIRGAS reference system is at present realized by the SIRGAS Continuously Operating Network (SIRGAS-CON) composed by about 200 stations distributed over Latin America and the Caribbean. SIRGAS member countries are improving their national reference frames by installing continuously operating GPS stations, which have to be consistently integrated into the continental network. As the number of these stations is rapidly increasing, the analysis strategy of the SIRGAS-CON network is based on two hierarchy levels: a) A core network with homogeneous continental coverage and stable site locations ensures the long-term stability of the reference frame. This network is processed by DGFI (Germany) as the IGS RNAAC SIR. b) Several densification sub-networks (corresponding to the national reference networks) improve the accessibility to the reference frame in the individual countries. Currently, the SIRGAS-CON stations are classified in three densification sub-networks (a southern, a middle, and a northern one), which are processed by the SIRGAS Local Processing Centres CIMA (Argentina), IBGE (Brazil), and IGAC (Colombia). These four Processing Centres deliver loosely constrained weekly solutions for the assigned sub-networks, which are integrated in a unified solution by the SIRGAS Combination Centres (DGFI and IBGE). The main SIRGAS products are: loosely constrained weekly solutions in SINEX format for further combinations of the

C. Brunini (*) Universidad Nacional de La Plata (UNLP), La Plata, Argentina e-mail: [email protected] L. Sanchez H. Drewes W. Seemuller Deutsches Geod€atisches Forschungsinstitut (DGFI), Munich, Germany S. Costa A. da Silva Instituto Brasileiro de Geografia e Esta´tistica (IBGE), Rio de Janeiro, Brazil V. Mackern Universidad Nacional de Cuyo, Mendoza (CIMA), Argentina W. Martı´nez Instituto Geogra´fico Agustı´n Codazzi (IGAC), Bogota, Colombia S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_1, # Springer-Verlag Berlin Heidelberg 2012

3

4

C. Brunini et al.

network, weekly positions aligned to the ITRF as reference for GPS positioning in Latin America; and multi-year solutions (positions + velocities) for practical and scientific applications requiring time-dependent coordinates. This paper describes the analysis of the SIRGAS-CON network as the current realization of the SIRGAS reference system, its quality and consistency, as well as the planned activities to continue improving this reference frame.

1.1

Introduction

SIRGAS (Sistema de Referencia Geoce´ntrico para las Ame´ricas) as a reference system is defined identical with the ITRS (International Terrestrial Reference System). It is realized by means of a regional densification of the global ITRF (International Terrestrial Reference Frame) in Latin America and the Caribbean. The SIRGAS reference frame is in the same way extended to the countries through national densifications, which provide accessibility to the reference frame at national and local levels (Sanchez and Brunini 2009). SIRGAS has three realizations: two by means of episodic GPS campaigns and one by means of a network of continuously operating GPS stations. The first realization of SIRGAS (SIRGAS95) refers to the ITRF94, epoch 1995.4. It is given by a high-precision GPS network of 58 points distributed over South America (SIRGAS, 1997). In 2000, this network was re-measured and extended to the Caribbean, Central and North American countries. This second realization (SIRGAS2000) includes 184 GPS stations and refers to the ITRF2000, epoch 2000.4 (Drewes et al. 2005). The third realization of SIRGAS is the SIRGAS Continuously Operating Network (SIRGAS-CON), which is at present composed by more than 200 permanently operating GPS sites. The SIRGAS-CON network is weekly computed by the SIRGAS Analysis Centres; main products of this computation are: loosely constrained weekly solutions for station positions to be included in the IGS (International GNSS Service) global polyhedron and in multi-year solutions of the network; weekly station positions aligned to the ITRF for further applications in Latin America; and multi-year solutions providing station positions and velocities for high-precise practical and scientific applications. The SIRGASCON weekly positions refer to the observation epoch and to the current frame in which the GPS satellite orbits (i.e. IGS final orbits, Dow et al. 2009) are

given, at present the IGS05, the IGS realization of the ITRF2005 (see IGSMAIL 5447, http://igscb.jpl.nasa. gov/). The coordinates of the multi-year solutions refer to the latest available ITRF and to a specified epoch, e.g. the most recent SIRGAS-CON multi-year solution SIR09P01 refers to IGS05, epoch 2005.0 (Seemuller et al. 2009). The relationship between the different SIRGAS realizations is given by the transformation parameters between the corresponding ITRF solutions they refer and by taking into account the station position variations with time through a velocity (deformation) model (Drewes and Heidbach 2005). In this way, realizations or densifications of SIRGAS associated to different ITRFs and reference epochs materialize the same reference system and, after reducing them to the same frame and epoch, their positions are compatible at the mm-level. The present paper summarizes the improvement of the SIRGAS realization by means of the SIRGASCON network, the applied procedures for generating weekly solutions of this reference frame, quality and consistency of the obtained coordinates, as well as ongoing activities to avoid some disadvantages of the actual analysis strategy. The multi-year solutions for the SIRGAS-CON reference frame are presented by e.g. Seemuller et al. 2008, 2009.

1.2

The SIRGAS-CON Reference Frame

The initial realizations of SIRGAS based on pillars have been replaced by an increasing number of continuously operating GPS stations (Fig. 1.1), which all together constitute the SIRGAS-CON network (Fig. 1.2). 48 of these stations belong to the IGS global network, while the others (about 160) correspond to the national reference frames.

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame

Fig. 1.1 SIRGAS continuously operating stations since 2000

5 IGS

No. of stations

Regional

Total

200 150 100 50 0 2000

2001

2002

2003

2004

2005

2006

2007

2008 06/2009

Year

To guarantee the consistency of the national reference frames with the global reference frame in which the GPS orbits are computed, the national reference stations are integrated into the SIRGAS-CON network and all together are processed in a common way. This provides homogeneous consistency and accuracy of their coordinates on a continental level. Until GPS week 1495 (August 2008), the Deutsches Geod€atisches Forschungsinstitut (DGFI, Germany), as the IGS RNAAC SIR (IGS Regional Network Associate Analysis Centre for SIRGAS), processed the entire SIRGAS-CON network in one block only (Seemuller and Drewes 2008). However, given the large number of SIRGAS-CON stations, this usual one-block processing became unfeasible and it was necessary to redefine the analysis strategy of the network. The new analysis strategy is based on (1) defining a core continental network (SIRGAS-CON-C) as the primary densification of the ITRF in Latin America, and (2) improving the geographical density of this core network by means of densification sub-networks (SIRGASCON-D). The core network ensures the long-term stability of the continental reference frame, while the densification sub-networks make it available at national and local levels. Although, they appear as two different categories, core and densification stations match requirements, characteristics, performance, and quality of the ITRF stations. The SIRGAS-CON-D sub-networks shall correspond to the national reference frames, i.e., as an optimum there shall be as many sub-networks as countries in the region. Since at present not all of the countries are operating a Processing Centre, the existing stations are classified in three densification sub-networks (Fig. 1.2): a northern one covering Mexico, Central America, the Caribbean, Colombia, and Venezuela; a

middle one comprising stations installed on Brazil, Ecuador, Bolivia, Suriname, French Guyana, Guyana, Peru, and Bolivia; and a southern one including the stations located in Uruguay, Paraguay, Argentina, Chile, and Antarctica. Each densification sub-network includes a minimum number of IGS and SIRGASCON core stations as overlapping points for the combination.

1.3

Analysis of the SIRGAS-CON Network

The SIRGAS-CON-C network is computed by DGFI. The densification sub-networks are processed by the active SIRGAS Local Processing Centres until new ones become operational. At present, they are: Centro de Procesamiento Ingenierı´a-Mendoza-Argentina at the Universidad Nacional del Cuyo (CIMA, Argentina), Instituto Brasileiro de Geografia e Estatistica (IBGE, Brazil), and Instituto Geogra´fico Agustı´n Codazzi (IGAC, Colombia). These four Processing Centres apply a common procedure established by SIRGAS (in agreement with the standards of the IGS and the IERS – International Earth Rotation and Reference Systems Service) to generate loosely constrained weekly solutions for station positions (see e.g. Natali et al. 2009; Seemuller and Sanchez 2009). In these solutions satellite orbits, satellite clock offsets, and Earth orientation parameters are fixed to the final weekly IGS solutions (Dow et al. 2009) and all station positions are constrained to 1 m. The individual contributions are integrated in a unified solution by the SIRGAS Combination Centres DGFI and IBGE (Fig. 1.3). The DGFI combinations are provided to the users as the SIRGAS official

6

C. Brunini et al.

Fig. 1.2 SIRGAS-CON network (status August 2009)

products (Sanchez et al. 2011), while the IBGE combinations assure redundancy and control for those products (Costa et al. 2009). At present, all SIRGAS Analysis Centres use the Bernese GPS Software (Dach et al. 2007) for processing the individual sub-networks and for their combination. Before combining the individual solutions, the constraints included in the delivered normal equations

are removed and the solutions are separately aligned to the IGS05 reference frame. The obtained standard deviations are analysed to establish the quality of the individual solutions and to determine variance factors, when it is necessary to compensate differences in the stochastic models of the Processing Centres. The station positions computed from each solution are compared by means of a similarity transformation to the

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame

7

Fig. 1.3 Data flow in the weekly processing of the SIRGASCON reference frame

IGS weekly values and to each other to identify possible outliers. Once inconsistencies and outliers are reduced from the individual free normal equations, a combination for a loosely constrained weekly solution for station positions (all of them constrained to 1 m) is computed. This solution is submitted in SINEX format to IGS for the global polyhedron and it is stored to be included in the next multi-year solution of the SIRGAS-CON network. A solution aligned to the IGS05 reference frame is also computed to provide weekly positions of all SIRGAS-CON stations for further applications (Fig. 1.4). Different criteria are applied to establish the quality of the contributing solutions delivered by the SIRGAS Processing Centres. The first one relates to the determination of mean standard deviation of station positions by solving the individual normal equations with respect to the IGS05 frame. These standard deviations represent the formal errors of the individual solutions. Secondly, the analysis of station position time series allows ascertaining the consistency of the individual contributions from week to week (repeatability). Then, the comparison by means of a similarity transformation of the individual solutions referring to the IGS05 with the IGS weekly positions provides information about their compatibility with the IGS global network. Figure 1.5 presents the mean values

Fig. 1.4 Combination procedure applied to generate the weekly solution of the SIRGAS-CON reference frame

for each criterion and for each sub-network for the period between the GPS weeks 1495 and 1538. These results indicate that the individual solutions are at the same level of precision: the formal error of the station positions is about 1.6 mm and the repeatability of the weekly coordinates is estimated to be ~ 2.0 mm for the horizontal component and 4.0 mm in the height.

1.4

Weekly Processing of the SIRGAS-CON Reference Frame

Regional and national reference frames supporting GNSS positioning must be consistent with the reference frame in which the GPS orbits are determined. For that reason, the IGS RNAAC SIR yearly generates a new multi-year solution referred to the current ITRF realization and including the SIRGAS-CON stations operating more than 2 years (e.g. Seemuller et al 2008, 2009). The latest solution SIR09P01 contains 128 stations with positions and velocities referring to IGS05, epoch 2005.0 (Seemuller et al. 2009). However, as mentioned before, the SIRGAS-CON network

8

Fig. 1.5 Evaluation of the solutions computed for the SIRGASCON individual sub-networks (mean values for the period GPS weeks 1495–1538)

is composed by more than 200 stations and those stations (about 80) that are not included in SIR09P01 can be used as reference points only, if their weekly positions linked to the ITRF (i.e. IGS05) are available. In this way, weekly solutions of the SIRGAS-CON network aligned to the IGS05 frame are necessary. Usually, epoch solutions (daily, weekly, multi-year) of regional reference networks are aligned to the ITRF using a set of fiducial stations with known positions and constant velocities; i.e. they consider linear coordinate changes only. However, GPS stations show significant seasonal position variations (mainly in the up component) resulting from a combination of geophysical loading and systematic errors. Ignoring these seasonal variations at reference stations can introduce systematic errors in the datum realization and the reference networks can be significantly deformed. These effects are larger in regional networks than in global ones, especially in zones with strong seasonal variations as the SIRGAS region. In this way, with the objective of minimizing the influence of seasonal

C. Brunini et al.

variations in the weekly realization of the SIRGASCON frame, the SIRGAS Working Group I (Reference System) analyzed different strategies for the datum definition taken into account the minimal network deformation, the weekly repeatability of station positions, and the consistency with the IGS weekly solutions for the global network. This analysis basically consisted of solving the same free normal equations applying two different sets of reference coordinates for the datum definition: the first one corresponds to the IGS05 positions at epoch 2000.0 extrapolated to the observation epoch using the ITRF2005 constant velocities. The second set corresponds to the weekly positions determined for the IGS05 reference stations within the IGS weekly combination (igsyyPwwww. snx). After comparing the loosely constrained solutions (in which the network is not deformed) with the constrained ones, the main conclusion shows that applying constant velocities to the reference coordinates introduces the largest distortions (more than 5 mm) into the station positions, mainly at the fiducial points (Fig. 1.6). This is a consequence of constraining a seasonal signal to be a linear trend. In this way, the SIRGAS-CON weekly solutions are aligned to the IGS05 by constraining the positions of the reference IGS05 stations to the values resulting of the IGS weekly combinations (Sanchez et al. 2011). The quality control of the SIRGAS-CON weekly solutions takes into account (1) the mean standard deviation for station positions to estimate the formal error of the final values; (2) the coordinate repeatability after combining the individual solutions to evaluate the internal consistency of the combined network; (3) station position time series analysis to determine the consistency of the combined solutions from week to week; and the (4) comparison with the IGS weekly coordinates and the IBGE weekly combinations to ascertain the reliability of the weekly solutions as well as to guarantee the required redundancy for the generation of the final SIRGAS-CON weekly positions. Table 1.1 summarizes the mean values resulting from the evaluation criteria for the period covering the GPS weeks 1495–1538. The mean standard deviation of the combined solutions is very similar to those obtained for the individual contributions (Fig. 1.5), i.e. their quality is maintained and their combination does not generate distortions in the SIRGAS-CON weekly realization. The weekly repeatability of the resulting

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame

Fig. 1.6 3D residuals derived after comparing the primary loosely constrained solutions for the SIRGAS-CON network with the same solutions aligned to the IGS05 frame applying two different sets of reference coordinates for the datum definition. In (a) the IGS05 station positions at 2000.0 are

9

extrapolated to the observation epoch by means of the ITRF2005 constant velocities. In (b) the station positions computed within the IGS weekly combinations for the IGS05 stations are directly introduced as reference coordinates. Values presented on the maps are mean values for 117 weeks

Table 1.1 Evaluation of the SIRGAS-CON weekly realizations positions provides an estimate of the internal consis(mean values for the period between GPS weeks 1495–1538) tency of approximately 0.8 mm in the horizontal Criteria Component Value in components and 2.5 mm in the vertical one. The [mm] RMS values derived from the time series for station Mean standard deviation 1.64 coordinates and with respect to the IGS weekly 0.61 Mean RMS of residuals for coordinate N positions indicate that the external accuracy of the repeatability in the weekly combination E 0.87 network is about 1.5 mm in the horizontal position Up 2.51 and 3.8 mm in the height. Total 2.73 Mean RMS of residuals derived from N 1.50 time series of station positions E 1.36 Up 3.80 1.5 Closing Remarks and Outlook Total 4.33 RMS of residuals wrt IGS weekly N 1.39 The processing strategy described in this paper for the solutions E 1.75 SIRGAS-CON network is applied since GPS week Up 3.69 1495. As already mentioned, before (since June 1996 Total 4.35 to August 2008), the entire SIRGAS-CON network RMS of station coordinate differences N 1.10 was computed by DGFI in one adjustment. In order between DGFI and IBGE combinations E 1.10 to establish the consistency of the current combined Up 1.40 solutions with the previous computations, residual Total 2.20

position time series were generated from the weekly

10

solutions available between January 2000 and January 2009. Discontinuities or jumps at the epoch in which the analysis strategy was changed (last week of August 2008) are not identifiable. Results show that the current weekly combined solutions are at the same accuracy level and totally consistent with the previous computations (when the network was calculated in one block). Nevertheless, the present sub-network distribution has two main disadvantages: (1) Not all SIRGASCON stations are included in the same number of individual solutions, i.e., they are unequally weighted in the weekly combinations, and (2) since there are not enough Local Processing Centres, the required redundancy (each station processed by at least three processing centres) is not fulfilled. Therefore, SIRGAS promotes the installation of more Local Processing Centres hosted by Latin American countries. In this frame, institutions interested to install a SIRGAS Processing Centre shall pass a test period of one year. In this period, they have to align their processing strategies with the SIRGAS guidelines and meet the delivering deadlines. At present, there are five Experimental Processing Centres: Instituto Geogra´fico Militar of Ecuador (IGM, Ecuador), Laboratorio de Geodesia Fı´sica y Satelital at the Universidad del Zulia (LGFS-LUZ, Venezuela), Servicio Geogra´fico Militar of Uruguay (SGM, Uruguay), Instituto Nacional de Estadı´stica y Geografı´a (INEGI, Mexico), and Instituto Geogra´fico Nacional de Argentina (IGN, Argentina). They will become Official Processing Centres in the near future and a redistribution of the SIRGAS-CON stations between the operative SIRGAS Analysis Centres will allow including each regional station in the same number of individual solutions. This will significantly improve the reliability and quality control of the weekly solutions for the SIRGAS-CON reference frame.

References Costa SMA, da Silva AL, Vaz JA (2009) Report of IBGE Combination Centre. Period of SIRGAS-CON solutions:

C. Brunini et al. from week 1495 to 1531. Presented at the SIRGAS 2009 General Meeting. Buenos Aires, Argentina. September. Available at www.sirgas.org. Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS Software Version 5.0 – Documentation. Astronomical Institute, University of Berne, p 640 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a hanging landscape of Global Navigation Satellite Systems. J Geodesy 83:191–198. doi:10.1007/s00190-0080300-3 Drewes H, Kaniuth K, Voelksen C, Alves Costa SM, Souto Fortes LP (2005) Results of the SIRGAS campaign 2000 and coordinates variations with respect to the 1995 South American geocentric reference frame. In: Sanso F (ed) A window on the future of geodesy, vol 128, IAG symposia. Springer, Heidelberg, pp 32–37 Drewes H, Heidbach O (2005) Deformation of the South American crust estimated from finite element and collocation methods. In: Sanso F (ed) A window on the future of geodesy, vol 128, IAG Symposia. Springer, Heidelberg, pp 544–549 Natali MP, M€uller M, Ferna´ndez L, Brunini C (2009) CPLat: the pilot processing center for SIRGAS in Argentina. In: Drewes H (ed) Geodetic reference frames, vol 134, IAG Symposia. Springer, Heidelberg, pp 179–184 Sanchez L, Brunini C (2009) Achievements and challenges of SIRGAS. In: Drewes H (ed) Geodetic reference frames, vol 134, IAG symposia. Springer, Heidelberg, pp 161–166 Sanchez LW, Seem€uller MS, Seitz M (2011) Combination of the weekly solutions delivered by the SIRGAS Processing Centres for the SIRGAS-CON reference frame. In: Kenyon S et al (eds) Geodesy for planet Earth, Buenos Aires Argentina, vol 136, IAG symposia. Springer, Heidelberg Seemuller W, Drewes H (2008) Annual Report 2003–2004 of IGS RNAAC SIR. In: IGS 2001–02 Technical Reports, IGS Central Bureau, (eds) Jet Propulsion Laboratory, Pasadena, CA. Available at http://igscb.jpl.nasa.gov/igscb/resource/ pubs/2003-2004_IGS_Annual_Report.pdf Seemuller W, Krugel M, Sanchez L, Drewes H (2008) The position and velocity solution DGF08P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 79. DGFI, Munich. Available at www.sirgas.org Seemuller W, Seitz M, Sanchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at www.sirgas.org Seemuller W, Sanchez L (2009) SIRGAS Processing Centre at DGFI: report for the SIRGAS 2009 General Meeting. Presented at the SIRGAS 2009 General Meeting. Buenos Aires, Argentina, September. Available at www.sirgas.org SIRGAS (1997). SIRGAS Final Report; Working Groups I and II IBGE, Rio de Janeiro, p 96. Available at www.sirgas.org

2

Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring Alvaro Santamarı´a-Go´mez, Marie-Noe¨lle Bouin, €ppelmann and Guy Wo

Abstract

The University of La Rochelle (ULR) TIGA Analysis Center (TAC) completed a new global reprocessed solution spanning 13 years with more than 300 GPS permanent stations, 216 of them being co-located with tide gauges. A state-of-theart GPS processing strategy was applied, in particular, the station sub-networks used in the daily processing were optimally built. Station vertical velocities were estimated in the ITRF2005 reference frame by stacking the weekly position estimates. Outliers, offsets and discontinuities in time series were carefully examined. Vertical velocities uncertainties were assessed in a realistic way by analysing the type and amplitude of the noise content in the residual position time series. The comparison shows that the velocity uncertainties have been reduced by a factor of 2 with respect to previous ULR solutions. The analysis of this solution and its by-products shows the high geodetic quality achieved in terms of homogeneity, precision and consistency with respect to other top-level geodetic solutions.

2.1

A. Santamarı´a-Go´mez (*) Instituto Geogra´fico Nacional, c/ General Iban˜ez Ibero 3, 28071, Madrid, Spain Institut Ge´ographique National, LAREG/GRGS, 6–8 av. Blaise Pascal, 77455, Champs-sur-Marne, France e-mail: [email protected] M.-N. Bouin Institut Ge´ographique National, LAREG/GRGS, 6–8 av. Blaise Pascal, 77455, Champs-sur-Marne, France CNRM/CMM, Me´te´o France, 13 rue du Chatellier, CS 12804, 29228, Brest, France G. W€oppelmann Universite´ de La Rochelle-CNRS, UMR 6250 LIENSS, 2 rue Olympe de Gouges, 17000, La Rochelle, France

Introduction

In order to estimate long-term geocentric sea level rise, tide gauges trends must be corrected for the long-term vertical displacements of the land upon which they are settled. In addition, for proper satellite altimeter calibration purposes, tide gauges trends must be referred to a common, global and stable reference frame, such as the latest realization of the International Terrestrial Reference Frame (ITRF) (Altamimi et al. 2007). These long-term vertical displacements can be corrected by modelling geological processes as the Global Isostatic Adjustment (GIA) (e.g. Douglas 2001) or directly from continuous geodetic observations at or near tide gauges. This second method should be preferred as it takes into account local displacements (geological, anthropogenic or whatever), not accounted

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_2, # Springer-Verlag Berlin Heidelberg 2012

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A. Santamarı´a-Go´mez et al.

12

Fig. 2.1 Number of daily available stations (grey) and percentage of daily resolved ambiguities (black)

for in the GIA models. Within the different geodetic techniques used for this purpose (GPS, DORIS and absolute gravity), GPS is the most widespread. Recent studies (W€ oppelmann et al. 2009, Bouin and W€oppelmann 2010) have shown that correcting the tide gauge trends using continuous GPS stations ([email protected]) improves the consistency of the sea level rates. To this aim, the International GNSS Service (IGS) Tide Gauge Benchmark Monitoring Pilot Project (TIGA) was established in 2001 (Sch€ one et al. 2009). Since 2002, the ULR consortium contributes to the TIGA project as an Analysis and Data Center (W€oppelmann et al. 2004). Several global vertical velocity field solutions (ULR solutions hereafter) were released with different station networks, time spans and processing strategies (W€oppelmann et al. 2007, 2009). In this paper, we present the fourth ULR solution based on an homogeneous reprocessing of a larger global network of 316 stations, spanning an increased period of 13 years (January 1996 to December 2008). This solution comes out with a new data analysis strategy, including a new sub-network design and combination. The troposphere and ocean tide modelisation were also improved. Both GPS processing and vertical velocity estimation strategies are described; realistic uncertainties are estimated by analysing the noise content of time series. Finally, the quality of the solution is assessed and discussed.

2.2

Data Analysis Strategy

2.2.1

Data

The global tracking network consists of 316 GPS stations. 216 of them are [email protected], including 81 stations committed to TIGA. Also 124 of them are

IGS reference frame (RF) stations used for realizing the reference frame (Kouba et al. 1998) and for improving the network geometry. This network was processed over the period 1st January 1994 to 31st December 2008. Small RINEX files (less than 5 h of observation) were rejected. This quality check procedure yielded a number of daily available stations between a minimum of 25 in 1994 (53 in 1996) and a maximum of 239 in 2006 (grey line in Fig. 2.1). 1994 and 1995 were finally not retained in the solution due to a lack of fixed ambiguities and therefore quality (black line in Fig. 2.1) and they will not be further considered.

2.2.2

Improved Network Geometry

GPS processing time increases exponentially with the number of stations. To overcome this limitation, it is usual to split the whole network in several subnetworks, to process each sub-network independently and then to combine the sub-network solutions into a unique daily solution. Historic ULR solutions (ULR1 to ULR3 solutions) used five global, manually-selected, permanent subnetworks over the entire data span (“static subnetworks” hereafter). Using this approach, the a priori stations included in each sub-network were always the same, whether or not their data were available for a specific day, making the geometry worse when their data were missing, and therefore, possibly yielding an unnecessary large number of sub-networks in the processing (always five). This static configuration was changed in the ULR4 solution into a new station distribution approach resulting in global, automatic, daily-variable sub-networks (“dynamic sub-networks” hereafter), with up to 50 stations per sub-network.

2

Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring

Shorter baselines improves ambiguity resolution (Steigenberger et al. 2006). With the dynamic approach, all daily available stations were distributed into the strictly necessary number of sub-networks, ensuring optimal dense sub-networks. Thus, the number of dynamic sub-networks used grows from 1 in 1996 to 6 in 2003. Moreover, to obtain global geometrically well-distributed sub-networks for optimal orbit estimation, each station is assigned to the sub-network where it is more isolated, i.e. reducing the baselines. In this way, “deserted” areas of each sub-network are iteratively being “populated”. In addition, six daily-variable common IGS RF stations, with more than 12 h of observation, are included in each dynamic sub-network to combine the solutions. Northernmost and southernmost stations are always selected and then four other globally welldistributed stations are added. Static versus dynamic approaches were compared by processing two solutions using the same stations and processing strategy except for the stations distribution. Figure 2.2 shows that using dynamic sub-networks clearly increases the percentage of resolved ambiguities as the number of available stations decreases, up to 20% in 1997 (Fig. 2.2). The 10% offset in the percentage of resolved ambiguities observed at the end of 1999 for both approaches is related to the use of code bias corrections (see Sect. 2.2.3), only available for post2000 year period when the test was performed.

2.2.3

Models and Parameterization

Double-differenced ionosphere-free carrier phase data is analysed using GAMIT software version

Fig. 2.2 Resolved ambiguities for static (grey), dynamic subnetworks (top black) and the difference (bottom black)

13

10.34 (Herring et al. 2006a). The elevation cut-off angle is set to 10 , avoiding mismodelling of lowelevation troposphere and phase center variations (PCV) of relative-to-absolute antenna calibration. Sampling rate is set to 3 minutes. Carrier phase observations are weighted in two iterations: by elevation angle first and then by elevation angle and by station, accounting for the station phase residuals from the first iteration. Code bias corrections are applied for the whole period using monthly tables from the Astronomical Institute of the University of Bern (AIUB) [IGSMAIL-2827 (2000) at http:// igscb.jpl.nasa.gov/mail/). Real-valued double differenced phase cycle ambiguities are adjusted except when they can be resolved confidently. In this case, they are fixed using the MelbourneW€ubbena wide-lane to resolve L1–L2 cycles and then estimation to resolve L1 and L2 cycles. For satellite antennas, satellite-specific z-offsets (Ge et al. 2005) and block-specific nadir angledependent absolute PCV (Schmid et al. 2007) are applied. For receiver antennas, L1/L2 offsets and azimuth-dependent, when available, and elevationdependent absolute PCV are applied. A priori zenith hydrostatic (dry) delay values are extracted by station from the ECMWF meteorological model through the VMF1 grids (Boehm et al. 2006). Residual delays are adjusted for each station assuming mostly dominated by the wet component and parameterized by a piecewise linear, continuous model with 2 h intervals. Both dry and wet VMF1 mapping functions are used. One gradient is estimated for each day and each station. Solid Earth tides are corrected following IERS Conventions (2003) (McCarthy and Petit, 2004). Ocean tide loading is corrected using FES2004 model (Lyard et al. 2006). No atmospheric tide nor non-tidal corrections were applied. Earth orientation parameters (EOP) are daily estimated as a piecewise, linear model with a priori values from IERS Bulletin B. UT1–UTC offsets are highly constrained to their a priori values. Satellite positions and velocities are adjusted in 24 h arcs taking IGS final orbits (Dow et al. 2005) as a priori. Solar radiation pressure parameters are estimated using the Berne model (Beutler et al. 1994).

A. Santamarı´a-Go´mez et al.

14

2.2.4

Data Processing Scheme and Reference Frame

Each dynamic sub-network is processed independently using GAMIT software. The daily sub-network solutions are combined into a daily solution (by estimating only translations and rotations) using GLOBK (Herring et al. 2006b) by means of the estimated orbital parameters, the estimated positions of the six common stations and their estimated zenith tropospheric path delays. Daily loose solutions are constrained by no-net-rotation (NNR) constraints with respect to ITRF2005 and combined into a weekly solution using CATREF software (Altamimi et al. 2007). These weekly solutions are aligned to ITRF2005 using NNR constraints with all IGS RF stations available, whereas inner constraints (Altamimi et al. 2007) are used for scale and translation, in order to preserve the weekly apparent geocenter motion information. All the weekly solutions for the whole period (GPS weeks 0834–1512), are then combined into a longterm solution using CATREF. This long-term solution (ULR4) is aligned to ITRF2005 using minimal constraints over all the transformation parameters with a selected set of IGS RF stations called datum. The 68 stations retained in the datum were selected based on their data availability (at least present in 80% of the whole processed period) and their quality as follows. Firstly, stations with known or suspected velocity discontinuities were rejected, and secondly, in an iterated process, stations showing large position and velocity residuals with respect to ITRF2005 values were also rejected. Thresholds for positions were set to 0.5 cm in horizontal and 1.5 cm in vertical. The larger value in the vertical component is due to the fact that ITRF2005 GPS coordinates were estimated with a relative PCV model. Station differences using the absolute PCV model are estimated to be within this range. Thresholds for velocity residuals were set to 1.5 mm/year and 2 mm/year respectively. The residual position time series of each station were visually examined. To avoid biased velocities, all discontinuities (significant offsets and velocity changes) were detected, identified if possible, and removed using ITRF2005 discontinuities as a priori. Then, all outliers were removed in an iterative process, from bigger to smaller magnitude (depending on the time series noise), down to a minimum of 2 cm for residuals and 4 for normalized residuals.

2.3

Results

2.3.1

Vertical Rates

The vertical velocity fields of ULR4 and ULR3 (W€oppelmann et al. 2009)) solutions were compared using a common set of 170 stations with more than 4.5 years of data. Figure 2.3 shows that most of the velocity differences are below 1 mm/year (RMS of 0.8 mm/year), except some stations for which larger differences are due to different discontinuities on their time series. The mean difference between both velocity fields is 0.16 0.06 mm/year which is related to the different datum used to aling the solutions. This misalignment is under the internal precision of the ITRF2005. From the complete ULR4 solution, 224 stations with more than 4.5 years of data were retained. For these stations, their estimated velocities are confidently not influenced by seasonal signals (Blewitt and Lavalle´e 2002). Nevertheless, the rate uncertainties estimated with a standard least squares algorithm (based on a Gaussian white noise process) are clearly optimistic by a factor of 3–11 (Zhang et al. 1997; Mao et al. 1999). More realistic uncertainties of the estimated velocities must account for correlated noise present in the time series. A noise analysis was performed using the Maximum Likelihood Estimation (MLE) technique (CATS software, (Williams, 2008)). Vertical velocity uncertainties were estimated using a white noise plus power law noise model. To avoid biased adjustments, time series were previously examined for periodic signals. Besides the annual and semi-annual terms, we also found and removed up to six harmonics of

Fig. 2.3 Vertical velocity difference between ULR3 and ULR4. Dashed lines represent 1 mm/year

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Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring

15

Fig. 2.4 Histogram of estimated uncertainties for ULR4 (grey) and ULR3 (black) solutions and their median values

the GPS “draconitic” period described by Ray et al. (2007). Figure 2.4 shows the histogram of the realistic vertical velocity uncertainties of the ULR4 solution with respect to the realistic uncertainties estimated for the ULR3 solution also using CATS. The improvement is close to a factor of 2. Also the factor of optimism of the formal uncertainties with respect to the realistic ones is 2–3, quite smaller than the abovementioned values. This is due to the improvement and consistency of the processing strategy presented here, which results in a noticeable reduction of the correlated noise content for the ULR4 solution compared to previous solutions.

2.3.2

Weekly Repeatability

The internal quality of the ULR4 solution was assessed by analysing the repeatability of the weekly position solutions. Figure 2.5 shows the repeatability of the time series (mean values of the weighted RMS of the weekly positions with respect to the long-term combined positions) for ULR4 and ULR3 solutions. Horizontal and vertical repeatabilities are improved in the ULR4 solution. Moreover, for the whole reprocessed period vertical repeatabilities are more stable, showing the improved ULR4 time consistency. ULR4 repeatability values are between 1 and 3 mm for the horizontal and between 4 and 6 mm for the vertical component (3D weighted RMS between 2 and 4 mm). These values are fully consistent with those of the IGS

Fig. 2.5 Horizontal (bottom) and vertical (top) weighted RMS of the weekly solutions with respect to the long-term solution for both ULR4 (black) and ULR3 (grey) solutions

combined solution (Altamimi and Collilieux, 2008), showing that ULR4 solution is comparable in quality with the ITRF2005.

2.3.3

Origin and Scale

As a satellite technique, GPS estimated origin should be coincident with the Earth’s center of mass. However this affirmation is not completely fulfilled due to remaining GPS-specific systematic errors, as the modelling of the solar radiation pressure coefficients or the unaccounted effect of higher ionospheric orders (Herna´ndez-Pajares et al. 2007). We have estimated here apparent geocenter motion using the network shift or geometric approach

A. Santamarı´a-Go´mez et al.

16

(Lavalle´e et al. 2006). Figure 2.6 shows the translation and scale parameters of the weekly solutions with respect to the long-term combined solution aligned to the ITRF2005. Translation trends are not significant, showing the consistency of the secular origin definition with respect to the ITRF2005. The scale shows no trend either, as this parameter is completely dependent on the ITRF2005 scale definition through the satellites antenna z-offset corrections. For intercomparison purposes, an annual signal was estimated for each transformation parameter (Table 2.1).

Compared to SLR results (Collilieux et al. 2009), the annual amplitudes of the equatorial components (X and Y) and the scale are fully consistent. However, the amplitude of the Z component is twice larger. Regarding the annual phase, the scale parameter is fully consistent, but all translational parameters show a shift of about 137º (4.5 months). Compared to other GPS results (Lavalle´e et al. 2006), the amplitude of the Z component and both equatorial phases are consistent. The phase of Z component exhibits larger solution-dependent variations. Both issues point probably at the above-mentioned GPS systematic errors and also at the poor performance of the network shift method used with a not-well distributed global network (Lavalle´e et al. 2006).

2.3.4

Orbits

The estimated ULR4 orbits were compared with the current official non-reprocessed IGS final orbits (Dow et al. 2005). A classic 7-parameter Helmert transformation was applied between both 24 h-arc sets. 1D RMS differences (the average of the three RMS components) were estimated for each common observed satellite and then the median daily RMS value was extracted and traced (black line, Fig. 2.7). We show that ULR and IGS orbits are in good agreement with each other, from 8.5 cm in 1996 to 1.5 cm in 2009. The same range of differences was obtained between IGS orbits and reprocessed orbits from SIO/SOPAC IGS Analysis Center (light grey line). Some smaller differences were obtained with

Fig. 2.6 Weekly translation and scale parameters with respect to the ITRF2005. Also their trends and annual signal are traced

Table 2.1 Annual signal of apparent geocenter and scale TX TY TZ Scale

Amplitude (mm) 2.3 0.2 4.2 0.3 9.9 0.8 1.8 0.1

Phase (deg) 164.6 5.4 122.2 3.5 171.3 3.5 243.2 1.6

Fig. 2.7 7-day smoothed daily RMS between final IGS orbits and ULR (black), SIO/SOPAC (light grey) and CODE/AIUB (dark grey) reprocessed orbits

2

Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring

reprocessed CODE/AIUB IGS Analysis Center (dark grey line) for the post-2000 period. This demonstrates that the ULR4 orbits are of the same quality as the reprocessed orbits of some of the IGS Analysis Centers.

2.4

Concluding Remarks

The new ULR4 solution is based on an homogeneous reprocessing of a global GPS network of 316 stations spanning up to 13 years of data. The processing strategy was improved with respect to past ULR solutions. Special attention was paid to the sub-network geometry distribution, which clearly improves the quality of the reprocessing by increasing the number of resolved ambiguities. The analysis of the results and by-products of this solution (vertical velocities, repeatability, transformation parameters and orbits) shows the high geodetic quality achieved. The stateof-the-art GPS processing strategy implemented fulfils the IGS requirements and recommendations. Thereby, in addition to the IGS TIGA project, the ULR consortium is participating with its latest solution to the first IGS reanalysis campaign, enabling an invaluable extension of IGS and ITRF reference frames towards tide gauges. Also, the ULR consortium is contributing to the Working Group on Regional Dense Velocity Fields of the International Association of Geodesy Subcommision 1.3. (Bruyninx 2011). Further studies will be carried out in order to assess the geophysical usefulness of this solution. For example, this global and accurate vertical velocity field may be used to separate vertical land motion trends from relative sea level trends as recorded by tide gauges. Acknowledgements The authors acknowledge two unknown reviewers who contributed to an improved paper. We also thank the invaluable technical support given by Mikael Guichard, Marc-Henri Boisis-Delavaud and Frederic Bret from the IT centre of the University of La Rochelle (ULR). The ULR computing infrastructure used for the reprocessing of the GPS data was partly funded by the European Union (Contract 31031-2008, European regional development fund). This work was also feasible thanks to all institutions and individuals worldwide that contribute to make GPS data and products freely available.

17

References Altamimi Z, Collilieux X (2008) IGS contribution to ITRF. J Geod 83(3–4):375–383 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401 Beutler G, Brockmann E, Gurtner W, Hugentobler U, Mervart L, Rothacher M (1994) Extended orbit modeling techniques at the CODE Processing Center of the International GPS Service for Geodynamics (IGS): theory and initial results. Manuscripta Geodaetica 19:367–386 Blewitt G, Lavalle´e D (2002) Effect of annual signals on geodetic velocity. J Geophys Res 107:B02145 Boehm J, Werl B, Schuh H (2006) Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for Medium-Range Weather Forecasts operational analysis data. J Geophys Res 111:B02406 Bouin M-N, W€oppelmann G (2010) Land motion estimates from GPS at tide gauges: a geophysical evaluation. Geophys J Int 180:193–209 Bruyninx C (2011) A dense global velocity field based on GNSS observations: preliminary results. In: Kenyon S et al (eds) Geodesy for planet Earth. Springer, Heidelberg Collilieux X, Altamimi Z, Ray J, van Dam T, Wu X (2009) Effect of the satellite laser ranging network distribution on geocenter motion estimation. J Geophys Res 114:B02145 Douglas B (2001) Sea level change in the era of the recording tide gauge, vol 75, International geophysics series. Academic, San Diego, CA Dow JM, Neilan RE, Gendt G (2005) The International GPS Service: celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36:320–326 Ge M, Gendt G, Dick G, Zhang F, Reigber C (2005) Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys Res Lett 32:L06310 Herna´ndez-Pajares M, Juan JM, Sanz J, Oru´s R (2007) Secondorder ionospheric term in GPS: Implementation and impact on geodetic estimates. J Geophys Res 112:B08417 Herring TA, King RW, McClusky SC (2006a) GAMIT: Reference Manual Version 10.34. Internal Memorandum, Massachusetts Institute of Technology, Cambridge Herring TA, King RW, McClusky SC (2006b) GLOBK: Global Kalman filter VLBI and GPS analysis program Version 10.3. Internal Memorandum, Massachusetts Institute of Technology, Cambridge Kouba J, Ray J, Watkins M (1998). IGS Reference Frame realization. IGS 1998 Analysis Center Workshop – Proceedings Darmstadt. p 139 Lavalle´e D, van Dam T, Blewitt G, Clarke P (2006) Geocenter motions from GPS: A unified observation model. J Geophys Res 111:B05405 Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56:394–415 Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104:2797–2816

18 McCarthy D, Petit G (2004) IERS Technical Note 32 – IERS Conventions (2003) Technical report, Verlag des Bundesamts fur Kartographie und Geodasie, Frankfurt am Main, Germany Ray R, Altamimi Z, Collilieux X, Van Dam T (2007) Anomalous harmonics in the spectra of GPS position estimates. GPS Solut 12(1):55–64 Schmid R, Steigenberger P, Gendt G, Ge M, Rothacher M (2007) Generation of a consistent absolute phase-center correction model for GPS receiver and satellite antennas. J Geod 81:781–798 Sch€one T, Sch€on N, Thaller D (2009) IGS Tide Gauge Benchmark Monitoring Pilot Project (TIGA): scientific benefits. J Geod 83:249–261 Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111:B05402 Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solut 12(2):147–153

A. Santamarı´a-Go´mez et al. W€oppelmann G, McLellan S, Bouin M-N, Altamimi Z, Daniel L (2004) Current GPS data analysis at CLDG for the IGS TIGA Pilot Project. Cahiers du Centre Europe´en Ge´odynamique & de Sismologie 23:149–154 W€oppelmann G, Martı´n Mı´guez B, Bouin M-N, Altamimi Z (2007) Geocentric sea-level trend estimates from GPS analyses at relevant tide gauges world-wide. Glob Planet Change 57(3–4):396–406 W€oppelmann G, Letretel C, Santamarı´a A, Bouin M-N, Collilieux X, Altamimi Z, Williams S, Martı´n Mı´guez B (2009) Rates of sea-level change over the past century in a geocentric reference frame. Geophys Res Lett 36:L12607 Zhang J, Bock Y, Johnson H, Fang P, Williams S, Genrich J, Wdowinski S, Behr J (1997) Southern California Permanent GPS Geodetic Array: error analysis of daily position estimates and site velocities. J Geophys Res 102:18035–18056

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results C. Bruyninx, Z. Altamimi, M. Becker, M. Craymer, L. Combrinck, A. Combrink, J. Dawson, R. Dietrich, R. Fernandes, R. Govind, T. Herring, A. Kenyeres, R. King, C. Kreemer, D. Lavalle´e, J. Legrand, L. Sa´nchez, G. Sella, Z. Shen, A. Santamarı´a-Go´mez, and €ppelmann G. Wo

Abstract

In a collaborative effort with the regional sub-commissions within IAG subcommission 1.3 “Regional Reference Frames”, the IAG Working Group (WG) on “Regional Dense Velocity Fields” (see http://epncb.oma.be/IAG) has made a first attempt to create a dense global velocity field. GNSS-based velocity solutions

C. Bruyninx (*) Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium e-mail: [email protected] Z. Altamimi Institut Ge´ographique National, Service de la recherche/ LAREG, c/o ENSG, 6 - 8 Ave. Blaise Pascal, 77455 Champssur-Marne, France M. Becker Institute of Physical Geodesy, Technische Universit€at Darmstadt, Petersenstr, 13, 64287 Darmstadt, Germany M. Craymer Geodetic Survey Division, Natural Resources Canada, 615 Booth Str., Ottawa Ontario K1A 0E9, Canada L. Combrinck HartRAO, PO Box 443, 1740 Krugersdorp South Africa, Canada A. Combrink HartRAO, PO Box 443, 1740 Krugersdorp South Africa, Canada J. Dawson Geoscience Australia, Cnr Jerrabomberra Ave. and Hindmarsh Drive, Symonston, ACT 2609, Australia R. Dietrich TU Dresden, Institut f€ ur Planetare Geod€asie, Mommsenstr, 13, 01062 Dresden, Germany R. Fernandes University of Beira Interior, IDL, R. Marqueˆs d’Aacute;vila e Bolama, 6201-001 Covilha˜, Portugal and Delft University of Technology, DEOS – PSG, PO Box 5058, 2600 GB Delft, The Netherlands

R. Govind Geoscience Australia, Cnr Jerrabomberra Ave. and Hindmarsh Drive, Symonston, ACT 2609, Australia T. Herring Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA A. Kenyeres FOMI Satellite Geodetic Observatory, POBox 585 1592 Budapest, Hungary R. King Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA C. Kreemer Nevada Bureau of Mines and Geology, and Seismological Laboratory, University of Nevada, 1664 N. Virginia Str., MS178, Reno, NV 89557, USA D. Lavalle´e Nevada Bureau of Mines and Geology, and Seismological Laboratory, University of Nevada, 1664 N. Virginia Str., MS178, Reno, NV 89557, USA J. Legrand Delft University of Technology, Faculty of Aerospace Engineering, DEOS – PSG, PO Box 5058 2600 GB Delft, Netherlands L. Sa´nchez Royal Observatory of Belgium, Ave. Circulaire 3, 1180, Brussels, Belgium G. Sella Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_3, # Springer-Verlag Berlin Heidelberg 2012

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for more than 6,000 continuous and episodic GNSS tracking stations, were proposed to the WG in reply to the first call for participation issued in November 2008. The combination of a part of these solutions was done in a two-step approach: first at the regional level, and secondly at the global level. Comparisons between different velocity solutions show an RMS agreement between 0.3 and 0.5 mm/year resp. for the horizontal and vertical velocities. In some cases, significant disagreements between the velocities of some of the networks are seen, but these are primarily caused by the inconsistent handling of discontinuity epochs and solution numbers. In the future, the WG will re-visit the procedures in order to develop a combination process that is efficient, automated, transparent, and not more complex than it needs to be.

3.1

Introduction

The Working Group “Regional Dense Velocity Fields” has been created in 2007 at the IUGG (International Union of Geodesy and Geophysics) General Assembly in Perugia, Italy. It is embedded within IAG (International Association of Geodesy) sub-commission 1.3 on “Regional Reference Frames” where it co-exists with the regional sub-commissions for Europe, South and Central America, North America, Africa, South-East Asia and Pacific, and Antarctica (Drewes et al. 2008). The long-term goal of the Working Group is to provide a globally referenced dense velocity field based on GNSS observations and linked to the multitechnique global conventional reference frame, the ITRF (International Terrestrial Reference Frame, Altamimi et al. 2007a).

Z. Shen NOAA-National Geodetic Survey (NGS), 1315 East-West Hwy, Silver Spring, MD 20910, USA A. Santamarı´a-Go´mez University of California, Department of Earth and Space Sciences, 595 Charles E Young Drive, Los Angeles, CA 900951567, USA G. W€oppelmann Instituto Geogra´fico Nacional, c/ General Iban˜ez Ibero 3, 28071 Madrid, Spain and Institut Ge´ographique National, Service de la recherche/LAREG, c/o ENSG, 6–8 Ave. Blaise Pascal, 77455 Champs-sur-Marne, France

3.2

Working Group Objectives and Work Plan

3.2.1

Objectives

The Working Group on “Regional Dense Velocity Fields” joins the efforts of groups processing local/ regional/global CORS or repeated GNSS campaigns and set up the following action items: – Define specifications and quality standards for the regional SINEX solutions and relevant meta-data (e.g. description of GNSS equipment and position/ velocity discontinuities) – Collect SINEX solutions and their meta-data – Study in-depth the individual strengths and shortcomings of local/regional and continuous/ epoch GNSS solutions to determine site velocities – Define optimal strategies for the combination of regional and global SINEX solutions – Provide dense regional velocity fields – Provide the densification of the ITRF2005 (or its successors) – Encourage participation in related symposia – Implement a web site in order to provide information on the activities and access to the products of the WG – And prepare recommendations and a comprehensive final report on the WG activities at the next IUGG General Assembly in 2011

3.2.2

Work Plan

The work plan of the WG has been divided into two major parts. During the first part, covering 2007–2009, the WG set up the initial strategy and submission

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results

guidelines, collected a first set of test solutions, and performed a first preliminary velocity combination. The working group closely links its activities with the regional sub-commissions within IAG sub-commission 1.3. Their expertise, coordination role for their region, and their capability to generate a unique cumulative solution for their region including velocity solutions from third parties (even campaigns) is essential for the WG. The initial WG strategy consisted therefore in a two-step approach. First, region coordinators (one for each region corresponding to the regions of the different regional sub-commissions) gathered sub-regional velocity solutions for their region (in accordance with the WG requirements) and combined these with, where available, the velocity solution from the regional subcommissions (e.g. EUREF, SIRGAS. . .) in order to produce one regional combined velocity solution in the SINEX format. Secondly, two combination coordinators -T. Herring (MIT, US) and D. Lavalle´e (TU Delft, Netherlands)- combined these regional SINEX solutions with the long-term solutions from global networks to generate a preliminary velocity solution tied to the ITRS. The main goal of this preliminary solution was to identify the problems that would arise and help to set strategic choices and guidelines for the future. These guidelines will be used to issue a new solution in the second part of the WG term, 2010–2011. The results of the 2007–2009 period will be presented in this paper. As mentioned in the introduction, the WG accepts velocity solutions based on CORS and repeated GNSS campaigns (under specific conditions, see Sect. 3.1). One of the strategic choices the WG group had to make from the start was to decide whether to (1) stack weekly combined regional and global (position) SINEX solutions to compute the velocities or to (2) combine cumulative regional and global (position + velocity) SINEX solutions. Considering that the WG does not have access to the weekly SINEX of many cumulative velocity solutions, it was decided to go for approach (2). This will allow us combining, if necessary, only velocities (without the positions). In addition, it will allow us a stepwise combination of regional and global solutions; it will also facilitate meta-data management and outlier detection, as these will be done at regional level. And finally, it perfectly fits in the initial frame (using region and combination coordinators) that was set up. The disadvantages of combining cumulative (position+) velocity solutions are

21

however that no coordinate time series will be available to the WG and that it will be necessary to consistently handle discontinuities, especially on frame-attachment sites.

3.3

Call for Participation

3.3.1

Initial Submission Guidelines

In order to allow inclusion of a maximum number of velocity solutions, the WG set up the following guidelines for the contributing solutions: – Minimum 2 years of continuous data or 2 campaign epochs over a 4 year period – Minimum 2.5 years of continuous data if significant seasonal signals are present – Significant number of “frame-attachment” sites, preferably observed over a period exceeding 5 years – Position/velocity discontinuities should be identical to the ones used by the (IGS) International GNSS Service (Dow et al. 2009) – Velocity constraints should be minimal or removable – SINEX format should contain full covariance information (an exception is allowed for PPP solutions only providing correlations between individual station coordinates) The detailed submission guidelines are available from the Working Group web site: http://www. epncb.oma.be/IAG/.

3.3.2

Call for Participation

A first Call for Participation (CfP) was issued at the end of 2008. Analysts, producing regional and global velocity solutions, were invited to submit their SINEX files to the Working Group. Figure 3.1 shows the map with the sites for which solutions that have been proposed following this CfP (black dots); in total more than 6,000 sites were proposed.

3.4

Input for Preliminary Combination

A preliminary velocity combination has been computed in the summer of 2009. This solution contained contributions from the region coordinators as well as one global solution.

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Fig. 3.1 Map with the sites for which velocity solutions have been proposed to the Working Group up to July 2009 (black dots). In total more than 6,000 sites have been proposed. In red: sites used in the first preliminary combination (see Sect. 3.4)

3.4.1

Regional Contributions

Several of the region coordinators prepared a velocity solution for their region to be included in the preliminary combination. The African and South & Central American contributions are based on the contribution of a single analysis center, while the solutions from Europe and South-East Asia & Pacific are combined solutions based on input from several analysis groups. More details are given below: – Africa (see Fig. 3.2): The solution includes 93 CORS and covers the period from Jan. 1996 till June 2009 (Fernandes et al. 2007). The GNSS data analysis, has been done using GIPSY/OASIS II (Zumberge et al. 1997) by applying the PPP strategy with ambiguity resolution (Blewitt 2008). GIPSY tools were also used to derive the velocity solutions. – Europe (see Fig. 3.3): The solution includes the velocities estimated from a reprocessing of the EUREF Permanent Network (EPN), maintained by the regional sub-commission for Europe (Bruyninx et al. 2009), complemented with several sub-regional velocity solutions. In total, velocity estimates for 525 sites were obtained, which is more than twice the number of the sites presently included in the EPN. All of the contributing sub-regional solutions were available in the SINEX format and the combination was done with the CATREF software (Altamimi et al. 2007b). The main problem encountered during the combination was the fact that some of the submitted sub-regional solutions did not use any discontinuities at all (more about this in Sect. 3.6).

Fig. 3.2 Sites contributing to the African velocity solution

Fig. 3.3 Sites and solutions included in the European solution: reprocessed EPN (‘96-’09) in black, AGNES (‘98-09’) in pink, AMON (‘01-’09) in green, ASI (’97-’09) in red, IGN (‘98-’09) in brown, and CEGRN (‘94-’07) blue triangles

– South and Central America (see Fig. 3.4): The solution includes about 128 CORS from the

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results

23

In addition, as indicated in Fig. 3.5, not all solutions were available in full SINEX format.

3.4.2

Fig. 3.4 Sites contributing to the South and Central America velocity solution

One global solution was included in this first test combination. This solution, from the ULR consortium (Universite´ de La Rochelle and IGN/LAREG) is based on a reprocessing of 299 CORS from Jan. 1996 till Jan. 2009 using the GAMIT software (Herring et al. 2007), see Fig. 3.6. Its main objective is the correction of vertical land movements that affect the tide gauge records (W€oppelmann et al. 2009). A key issue to achieve the accuracy requirement of the sea-level application (submm/year) is the realization of a stable and accurate reference frame. The ULR solution therefore includes a global set of reliable reference frame stations from the IGS. It includes three additional years of data and an improved data analysis strategy with respect to the previous solution (W€oppelmann et al. 2009). See details in Santamaria-Gomez et al. (this issue). The stacking of the solutions was done with CATREF. The ULR network has several sites common to the regional solutions.

3.5 Fig. 3.5 Sites and solutions contributing to the East-Asia and Pacific velocity solution. Solutions with full SINEX information are indicated with circles: PCGIAP (‘97-’06) in dark blue, SW Austr. seism. zone (‘02-’06) in light blue, and GeoScience Australia TIGA (‘97-’09) in black. The triangles indicate the stations belonging to networks providing velocity-only solutions: Tibet (‘98-’04) in green, Asia (‘94-04) in pink, Global (‘95-’07) in orange, and Indonesia (‘91-’01) in red

SIRGAS network (Seem€ uller et al. 2009) and covers the period from 2000 till 2009. The GNSS data processing, as well as the cumulative SINEX solution, have been computed using the Bernese V5.0 software (Beutler et al. 2007). – South-East Asia and Pacific (see Fig. 3.5): The solution comprises 1,156 sites resulting of a combination of several sub-regional networks. The combination was done using the CATREF software. In this solution, ensuring the consistent use of station names, particularly four-character identifiers, was a major struggle for many stations.

Global Contribution

Test Combination

The submitted regional networks and the global network were combined using two different approaches. First approach was a step-wise one: first a combination of sites that are present in at least three solutions was done. Based on the common sites from this combination, re-weighting factors for each SINEX file were estimated. The solution is then iterated with the final weights coming from the w2 of the individual solution velocity estimates with respect to the combination, using sites common to at least two solutions. The variance weights vary from about 200 to 1.6. The differences are most probably caused by the usage of different software packages and will be investigated in more detail in the future. The second approach consisted in attaching each regional network to the global network (ULR) using frame-attachment sites. This means that the global sites are not changed by the attachment of the regional sites but the regional networks are adjusted. More details on this approach are given in Davis and Blewitt (2000).

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Fig. 3.6 Sites contributing to the global ULR velocity solution

Figure 3.7 and Table 3.1 present the first comparisons between each of the regional velocity solutions and the global ULR solution. The comparison was done after performing a Helmert transformation on both positions and velocities (estimating translations and scale, together with their rate). The results clearly show that the European and South-Asia & Pacific solutions agree better with the ULR solution than the SIRGAS and African solutions. This does not necessarily mean that the quality of the latter solutions is worse than the first ones, but reflects more the fact that inconsistent discontinuity epochs and solution numbers are used between the last two solutions and the ULR. This is confirmed by the fact that the SIRGAS and African solutions also have a significantly larger position RMS w.r.t. to the ULR solution compared to the European and South-Asia & Pacific solution. Detailed maps of the comparison between the different solutions are available from the WG web site at http://epncb.oma.be/IAG/

3.6

Difficulties

Not all sites included in the contributing solutions have official DOMES numbers assigned by the IERS (International Earth Rotation and Reference Systems Service) and this can make SINEX combination software fail. As a large number of these sites are third party sites without detailed monumentation information, it is impossible to request official IERS DOMES numbers for them. Therefore, the WG implemented a coordinated approach for attributing virtual DOMES

Fig. 3.7 Differences between global ULR velocity solution and each of the regional velocity solutions after a Helmert transformation. In purple: vertical velocity differences and in blue: horizontal velocity differences. Top: Europe, middle-left: Africa, middleright: South and Central America, bottom: East-Asia and Pacific

numbers. Moreover, in the case of duplicate station names, a new station identification and virtual DOMES number was assigned in a coordinated way,

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results

avoiding overlaps and inconsistencies between the different regions. Typically, when a position change occurs at a CORS, two different positions are estimated: one before the discontinuity epoch and one after it. Independently of the fact that the position change is associated with a velocity change or not, most stacking is done in a way to estimate, in addition to the two position solutions, also two velocity solutions. Separate velocities may also be estimated but can be linked through constraints if they are statistically compatible, as usually the case for instrument-related discontinuities. This principle is illustrated in Fig. 3.8. When discontinuities occur at reference frame sites or at sites common to different solutions, it is imperative that the same discontinuity epochs and solution numbers are applied during the analysis Table 3.1 Agreement between global ULR velocity solution and each of the regional velocity solutions after a Helmert transformation Solution

RMS Pos. (mm) Hor. Up. Europe 1.68 2.58 Africa 4.54 4.14 South & Central America 3.85 4.41 South-Asia & Pacific 2.12 3.83 1997

1998

1999

2000

North-component

15 7.5 0 -7.5 -15

# Common (excluded sites) 43(10) 12(2) 25(3) 26(13)

2001

2002

before combining the solutions. As in this first test, cumulative position + velocity solutions have been combined, the WG asked the analysts producing these solutions to apply the station discontinuities identified by the IGS/ITRF (ftp://macs.geod.nrcan.gc.ca/ pub/requests/sinex/discontinuities/ALL.SNX). However, the treatment of discontinuities and velocity changes are subject to interpretation and analysis groups may not necessarily agree on discontinuity epochs. In practice, the different contributors did not strictly follow the IGS discontinuities, obviously influencing the estimated positions as well as velocities and most probably causing some of the outliers seen in Fig. 3.8. This exercise demonstrated the need to come up with a consensus on the discontinuities. To test a collaborative approach, a first attempt was made to merge the discontinuities reported by the different groups in one file which can be edited and maintained by the different groups. In addition, to the “bookkeeping” problems described above, some sub-regional solutions consisted of precise velocity estimates with only approximate coordinates. The implication is that inter-site correlations (not always negligible) are neglected which caused failure of some combination software. Other numerical instabilities were seen due to the equating (or heavily constraining) of velocities before and after a position jump. 2003

2004

2005

2006

2007

2008

2009

15 7.5 0 -7.5 -15

Discontinuity Epoch

[mm]

[mm]

1996

15 7.5 0 -7.5 -15

RMS Vel. (mm/year) Hor. Up 0.28 0.44 0.92 1.24 0.74 1.26 0.22 0.47

25

East-component

15 7.5 -7.5 -15

25

25

0

0

[mm]

-25

-25

-50

Position Solution 1

-50

Position Solution 2

-75

-75

-100

Up-component

-125 850

900

950

1000

-100

Heavily constrained velocities

-125 1050

1100

1150

1200

1250

1300

GPS WEEK

Fig. 3.8 Principle of the introduction of discontinuity epochs and solution numbers

1350

1400

1450

1500

1550

26

3.7

C. Bruyninx et al.

Summary and Outlook

The IAG Working Group on “Regional Dense Velocity Fields” preformed a first test combination of a set of cumulative velocity solutions from regional and global networks in order to identify the main problems when producing a dense velocity field based on multiple cumulative position and velocity solutions. The test identified the urgent need for a consensus on the attribution of discontinuity epochs for stations common to several solutions. Due to the use of different analysis strategies and software packages by the individual contributors, finding such a consensus is a challenge as most probably not the same discontinuities are seen by different people. In addition, the treatment of the post-seismic signals is also subject to interpretation. A possible way to go ahead for the Working Group could be to combine solutions at the weekly level and only deal with the attribution of the discontinuity epochs at the combination level. This would mean that in a first step the weekly global solutions would be combined with to generate a global core network with reliable velocity solutions. In a second step, weekly combined regional solutions (including the sub-regional solutions providing weekly contributions) could be added to this global core network on a weekly basis resulting in a densified core network which could then be used for velocity estimation using one single set of agreed-upon discontinuities. Finally, the remaining cumulative velocity solutions (for which the WG does not have access to weekly position solutions) could be attached to the cumulative densified core network. This approach is one of the alternative procedures which are presently under discussion within the WG. Acknowledgements The authors of this paper would like to thank the groups who submitted velocity solutions. The full list of contributors is available from http://epncb.oma.be/IAG/

References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007a) ITRF2005: a new release of the international terrestrial reference frame based on time series of station positions and earth orientation parameters. J Geophys Res 112: B09401. doi:10.1029/2007JB004949

Altamimi Z, Sillard P, and Boucher C (2007b). CATREF software: Combination and analysis of terrestrial reference frames. LAREG, Technical, Institut Ge´ographique National, Paris, France Beutler G, Bock H, Brockmann E, Dach R, Fridez P, Gurtner W, Habrich U, Hugentobler D, Ineichen A, Jaeggi M, Meindl L, Mervart M, Rothacher S, Schaer R, Schmid T, Springer P, Steigenberger D, Svehla D, Thaller C, Urschl R, Weber (2007) Bernese GPS software version 5.0. ed. Urs Hugentobler, R. Dach, P. Fridez, M. Meindl, Univ. Bern Blewitt G (2008) Fixed point theorems of GPS carrier phase ambiguity resolution and their application to massive network processing: Ambizap. J Geophys Res 113:B12410. doi:10.1029/2008JB005736 Bruyninx C, Altamimi Z, Boucher C, Brockmann E, Caporali A, Gurtner W, Habrich H, Hornik H, Ihde J, Kenyeres A, M€akinen J, Stangl G, van der Marel H, Simek J, S€ohne W, Torres JA, Weber G (2009). The European Reference Frame: Maintenance and Products, IAG Symposia Series, “Geodetic Reference Frames”, Springer, vol 134, pp 131–136, DOI: 10.1007/978-3-642-00860-3_20 Davis Ph, Blewitt G (2000) Methodology for global geodetic time series estimation: A new tool for geodynamics. J Geophys Red 105(B5):11083–11100 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. J Geod 83:191–198. doi:10.1007/s00190008-0300-3 ´ da´m J, Ro´zsa S (eds) (2008). The Drewes H, Hornik H, A geodesist handbook 2008. Journal of Geodesy, Springer, 82 (11):661–846 Fernandes RMS, Miranda JM, Meijninger BML, Bos MS, Noomen R, Bastos L, Ambrosius BAC, Riva REM (2007) Surface velocity field of the Ibero-Maghrebian Segment of the Eurasia-Nubia plate boundary. Geophys J Int 169:315–324. doi:10.1111/j.1365-246X.2006.03252.x Herring TA, King RW, McClusky SC (2007) Introduction to GAMIT/GLOBK, Release 10.3, Mass. Instit. of Tech., Cambridge Santamaria-Gomez A, Bouin M-N, and W€oppelmann G. An improved GPS data analysis strategy for tide gauge benchmark monitoring. (Submitted) Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at http://www.sirgas.org/index.php?id¼97 W€oppelmann G, Letetrel C, Santamaria A, Bouin M-N, Collilieux X, Altamimi Z, Williams SDP, Martin Miguez B (2009) Rates of sea-level change over the past century in a geocentric reference frame. Geophys Res Lett 36:L12607. doi:10.1029/2009GL038720 Zumberge J, Heflin M, Jefferson D, Watkins M, Webb F (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102:5005–501

4

Enhancement of the EUREF Permanent Network Services and Products €hne, A. Kenyeres, G. Stangl, C. Bruyninx, H. Habrich, W. So €lksen and C. Vo

Abstract

This paper describes the EUREF Permanent Network (EPN) and the efforts made to monitor and improve the quality of the EPN products and services. It is shown that the EPN is becoming a multi-GNSS tracking network and that the EPN Central Bureau and the Analysis Centers are preparating to include the new satellite signals in their routine operations. Thanks to the EPN Special Project on “Reprocessing”, set up early 2009, EPN products with much better quality and homogeneity will be generated. The Special Project on “Real-time analysis” will improve the reliability of the EPN real-time data streams and develop new EPN real-time products.

4.1

C. Bruyninx (*) Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium e-mail: [email protected] H. Habrich W. S€ ohne Bundesamt f€ur Kartographie und Geod€asie, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany A. Kenyeres FOMI Satellite Geodetic Observatory, P.O. Box 585, 1592 Budapest, Hungary G. Stangl Department of Satellite Geodesy, Institute of Space Research, Schmiedlstrasse 6, 8042 Graz, Austria C. V€olksen Bayerische Kommission f€ ur die Internationale Erdmessung, Bayerische Akademie der Wissenschaften, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany

Introduction

The IAG (International Association of Geodesy) Regional Reference Frame sub-commission for Europe, EUREF, is responsible for defining, providing access and maintaining the European Terrestrial Reference System (ETRS89), which is recommended by the European Commission for use in all EU member states (Bruyninx et al. 2009a). In 1996, EUREF created the EUREF Permanent Network (EPN) based on a partnership with site operators of permanent GNSS sites who are willing to share their data with the public. The EPN cooperates closely with the International GNSS Service (IGS, Dow et al. 2005); EUREF members are participating to the IGS Real-Time Pilot Project, the IGS GNSS Working Group, the IGS Antenna Calibration Working Group and the IGS Infrastructure Committee. The EPN network contains more than 220 stations (as of October 2009) and its GNSS observations are used extensively by the public, national mapping agencies, and researchers (Bruyninx 2004). EPN

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C. Bruyninx et al.

stations provide daily (mandatory) and hourly (93% of the stations) data in the RINEX format as well realtime data (48% of the stations) in RTCM, RTIGS or raw formats where 15-min high-rate RINEX files are derived from. Two Regional Data Centres (at BKG and OLG) provide access to all hourly and daily data, one data centre (at EPN Central Bureau) to provides access to historical daily EPN observation data, and one regional broadcaster (www.euref-ip.net at BKG) broadcasts all real-time EPN data streams. 16 Local Analysis Centers (LAC) deliver weekly (some also daily and hourly) site position estimates in SINEX format and hourly zenith path delays. These results are the basis to generate the following EPN products: – Weekly combined site positions (in SINEX format) – Daily combined site zenith path delays – Cumulative site positions and velocities, in both the ITRS (International Terrestrial Reference System) and ETRS89, and a discontinuity table – Fully analysed residual position time series The day-to-day management of EPN is performed by the EPN Central Bureau (http://epncb.oma.be). In this paper we describe the efforts recently undertaken within the EPN to keep the pace with new user demands and GNSS modernization, and to continuously improve the EPN products and services.

4.2

Modernization of the EPN

4.2.1

Tracking Network

Since 1996, the EPN has grown from a network of continuously observing dual frequency GPS stations to a network where 49% of the stations (Oct. 2009) are observing GPS as well as GLONASS satellites. In addition, 48 EPN stations have the GPS L2C tracking capability and 31 are equipped with an L5-capable receiver. The new observation types imply switching to RINEX 3 (e.g. ftp://ftp.unibe.ch/aiub/rinex/rinex300. pdf) and an update of the quality checking routines. Since Dec. 2006, each antenna/radome introduced in the EPN (new stations or replacements at existing stations) must have true absolute calibrations. Thanks to this requirement, the number of antenna/radome combinations without true absolute calibrations decreased from 31 to 18%. In addition, more than 90% of the new antennas installed within the EPN are multi-GNSS antenna. From these, 75% are designed for observing GPS and GLONASS signals

and 25% are in addition Galileo-ready. Taking into account that due to local site conditions and near-field multipath antenna/radome replacements continue to introduce jumps in the station positions (even with absolutely calibrated antenna/radome combinations) and that the large majority of new EPN antennae are multi-GNSS capable, the EUREF Technical Working Group (TWG) -which is the EUREF Steering Committee- strongly recommends all station managers, in the case an antenna has to be replaced, to replace it directly with a multi-GNSS antenna.

4.2.2

Data Analysis

The large majority of the EPN Analysis Centers uses data analysis software from third parties and therefore depends on the multi-GNSS policy of these software providers. In practice, 15 of the 16 LAC use GNSSanalysis software that is GLONASS-ready. However, even while EPN stations have been observing GLONASS signals since several years, only 4 of these 15 EPN Analysis Centers presently include GLONASS observations in their data analysis. But, the majority of the LACs plan to start with GPS +GLONASS data analysis within the next year. To guarantee consistency with the IGS, EUREF asks the EPN LACs to use IGS products (e.g., satellite orbits & clocks, and antenna calibrations) within their data analysis. However, to accommodate the missing combined GPS/ GLONASS IGS orbits, most of the four LACs processing GLONASS use the fully consistent combined GPS/GLONASS orbit files from individual IGS ACs, such as CODE (Dach et al. 2009). In addition, contrary to the IGS, the EPN has allowed the introduction of individual receiver antenna calibrations. They allow to introduce new antenna-specific calibrations while, to maintain the consistency of the reference frame, the type calibrations cannot always be updated.

4.3

Quality of the EPN Products

4.3.1

IGS Global Combination of Regional Networks

The Massachusetts Institute of Technology (MIT), Cambridge, USA is an IGS Analysis and Associate Analysis Centre. It generates the global MIT T2 solution as a combination of eight IGS ACs solutions

4

Enhancement of the EUREF Permanent Network Services and Products

(Herring 2008). In addition, MIT also generates the MIT T2 RNAAC solution by adding three regional solutions (EPN, SIRGAS -Sistema de Referencia Geoce´ntrico para Las Ame´ricas- and GSI –Geographical Survey Institute-) and one global solution (CNES Centre National d’Etudes Spatiales- Toulouse) to the MIT T2 solution. The delay of the publication of the mentioned two products is 8 weeks after the end of observations and it requires publishing the weekly EPN combination well in time. The global solution from CNES largely overlaps with the MIT T2 solution. The EPN, SIRGAS and GSI solutions densify the IGS network in Europe, South America and Japan. The coordinate comparison between the EPN and the MIT T2 solutions may be considered as a kind of accuracy estimate and interpreted as a measure of the global alignment of the EPN network. The computation of such r.m.s. numbers for 10 weeks (week 1,511–1,520) confirm that the global alignment is in the range from 2 to 4 mm.

4.3.2

Residual Position Time Series

The main target of the EPN coordinate time series analysis and monitoring is to strengthen the EPN as a geodetic reference network and to offer various products for geodesists and geophysicists. Using the CATREF software (Altamimi et al. 2007b), each 15 weeks, updated EPN cumulative solutions are released (while each 5 weeks an internal solution is created) based on the weekly combined EPN SINEX solutions. During the time series analysis all station specific events (coordinate outliers and discontinuities), confirmed by the log files are identified and considered. A cumulative solution is associated with the following products: – The EPN cumulative solution in SINEX format tied to the actual ITRS reference frame realization (currently ITRF2005) using minimum constraints – EPN station positions and velocities, as the most accurate and up-to-date solutions for the EPN sites. They are used for the maintenance of the regional densification of the ITRFyy between two releases and also for the maintenance of the ETRS89 (see Sect. 4.2) – An up-to-date list of station discontinuities fully harmonized with the IGS/ITRF discontinuity table – Residual coordinate time series as the Helmert difference between the positions in the cumulative

29

solution and the ones in the weekly input SINEX solutions – Harmonic analysis of the time series to detect seasonal coordinate variations – Sophisticated statistical analysis of the position residuals using the CATS software (Williams 2008) to estimate reliable velocity uncertainties and station-specific noise characteristics More information on these products is available from the EPN CB web page at http://www.epncb.oma.be/ _dataproducts/products/timeseriesanalysis/ (Fig. 4.1).

4.3.3

Tropospheric Zenith Path Delays

Since June 2001 (GPS week 1108) the EPN LACs are delivering the estimated Zenith Path Delay (ZPD) parameters to the BKG Data Centre. Daily files (in the so-called SINEX TRO format) are produced on a weekly basis, together with the weekly coordinate solution. These ZPD-estimates were originally combined using a procedure based on the strategy developed for the IGS (Gendt 1997), but since then several changes have been introduced regarding, e.g., reference frame realization, software versions, processing options, absolute antenna calibration, etc. The results have been improved step by step, leading to an internal precision of 2–3 mm ZPD (Fig. 4.2). The most significant progress can be seen from November 2006 on (GPS week 1400) when the absolute antenna models were introduced, especially for stations in the south of Europe. Thanks to the Memorandum of Understanding between EUREF and EUMETNET (Pottiaux et al. 2009) EUREF has now access to radiosonde observations and synoptic data. Applying these data, GNSS processing and analysis may be improved in the future. Moreover, a validation of the GNSS-derived ZPD parameters by those derived from radiosonde observations and vice versa can be performed. Figure 4.3 shows the time series of the differences between ZPD parameters derived from radiosonde data and the EPN combined solution for the station YEBE. One can see that the jump in November 2006 reduced the bias between both observables and the scattering increased during summer time. In 2008 the EPN Special Project “Troposphere Parameter Estimation” was closed and the ZTD processing moved towards routine operation.

30

C. Bruyninx et al.

Fig. 4.1 Residual position time series of the EPN station CHIZ. Top: raw time series; bottom: cleaned time series after removal of seasonal signals and introduction of multiple position and velocity estimates

4.4

Maintenance of and Access to ETRS89

4.4.1

Historical EUREF Campaigns

Immediately after defining the ETRS89 and realizing the ETRF89 (with ETRF, the EUREF Terrestrial Reference Frame), European countries started the

densification of the ETRF using GPS campaigns. These campaigns had typically a minimal duration of three full days so that the comparison of the daily position estimates allowed to get an impression of the precision of the positions. Based on this, EUREF campaigns are divided in three classes: Class A is (only for permanent stations) requiring a 1 cm r.m.s. for the position over a time period exceeding the

4

Enhancement of the EUREF Permanent Network Services and Products

Fig. 4.2 Daily ZPD bias time series from EPN combination for YEBE

Fig. 4.3 Time series of differences between radiosonde and GNSS ZPD parameters for YEBE

campaign period. A class B campaign requires an r.m.s. of 1 cm for the estimated positions at the epoch of observations, whereas Class C means the precision is worse than 1 cm. Not all of these campaigns have been accepted by EUREF, but most of them have been adopted by the National Mapping Agencies of the different countries (Fig. 4.4). Some countries have in the mean time replaced the old campaigns of the 1990s by newer ones, using a set of permanent stations. These pseudo-campaigns are usually based on 1 week of permanent observations. Campaign coordinates are valid for the mean epoch of the measurements within the then used reference frame, e.g., ETRF2000. The advantage of campaigns is that a number of national markers get coordinates validated by an international organization.

4.4.2

Densification of the ETRF

Only a selected number of EPN sites (mostly the ones belonging to the IGS) have coordinates included in recent ITRF realizations released by the IERS

31

(International Earth Rotation and Reference Systems Service). In addition, the latest realization of the ITRS, ITRF2005 (Altamimi et al. 2007a) is based on observations from space geodetic techniques (GNSS, DORIS, VLBI, SLR) up to December 2005 and does not take into account any of the IGS/EPN data gathered after Jan 1st, 2006. The problem of relative antenna models, the limited number of stations, and the lack of frequent updates consequently restricts the use of the ITRF for EUREF densifications The antenna modeling problem will be resolved with the release of the ITRF2008 (expected for 2010) which will be compatible with absolute GNSS antenna models. To take full advantage of the EPN and its most recent GNSS observation data, the EUREF TWG decided, to release regular official updates of the ITRS/ETRS89 positions/velocities of the EPN stations. A first step in this process consisted in a densification of the ITRF2005 using all EPN data up to Dec. 2005 (the same observation period as covered by the ITRF2005). Since early 2009, this realization is updated each 15 weeks. The advantage of the regular update is that the most recent EPN results are taken as much as possible into account. In order to provide the most reliable products, the EPN stations are categorized taking the station quality and the length of available observation span into account (Kenyeres 2009) (Fig. 4.5): – Class A stations with positions at the 1 cm precision and velocities at the 1 mm/yr precision at all epochs. – Class B stations with positions at the 1 cm precision at the epoch of minimal position variance of each station; where no velocities are provided. Exclusively the Class A stations can be used as reference stations for the computation of EUREF campaigns and densification of the ETRF (see Bruyninx et al. 2009b).

4.5

Recent Initiatives

4.5.1

Real-Time Analysis Project

At the end of 2007, the EUREF-IP pilot project for real-time data streaming was successfully transferred to routine operation. Within this pilot project the EPN

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C. Bruyninx et al.

Fig. 4.4 Geographical distribution of accepted EUREF campaigns Fig. 4.5 In green: EPN class A stations; in red: EPN class B stations (status Sept. 2009)

guidelines for real-time data have been developed (Dettmering et al. 2006) and disseminating streams via NTRIP (Weber et al. 2005) became a widely used and implemented feature. From Jan. 2006 to

May 2009 the number of EPN real-time stations grew from 16 to 103 and the number of registered users at EUREF-IP broadcaster at BKG grew from 500 to 1,250 . To pursue this success, the EUREF

4

Enhancement of the EUREF Permanent Network Services and Products

TWG launched a new Special Project “EPN RealTime Analysis” with the primary goals to develop and set-up a re-dissemination of GNSS real-time data/products in Europe via NTRIP broadcasters, validate satellite orbit and clock correctors to broadcast ephemeris and establish backups for all critical realtime service components. In that frame, a new realtime dissemination concept was developed (S€ohne et al. 2009). While today EPN stations stream realtime data to EUREF-IP broadcaster maintained by BKG, the goal is to maintain a series of top-level casters distributing the real-time data and sharing workload. This activity will be supported by a number of relay casters distributing real-time data to, e.g., special communities. In the final stage it is envisaged that almost every reference station is streaming realtime data to at least two different broadcasters to overcome the “single point of failure” issue.

4.5.2

Reprocessing Special Project

Reprocessing of regional networks (V€ olksen 2008) or the complete EPN became an important issue during recent years due to the availability of reprocessed orbit and clock products (Steigenberger et al. 2006). A reprocessing of the complete EPN has been done in 2008 by the MUT (Warsaw) and ROB (Brussels) LAC, clearly demonstrating an improvement of the EPN time series which were previously inconsistent due to analysis and modeling changes (Kenyeres et al. 2009). In order to coordinate the reprocessing of the full EPN between all EPN LAC a new EPN Special Project was created at the EUREF TWG meeting held in Budapest on Feb. 26–27, 2009 (V€ olksen 2009). During its pilot phase, from 2009 till 2010, a test reanalysis of the year 2006 will be performed in order to test and develop new strategies and standards for the data analysis. It is expected that each LAC setup the facilities for the reprocessing of the 2006 data before the end of 2009. At the end of the pilot phase, it is expected to have identified suitable sets of reprocessed products, which will be generated during re-analysis as well as the day-to-day analysis. The next step will then consist of a re-analysis of the complete EPN data set (1996-200x) applying the new strategies and standards. The combination of the different solutions by the analysis coordinator will provide

33

reports and feedback to the working group and the LACs.

4.6

Summary

In response to evolving user needs and new satellite signals, the EPN is continuously improving its tracking network, products and services. Examples are the new EPN real-time analysis and reprocessing special projects, and the fact that new cumulative EPN coordinates are now updated each 15 weeks. While the EPN tracking network is preparing to track new and modernized GNSS signals, monitoring and processing the new and modernized GNSS signals imposes the usage of updated exchange formats, antenna calibrations, consistent multi-GNSS satellite orbits and clocks, and requires additional software developments. For several reasons, the EPN Central Bureau (resp. Analysis Centers) are not yet ready for a real multi-GNSS data monitoring resp. analysis, but this situation is expected to improve dramatically in the next year.

References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007a) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401. doi:10.1029/2007JB004949 Altamimi Z, Sillard P, Boucher C (2007b) CATREF software: combination and analysis of terrestrial reference frames. LAREG, Technical, Institut Ge´ographique National, Paris, France Bruyninx C (2004) The EUREF Permanent Network: a multidisciplinary network serving surveyors as well as scientists. GeoInformatics 7:32–35 Bruyninx C, Altamimi Z, Boucher C, Brockmann E, Caporali A, Gurtner W, Habrich H, Hornik H, Ihde J, Kenyeres A, M€akinen J, Stangl G, van der Marel H, Simek J, S€ohne W, Torres JA, Weber G (2009a) The European reference frame: maintenance and products, IAG Symposia Series, “Geodetic Reference Frames”, vol 134. Springer, Heidelberg, pp 131–136. doi: 10.1007/978-3-642-00860-3_20 Bruyninx C, Altamimi Z, Caporali A, Kenyeres A, Lidberg M, Stangl G, Torres JA (2009b) Guidelines for EUREF Densifications, ftp://epncb.oma.be/pub/general/Guidelines_ for_EUREF_Densifications.pdf Dach R, Brockmann E, Schaer S, Beutler G, Meindl M, Prange L, Bock H, J€aggi A, Ostini L (2009) GNSS processing at CODE: status report. J Geodesy 83(3–4):353–366. doi:10.1007/s00190-008-0281-2

34 Dettmering D, Weber G, Bruyninx C, v.d.Marel H, Gurtner W, Torres J, Caporali A (2006) Real-time GNSS in routine EPN operations: concept. http://epncb.oma.be/_organisation/ guidelines/EPNRT_WhitePaper.pdf Dow JM, Neilan RE, Gendt G (2005) The International GPS Service (IGS) celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36(3):320–326. doi:10.1016/j.asr.2005.05.125 Gendt G (1997) IGS Combination of Tropospheric Estimates (Status Report). In: Proceedings of the IGS Analysis Centre Workshop, Pasadena, CA Herring T (2008) RNAAC combinations in the IGS, presented at IGS Analysis Center Workshop 2008, Miami Beach, FL, 2–6 June 2008 Kenyeres A (2009) Maintenance of the EPN ETRS89 coordinates. In: Minutes of EUREF TWG meeting, Budapest, Hungary, 26–27 Feb 2009. http://www.euref.eu/ Kenyeres A., Figurski M, Legrand J, Kaminski P, Habrich H (2009) Homogeneous reprocessing of the EPN: first experiences and comparisons. Bull Geod Geomatics 3:207–218 Pottiaux E, Brockmann E, .S€ ohne W, Bruyninx C (2009) The EUREF – EUMETNET collaboration: first experience and potential benefits. Bull Geod Geomatics 3:269–288

C. Bruyninx et al. S€ohne W, St€urze A, Weber G (2009) Increasing the GNSS stream dissemination capacity for IGS and EUREF. http://epncb.oma. be/_dataproducts/data_access/real_time/BroadcastConcept.pdf Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111(B5):B05402. doi:10.1029/ 2005JB003747 V€olksen C (2008) Reprocessing of a regional GPS network in Europe. In: Sideris MG (eds) Observing our changing Earth: proceedings of the 2007 IAG General Assembly, Berlin, pp 57–64 V€olksen C (2009) Draft Charter for the EUREF Working Group on reprocessing of the EPN. http://epn-repro.bek.badw.de/ Documents/charter_repro.pdf Weber G, Dettmering D, Gebhard H (2005) Networked Transport of RTCM via Internet Protocol (NTRIP). In: Sanso F (ed) Proceedings of the IAG symposia ‘A window on the Future of Geodesy’, vol 128. Springer, Heidelberg, pp 60–64 Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solut 12(2):147–153. doi:10.1007/s1029-0070086-4

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna Ulla Kallio and Markku Poutanen

Abstract

For the next-generation geodetic VLBI network a 1 mm positioning accuracy is anticipated. The accuracy should be site-independent consistent, reliably controlled, and traceable over long time periods. There are a number of remaining limitations. These include random and systematic components of the delay observable itself, various antenna-related errors, and especially a proper handling of local ties at multi-technique sites. At the Mets€ahovi Fundamental Station operated by the Finnish Geodetic Institute there are a CGPS and Glonass receivers, both in IGS network, a SLR (currently under renovation), a DORIS beacon, a superconducting gravimeter, and a 14.5 m radio telescope owned by Mets€ahovi Radio Observatory of the Helsinki University of Technology. Between five and eight geodetic VLBI sessions are conducted annually. We tested a method to simultaneously measure the tie of the VLBI antenna to the GPS network by tracking the movement of two GPS antennas attached to the radio telescope during geodetic VLBI sessions. We used kinematic trajectory solutions of the two GPS antennas to calculate the orientation and the reference point of the VLBI antenna. In this paper we describe the data acquisition, calculation model, some error sources and test results of four campaigns. The position of the reference point is time, temperature, antenna elevation and azimuth dependent. We propose that in the future, the position should be tracked permanently during geo-VLBI campaigns with attached GPS antennas.

5.1

U. Kallio M. Poutanen (*) Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02431 Masala, Finland e-mail: [email protected]

Introduction

The IERS Working Group on co-location and local ties gives recommendations and guidelines for tie vector measurements, computation and transformation for multi-technique sites in the global reference frame (IERS 2005, 2009). The idea of local tie measurements is to solve for the reference points and orientations of the all instruments in a fundamental station with

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35

36

respect to one common marked point or solve for the reference point and orientation of one instrument relative to another instrument in a local reference frame. The tie vector must then be transformed to the global reference frame. Local ties should be determined at regular intervals. There are different practices and recommendations on the repeat rate, ranging from twice per year up to several years. How often ties can be completed in practice depends on the workload and difficulties of the measurement. It is a totally different question, how often such ties should be made to reliably detect possible movement or deformation of an instrument. The measurement methods in different ties depends on the case and the instruments used in tracking have varied from tachymeters and levelling instruments or laser trackers to GPS (Dawson et al. 2007; IERS 2005; ITRF 2009; Sarti et al. 2004; L€ osler and Haas 2009). The methods have been space intersection with two or more theodolites or kinematic or static GPS measurements, and the use of a laser tracker. As pointed out e.g., in (Sarti et al. 2008), there are basically three surveying approaches: a direct method, a hybrid method and an indirect method. In the direct method the reference point of an instrument is an actual geodetic marker which can be surveyed directly. In the hybrid method (mainly applicable to some VLBI and SLR telescopes) the reference point itself is not identified by a geodetic marker but defined by the axes of rotation of the telescope. Some surveying targets are fixed on the telescope to make the axis visible for pointing of the surveying instrument. The indirect method is entirely based on observed positions of targets fixed on solid structure of the telescope. The positions of the targets are measured with surveying techniques when the telescope is turned in different positions. In our approach, the indirect method was applied. Actually there is the same local tie problem with reference points and orientations of the measuring instruments as there is with the VLBI antenna itself. One needs a local reference network around the fundamental station. Another question is how to connect the local network to the global reference frame reliably with controlled error estimates, and how to combine different measurements together. Additionally, a local geoid model is needed to tie the local measurements in a global frame.

U. Kallio and M. Poutanen

In our measurements the main goal was to determine the vector between the VLBI antenna reference point and the IGS GPS point. Beside that we tied these two points to the benchmarks of the local network. We also determined the orientation and the offset of axes of the radio telescope.

5.2

Modelling the Antenna Rotation

The IVP (Invariant Point) of a VLBI antenna is defined as a fixed point (in a reference frame) to which VLBI observations are referred. The IVP is the intersection of the primary axis with the shortest vector between the primary and secondary axis (Dawson et al. 2007). The radio telescope is an instrument with two rotation axes, and like the other instruments with axes (e.g., tachymeters, laser trackers, cameras, SLR) has errors such as eccentricity or offset between axes and axes misalignment. In general the directions of axes are not perpendicular. The secondary axis (elevation) rotates about the primary axis (azimuth) and they don’t intersect. In this case, the reference point is the projection of the secondary axis on the primary axis. IVP is not a marked point and no line of sight from outside the telescope to the reference point is possible. The determination of the IVP coordinates must be done indirectly by tracking the positions of some points on the antenna structure in different antenna positions. Let’s assume that the antenna is rotated around its axes. If the azimuth is kept fixed then a point in the antenna structure rotates about the elevation axis and the track forms a circle in space. Similarly, if the elevation is locked then the point rotates about the azimuth axis. In an ideal case, all the tracks together form a sphere. In recent determinations of the IVP, the model of three dimensional circles has been the most popular. (Dawson et al. 2007; ITRF 2009) The order of observations has been planned so that the circle fitting is possible. Observing all circles which are needed for reliable fitting means a large number of VLBI antenna positions. The whole process is typically very time consuming. A different approach for calculating the reference point was presented by L€osler (L€osler and Hennes 2008; L€osler and Haas 2009). In the L€osler’s model

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna

37

one computes the transformation of the tracked coordinates from the instrument coordinate system to the VLBI-antenna coordinate system. It uses the telescope angle readings as observations but the order of observations is arbitary. Therefore the model can be applied also to the sparse and scattered data and it includes antenna axes errors.

5.3

€hovi Model The Metsa

We developed a model to be used in connection of kinematic GPS measurements which were made during the normal use of the VLBI telescope. GPS data were collected during an ordinary 24-h geodetic VLBI session. Therefore, paths of the two GPS-antennas attached on the radio telescope dish do not form such kind of circles that could be used in three dimensional circle fitting model to solve for the azimuth and the elevation axes and the IVP coordinates. The idea of using telescope angles as observations is the same as in L€ osler’s model, but we reformulated the movement of the antenna points and the rotation matrices. Our model is suitable also for different types of telescope mounting other than the elevationazimuth mount described here. The basic assumptions of our model are that points in antenna structure rotate about the elevation axis and the elevation axis rotates about the azimuth axis. The axes need not intersect or be perpendicular. The position vector of the GPS-antenna X (in an arbitrary reference frame) is the sum of three vectors: the position vector of the reference point X0, the axis offset vector (E-X0) rotated by angle a about the azimuth axis a and a vector from the eccentric point E to the antenna point p rotated about the elevation axis e by angle b and about the azimuth axis by angle a (Fig. 5.1). Unknown parameters are X0, E, a, e, and p. Observations are coordinates X for each antenna point and epoch, and VLBI antenna angle readings a and b for every epoch. The estimated values of E, e and p are those of an antenna initial position which may be zero for both angles. The basic equation of our model is X0 þ Ra;a ðE X0 Þ þ Ra;a Rb;e p X ¼ 0

(5.1)

Fig. 5.1 The model parameters in a local reference frame. IVP denotes the Invariant Point, to which VLBI observations are referred. Vectors X, p1 and p2 denote the position vectors of the two GPS antennas attached on side of the VLBI antenna

Rotation matrices Ri,j are formed by applying the Rodrigues’ rotation formula which uses rotation axis and angle. The rotated X is Xr ¼ cosðaÞX þ ð1 cosðaÞÞaaT X þ sinðaÞa X (5.2) The rotation matrix is then 0

1 Rða; aÞ ¼ cosðaÞ@ 0 0 0 xx þ ð1 cosðaÞÞ@ xy xz 0 0 z þ sinðaÞ@ z 0 y x

1 0 0 1 0A 0 1 1 xy xz yy yz A yz zz 1 y x A 0

(5.3)

where a is the rotation angle, and the components of the unit vector of a rotation axis a are x, y and z. In our model the angles in the rotation matrices are the VLBI antenna azimuth and elevation angle readings. Azimuth and elevation axes are described with unknown unit direction vectors. There are four parameters in rotation matrix: three for axis and one

38

U. Kallio and M. Poutanen

for the angle, although only three are independent parameters because the axis is a unit vector. Therefore, we need two condition equations between parameters in our adjustment, one for each axis. aT a 1 ¼ 0

(5.4)

e e1¼0

(5.5)

T

The other two conditions between parameters are needed for finding the shortest vector between the azimuth and elevation axes. The offset vector between the axes is perpendicular to both axes. ðE X0 ÞT a ¼ 0

(5.6)

ðE X0 ÞT e ¼ 0

(5.7)

We do not need any condition equation between the GPS-antenna points: the model itself includes the condition of the constant distance between the points. The number of unknown parameters in the model is u ¼ 12 þ m 3

(5.8)

and the number of observations n¼tm3þt2

1 0 0 0 0 0 FðX0 ; E ; a ; e ; p ; ai ; bi ; X1 Þ C B .. yi ¼ @ A . 0

0

(5.10)

The solution of unknown parameters is reached by iteration in a linearized least squares mixed model with conditions between the parameters: 0 t h 11 i P T 1 T 1 T x D A ðB P B Þ A i i i i i A ¼@ i k h D 0 0 t h i1 P T 1 T Ai ðBi P1 i Bi Þ yi A @ i GðX0 0 ; E0 ; a0 ; e0 ; p0 Þ (5.11)

0

0

0

(5.12)

0

FðX0 ; E ; a ; e ; p ; ai ; bi ; Xm Þ

The matrix G includes condition equations with approximate values of parameters. 0

1 g1 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ B g2 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ C C G¼B @ g3 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ A g4 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ

(5.13)

Jacobian matrices A and B for epoch i are 0 B Ai ¼ B @

(5.9)

where t is the number of epochs and m is the number of points to be tracked. The degree of freedom in our mixed model least squares adjustment with four conditions between parameters is r ¼ 3m t þ 4 u

where xh is the correction to the approximate values of parameters after h iteration and yi includes the basic equation for all points in epoch i with approximate values of parameters. The apostrophes here denote the approximate value.

0 B Bi ¼ B @

@F @X0

@F @X0

@F @azi

@F @azi

@F @E

@F @a

.. .

@F @e

@F @p1

@F @E

@F @a

@F @e

@F @eli

@F @X1

.. .

.. .

@F @eli

0

.. .

0

1 0 .. C . C A (5.14) @F @pm

1 0 .. C . C A

(5.15)

@F @Xm

and D¼

@G @G @G @G @G @G ... @X0 @E @a @e @p1 @pm

(5.16)

The weight matrix of the epoch i is 0

s2a @ Pi ¼ 0 0

0 s2b 0

11 0 0A Ci

(5.17)

s2a and s2b are variances of azimuth and elevation observations, respectively, and Ci is the covariance matrix of coordinate observations in epoch i.

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna

5.4

Tie Measurements Using Kinematic GPS

5.4.1

Installation of the GPS Antennas

Due to restrictions on telescope time and the anticipated time-consuming procedure with traditional surveying methods or static GPS, we ended up testing a kinematic approach. We had two Ashtech Dorne Margolin-type GPS antennas attached to the edge of theVLBI antenna and we gathered data with two Ashtech Z-12 receivers during the following geo-VLBI campaigns: EUROPE-96 (27–28. 11.2008), IVS-T2059 (16–17.12.2008), EUROPE-97 (19–20.1.2009), EUROPE-98 (25–26.3.2009), and IVST2061 (7–8.4.2009). The GPS-antennas were attached to rotating holders on both sides of the radio telescope dish, and they had the counterweights that forced them to point to the zenith regardless of the position of the radio telescope. The GPS antenna reference point (ARP) was in the rotation axis of the antenna holder (Fig. 5.2). The antennas were assembled so that the rotation axis of the holder of a GPS-antenna pointed to the second antenna and this direction was approximately the same as the elevation axis. We measured the angle between the antenna north arrow and the direction of the elevation axis. It is possible to calculate the GPS antenna orientation and phase centre corrections for every epoch. For this we can use the results of an individual antenna calibration.

5.4.2

39

same time we gathered GPS data on six pillar points to form a local network. In this paper we concentrate on the tie to the IGS point. Kinematic co-ordinate solutions were computed with the Trimble TTC software on the fly strategy. We accepted only the fixed solutions with the pdop values less than 8 and the standard deviations of coordinates less than 0.1 m. If the left hand side GPS antenna was rejected we also rejected the right hand side of the same epoch for symmetry. Because the mutual distance of the antennas did not change, we rejected both observations where the distance between the antennas differed more than 0.05 m from the median value. The height difference between the antennas was allowed to differ from the median not more than 0.1 m. Azimuth and elevation angles of the VLBI antenna were dumped into binary disk files every 15 s and v tagged with UTC time, one file for each epoch. Subsequently, those files were afterwards combined into one ASCII file. Due to a system error during the first campaign, the actual VLBI antenna information was missing. The telescope angle readings are used as observations. The trajectory co-ordinate data of the GPSantennas and the VLBI-antenna angle readings must be combined. The synchronization error of one second in time may cause up to 0.01 (depending on the angular velocity) orientation error to the estimated elevation axis. We obtained good data only when the VLBI-antenna was tracking a radio source. When it changed the source the movement was too quick for fixed solutions.

Preparation of the Kinematic Data

The GPS observation interval was 30 s. This is also the interval for the Mets€ahovi IGS GPS receiver which was our main target for the tie measurement. At the

Fig. 5.2 GPS antenna holder on the VLBI antenna

5.4.3

GPS Antenna Phase Center Variation

The orientation of the GPS antennas on the VLBI antenna was changed when the VLBI antenna rotated. We used individually calibrated antennas to eliminate the effect of antenna phase center variation in the computation. Only in one azimuth position the orientation of the GPS antennas was the same as used in the calibration. If we were using the existing antenna calibration tables in the kinematic solution in the ordinary way it would cause systematic errors in the coordinates of GPS antennas. This would propagate

40

U. Kallio and M. Poutanen

to the solution of the reference point coordinates, and especially to the axis offset. One possible solution for the problem is to use only elevation dependent tables for phase variations and let the north and east components of the offset vectors be zeros. If the distribution of VLBI antenna positions over azimuth angles is homogeneous and number and directions of satellites is equal to every direction, then the systematic error will be canceled in the calculation of the reference point. However, it affects still in the solution of axis offset. Unfortunately the distribution of VLBI antenna positions was not equal because of the distribution of the tracked radio sources during the geodetic VLBI session. The VLBI antenna dish also blocked observations so that it was not possible to get fixed solution for trajectory points in every azimuth and elevation positions. We tested the possibility to apply antenna correction tables from the absolute calibration directly to the RINEX data and then use the antenna calibration tables with zero offsets and zero phase variations during the GPS data processing. The GPS antenna orientation was calculated for every epoch from known angle between elevation axis and the north arrow of the GPS antenna and the azimuth angle reading of the telescope. The antenna offset in the direction of a satellite was then subtracted in the RINEX code and phase data.

about 2/3, but we had still data enough for the final adjustment (Table 5.1).

5.4.4

Table 5.2 The local tie vectors between METS GPS and VLBI reference point and the axis offsets in the four campaigns in meters

Computing and Outlier Detection

We programmed the linearized least squares mixed model with conditions between parameters using Octave language. Over two iterations the residuals were checked and outliers removed. Because there still existed some bad data after the preprocessing and filtering we rejected those observations which had associated residuals larger than 1 m. Also angle readings with residuals larger than 0.1 were rejected. After two more iterations we repeated the rejection procedure with limit values 0.2 m and 0.001 . After this we repeated again the adjustment and rejected those observations which had a standardized residual larger than 3. For symmetry we rejected observations from both sides of the VLBI antenna. The loss of the data during the process was

5.5

Results

The maximum difference between the solutions of the reference point in the four campaigns was 2 mm in each vector component. The dependence of the axis offset values on the GPS antenna orientation changes is clear. The VLBI axis offset vector E-X0 rotates about the azimuth axis at the same time and the same amount as the GPS antenna phase center offset vector. In our basic equation it is impossible to distinguish these offsets from each other. In the Trimble TTC processing the control point was METS with coordinates in ITRF 2005 at epoch 1.1.2009. In the results presented in the Table 5.2. the Table 5.1 The loss of the data during the processing Campaign IVS-T2059 EUROPE-97 EUROPE-98 IVS-T2061

GPS antenna Left Right Left Right Left Right Left Right

IVS- T2059

Trajectory points 2,495 2,674 2,642 2,810 2,676 2,823 2,807 2,787

Points in the final adjustment 809 809 642 642 698 698 499 499

EUROPE-97 EUROPE-98 IVS- T2061

N 37.6086 2 37.6080 3 37.6071 3 37.6094 3 E 122.4009 2 122.4006 2 122.4019 2 122.3995 3 U 14.6781 4 14.6780 5 14.6776 6 14.6786 8 Axis 0.0045 6 0.0050 7 0.0052 9 0.0020 11 offset Differences to ITRF2005 (2009.0) N 0.0021 0.0027 0.0035 0.0012 E 0.0005 0.0008 0.0005 0.0019 U 0.0239 0.0239 0.0234 0.0245 Differences to the determination by Jokela N 0.0061 0.0067 0.0076 0.0053 E 0.0032 0.0028 0.0041 0.0018 U 0.7262 0.7262 0.7267 0.7256

Single numbers denote the standard error, in 0.0001 m. We also show a comparison to the vectors from ITRF2005 (2009.0), and to the earlier determination by Jokela

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna

GPS antenna phase center variations have been corrected in the RINEX data epoch by epoch before the kinematic trajectory computing. For comparison, the vector from the METS GPS ARP to the Mets€ahovi VLBI reference point in ITRF 2005 coordinates at epoch 1.1.2009 was calculated from the ITRF data set. Differences to our results are presented in Table 5.2. The agreement in horizontal components is good but the vertical component differs more than 2 cm. The comparison to the earlier direct determination by Jorma Jokela in 2004 (Jokela et al. 2009) is shown in Table 5.2. The vertical component differs more than 70 cm because the directly achievable point in Jokela’s determination was not the intersection point of the axes but the pinhole on the floor of the upper platform of the VLBI telescope. Beside the local tie vector we computed the antenna axis offset and orientation. The orientation was modeled as unit direction vectors of the axes. The orientation angles of the telescope, non-perpendicularity of axes and the tilt of the azimuth axis was calculated. Because our model doesn’t take into account epoch-wise variation of the azimuth axis tilt, the direction vector of azimuth axis varies between campaigns. The antenna axes offset differs from the offset determined by Leonid Petrov in 2007: 0.0051 0.0037 m (Petrov 2007). The absolute value from the first three campaigns is about the same, but the sign is opposite to that of Petrov.

41

expect more reliable results for the axes orientation and alignment. We have also used traditional terrestrial surveying methods combined with space intersection technique by measuring reflecting targets on antenna construction. This method was very time consuming and we had telescope for that purpose only during bad weather conditions (e.g., heavy rain, snowing) when it was not in normal use. The results will be published in a future paper. The kinematic method has proved to be suitable for monitoring the IVP of a radio telescope simultaneously during a VLBI session. Especially for sites without a radome, there will be less noisy data. Based on our experience, we may propose that GPS tracking should be done permanently during VLBI sessions. This will give an instantaneous tie to the co-located CGPS antenna. This technique should be considered especially for the VLBI2010 plan. Acknowledgements We would like to thank T. Lindfors, A. Mujunen, M. Tornikoski, J. Kallunki, E. Oinaskallio and H. R€onnberg of the Mets€ahovi Radio Observatory of the Aalto University for their invaluable help. Elevations and azimuths of GPS satellites were computed using “wheresat” of GPStk. (http://www.gpstk.org/bin/view/ Documentation/WebHome). The Mets€ahovi model was programmed with Octave language (http://www.octave.org). This work was partly supported by the Academy of Finland project 134952.

References 5.6

Conclusions and Future Actions

The test shows that a millimeter accuracy is possible to achieve in local tie vector determination with kinematic GPS. Our calculation model is suitable for sparsely scattered data as well as for the data with preplanned circles. The angle readings of the telescope must be synchronized carefully with the GPS observations. Systematic errors like GPS antenna phase center variations must be taken into account although they are smaller than the accuracy of the trajectory point in kinematic solution. Future studies will continue with the analysis of the static GPS observation with two hour sessions of each 90 pre-planned VLBI antenna position. Observations that have been made in March and April 2009. We

Dawson J, Sarti P, Johnston GM, Vittuari L (2007) Indirect approach to invariant point determination for SLR and VLBI systems: an assessment. J Geodesy 81(6–8):433–441 IERS (2005) In: Richter B, Schwegmann W, Dick WR (eds) Proceedings of the IERS workshop on site co-location, Matera, Italy, 23–24 Oct 2003. IERS technical note No. 33. Verlag des Bundesamts f€ur Kartographie und Geod€asie, Frankfurt am Main, p 148 IERS (2009) http://www.iers.org/MainDisp.csl?pid¼68-38. (8.10.2009) ITRF (2009) Co-location survey: online reports, in ITRF web site: http://itrf.ensg.ign.fr/local_surveys.php. (8.10.2009) Jokela J, H€akli P, Uusitalo J, Piironen J, Poutanen M (2009) Control measurements between the Geodetic Observation Sites at Mets€ahovi. In: Drewes H (ed) Geodetic reference frames. IAG Symposium Munich, Germany. Springer, Heidelberg, pp 101–106, 9–14 Oct 2006 L€osler M, Haas R (2009) The 2008 local-tie determination at the Onsala Space Observatory. In: Proceedings of the EVGA, 2009, Bordeaux, France, 24–25 Mar 2009

42 L€osler M, Hennes M (2008) An innovative mathematical solution for a time-efficient ivs reference point determination. In: Measuring the changes, 2008. 4th IAG Symposium on Geodesy for Geotechnical and Structural Engineering, Lisbon, Portugal, 12–15 May 2008 Petrov L (2007) VLBI antenna axis offsets. http://geminig. sfc. nasa.gov/500/oper/solve_save_files/2007c.axo (27.10.2009)

U. Kallio and M. Poutanen Sarti P, Sillard P, Vittuari L (2004) Surveying co-located space geodetic instruments for ITRF computation. J Geod 78(3):210–222 Sarti P, Abbondanza C, Vittuari L (2008) Terrestrial Surveying Applied to Large VLBI Telescopes and Eccentricity Vectors Monitoring. 13th FIG symposium on deformation measurement and analysis. LNEC, Lisbon

6

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008, the ignwd08 Solution P. Willis, M.L. Gobinddass, B. Garayt, and H. Fagard

Abstract

In preparation for the computation of ITRF2008, the DORIS IGN analysis center has undertaken the task of a complete reprocessing of all DORIS data from 1993.0 to 2009.0, using all available DORIS data as well as the most recent models and estimation strategies. We provide here a detailed description of the major improvements recently made in the DORIS data processing, mainly in terms of solar radiation pressure, atmospheric drag, gravity field, and tropospheric correction. We address here the impact of the new IGN time series (ignwd08) on geodetic products using comparison to the previous IGN solutions (ignwd04). In particular, previous artifacts, such as 118-day or 1-year periodic errors in the TZ-geocenter solution or in the vertical component of high latitude DORIS tracking stations, have now disappeared, leading to more precise and reliable time series of DORIS station coordinates. Finally, possible future improvements are discussed proposing new investigations for the future.

6.1

P. Willis (*) Institut Ge´ographique National, Direction Technique, 2 avenue Pasteur, 94165 Saint-Mande´, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris Cite´, UFR STEP, 35 rue He´le`ne Brion, 75013 Paris, France e-mail: [email protected] M.L. Gobinddass Institut Ge´ographique National, LAREG, 6-8 avenue Blaise Pascal, 77455 Marne-la-Valle´e, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris Cite´, UFR STEP, 35 rue He´le`ne Brion, 75013 Paris, France B. Garayt H. Fagard Institut Ge´ographique National, Service de Ge´ode´sie et de Nivellement, 2 avenue Pasteur, 94165 Saint-Mande´, France

Introduction

DORIS (Doppler Orbitography and Radiopositioning Integrated on Satellite) is one of the four geodetic techniques participating in the realization of the International Terrestrial Reference System (ITRS) (Willis et al. 2006). Figure 6.1 presents the current DORIS permanent network, demonstrating a dense and homogenous geographical distribution of 57 tracking stations (Fagard 2006). Since 2003, an International DORIS Service (IDS) was created in order to foster international cooperation (Tavernier et al. 2002; Willis et al. 2010a). The Institut Ge´ographique National (IGN) is one of the seven IDS Analysis Service, providing products on a weekly basis (Willis et al. 2010b). In preparation of ITRF2008 (Altamimi and Collilieux 2010), a new DORIS time

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_6, # Springer-Verlag Berlin Heidelberg 2012

43

44

P. Willis et al. Table 6.1 Main differences in models and analysis strategies used for the two latest DORIS/IGN weekly solutions Model/strategy Gravity field Tropospheric mapping function Elevation cut-off Solar radiation pressure coefficient Atmospheric drag parameter

ignwd04 GGM01C Lanyi 15 estimated Reset/6h

ignwd08 GGM03S GMF 10 fixed Reset/1 h

Fig. 6.1 DORIS permanent tracking network (as of September 2009)

series (ignwd08) was reprocessed using data from 1993.0 to 2009.0. Since then, new IGN weekly solutions using the same processing strategy are regularly delivered at the IDS data center, on average once a week. The goal of this article is to present the major differences in terms of data analysis between this new reprocessed solution (ignwd08) and the previous DORIS solution (ignwd04), and to provide an overview of current available geodetic products from the IGN Analysis Center (AC): weekly time series of station coordinates, velocity field, terrestrial reference frame and polar motion.

6.2

DORIS Data Analysis

Table 6.1 summarizes the main differences between the ignwd04 and the ignwd08 analysis strategies. A more recent GGM gravity was used (Tapley et al. 2005). C21 and S21 rates were corrected to follow the IERS 2003 conventions (McCarthy and Petit 2004). No annual correction was taken into account in the gravity field coefficients. Using the more recent GMF mapping function (Boehm et al. 2006) allowed us to use DORIS data at a lower elevation without any drawback. Following recent investigations (Gobinddass et al. 2009a, 2009b), solar radiation pressure models were rescaled using a constant empirical parameter per satellite. Atmospheric drag parameters were reset every 1-hour for the SPOT and Envisat satellites (Gobinddass et al. 2010). This new strategy can be used for all days, even during high geomagnetic activity (Willis et al. 2005a). For the previous ignwd04 solution, a specific strategy was earlier required

(resetting the drag parameter for estimation every minute instead of every 6 h). This involved some non-automated processing for these few days, while the new procedure is fully automated.

6.3

Time Series of ignwd08 Station Coordinates

Unlike other IDS Analysis Centers, DORIS data are processed automatically as soon as they appear at the IDS data centers. Weekly station coordinates in SINEX format are available at these data centers within a few hours, and are provided in free-network (loosely constrained) for the IDS combination but also after projection and transformation in the latest ITRF solution (currently ITRF2005 but soon in ITRF2008 as this transformation is straightforward and does not require any DORIS data processing). These results are freely available to the scientific community at the following URL address for further geophysical investigations: http://ids.cls.fr/html/doris/ids-stationseries.php3. This technique provides SINEX results for geodesists (including full covariance information in a loosely constrained terrestrial reference frame). It also provides tabulated results in STCD format (Noll and Soudarin 2006) for geophysicists directly expressed in ITRF2005 (Altamimi et al. 2007). Figure 6.2 provides an example of such results for the Rio Grande station in Argentina. Different colors indicate the different occupations of this station, as related to equipment upgrades. The positions of successive occupation at the same site were tied together using geodetic local tie information. Smaller scatter after 2002.4 indicates an improvement in repeatability when 4 or 5 DORIS satellite are available. No large discontinuity can be noticed, showing a good

6

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008

45

Fig. 6.2 Weekly time series of ignwd08 Rio Grande station coordinates expressed in ITRF2005 (from STCD files). In North (a) and Up (b) (in mm)

agreement between the DORIS results and the geodetic local tie vectors. However a closer inspection may indicate possible problem after first occupation or after data gap (vertical). The East component (not displayed here) is noisier due to the North-South tracks of the sun-synchronous satellites (SPOTs and Envisat), especially for mid-latitude stations. Other authors already showed that previous artifacts in the DORIS time series at 118 days (TOPEX draconitic period) are no longer visible in the vertical component for the high-latitude station when solar radiation pressure models are empirically rescaled (Amalvict et al. 2009; Kierulf et al. 2009). The previous periodic effects visible in the ignwd04 time series were due to an improper handling of the solar radiation pressure (Gobinddass et al. 2009a, 2009b) and were observed in previous time series (le Bail 2006; Williams and Willis 2006) but were never fully appreciated. These problems were rather serious for altimetry as they affected mostly the Zcomponent of the stations, which is the major factor for mean sea level determination (Morel and Willis 2002, 2005; Beckley et al. 2007). Table 6.2 provides some information on annual signals present in DORIS/ IGN time series in the polar region. Table 6.2 demonstrates that the previous annual signals around 10 mm in the ignwd04 time series of vertical coordinates of high-latitude station are not present anymore in the new ignwd08 time series. Currently observed annual signals around 3 mm could easily be explained by real geophysical reasons, as geocenter motion is estimated to be at this level.

Table 6.2 Amplitude of annual signals in DORIS vertical time series (in mm) Station SPJB THUB ADEA ROTA SYOB BEMB

Lat (deg) 78 550 76 320 66 400 67 340 69 000 77 520

ignwd04 13.7 11.6 3.4 6.6 6.5 11.2

ignwd08 2.7 2.7 1.9 3.0 3.9 5.0

A similar problem was also previously detected at 1 year (draconitic periods of SPOT and Envisat, being sun-synchronous satellites) and also disappeared in this new ignwd08 solution (Willis et al. 2010b).

6.4

ign09d02 Derived Velocity Field

At regular intervals (every 6 months to 1 year), we also stack all the available DORIS weekly solutions and provide a cumulative position and velocity solution, making full use of geodetic local ties between successive DORIS occupations, using proper a priori variance, as provided by IGN. The latest IGN velocity field (ign09d02) available at both IDS data centers (CDDIS in USA and IGN in France). In most cases, these results can be explained by plate tectonics (Soudarin and Cre´taux 2006; Argus et al. 2010). However, in other cases, such as the Socorro Island (Mexico), the station displacement is not linear and can be attributed to local volcanic deformations (Briole et al. 2009). While GPS is the

46

P. Willis et al.

key player for geodynamics due to its easy densification, DORIS may still have a role to play in a few cases such as Africa where the geodetic infrastructure is still sparse (Nocquet et al. 2006; Argus et al. 2010). We do not estimate the DORIS-derived velocity more often than every 6 months, as a large number of DORIS observations is already available (16 years) For most stations, formal errors of 0.15–0.30 mm/yr are typical.

6.5

Terrestrial Reference Frame

As our DORIS weekly solutions are provided in free-network form, we can compute for each week the 7-parameters for the TRF (origin, orientation and scale), looking at the best transformation fit into the ign09d02 position/velocity solution already aligned on ITRF2005, but including all DORIS stations. Figure 6.3 displays results for the ignwd08 weekly scales with respect to ITRF2005. No antenna map correction was used for DORIS (Willis et al. 2005b) to map these results toward any ITRF. While the weekly scatter is small, a significant drift can be seen toward ITRF2005 realization. This is currently visible in all results from all DORIS Analysis Centers, sometimes with lower values (Valette et al. 2010) and no convincing explanation has yet been proposed.

Fig. 6.3 Weekly determination of the TRF scale between the ignwd08 solution and ITRF2005 (through ign09d02)

No significant discontinuity can be seen in these results. In our opinion, the discontinuities detected at the end of 2004 by Altamimi and Collilieux (2010) and Valette et al. (2010) may be related to results from other DORIS ACs that could map into the IDS-3 combination, which is the DORIS combination submitted for ITRF2008. Such problems could be related to the end of the TOPEX/DORIS data or to a software modification in the Envisat satellite (Willis et al. 2005b, 2007). The IGN weekly solutions also contain more DORIS stations than the IDS-3 solution, as a preselection was done for IDS-3, based on recommendation from DPOD2005 (Willis et al. 2009). Finally, when transforming into ITRF2005, we do not use the original ITRF2005 coordinates and velocities but our internal ign09d02 long-term solution. We also disregard some DORIS stations (six in total) on a week-byweek basis, taking into account possible temporary problems with these data, as done recently in a more sophisticated way using genetically modified networks (Coulot et al. 2009). It is certainly too early to have a definite conclusion on this difficult problem and more tests are required to understand the exact nature of such possible discontinuities. In Fig. 6.3, a different behavior may be observed for the very early data. This is still under investigation but it could be linked to data availability (only two DORIS satellites : TOPEX/Poseidon and SPOT-2) or

.

6

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008

to a preprocessing problem as detected earlier in the case of SPOT4 for most of the 1998 data and some early 1999 data (Willis et al. 2006).

6.6

Polar Motion

We also update every week polar motion estimation derived using the DORIS data. Tables 6.3 and 6.4 display direct comparisons of these daily DORIS results with the JPL/GPS time series. When considering the full data set (from 1993.0) in Table 6.3, a clear improvement can be seen for the most recent ignwd08 solution. When considering only the best DORIS results, when four or five satellites are available after 2002.4, the improvement is even more pronounced and RMS of 0.5 mas are now achievable (Table 6.4), even if a small bias of 0.3 mas in X still needs to be explained. This is a significant improvement when compared to earlier determination (Gambis 2006): RMS of 1.74 mas for XPole and 0.99 mas for YPole, with a 0.24 offset for recent data (2000.0–2004.0). Part of these improvements is due to the fact that no daily polar rates are estimated in the recent ignwd08 solution. Another improvement is related to systematic errors at the 5.2 day period (SPOT sub-cycle), linked again to the solar radiation pressure estimation as demonstrated in Willis et al. (2010).

Table 6.3 DORIS polar motion external precision compared to IGN/JPL time series (mean removed) XPole RMS (mas) ignwd04 1.864 ignwd08 1.228

YPole RMS (mas) 1.440 1.290

XPole mean (mas) 0.287 0.126

YPole mean (mas) 0.164 0.373

6.7

47

Discussion on Future Improvements

While the ignwd08 solution is still very new and regularly updated, future possible improvements are already considered: • Early analysis of Jason-2 data showed that it could improve the realization of the terrestrial reference frame (Zelensky et al. 2010) and that no effect related to the South Atlantic Anomaly (Willis et al. 2004) is observed in these data. • New DORIS satellites will be launched soon (Cryosat-2 from ESA and Altika from India and France). Direct use of the new DORIS phase and pseudorange (Mercier et al. 2010) should be investigated. • While only minor improvement is expected at the altitude of the DORIS satellite from a GOCEderived gravity field (Visser et al. 2009), timevarying effects , new tide models and AOD (Atmospheric and Ocean De-aliasing) corrections could improve the DORIS orbits. • For the tropospheric correction, VMF (Boehm 2004) could be used instead of GMF. Early tests showed that horizontal tropospheric gradients could be considered for DORIS data processing as well (Flouzat et al. 2009). • Previous studies using Laser data also demonstrated possible time tagging problems in the DORIS data files (Zelensky et al. 2010). • Finally, the recent inter-comparisons between the 7 IDS Analysis Centers (Valette et al. 2010) should certainly lead to new investigations, for example for problems related to the South Anomaly (Bock et al. 2010, Stepanek et al. 2010).

6.8

Conclusions

Period 1993.0–2002.4

Table 6.4 DORIS polar motion external precision compared to IGN/JPL time series (mean removed) XPole RMS (mas) ignwd04 1.387 ignwd08 0.584 2002.4–2008.7

YPole RMS (mas) 0.740 0.525

XPole mean (mas) 0.003 0.289

YPole mean (mas) 0.287 0.012

In conclusion, the new ignwd08 solution is a clear improvement over the previous ignwd04 solution. In particular, due to a better analysis strategy concerning the solar radiation pressure, previous artifacts at 118 days and 1 year have now disappeared in the Zgeocenter as well in the vertical component of highlatitude station time series. Significant improvements were also obtained for polar motion for which 0.5 mas

48

comparison with GPS results can be observed, when 4 or more DORIS satellites are available. Finally, with the use of more recent satellites (Jason-2, Cryosat-2, and Altika) equipped with new digital DGXX equipments, more improvements are already foreseen and currently under investigation. Acknowledgement This work was supported by the Centre National d’Etudes Spatiales (CNES). It is based on observations with DORIS embarked on SPOTs, TOPEX/Poseidon, ENVISAT and Jason satellites. This paper is IPGP contribution number 2593.

References Altamimi Z, Collilieux X (2010) DORIS contribution to ITRF2008. Adv Space Res 45(12):1500–1509 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005, a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth orientation parameters. J Geophys Res 112(B9), art. B09401 Amalvict M, Willis P, W€ oppelmann G, Ivins ER, Bouin MN, Testut L (2009) Isostatic stability of the East Antarctica station Dumont d’Urville from long-term geodetic observations and geophysical models. Polar Res 28 (2):193–202 Argus DF, Gordon R, Heflin M, Ma C, Eanes R, Willis P, Peltier WR, Owen S (2010) The angular velocities of the plates and the velocity of the Earth’s center from space geodesy. Geophys J Int 180(3):916–960. doi:10.1111/j.1365246X.2009.04463 Beckley BD, Lemoine FG, Luthcke SB, Ray RD, Zelensly NP (2007) A reassessment of global and regional mean sea level rends from TOPEX and Jason-1 altimetry ased on revised reference frame and orbits. Geophys Res Lett 34(14):L14608 Bock O, Willis P, Lacarra M, Bosser P (2010) An intercomparison of tropospheric delays estimated from DORIS and GPS data, Adv Space Res 46(12):1648–1660 Boehm J (2004) Vienna mapping functions in VLBI analysis. Geophys Res Lett 31:L01603 Boehm J, Niell A, Tregoning P, Schuh H (2006) Global Mapping Function (GMF), a new empirica mapping function based on numerical weather model data. Geophys Res Lett 33(7), art. L07304 Briole P, Willis P, Dubois J, Charade O (2009) Potential applications of the DORIS system. A geodetic study of the Socorro Island (Mexico) coordinate time series. Geophys J Int 178(1):581–590 Coulot D, Collilieux X, Pollet A, Berio P, Gobinddass ML, Soudarin L, Willis P (2009) Genetically modified networks. A genetic algorithm contribution to space geodesy, application to the transformation of SLR and DORIS EOP time series. In: European Geocience Union meeting, Vienna, Austria, EGU2009-7988 Fagard H (2006) Twenty years of evolution for the DORIS permanent network, from its initial deployment to its renovation. J Geod 80(8–11):429–456

P. Willis et al. Flouzat M, Bettinelli P, Willis P, Avouac JP, Heriter T, Gautam U (2009) Investigating tropospheric effects and seasonal position variations in GPS and DORIS time series from the Nepal Himalaya. Geophys J Int 178(3):1246–1259 Gambis D (2006) DORIS and the determination of the Earth’s polar motion. J Geod 80(8–11):649–656 Gobinddass ML, Willis P, de Viron O, Sibthorpe AJ, Zelensky NP, Ries JC, Ferland R, Bar-Sever YE, Diament M (2009a) Systematic biases in DORIS-derived geocenter time series related to solar radiation pressure mis-modelling. J Geod 83 (9):849–858 Gobinddass ML, Willis P, de Viron O, Sibthorpe A, Zelensky NP, Ries JC, Ferland R, Bar-sever YE, Diament M, Lemoine FG (2009b) Improving DORIS geocenter time series using an empirical rescaling of solar radiation pressure. Adv Space Res 44(11):1279–1287 Gobinddass ML, Willis P, Diament M, Menvielle M (2010). Refining DORIS atmospheric drag estimation in preparation of ITRF2008. Adv Space Res 46(12):1566–1577 Kierulf HP, Pettersen BR, MacMillan DS, Willis P (2009) The kinematics of Ny-Alesund from space geodetic data. J Geodyn 48(1):37–46 le Bail K (2006) Estimating the noise in space-geodetic positioning, the case of DORIS. J Geod 80(8–11):541–565 McCarthy D, Petit G (eds) (2004) IERS 2003 Conventions. In: IERS Techn Note 32, Frankfurt-am-Main, Germany Mercier F, Cerri L, Berthias JP (2010) Jason-2 DORIS phase measurement processing. Adv Space Res 45 (12):1441–1454. doi:10.1016/j.asr.2009.12.002 Morel L, Willis P (2002) Parameter sensitivity of TOPEX orbit and derived mean sea level to DORIS stations coordinates. Adv Space Res 30(2):255–263 Morel L, Willis P (2005) Terrestrial reference frame effects on global sea level rise determination from TOPEX/Poseidon altimetric data. Adv Space Res 36(3):358–368 Nocquet JM, Willis P, Garcia S (2006) Plate kinematics of Nubia-Somalia using a combined DORIS and GPS solution. J Geod 80(8–11):591–607 Noll C, Soudarin L (2006) On-line resources supporting the data, products, and information infrastructure for the International DORIS Service. J Geod 80(8–11):419–427 Soudarin L, Cre´taux JF (2006) A model of present-day tectonic plate motions from 12 years of DORIS measurements. J Geod 80(8–11):609–624 Stepanek P, Dousa J, Filler V (2010) DORIS data analysis at Geodetic Observatory Pecny using single-satellites and multisatellites geodetic solutions. Adv Space Res 46 (12):1578–1592 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F (2005) GGM02, an improved Earth gravity field model from GRACE. J Geod 79(8):467–478 Tavernier G, Soudarin L, Larson K, Noll C, Ries J, Willis P (2002) Current status of the DORIS Pilot experiment. Adv Space Res 30(2):151–156 Valette JJ, Lemoine FG, Ferrage P, Yaya P, Altamimi Z, Willis P, Soudarin L, (2010) IDS contribution to ITRF2008 Adv Space Res 46(12):1614–1632 Visser PNAM, van den IJssel J, van Helleputte T et al (2009) Orit determination for the GOCE satellite. Adv Space Res 43 (5):760–768

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Williams SDP, Willis P (2006) Error analysis of weekly station coordinates in the DORIS network. J Geod 80 (8–11):429–456 Willis P, Haines B, Berthias JP, Sengenes P, Le Mouel JL (2004) Behavior of the DORIS/Jason oscillator over the South Atlantic Anomaly. CR Geosci 336(9):839–846 Willis P, Deleflie F, Barlier F, Bar-Sever YE, Romans L (2005a) Effects of thermosphere total density perturbations on LEO orbits during severe geomagnetic conditions (Oct – Nov 2003). Adv Space Res 36(3):522–533 Willis P, Desai SD, Bertiger WI, Haines BJ, Auriol A (2005b) DORIS satellite antenna maps derived from long-term residuals time series. Adv Space Res 36(3):486–497 Willis P, Jayles C, Bar-Sever YE (2006) DORIS, from altimeric missions orbit determination to geodesy. CR Geosci 338 (14–15):968–979 Willis P, Haines BJ, Kuang D (2007) DORIS satellite phase center determination and consequences on the derived scale of the Terrestrial Reference Frame. Adv Space Res 39 (10):1589–1596 Willis P, Ries JC, Zelensky NP, Soudarin L, Fagard H, Pavlis EC, Lemoine FG (2009) DPOD2005: realization of a DORIS

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terrestrial reference frame for precise orbit determination. Adv Space Res 44(5):535–544 Willis P, Fagard H, Ferrage P, Lemoine FG, Noll CE, Noomen R, Otten M, Ries JC, Soudarin L, Tavernier G, Valette JJ (2010a) The International DORIS Service, toward maturity. Adv Space Res 45(12):1408–1420. doi:10.1016/j.asr.2009. 11.018 Willis P, Boucher C, Fagard H, Garayt B, Gobinddass ML (2010b) Contributions of the French Institut Ge´ographique National (IGN) to the International DORIS Service. Adv Space Res 45(12):1470–1480. doi:10.1016/j.asr.2009.09.019 Zelensky NP, Berthias JP, Lemoine FG (2006) DORIS time bias estimated using Jason-1, TOPEX/Poseidon and Envisat orbits. J Geod 83(9):497–506 Zelensky NP, Berthias JP, Lemoine FG (2006) DORIS time bias estimated using Jason-1, TOPEX/Poseidon and ENVISAT orbits. J Geod 80(8–11):497–506 Zelensky NP, Lemoine FG, Chinn DS, Rowlands DD, Luthcke SB, Beckley D, Pavlis D, Ziebart A, Sibthorpe A, Willis P, Luceri V (2010) DORIS/SLR POD modeling improvements for Jason-1 and Jason-2. Adv Space Res 46(12): 1541–1558

7

Towards a Combination of Space-Geodetic Measurements A. Pollet, D. Coulot, and N. Capitaine

Abstract

The International Terrestrial Reference Frame (ITRF), the Earth Orientation Parameter (EOP) time series, and the International Celestial Reference Frame are obtained separately and may thus present inconsistencies. To solve this problem, a first step has been made with the latest ITRF realization (ITRF2005), which has been computed, for the first time, with consistent EOP time series. Another approach to better understand this issue is to directly estimate, in the same process, station positions and EOP time series, from all the space-geodetic measurements. In the framework of the French Groupe de Recherche de Ge´ode´sie Spatiale (GRGS) activities, this latter approach has been studied for several years. For this purpose, the observations of VLBI, SLR, GPS, and DORIS techniques are combined using the same models and software for all the individual data processing. In this paper, we study methodological issues regarding the definition and the consistency of the weekly combined terrestrial frames.

7.1

Introduction

The International Earth rotation and Reference systems Service (IERS) provides different geodetic products as the International Terrestrial Reference Frame (ITRF), the Earth Orientation Parameters (EOP), and the International Celestial Reference Frame (ICRF), the EOP providing the link between

A. Pollet (*) D. Coulot Institut Ge´ographique National, LAREG & GRGS, 6-8 Avenue Blaise Pascal, 77455 Champs-sur-Marne, Marne-la-Valle´e, France e-mail: [email protected] N. Capitaine SYRTE, Observatoire de Paris, CNRS, UPMC & GRGS, 61 Avenue de l’Observatoire, 75014 Paris, France

ITRF and ICRF. These products are computed separately; this may introduce inconsistencies. The latest ITRF realization, ITRF2005 (Altamimi et al. 2007), has been a major step toward the consistency between IERS products. Indeed, for the first time, the ITRS Product Center (PC) has provided the ITRF2005 together with consistent EOP time series. Another approach is under investigation, namely the combination of space-geodetic techniques (DORIS/GPS/SLR/VLBI) at the measurement level. This method permits us to use the same models and software for all techniques in order to obtain consistent results. Furthermore, such a combination enables the introduction of new common parameters and technique links, in addition to the local ties. It should thus allow us to evidence the possible systematic errors of each technique in order to understand and

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_7, # Springer-Verlag Berlin Heidelberg 2012

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A. Pollet et al.

Table 7.1 Configurations of GPS, VLBI, DORIS, and SLR processing Data

GPS Double differences

Satellites

GPS constellation

Sampling rate Elevation cutoff angle Network

10 min 10 10 150 stations (IGS core sites + colocated stations) Absolute ITRF2005 IERS 05 C04 (Bizouard and Gambis 2009) IERS conventions (McCarthy and Petit 2004) FES2004 (Lyard et al. 2006) FES2004 (Lyard et al. 2006) Not modeled ECMWFa (Guo and Langley 2004) 48 h with a 24-h overlapping

Antenna model Stations EOP Solid Earth tides Ocean tides Ocean loading Atm. loading Tropospheric model Mapping function Orbital arc length a

VLBI IVS-R1, IVS-R4, intensive sessions

DORIS

SLR

SPOT 2,4,5 ENVISAT Jason 1

LAGEOS 1 LAGEOS 2

12 8 All available stations

ITRF2005 rescaled

30 h with a 6-h overlapping

(Mendes and Pavlis 2004) (Mendes et al. 2002) 9 days with a 2-day overlapping

http://www.ecmwf.int

to reduce them. Several studies have been carried out on this subject. Andersen (2000) has applied this kind of combination with a stochastic approach and a square root formation filtering and smoothing to VLBI sessions. Thaller et al. (2007) have combined GPS and VLBI data and got promising results regarding EOP and Zenithal Tropospheric Delays (ZTD). A proof of the great interest aroused by such combination is the creation of a new IERS working group (COL, for Combination at the Observation Level) in 2009. The present study is the continuation of Coulot et al. (2007), who have combined DORIS, GPS, SLR, and VLBI normal systems to evidence the interest of this approach for EOP (and, in particular, Universal Time – UT). We focus here on the definition and the homogeneity of the combined terrestrial frame. In the first section, we present the data used and the different parameters estimated during the combination. Then, we test different possible approaches to combine the observation systems with a particular emphasis on the consistency of the combined frame. Finally, we underline the major role of the local ties in the computation and we provide some conclusions and prospects.

7.2

Data, Software, and Parameters

We carry out tests over nearly 3 months of data obtained by the four space-geodetic techniques, available at the IVS, ILRS, IGS, and IDS data centers (between January 9 and March 19, 2005). The CNES1/GRGS software GINS provides the observation systems for each technique. Indeed, this software, which is also designed for gravity field determination (Bruinsma et al. 2009), has the capability to process DORIS, GPS, SLR, and VLBI data (cf. Bourda et al. 2007). It is used by the IDS LCA-CNES/CLS, the IGS CNES/CLS, and the ILRS GRGS Analysis centers (AC), and by the Bordeaux Observatory for VLBI data processing. Table 7.1 details the processing configurations for each technique. These configurations are close to those applied by the AC for the satellite techniques and by the Bordeaux Observatory team for VLBI. The LAREG/GRGS LOCOMOTIV software combines weekly individual observation systems, uses the degree of freedom method (Sillard 1999) in

1

Centre National d’Etudes Spatiales, French institute.

7

Towards a Combination of Space-Geodetic Measurements

order to provide optimal weights for each satellite technique (one weight per satellite observation set) and each VLBI session observation set in the combination, and estimates the following parameters: – Weekly station positions, at the middle of the GPS week. – Daily EOP (polar motion and UT), at noon. – Orbital parameters in agreement with the three GRGS AC configurations. – ZTD, every 2 h, except for SLR. – Technique specific biases in agreement with the three GRGS AC configurations and with the Bordeaux Observatory VLBI configuration. In addition, depending on the model applied for the combination, we estimate or not estimate Helmert parameters (translations and scale factor for each satellite technique and only scale factor for VLBI, cf. Sect. 7.2).

7.3

Combination, Referencing, and Consistency

Coulot et al. (2007) directly combined the normal equation systems derived from the technique observations. In this work, only the EOP were used as common parameters. The minimum constraints were applied over four sub-networks (one per technique). These minimum constraints were related to the parameters evidenced, per technique, in loose constrained combined solutions, by the reference system effect criterion (Sillard and Boucher 2001). This combination provided heterogeneous combined frames. Indeed, each technique realized its own reference frame. This may be caused by the lack of links between the technique networks; common EOP indeed only link techniques regarding orientation. In the next sections, we thus introduce the local ties and we carry out different combinations (from C1 to C4) in order to obtain a solution for which the systematic errors of the techniques are well handled.

7.3.1

Model for an Ideal Case

We introduce local ties in the combination through observation equations:

53

X01 þ dXc1 ðX02 þ dXc2 Þ ð12Þ

¼ LocTieX

ðSLocTie12 Þ ;

(7.1)

with X01 (resp. X02 ) being the station 1 (resp. 2) a priori positions, dXc1 (resp. dXc2 ) the estimated position ð12Þ the local tie offsets of station 1 (resp. 2), LocTieX vector between the stations 1 and 2 and SLocTie1–2 its variance-covariance matrix. Theoretically speaking, each technique is sensitive to the scale of the terrestrial frame. Each satellite technique is sensitive to the terrestrial frame origin (the geocenter), via the dynamical orbits of its dedicated satellites. Through local ties, we can transmit this definition of the origin to the VLBI station network. Finally, we must conventionally define the orientation of the frame; no technique is sensitive to this orientation (EOP are estimated). We should thus theoretically obtain a homogeneous combined solution by gathering the normal equation systems per technique, with local ties and three minimum constraints (one per rotation) applied over a GPS sub-network to define the orientation of the combined frame w.r.t. ITRF2005. However, as shown in Table 7.2 (C1 test), we still have inconsistent results. Indeed, we estimate the transformation parameters between different networks in the Fc combined frames (all the stations and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005 in order to evaluate the homogeneity of the combined solutions. Each technique network realizes its own reference frame. For example, we notice a 3D discrepancy of about 9.6 mm between the DORIS and GPS translations and a scale difference of about 0.7 ppb (4 mm) between SLR and VLBI. Two reasons may explain these heterogeneities. Systematic errors exist between the techniques, due to problems in the models used, in particular the models specific to a particular technique [antenna model for GPS (Ge et al. 2005), solar radiation pressure for DORIS (Gobinddass et al. 2009), etc.] and/or introducing local ties on a weekly basis may be problematic. Indeed, on a weekly basis, we can get poor network distributions, especially regarding the co-located station networks for VLBI or SLR. To investigate these inconsistencies, we directly introduce Helmert parameters to take into account possible mismodellings.

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A. Pollet et al.

Table 7.2 Transformation parameters – mean value (formal error of the mean value) standard deviation value in mm for translations and scale factor Tx, Ty, Tz, and D and m as for rotations Rx, Ry, and Rz – estimated between different networks in the Fc combined frames (all the stations and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005, for C1, C2, C3, and C4 combinations over 3 months of data (see text) Combination C1 C2 Helmert parameters No Yes Local tie handling Equation (7.1) Equation (7.1) Local tie weights Provided by ITRF PC Provided by ITRF PC Transformation parameters between all stations and ITRF2005 Tx 5.2 (0.7) 3.5 0.1 (0.2) 0.1 Ty 1.5 (0.5) 1.6 0.0 (0.2) 0.1 Tz 6.9 (0.9) 5.3 0.2 (0.3) 0.3 D 2.1 (0.4) 1.1 0.0 (0.3) 0.3 Rx 40 (5) 28 2 (3) 2 Ry 36 (17) 34 2 (5) 2 Rz 32 (15) 26 2 (6) 3 Transformation parameters between DORIS network and ITRF2005 Tx 2.6 (0.7) 3.5 5.9 (0.6) 2.1 Ty 0.1 (0.7) 2.5 1.7 (0.6) 1.0 Tz 0.8 (0.8) 3.7 2.9 (0.7) 3.0 D 5.9 (0.5) 3.7 0.5 (0.7) 3.3 Rx 63 (22) 60 29 (27) 73 Ry 105 (25) 72 298 (37) 139 Rz 175 (28) 93 146 (24) 58 Transformation parameters between GPS network and ITRF2005 Tx 5.1 (0.7) 3.2 0.1 (0.3) 0.1 Ty 2.3 (0.7) 2.2 0.0 (0.2) 0.0 Tz 9.8 (1.0) 5.3 0.1 (0.3) 0.1 D 1.3 (0.4) 0.9 0.0 (0.2) 0.1 Rx 20 (16) 31 2 (2) 0 Ry 29 (16) 30 1 (2) 0 Rz 25 (14) 24 3 (3) 0 Transformation parameters between SLR network and ITRF2005 Tx 3.9 (0.7) 3.9 0.5 (0.7) 1.6 Ty 2.4 (0.5) 1.9 3.4 (0.6) 1.6 Tz 0.8 (0.9) 4.1 3.9 (0.7) 0.6 D 16.2 (0.6) 2.4 2.4 (0.7) 1.2 Rx 34 (23) 85 36 (24) 86 Ry 16 (26) 92 29 (27) 91 Rz 47 (24) 66 51 (19) 57 Transformation parameters between VLBI network and ITRF2005 Tx 3.8 (0.9) 3.5 1.5 (0.9) 3.2 Ty 5.5 (0.9) 4.8 6.9 (0.9) 4.6 Tz 7.2 (0.9) 3.5 10.8 (1.0) 4.1 D 12.1 (0.7) 3.8 6.2 (0.9) 4.4 Rx 126 (25) 75 126 ( 25) 62 Ry 76 (28) 94 36 (31) 95 Rz 87 (38) 172 64 (43) 181

C3 No Equation (7.7) Equation (7.8)

C4 Yes Equation (7.7) Equation (7.8)

5.3 (0.7) 3.6 1.9 (0.5) 1.5 5.9 (0.8) 5.3 3.3 (0.4) 1.5 36 (5) 27 41 (18) 39 36 (16) 30

0.1 (0.2) 0.2 0.0 (0.2) 0.2 0.4 (0.3) 0.4 0.1 (0.2) 0.3 5 (3) 6 2 (5) 3 1 (7) 5

4.9 (0.6) 3.0 1.8 (0.6) 2.0 5.3 (0.8) 4.0 3.1 (0.4) 1.3 26 (19) 42 42 (17) 36 31 (16) 30

0.3 (0.4) 0.6 0.7 (0.4) 0.5 0.3 (0.4) 0.7 0.5 (0.4) 1.5 4 (13) 22 8 (8) 8 1 (12) 16

5.1 (0.7) 3.3 0.1 (0.6) 1.4 7.3 (0.9) 5.0 1.8 (0.4) 1.0 30 (15) 28 33 (17) 34 29 (15) 26

0.0 (0.2) 0.1 0.0 (0.2) 0.0 0.0 (0.2) 0.0 0.0 (0.2) 0.1 0 (2) 0 0 (2) 0 0 (2) 0

5.2 (0.6) 3.2 1.2 (0.5) 1.7 0.3 (0.8) 3.6 12.0 (0.4) 2.4 13 (17) 40 36 (17) 39 40 (16) 39

1.4 (0.4) 0.3 0.2 (0.4) 1.2 0.4 (0.4) 1.0 0.6 (0.5) 1.2 9 (14) 33 13 (14) 29 10 (11) 25

5.0 (0.7) 2.8 1.5 (0.7) 2.5 2.0 (0.8) 3.3 6.3 (0.5) 1.8 6 (22) 58 64 (19) 44 51 (19) 45

0.9 (0.5) 1.3 1.5 (0.5) 1.5 1.5 (0.5) 1.2 2.2 (0.5) 1.8 8 (19) 43 32 (19) 45 24 (18) 39

7

Towards a Combination of Space-Geodetic Measurements

7.3.2

55

3 dXc 6 dEOPc 7 7 6 6 a 7 7 6 7 Y ¼ B6 6 yDORIS 7; 6 yGPS 7 7 6 4 ySLR 5 yVLBI 2

Improved Model

A possible way to ensure the homogeneity of the combined solution is to introduce the Helmert (transformation) parameters in the observation systems of each technique. We consider the following observation system: 2

3 dX Y ¼ A4 dEOP 5; a

(7.2)

with Y being the pseudo-observations, A the design matrix, dX the station position offsets w.r.t. a frame Fk, dEOP the EOP offsets consistent with the orientation of Fk and a the other parameter offsets such as orbital parameters, ZTD, etc. We then introduce the transformation parameters, using the following equations: dX ¼ dXc þ T þ DX0 þ RX0 ; dEOP ¼ dEOPc þ R0 ;

(7.3)

with dXc being the station position offsets w.r.t. Fc, the combined frame, T, D, R, and R’ the scalars, vectors, and matrices related to the transformation parameters between the frames Fc and Fk, X0 the a priori station positions, and dEOPc the EOP offsets consistent with the orientation of Fc. In practice, no technique is sensitive to the orientation of Fc (we estimate EOP). The estimated EOP offsets thus align w.r.t. the orientation we define (ibid regarding the frame origin for the VLBI technique). We thus estimate translations and scale factor for satellite techniques: dX ¼ dXc þ T þ DX0 ; dEOP ¼ dEOPc ;

where ytech (tech corresponding respectively to DORIS, GPS, SLR or VLBI) is the vector of the transformation parameters per technique (a scale factor for each technique and three translation parameters for each satellite technique) and B is the new design matrix, deduced from the matrix A. These transformation parameters are estimated in addition to all the other parameters. Due to the lack of information regarding the Fc combined frame definition, the normal system deduced from this observation system presents rank deficiency. As local ties link the technique networks, this rank deficiency correspond to the seven parameters needed to define the combined frame. This definition is obtained by using constraints: either constraints on the estimated transformation parameters and/or minimum constraints. Indeed, we can apply seven minimum constraints w.r.t. ITRF2005 to define Fc. We can also take advantage of the Helmert parameters. For example, if we want to define the origin of Fc as the SLR origin, we do not estimate the SLR translations. In this case, we define the origin of the combined frame independently of ITRF.

7.3.3

Numerical Tests

(7.4)

and only scale factor for VLBI technique: dX ¼ dXc þ DX0 ; dEOP ¼ dEOPc :

(7.6)

(7.5)

We thus obtain the following global observation system:

We compare here two combinations: the “ideal case” combination, called C1 (cf. Sect. 7.3.1), and a similar one with Helmert parameters, called C2 (cf. Sect. 7.3.2). For local ties, we use the values and the associated standard deviations provided by the ITRF PC.2 More than 50 local ties are used per week. For the

2

See tab.

http://itrf.ensg.ign.fr/ties/ITRF2005/ITRF2005-localties.

56

A. Pollet et al.

combination C1, we apply three minimum constraints w.r.t. ITRF2005 over a GPS sub-network to define the orientation of Fc. Consequently, Fc has the orientation of ITRF2005 but not necessarily the same origin and scale. For the combination C2, we use seven minimum constraints w.r.t. ITRF2005 over the same GPS subnetwork to define Fc, which is thus theoretically fully expressed in ITRF2005. To check the homogeneity of the combined solutions, we compute the transformation parameters between different networks in the Fc combined frame (all the stations, and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005. Table 7.2 shows that the estimation of the Helmert parameters in the combination increases the homogeneity of the solution. For example, the 3D discrepancy between the DORIS and GPS translations decreases from 9.6 to 6.7 mm. However, there are still some heterogeneities. The VLBI and SLR networks present the largest ones, probably due to poor co-located networks with a weekly sampling.

7.4

A Possible Approach of Referencing

In order to obtain an homogeneous solution, we present another way of introducing local ties in the combination by the use of equality constraints between estimated co-located station position offsets: dXc1 dXc2 ¼ 0 ðSTie12 Þ :

(7.7)

with dXc1 (resp. dXc2 ) the estimated position offsets of station 1 (resp. 2). Doing so, we consider that the value of the local tie is the difference between the ITRF2005 a priori station ð12Þ positions [X01 X02 ¼ LocTieX – cf. (7.1)]. We thus switch from rescaled ITRF2005 to ITRF2005 for SLR a priori station positions, in order to get consistent values. The standard deviations sTie12 (on which STie12 is based) are computed from the variancecovariance matrix of the ITRF2005 a priori station positions X01 andX02 , at the considered epoch t:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Var X01 ðtÞ X02 ðtÞ ¼ Var X01 ðtÞÞ þ VarðX02 ðtÞ 2 cov X01 ðtÞ; X02 ðtÞ

sTie12 ¼

i X0i ðtÞ ¼ X0i ðt0 Þ þ dt X_ 0 ðt0 Þ with dt ¼ t t0 hence : Var X0i ðtÞ ¼ Var X0i ðt0 Þ þ dt2 Var X0i ðt0 Þ i þ 2 dt cov X0i ðt0 Þ; X_ 0 ðt0 Þ and : cov X01 ðtÞ; X02 ðtÞ ¼ cov X01 ðt0 Þ; X02 ðt0 Þ 1 2 2 þ dt2 cov X_ 0 ðt0 Þ; X_ 0 ðt0 Þ þ dt cov X01 ðt0 Þ; X_ 0 ðt0 Þ 1 þ dt cov X_ 0 ðt0 Þ; X02 ðt0 Þ

(7.8) In practice, sTie12 sLocTie12 . Furthermore, this approach is ITRF-dependent: if an event has changed a co-located station position after 2005, this kind of local ties cannot be used for the considered co-location site. A secular solution, estimated from our data and used as the a priori solution, could avoid this dependence. Due to our short data span, here we have used the ITRF2005 to evaluate these constraints. We compute two combinations based on this approach, C3 and C4. Regarding the estimated parameters and the minimum constraints used, C3 (resp. C4) is similar to C1 (resp. C2). In addition to C1 and C2, Table 7.2 also provides the statistics for the transformation parameters for the C3 and C4 combinations. The homogeneity of the solutions is improved for C3 and C4 but only the C4 combination gives a real consistent combined solution. Indeed, the heterogeneities for the scale and Tz still exist in the C3 combination. Regarding C4, these heterogeneities are embedded in the estimated Helmert parameters. For example, the difference between the VLBI and SLR estimated scale factors is about 0.7 ppb over the 3 months of studied data (Mean of estimated VLBI/SLR scale factor: 2.1/2.8 ppb). In the same way, we find a 3D discrepancy of about 10.2 mm between the DORIS and GPS translations in the estimated Helmert parameters. They consequently do not disturb the referencing of the solution anymore. On the one hand, the local ties are essential to link the technique networks in the combination but, on the another hand, they must be used carefully as they have a great impact.

7

Towards a Combination of Space-Geodetic Measurements

7.5

Conclusions and Prospects

Through the direct estimation of transformation parameters in the combination process, we get an homogeneous combined solution. With our new approach, the problem of datum inconsistency evidenced by (Coulot et al. 2007) is solved. But the introduction of the colocation information on a weekly basis appears to be problematic and it can twist the combined frame, due to poor weekly co-located networks. From the tests carried out in the present study, we recommend to use equality constraints between estimated co-located station position offsets together with a rigorous weighting [cf. (7.7) and (7.8)] to obtain an homogeneous result. Even with this method, the definition of the combined frame will still inevitably depend on this link between the technique networks. Consequently, the introduction of supplementary common parameters and links could be helpful to decrease this dependence. At a terrestrial level, we could use common geodynamic signals (such as loading effects, for instance), ZTD, etc., and, at a space level, the use of multi-technique satellites should be of great interest. With our combination model, the inconsistencies between the techniques are embedded in the estimated Helmert parameters. The understanding of these inconsistencies should help to improve some models. Gradually, such improvements should lead to more consistency and, consequently, to definitions of combined frame more independent of any external terrestrial reference frame. However, as long as technique discrepencies exist, the estimation of the Helmert parameters is essential to insure the homogeneity of the combined solutions.

References Altamimi Z, Collilieux X, Legrand J, Boucher C (2007) ITRF2005: a new release of the international terrestrial reference frame based on time series of station positions and earth orientation parameters. J Geophys Res 112:B09401. doi:10.1029/2007JB004949 Andersen PH (2000) Multi-level arc combination with stochastic parameters. J Geod 74:531–551. doi:10.1007/s001900000115

57 Bizouard C, Gambis D (2009) The combined solution C04 for Earth orientation parameters consistent with International Terrestrial Reference Frame 2005. In: Proceedings of IAG Symposia, 134, pp 265–270. doi: 10.1007/978-3-64200860-3 Bourda G, Charlot P, Biancale R (2007) GINS: a new tool for VLBI geodesy and astrometry. In: Proceedings of the 18th European VLBI for Geodesy and Astrometry (EVGA) Working Meeting, 79, Vienna, pp 59–63, ISSN 1811–8380 Bruinsma S, Lemoine J-M, Biancale R, Vale`s N (2009) CNES/GRGS 10-day gravity field models (release 2) and their evaluation. Adv Space Res 45(2010):587–601. doi:10.1016/j.asr.2009.10.012 Coulot D, Berio P, Biancale R, Loyer S, Soudarin L, Gontier A-M (2007) Towards a direct combination of space-geodetic techniques at the measurement level: methodology and main issues. J Geophys Res 112:B05410. doi:10.1029/ 2006JB004336 Ge M, Gendt G, Dick G, Zhang FP, Reigber C (2005) Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys Res Lett 32:L06310. doi:10.1029/2004GL0222241414 Gobinddass ML, Willis P, De Viron O, Sibthorpe A, Zelensky NP, Ries JC, Ferland R, Bar-Sever Y, Diament M, Lemoine FG (2009) Improving DORIS geocenter time series using an empirical rescaling of solar radiation pressure models. Adv Space Res 44(11):1279–1287. doi:10.1016/j. asr.2009.08.004 Guo J, Langley RB (2004) A new tropospheric propagation delay mapping function for elevation angles down to 2 degrees. In: Proceedings of ION GPS 2003, Portland, OR Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56:394–415. doi:10.1007/s10236-006-0086-x McCarthy DD, Petit G (2004) IERS conventions (2003). IERS technical note 32, Verlag des Bundesamts f€ur Kartographie und Geod€asie, Frankfurt, ISBN 3-89888-884-3 Mendes VB, Pavlis EC (2004) High-accuracy zenith delay prediction at optical wavelengths. Geophys Res Lett 31: L14602. doi:10.1029/2004GL020308 Mendes VB, Prates G, Pavlis EC, Pavlis DE, Langley RB (2002) Improved mapping functions for atmospheric refraction correction in SLR. Geophys Res Lett 29(10):1414. doi:10.1029/ 2001GL014394 Sillard P (1999) Mode´lisation des syste`mes de re´fe´rence terrestres. Contribution the´orique et me´thodologique. PhD thesis, Observatoire de Paris, Paris Sillard P, Boucher C (2001) A review of algebraic constraints in terrestrial reference frame datum definition. J Geod 75:63–73 Thaller D, Kr€ugel M, Rothacher M, Tesmer V, Schmid R, Angermann D (2007) Combined Earth orientation parameters based on homogeneous and continuous VLBI and GPS data. J Geod 81:529–541

.

8

Improving Length and Scale Traceability in Local Geodynamical Measurements J. Jokela, P. H€akli, M. Poutanen, U. Kallio, and J. Ahola

Abstract

Traceability is a feature that is required more frequently in local geodetic highprecision measurements. This basic term of metrology, a measurement science, describes the property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty (BIPM International vocabulary of metrology – basic and general concepts and associated terms (VIM). JCGM 200:2008. Joint Committee for Guides in Metrology, 2008b). GPS measurements are widely used in local geodynamical research. From the viewpoint of metrology, their traceability is uncontrollable because the scale cannot be unambiguously conducted based on the definition of the metre. In particular, atmospheric effects on a GPS signal cannot be modelled or calibrated along the path of the signal. We are testing a method to bring the traceable scale to small GPS networks using high-precision electronic distance measurement (EDM) instruments, the scales of which have been corrected and validated in calibrations at the Nummela Standard Baseline. The traceable scale of EDM is expected to explain the annual scale variations that have been found in GPS time series and to improve results of episodic GPS campaigns. The scale of a standard baseline is validated and maintained through regular interference measurements with the V€ais€al€a interference comparator, in which a quartz gauge conveys the traceable scale. The results from the interference measurements in 2005 and 2007 in Nummela are presented here together with a brief description of the present state of the renowned measurement standard. A standard uncertainty of 0.08 ppm was obtained again for the baseline length of 864 m, and the results confirm the good long-term stability of the baseline. The scale is transferred further to geodetic and geophysical applications by using calibrated high-precision EDM instruments as transfer standards.

J. Jokela (*) P. H€akli M. Poutanen U. Kallio J. Ahola Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland e-mail: [email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_8, # Springer-Verlag Berlin Heidelberg 2012

59

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Local geodynamical measurements will profit from the reduced and accurately estimated uncertainty of the measurement, and therefore we seek further innovations to improve their traceability. We present here a topical example of calibrations and scale transfer for a baseline and monitoring network around a nuclear power plant. We compare simultaneously measured EDM and GPS results and show a scale bias of approximately 1 ppm between them. By using a traceable length in the network, the bias could be reduced, e.g. by improving the processing strategy of GPS observations. This paper focuses on the metrological part of EDM. Some related results and analysis of GPS measurements are discussed in another paper in this volume (Koivula et al. GPS metrology – bringing traceable scale to local crustal deformation GPS network. IAG Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, 2010).

8.1

The Nummela Standard Baseline

The length and scale traceability is based on the Nummela Standard Baseline of the Finnish Geodetic Institute (FGI). The baseline is widely known as the longest, most accurate and stable measurement standard in the world for traceable geodetic length measurements. It serves in national and international scale transfer measurements for other national or accredited calibration services for EDM instruments, as well as for scientific purposes. The accuracy of the baseline, in terms of standard uncertainty, is better than 0.1 mm/km according to the interference measurements with the V€ais€al€a comparator. Since 1997 the traceable scale has been transferred from Nummela to about 20 baselines or test fields in more than ten countries. The FGI is one of the National Standards Laboratories in Finland.

8.1.1

The 80-Years History

The 864-m geodetic baseline in Nummela, northwest of Helsinki, Finland, was established in 1933, replacing an older measurement standard in Santahamina, southeast of Helsinki. Since 1947, when the first successful interference measurement with the wellknown V€ais€al€a (white light) interference method was performed, it has been called the Nummela Standard Baseline. Originally it was used for the calibration of invar wires for triangulation, and later on for EDM instruments. The scale of the Finnish first-order triangulation network was conducted from Nummela.

The importance of the V€ais€al€a interference method was already recognized in an IAG motion in 1951 and in an IUGG resolution of 1954, which recommended the use of similar methods for assuring a uniform scale in geodetic networks. Since then, the FGI has measured 12 similar baselines on almost all continents. Interference measurements of a standard baseline are still performed using the classical V€ais€al€a comparator. Even today, the method is superior to modern techniques in terms of accuracy for outdoor baselines up to a few kilometres. The scale is traceable to the definition of the metre through a quartz gauge system. The lengths of one-metre-long quartz gauges are known from comparisons and absolute calibrations with better than 40 nm standard uncertainty and multiplied with the V€ais€al€a comparator. Also, the history of the quartz metres and their comparisons and calibrations dates back more than 80 years.

8.1.2

The Present State

Between 1947 and 2007 the Nummela Standard Baseline has been measured 15 times using the V€ais€al€a method, complemented by regular comparisons and absolute calibrations of quartz gauges. The latest absolute calibrations have been performed by the PTB, Braunschweig, in Germany, and MIKES, Helsinki, in Finland (Lassila et al. 2003), showing equal results with smaller than 40 nm standard uncertainties for a set of quartz gauges. Standard uncertainties of comparisons, performed at the Tuorla Observatory of the University of Turku and, since 2005, by the FGI, are a few nm (Fig. 8.1).

8

Improving Length and Scale Traceability in Local Geodynamical Measurements

61

151,5 mm + 151,0 1m 150,5 1950

1960

1970

1980

1990

2000

2010

Year

Fig. 8.1 Length of quartz metre no. VIII, which conducts the scale in interference measurements at the Nummela Standard Baseline, from comparisons at Tuorla. The black spot at the year 2000 is the latest absolute calibration of this particular gauge Table 8.1 Baseline lengths at the Nummela Standard Baseline from the 15 interference measurements from 1947 to 2007 Epoch 1947.7 1952.8 1955.4 1958.8 1961.8 1966.8 1968.8 1975.9 1977.8 1983.8 1984.8 1991.8 1996.9 2005.8 2007.8

024 mm + 24 m – – – – – – – – 33.28 0.02 33.50 0.02 33.29 0.03 33.36 0.04 33.41 0.03 33.23 0.04 33.22 0.03

072 mm + 72 m – – – – – – – – 15.78 0.02 15.16 0.02 15.01 0.03 14.88 0.04 14.87 0.04 14.98 0.04 14.95 0.02

0216 mm + 216 m – – – – – – – – 54.31 0.02 53.66 0.04 53.58 0.05 53.24 0.06 53.21 0.04 53.20 0.04 53.13 0.03

0432 mm + 432 m 95.46 0.04 95.39 0.05 95.31 0.05 95.19 0.04 95.21 0.04 95.16 0.04 95.18 0.04 94.94 0.04 95.10 0.05 95.03 0.06 94.93 0.06 95.02 0.05 95.23 0.04 95.36 0.05 95.28 0.04

0864 mm + 864 m 122.78 0.07 122.47 0.08 122.41 0.09 122.25 0.08 122.33 0.08 122.31 0.06 122.37 0.07 122.33 0.07 122.70 0.08 – 122.40 0.09 122.32 0.08 122.75 0.07 – 122.86 0.07

The number following the symbol is the numerical value (in mm) of the combined standard uncertainty

The results of our latest interference measurements in 2005 and 2007 are presented in Table 8.1, together with the previous results. In calibrations they are used as true values with a known uncertainty in the traceability chain. They are lengths between underground benchmarks, to which (and from which, for calibrations) the lengths between reference points on less permanent observation pillars are projected using precision tacheometry and mechanical plumbing and probing. To maintain sub-mm uncertainties, optimal measurement geometry is essential in these studies. The standard (1-s) uncertainties of the lengths from 24 to 864 m ranged from 0.02 to 0.07 mm in the 2007 measurements, which means they were about the same as previous measurements. The difference between the first interference measurement in 1947 and the latest one in 2007 is 0.08 mm for the full length of 864 m, and the variation during the

60-year time span is 0.6 mm. This proves in excellent fashion the stable location of the baseline on a forested non-frozen sandy ridge. The small, but sometimes significant variations in the results are impossible to detect with any other metrological or geodetic method in field conditions, and they are small enough not to disturb calibrations of the most precise EDM instruments. The variations are caused by the settling of the markers after they have been cast in the ground, by later construction projects and other disturbing activities in the neighbourhood, and by slightly improved methods in processing the measurements. Results for the distances up to 216 m before 1977 have not been published. In 1983 and 2005 measurements for the length 864 m were not possible because of unfavourable weather conditions. The latest publications giving more detailed information on the baseline are by Jokela and Poutanen (1998), Jokela et al. (2009) and Jokela and H€akli (2010).

62

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Working conditions and security at the unique measurement standard were greatly improved by construction projects in 2004. The old office and store buildings were replaced with a new house. A part of the 0.12 km2 baseline property was fenced, including all observation pillars and underground benchmarks. Roofs were built over all observation pillars. Most of the observation pillars were reconditioned before the interference measurements in 2007, and a new subsurface drainage system keeps the soil dry around the comparator house, preventing possible “floods” caused by melting snow. The old baseline and the interference method were acknowledged again in 2008 when the FGI, together with eight other European institutes, entered a joint research project called “Absolute long distance measurement in air” (Wallerand et al. 2008). This project, which is a part of the European Metrology Research Programme (EMRP), is financially supported by the European Union. The baseline will be utilized in testing and validating new absolute distance measurement (ADM) techniques and instruments, including improved determination of refraction. New instruments are expected to improve distance measurement traceability up to several kilometres, which will allow for more reliable measurements in local deformation networks or other geodetic applications requiring a traceable scale.

measured - true, mm

measured - true, mm

0.0

–0.5

216

432

648

2008-10-31... 2008-11-06

1.0

0.5

0

Calibration of Transfer Standards

High-precision EDM instruments (typically a Kern Mekometer ME5000, since other suitable instruments are even fewer) are used as transfer standards from the Nummela Standard Baseline to other applications. The principles for how to perform EDM observations and make the necessary reductions and corrections to them are well known, and only a few details noteworthy in our scale transfer measurements are discussed here. All measurements are performed in field conditions. The calibration of EDM equipment for a scale transfer takes a few days, including several measurements of all available baseline distances in both directions. This allows for possible daily changes in the equipment and makes the estimation of uncertainty more reliable. Projection measurements at the Nummela Standard Baseline are performed before and after every important calibration, usually a few times a year. More frequent projections are needed during interference measurements, which typically last about three months. Calibrations are performed immediately before and after the transfer, which reveals possible changes in the equipment during the procedure (Fig. 8.2). In addition to the calibration measurements, similar possible sources of uncertainty have to be taken into account at the transfer site. One of them is the centring

2008-08-28... 2008-09-03

1.0

–1.0

8.2

864

m

Fig. 8.2 Example of calibration of transfer standard (Kern ME5000) before and after a scale transfer. Scale correction (+0.151 mm/km 0.049 mm/km, 1-s) is determined from the

0.5

0.0

–0.5

–1.0

0

216

432

648

864

m

differences between measured and true distances. The additive constant (+0.079 mm 0.014 mm, 1-s) has been corrected. Variation and corrections for this equipment are especially small

8

Improving Length and Scale Traceability in Local Geodynamical Measurements

of instruments. Common commercial fixing methods are widely used with many kinds of surveying instruments and also at geodetic baselines, but in integrated geodynamical measurements self-made solutions are often needed. The GPS antenna is often centred and adjusted with a standard forced-centring plate or directly attached to a pillar. But does the uncertainty really remain smaller than 1 mm? Local monitoring networks and tie measurements that include reference points of, for example, SLR or VLBI are even more challenging. Weather correction is another major source of uncertainty. At local networks the distances to be measured may be short, but at least ambient temperature, air pressure and relative humidity must be observed with sufficient accuracy and observed distances corrected with proper formulas, e.g. as recommended by the IAG (1999). The influences of all sources of uncertainty in the traceability chain must be carefully estimated and summarized in the estimate of combined uncertainty as described in metrological regulations (BIPM 2008a). This also includes of course uncertainties in the quartz gauge system and in interference measurements. In favourable conditions 0.5 to 1.0 mm/km values of expanded (k ¼ 2) uncertainty can be expected at the transfer site, depending on the application. By using traceable baseline lengths to adjust the geodetic networks, we can extend these values to a traceable scale of the network, as was done in triangulation. The method as such is probably not reasonable for GPS networks, though it is an interesting addition for estimating the uncertainty of measurements.

8.3

Traceable Scale in Local Geodynamical Measurements

GPS is an excellent tool for crustal deformation research. There is no need for inter-station visibility, distances are not limited to local measurements and accuracies are generally superior to the traditional methods, especially over longer distances. Metrological traceability is a property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty. The basic requirement of calibrations

63

under specified conditions (BIPM 2008b) makes the traceability and scale of a GPS measurement uncontrollable from the viewpoint of metrology. Heights and height changes are more difficult to monitor than horizontal motions. This is mainly due to the influence of the atmosphere on the GPS signal and the effect of the observing geometry. Some sources of uncertainty are random variables, and can be estimated more reliably by increasing the number of observations or the length of time series. Others are systematic effects, which cannot be quantified, thus causing bias. Although the scale is based on time and the frequency carried by the GPS signal, one is unable to unambiguously conduct the measured distances from the definition of the metre. This is mainly due to the effects of the atmosphere and local conditions, such as multipath. There is no unique standardized set of parameters to process GPS observations (and to obtain the specified conditions), which leads to slightly different results. A more severe limitation is that there is no uniquely traceable documented chain from the computed distances to the definition of the metre. In geodynamical research, one is in many cases mainly interested in changes. In such cases the absolute scale is not critical, but it is sufficient to assume that the scale remains constant between measurements. However, this cannot be guaranteed in GPS observations, and under severe ionosphere conditions scale variations up to 1 ppm have been observed in the Olkiluoto network (Ollikainen et al. 2004). Such results, especially in episodic campaigns, beg for a proper scale in GPS measurements in order to obtain as reliable results as possible. Epoch to epoch variation in scale may lead to misinterpretation in deformation research and one cannot reliably determine the size of this deformation. One solution is to bring the scale into the network using high-precision EDM instruments. Local ties at multi-technique space geodetic sites are another interesting application of dimensional metrology. Quite often these ties are (at least partly) done using GPS, and current demands for sub-mm accuracies in the ties require calibrated EDM baselines for proper scale. Adjacent research projects on GPS/ EDM metrology in the FGI (local ties and another baseline) are reported in other papers in this volume (Kallio and Poutanen 2011; Koivula et al. 2011).

64

Olkiluoto Deformation Network

The FGI and Posiva, an expert organisation responsible for the final disposal of spent nuclear fuel, started GPS measurements in the 10-station control network at the Olkiluoto nuclear power plants in 1995 (Fig. 8.3). The purpose is to monitor possible local crustal deformations with regularly repeated GPS observations. The original network of 10 observation pillars has been slightly changed and expanded through adding new pillars, due to construction projects in the area (Chen and Kakkuri 1995, 1996; Ollikainen et al. 2004; Kallio et al. 2009). The extent of the network, 2 km 4 km, is quite optimal for combining EDM and GPS measurements. The interpretation of results from repeated GPS measurements has been complicated by obvious variations in scale, which are often explained by deficiencies in ionospheric modelling. Since autumn 2002 most GPS campaigns give 1 ppm longer distances than the spring campaign of the same year (Fig. 8.4). Annual variation is also visible in the 196 km long vector between the permanent GPS stations at Olkiluoto and Mets€ahovi (Fig. 8.5). To better monitor the scale and variations to it, the 511-m line between the observation pillars GPS7 and GPS8 was cleared and equipped with a geodetic baseline for high-precision EDM. The visibility for terrestrial observations is not obtainable between the other pillars. Between 1995 and 2009 the network has been measured with episodic GPS campaigns 28 times. Dual-frequency Ashtech Z-12 and Ashtech mZ receivers and Dorne Margolin choke ring antennas have been used to collect data and, to reduce uncertainties caused by antennas, the same antennas have been used at the same stations in each campaign. All observation campaigns have been processed using Bernese software (the latest with version 5.0) and with equal processing principles, e.g. by using network solution, independent L1 and L2 observables to obtain lower measurement noise, and a local ionosphere model computed from the observations (e.g. Kallio et al. 2009). Observation sessions have taken a minimum of 24 h, with an observing interval of 30 s. Since 2002, GPS and EDM observations have been simultaneously conducted twice a year, when possible, for about one week in April and October. The expected movements are extremely small. The largest detected, but

Fig. 8.3 The network for deformation research around the nuclear power plants at Olkiluoto

511258,0 Olkiluoto baseline

GPS

511257,5

EDM

511257,0

Length [mm]

8.4

J. Jokela et al.

511256,5 511256,0 511255,5 511255,0 2002

2003

2004

2005 2006 Year

2007

2008

2009

Fig. 8.4 The GPS and EDM results for the baseline between pillars GPS7 and GPS8 at Olkiluoto

statistically significant, movements are 0.18 mm/a, with a standard deviation of 0.06 mm/a (Kallio et al. 2009). The scale for EDM in the 15 measurements between 2002 and 2009 has been determined in 13 calibrations at the Nummela Standard Baseline. In the 13 measurements between 2002 and 2008 the result for EDM measurements has 12 times been shorter than the result for GPS measurements (Fig. 8.4), with the maximum differences being about 1 mm and the average about 1 ppm. The GPS result is longer in October than in April in six cases out of seven. The scale difference between the GPS and high-precision EDM is obvious. The probable reasons causing such behaviour in the GPS results are uncertainties in GPS antenna calibration values, atmospheric modelling and site-specific effects like multipath conditions. Thus far, the EDM scale has been used for comparison only. By applying the EDM scale directly to the

8

Improving Length and Scale Traceability in Local Geodynamical Measurements

Fig. 8.5 A typical GPS time series between two permanent GPS stations (Mets€ahovi and Olkiluoto). From top to bottom: change in height, East and North component. One can see the annual variation in the vector components. Vertical scale is in millimetres (Kallio et al. 2009)

GPS analysis, the traceable scale can be brought into the GPS network too. Some further research is currently being conducted to better understand the reason for the GPS/EDM difference, and to eliminate any site or antenna dependency (Koivula et al. 2011).

8.5

Conclusions and Further Activities

Scale transfer from a standard baseline (a national geodetic measurement standard) is a method for improving length and scale traceability, which can also be used for local geodynamical measurements. The traceability chain includes the maintenance of the quartz metre system, baseline measurements with the V€ais€al€a interference comparator, baseline maintenance with projection measurements, the calibration of transfer standards (high-precision EDM) and

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measurements at the transfer site. Most of these stages are included in the customary maintenance of the measurement standard and calibration service, and the effort needed for applications is simpler. The components of measurement uncertainty at every stage are estimated and, by combining them, an important part of a final measurement result, the uncertainty of measurement in the traceability chain, relative to the definition of the metre, is obtained. Up to 0.5 mm/km expanded uncertainty is obtained for distances of about 1 km. New results for the world-class measurement standard, the Nummela Standard Baseline of the FGI, are now available. They are frequently needed for scientific research and for scale transfers to other geodetic baselines and test fields and also for local geodynamical monitoring networks. The GPS/EDM network at the Olkiluoto nuclear power plants is presented as an example. Monitoring one GPS baseline with traceable EDM measurements cannot explain the scale variation in GPS results but can probably improve the processing strategy of episodic GPS campaigns in Olkiluoto. The scale difference between GPS and EDM has also provided motivation for further research. Measurements at Olkiluoto will be continued and there is a plan to expand the network. The amount of continuous GPS measurements may be increased; at the moment there is only one permanent station. A newly planned EastWest EDM/GPS baseline, in addition to the current North-South baseline, could be used to obtain better control over possible azimuth dependency. Acknowledgement For the scale transfer, we have used the Kern Mekometer ME5000 of Helsinki University of Technology’s (TKK) Laboratory of Geoinformation and Positioning Technology as a transfer standard. We thank Professor Martin Vermeer and the rest of the staff there for their willing cooperation.

References BIPM (2008a) Evaluation of measurement data – guide to the expression of uncertainty in measurement (GUM). JCGM 100:2008. Joint Committee for Guides in Metrology. http:// www.bipm.org/ BIPM (2008b) International vocabulary of metrology – basic and general concepts and associated terms (VIM). JCGM 200:2008. Joint Committee for Guides in Metrology. http:// www.bipm.org/

66 Chen R, Kakkuri J (1995) GPS Work at Olkiluoto for the Year of 1994. Work Report PATU-95-30e, Teollisuuden Voima Oy. Helsinki, p 11 Chen R, Kakkuri J (1996) GPS Operations at Olkiluoto, Kivetty, and Romuvaara in 1995. Work report PATU-96-07e, Posiva Oy. Helsinki, p 68 IAG (1999) IAG Resolutions adopted at the XXIIth General Assembly in Birmingham, 1999. http://www.gfy.ku.dk/~ iag/HB2000/part2/iag_res.htm Jokela J, H€akli P (2010) Interference measurements of the Nummela Standard Baseline in 2005 and 2007. Publications of the Finnish Geodetic Institute, no 144, p 85 Jokela J, Poutanen M (1998). V€ais€al€a baselines in Finland. Publications of the Finnish Geodetic Institute, no 127, p 61 Jokela J, H€akli P, Ahola J, Bu¯ga A, Putrimas R (2009) On traceability of long distances. XIX IMEKO World Congress Fundamental and Applied Metrology, Lisbon, Portugal, pp 1882–1887, 6–11 Sept 2009. http://www.imeko2009.it.pt/ Papers/FP_100.pdf Kallio U, Poutanen M (2011) Can we really promise a mmaccuracy for the local ties on a geo-VLBI antenna? (Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, Aug 31–Sept 4) Kallio U, Ahola J, Koivula H, Jokela J., Poutanen M (2009) GPS operations at Olkiluoto, Kivetty and Romuvaara in 2008.

J. Jokela et al. Working Report 2009–75. Posiva, Olkiluoto, p 216. http:// www.posiva.fi/en/databank/working_reports Koivula H, H€akli P, Jokela J, Buga A., Putrimas R (2011) GPS metrology – bringing traceable scale to local crustal deformation GPS network. (IAG Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, Aug 31–Sept 4) Lassila A,. Jokela J, Poutanen M, Xu J (2003) Absolute calibration of quartz bars of V€ais€al€a interferometer by white light gauge block interferometer. XVII IMEKO World Congress, Dubrovnik, Croatia, pp 1886–1889, 22–27 June 2003. http://www.imeko.org/publications/wc-2003/PWC-2003TC14-026.pdf Ollikainen M, Ahola J, Koivula H (2004). GPS operations at Olkiluoto, Kivetty and Romuvaara in 2002–2003. Working Report 2004–12. Posiva, Olkiluoto, p 268. http://www. posiva.fi/en/databank/working_reports Wallerand J-P, Abou-Zeid A, Badr T, Balling P, Jokela J, Kugler R, Matus M, Merimaa M, Poutanen M, Prieto E, van den Berg S, Zucco M (2008) Towards new absolute long distance measurement systems in air. 2008 NCSL International Workshop and Symposium, Orlando (USA), Aug 2008. http://www.longdistance project.eu/files/towards_ new_absolute.pdf

9

How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications? H. Drewes

Abstract

Geodetic parameters always correspond to a reference system defined by conventions and realized by a reference frame through materialized points with given coordinates. For the coordinate estimation one has to fix the geodetic datum, i.e. the origin and directions of the coordinate axes, and the scale unit. In geosciences applications, e.g. for geodynamics and global change research, the datum has to be fixed over a very long time period in order to refer time-dependent parameters to one and the same reference frame. The paper focuses on the methodology how to fix the datum by parameters independent of the measurements and deformations of the reference frame, and to hold it over a long time span. It is shown that transformations between reference frames at different epochs are not suited to realize the datum parameters because systematic network deformations may affect it. Independent parameters are in particular the first degree and order harmonic coefficients of the gravity field for fixing the origin, and external calibration for fixing the scale. The long-term stability is achieved by the permanent fixing of the datum parameters. Regional reference frames must refer to the global datum by using epoch station coordinates as fiducial values.

9.1

Introduction

The geodetic datum provides the origin, orientation and scale unit of a coordinate system with respect to the body of the Earth. All geodetic parameters refer to a given datum. Coordinates cannot be estimated from geodetic measurements without fixing the datum; there is a rank defect in the observation equation

H. Drewes (*) Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany e-mail: [email protected]

systems equal to the number of necessary datum parameters (datum defect). In a three-dimensional Cartesian coordinate system, the defect is seven: three to fix the position of the origin, three to fix the orientation of the coordinate axes, and one to fix the scale unit. The datum parameters cannot be measured; they must be given with the definition of the reference system (Latin “datum” ¼ given). If the geodetic datum is not unique or it changes in parameter estimations, the results cannot be compared among each other; e.g. the position coordinates referring to the origin in one location cannot be used together with coordinates referring to an origin in another location.

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_9, # Springer-Verlag Berlin Heidelberg 2012

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H. Drewes

The most challenging task of modern geodesy is the accurate measurement and the reliable representation of parameters of global phenomena and processes within the Earth System for applications and interpretations in geosciences, e.g. geodynamics and global change research. As these processes are very slow and long-lasting, the parameters must refer to one and the same datum over very long time intervals for the unique and unequivocal representation (stability over decades).

9.2

Terminology

In order to avoid misunderstandings we shall use the following nomenclature in this paper: – Datum parameters fix the origin, orientation and the scale unit of a coordinate system. – Network (or reference frame) motions are named shifts, rotations and dilatation given with respect to one and the same geodetic datum. – Network (similarity) transformation parameters provide the translations, orientation changes and scale factor of one network with respect to another one with a different datum. We further shall consider the following facts: – Transformation of one network to another changes the geodetic datum of the transformed network with respect to the original one. – Transformation between two different epochs of a geodetic network with identical datum provides the average motion of the whole network (shifts, rotations and dilatation) and not a datum change. – A deformed network is in general mathematically not similar to the original network, i.e. the deformed network cannot generally be expressed as a similarity transformation of the original one. An example is the change of the flattening of the Earth which requires affine transformations.

9.3

Current Status of Datum Realization

The International Terrestrial Reference System (ITRS) is defined as follows (McCarthy and Petit 2004):

– The origin of the coordinate system is in the Earth’s centre of mass (geocentre). – The orientation of coordinate axes follows the Earth Orientation Parameters of the BIH 1984.0. – The scale unit is the metre consistent with TCG. The International Reference Frame (ITRF) is a realization of the ITRS. In global and regional analyses, solutions are often aligned to the ITRF by transformations of observed station networks to a preexisting reference frame or to a previously observed network, e.g. by the conditions of “no net translation”, “no net rotation”, and/or “no net scale” (NNT, NNR, NNS) (e.g. Thaller 2008; Sa´nchez et al. 2011). Doing so between different epochs of an identical network, the transformation parameters provide the average motion of the network and not different realizations of the datum. The transformation parameters may not be taken as a datum correction. Coordinates transformed in this way cannot be used for applications in geosciences because common motions have gone to the transformation parameters and can no longer be identified in the coordinate differences. Let us demonstrate this effect by two examples: The Earth’s surface undergoes long-term global deformations. The largest vertical deformations are caused by post-glacial rebound (glacial isostatic adjustment), and horizontal ones caused by tectonic processes (plate tectonics, intra- and inter-plate deformations). The station network for observing these deformations is very inhomogeneous (e.g. Collilieux et al. 2009); we have an agglomeration of stations in North America and Europe, and sparse station distribution in other regions such as Africa and Asia (Fig. 9.1). To detect the deformations, we need long-lasting measurements and a reference frame with a stable geodetic datum over decades. Under these prerequisites we discuss the effect of a similarity transformation between the global station network observed at time t and the same network at time t + Dt, and its relevance for geosciences applications.

9.3.1

Vertical Deformation

We start with the example of an uplift caused by postglacial rebound, which produces changes in vertical station coordinates (heights h). If the vertical motions

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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

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Fig. 9.1 ITRF2005 station distribution (http://itrf.ensg.ign.fr)

Fig. 9.2 Vertical motions appear as transformation parameters

are the same in all observation points, they appear as a scale factor (a) in transformation parameters, not as displacements of individual stations (Fig. 9.2, above). If they are different in the northern and southern

hemispheres, they appear as a translation and a scale factor. Due to the inhomogeneous station distribution there may also appear changes in orientation (Fig. 9.2, below), although no network rotation occurred, because the transformation parameters become correlated if the origins of the coordinate systems to be transformed (geocentre) do not coincide with the centres of the networks. What is said here for displacements holds also for velocities in case of their transformation. The described effect can be seen in the results of ITRF2005 computations by different approaches. The computation at Institut Ge´ographique National (IGN), Paris (Altamimi et al. 2007) is based on similarity transformations of the epoch coordinate solutions (weekly from satellite techniques or daily from VLBI) to the combined solution, where the difference in epoch is taken into account by the estimated velocities. The Deutsches Geod€atisches Forschungsinstitut (DGFI), M€unchen, (Angermann et al. 2007, 2009) accumulates datum-free normal equations of the measurements. The comparison of both solutions shows systematic differences. These are predominantly negative in the northern and positive in the southern hemisphere (Fig. 9.3). The differences can be approximated by a sine function of latitude ’ (Fig. 9.4), which is equivalent to the Z-component (DZ ¼ Dh · sin ’). The associated differences in the Z-component are shown in Fig. 9.5. They are – as expected – predominately

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H. Drewes

Fig. 9.3 Differences of vertical velocities between ITRF2005 computations at DGFI and IGN

Fig. 9.4 Latitude dependence of vertical velocity differences and fit by a sine function, which indicates a translation of the network in Z-direction

negative with an average of 1.4 mm/a, and show a longitude dependence. There are, however, accumulated effects affecting the Z-component, which will be discussed below. Figure 9.6 shows the longitude dependence of the Z-component of velocity differences. We again see a sine function with even larger amplitude. The interpretation is more difficult, because there are additional effects from horizontal deformations.

9.3.2

Horizontal Deformation

Horizontal deformations in a global scope are mainly caused by plate tectonics. The kinematic datum of the ITRF is defined by the condition of no net rotation with regard to horizontal tectonic motions over the whole

Earth (McCarthy and Petit 2004). It is realized by transforming the geodetic velocities to those derived from plate tectonic models. The ITRF2005 computation at IGN uses the geophysical model NNR NUVEL-1A (DeMets et al. 1990, 1994). This model does not fulfil the NNR condition at present, because it is based on observations over geological time scales, includes only 16 rigid plates, and does not consider the extended deformation zones along the plate boundaries. Therefore, DGFI uses the Actual Plate Kinematic and Crustal Deformation Model (APKIM2005) based on the structure of the geophysical model PB2002 (Bird 2003) with 52 (micro-) plates, the ten major ones being identical with NUVEL-1A, and 13 deformation zones. The plate velocities are computed from present-day geodetic observations (Drewes 2009b) and show significant differences with respect to the geophysical model

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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

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Fig. 9.5 Differences of the Z-component of station velocities from ITRF2005 computations at DGFI and IGN

Fig. 9.6 Longitude dependence of the Z-component of station velocity differences, indicating a shift (1.4 mm/a) and a rotation of the network around an axis in the equatorial plane

(Fig. 9.7). The differences of the horizontal velocities from both ITRF computations based on these models are shown in Fig. 9.8 (mind the different velocity scales of Fig. 9.7 and Fig. 9.8). We clearly see the rotation around an axis from the Northern Atlantic to the Southern Indian Ocean which is nearly identical with the differences between NNR NUVEL-1A and APKIM2005. The global motions of all tectonic plates, which are modelled as rotations on a sphere, have a dominant trend towards northeast (Fig. 9.7). This causes a shift of the whole network with an average dX/dt ¼ 10 mm/a, dY/dt ¼ +13 mm/a, dZ/dt ¼ +15 mm/a (Drewes 2009b). This shift enters completely into the station velocities and does not affect the datum, if the origin of the reference frame is fixed to the geocentre and the NNR condition of the plate model is fulfilled. If we perform a transformation of the discrete geodetic

network with inhomogeneous point distribution to a slightly rotating geophysical plate model (like NNR NUVEL-1A with oX ¼ 0.04 mas/a, oY ¼ +0.03 mas/a, oZ ¼ 0.02 mas/a.), then the residual rotations appear also in the translation transformation parameters. The effect can be seen in the systematic change of all the ITRF translation parameters and its rates (Altamimi et al. 2002, 2007), in particular in the Y and Z components.

9.3.3

Regional Deformation

The examples presented so far refer to effects of global deformations on the global datum realization. Regional reference frames are normally defined as densifications of the ITRF, i.e. they are subject to the

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H. Drewes

Fig. 9.7 Comparison of the geophysical plate kinematic model PB2002 and the geodetic model APKIM2005

Fig. 9.8 Differences of horizontal velocities between ITRF2005 computations at DGFI and IGN

same effects. In addition, there are regional deformations due to tectonic, isostatic, sedimentary, atmospheric, hydrospheric, and other processes. Large effects are caused by seasonal loading. Figure 9.9 shows height variations of stations distributed over all the South American continent (from ’ ¼ 5 to ’ ¼ 35 ) computed from observations in the South

American reference frame (SIRGAS, Sa´nchez et al. 2011; Seem€uller et al. 2009). They clearly demonstrate a systematic periodic (seasonal) variation up to 2 cm. The datum of a regional reference frame is often realized by NNR, NNT, and NNS (i.e. similarity) transformations of the regional network to ITRF

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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

73

KOUR ( =5°, =-53°)

IMPZ ( =-5°, = - 47°)

AREQ ( =-16°, = - 71°)

CIUB ( =-16°, = -56°)

BRAZ ( =-16°, = - 48°)

LPGS ( =-35°, = -58°)

Fig. 9.9 Station height variations in South America [cm]

(or IGS) coordinates extrapolated with linear velocities from the ITRF (or IGS) reference epoch to the regional reference epoch. If the reference stations do not move linearly (Fig. 9.9), the extrapolated coordinates do not refer to actual geocentric station positions at every time. We get then the same type of effects as described in chapter 3, and we cannot use coordinate variations

estimated in this way for geosciences applications. To avoid this effect, the reference frames in some regional projects are realized by transforming the epoch (weekly) coordinates to the weekly IGS solutions (Craymer et al. 2007; Sa´nchez et al. 2011). This procedure accounts for the regional seasonal effects, but still includes the global motions of the network (secularly

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H. Drewes

due to post glacial rebound, seasonally due to different atmosphere and hydrosphere loads in the northern and southern hemispheres).

9.4

How to Fix the Geodetic Datum?

9.4.1

Fixing the Origin

The realization of the geodetic datum by similarity transformation to other networks or reference frames is not a suitable approach, because network deformations in the reference stations selected for the transformation change the datum definition. Fixing the origin to the geocentre must be done by a gravimetric approach as a network-independent method (Drewes 2009a). The Earth’s centre of mass is defined by (M ¼ total mass of the Earth) X0 X dm=M Y0 Y dm=M Z0 Z dm=M:

9.4.2 (9.1)

The first degree and order spherical harmonic coefficients of the Earth’s gravity field express the position of the geocentre with the semi-major axis a as a scaling factor: C11 X dm=a M S11 Y dm=a M C10 Z dm=a M:

do not change the geodetic datum. They must be interpreted as orbit errors and be reduced by improved orbit modelling. The geocentric coordinates of the ground stations are derived from the geocentric coordinates of the satellites using, e.g. SLR distance measurements. In differential approaches, like double differencing GNSS or Doppler methods, most of the relation to the geocentre is lost in the differencing. It can therefore not strictly be realized. As a consequence, SLR measurements should be included for any datum realization, also in regional reference frames. The fixing of the geocentre through the gravity field holds for any time. Thus it is clear that there is no time evolution of the origin. It remains always in the geocentre. If it is realized in this way, it guarantees the long-term stability of the reference frame for geosciences applications.

(9.2)

Using a gravity field model with C11¼S11¼C10¼0 in the coordinate estimation with dynamic satellite methods, as customary in satellite geodesy, fixes the origin of the coordinate system automatically to the geocentre. Satellite orbits are always geocentric unless additional constraints are introduced. Such constraints may enter by fixing the coordinates of tracking stations or an average of them (e.g. NNT constraints). Therefore, global orbit computations for reference frame determinations must not fix any terrestrial coordinates; the necessary minimum datum constraint is given by the gravity field. Wrong modelling of other gravitational or non-gravitational forces (e.g. solar radiation pressure) introduces additive constraints, too. They cause systematic errors in the orbits, but

Fixing the Orientation

The orientation of the coordinate system could also be realized by gravimetric methods through the principal axes of inertia, which are expressed by the second degree spherical harmonics of the gravity field (C21, S21, S22). These coefficients, however, can at present not be determined with the required accuracy. This may change by including data from future gravity field and global navigation satellite missions. Until these are available, the orientation must further be fixed conventionally (e.g. BIH84). The time evolution of the orientation of the coordinate system is characterized by the high correlation between station velocities and Earth Orientation Parameters (EOP). Systematic velocity changes may interchange with EOP. It is therefore indispensable to estimate velocities and EOP simultaneously in a global adjustment, and to use a present-day no-netrotation constraint by models derived from geodetic measurements instead of geologic-geophysical models. The orientation resulting from network transformations refers then to the same coordinate axes in velocities and EOP.

9.4.3

Fixing the Scale

The scale unit has to be fixed to the definition of the unit of length (metre) by calibrating the measuring

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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

instruments and reduction of atmospheric effects. There is no time evolution of the scale; it is always the same unit of length. We must never perform a similarity transformation between networks at different epochs with a scale factor as a parameter. This would only absorb the network dilatation and exclude geosciences interpretations (e.g. to prove or disprove the expansion theory of the Earth, or to detect global postglacial rebound). Conclusions

If we realize the geodetic datum by network transformations, we change the datum between the original and the transformed reference frame. A geocentric frame is then loosing the geocentricity. This behaviour is often discussed as geocentre motion. Doing so, one has to keep in mind that it is a motion w. r. t. the always deforming crust-based reference frame. There is no stable reference. A consequence is that the non-geocentric reference frame is no longer consistent with the satellite orbit computations, which normally use a geocentric gravity field (C11 ¼ S11 ¼ C10 ¼ 0). One would then have to re-compute all SLR, GPS and DORIS orbits with the changed lower spherical harmonic coefficients. For users of such a moving reference frame, one would have to provide the corresponding time-dependent gravity fields together with the station coordinates. It is by far more practicable to keep the origin fixed in the geocentre and to include all the global motions in the station velocities. Experiences with inconsistent global reference frames were reported, e.g. from satellite altimetry studies on global sea level changes (e.g. Beckley et al. 2007; Morel and Willis 2005). There are considerable differences in the interpretation of results when using different ITRF realizations as a reference frame. The datum of regional reference frames has strictly to be realized as a densification of the global reference frame in order to be consistent with the satellite orbits. True coordinates (ITRF or IGS) have to be used as fiducial values, and not the coordinates extrapolated from a reference epoch with linear velocities. In epoch solutions (e.g. weekly), the coordinates of the same epoch have to be taken from the superior frame. As the global

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network may undergo a common motion w. r. t. the geocentre, one has to fix the geocentric datum by external gravimetric parameters (C11 ¼ S11 ¼ C10 ¼ 0). This should be done by including always station coordinates derived from SLR in a combined network adjustment.

References Altamimi Z, Sillard P, Boucher C (2002) ITRF2000: a new release of the International Terrestrial Reference Frame for Earth science applications. J Geophys Res 107(B10) 2214:19. doi:10.1029/2001JB000561 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial reference frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:19. doi:10.1029/2007JB004949, B09401 Angermann D, Drewes H, Kr€ugel M, Meisel B (2007) Advances in terrestrial reference frame computations, vol 130, IAG Symposia. Springer, Heidelberg, pp 595–602 Angermann D, Drewes H, Gerstl M, Kr€ugel M, Meisel B (2009) DGFI combination methodology for ITRF2005, vol 134, IAG Symposia. Springer, Heidelberg, pp 11–16 Beckley BD, Lemoine FG, Luthcke SB, Ray RD, Zelensky NP (2007) A reassessment of global and regional mean sea level trends from TOPEX and Jason-1 altimetry based on revised reference frame and orbits. Geophys Res Lett 34:14608. doi:10.1029/2007GL030002, 5pp Bird P (2003) An updated digital model for plate boundaries. G3 – Geochem Geophys Geosyst 4(3):52. doi:1010.1029/ 2001GC000252 Collilieux X, Altamimi Z, Ray J, Van Dam T, Wu X (2009) Effect of the satellite laser ranging network distribution on geocenter motion estimation. J Geophys Res 114:B04402. doi:10.1029/2008JB005727, 17pp Craymer MR, Piraszewski M, Henton JA (2007) The North American Reference Frame (NAREF) project to densify the ITRF in North America. Proceedings ION GNSS 20th International Technical Meeting of the Satellite Division, pp 2145–2154 DeMets C, Gordon RG, Argus DF, Stein S (1990) Current plate motions. Geophys J Int 101:425–478 DeMets C, Gordon R, Argus DF, Stein S (1994) Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophys Res Lett 21:2191–2194 Drewes H (2009a) Reference systems, reference frames, and the geodetic datum – basic considerations, vol 133, IAG Symposia. Springer, Heidelberg, pp 3–9 Drewes H (2009b) The Actual Plate Kinematic and crustal deformation Model (APKIM2005) as basis for a non-rotating ITRF, vol 134, IAG Symposia. Springer, Heidelberg, pp 95–99 McCarthy DD, Petit G (2004) IERS Conventions 2003. IERS Technical Note No. 32 Morel L, Willis P (2005) Terrestrial reference frame effects on sea level rise determined by TOPEX/Poseidon. Adv Space Res 36:358–368. doi:10.1016/j.asr.2005.05.113

76 Sa´nchez L, Seem€uller W, Seitz M (2011) Combination of the weekly solutions delivered by the SIRGAS Processing Centres for the SIRGAS-CON Reference Frame. In: Kenyon S et al (eds) Geodesy for Planet Earth, IAG Symposia. Springer, Heidelberg Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional

H. Drewes Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85 Thaller D (2008) Inter-technique combination based on homogeneous normal equation systems including station coordinates, Earth orientation and troposphere parameters. GFZ Potsdam Sci Rep STR08/15

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence of Postglacial Rebound: An Evaluation of the Nordic Geodynamical Model in Finland P. H€akli and H. Koivula

Abstract

The IAG Reference Frame Sub-Commission for Europe (EUREF) created the European Terrestrial Reference System 89 (ETRS89) and fixed it to the Eurasian plate in order to avoid time evolution of the coordinates due to plate motions. However, the Fennoscandian area in Northern Europe is affected by postglacial rebound (PGR), causing intraplate deformations with respect to the stable part of the Eurasian tectonic plate. The Nordic countries created their national ETRS89 realizations in the 1990s and have adopted them as the basis for geospatial data. As the most accurate GNSS processing is done in ITRS realizations, an accurate connection to national ETRS89 realizations is required. If the official EUREF transformation is used, residuals are up to 10 cm in the Nordic countries. Therefore, the Nordic Geodetic Commission (NKG) has created a 3-D intraplate velocity model NKG_RF03vel over Fennoscandia and a new transformation procedure to correct for the deformations caused by PGR. This paper evaluates the NKG approach and compares it to the current recommendation given by EUREF with a 100-point ETRS89 realization in Finland. The results show that, by using a high-quality intraplate velocity model, the transformation residuals are reduced to the cm-level.

10.1

Introduction

National reference frames are usually fixed to a global or regional terrestrial reference frame (TRF) that in most cases is one of the ITRS realizations, ITRFyy. ITRF solutions are the most accurate global realizations, but the coordinates are also time-

P. H€akli (*) H. Koivula Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland e-mail: [email protected]

dependent due to site velocities. Site velocities are needed if the Earth’s dynamics (e.g., plate tectonic motion) are to be taken into account at the cm-level. Time variable coordinates, however, are not useful for geodetic practice. To avoid time evolution of the coordinates, the IAG Reference Frame Sub-Commission for Europe, EUREF, has created the European Terrestrial Reference System 89 (ETRS89), which is fixed to the stable part of the Eurasian tectonic plate and coincides with the ITRS at the epoch 1989.0 (Boucher and Altamimi 1992). However, the Fennoscandian area in Northern Europe is affected by postglacial rebound (PGR), causing intraplate

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78

deformations. The uplift part of the PGR in Fennoscandia has been observed for centuries, first with tide gauges, later with repeated levellings and gravity observations and, recently, with permanent GPS stations as well. Intraplate deformations in Europe and Fennoscandia have been studied, for example, by Nocquet et al. (2001, 2005), Johansson et al. (2002) and Lidberg et al. (2007). The land uplift has a maximum of approx. 10 mm/year (from the reference ellipsoid). The horizontal part of the PGR, from the time series of permanent GPS stations, is up to 2.5 mm/year. These deformations can be seen, for example, in EPN (EUREF Permanent Network) velocities expressed in ETRS89, see Figs. 10.1 and 10.2. The figures were plotted from the EPN cumulative solution up to GPS week 1,540 of class A stations (EPN 2010). Class A stations are categorized as highquality fiducial stations meaning that the velocity estimates are known better than 0.5 mm/yr (Kenyeres 2010). The Nordic countries created their national ETRS89 realizations in the 1990s using the ITRF of that time (Denmark ITRF92, Norway ITRF93, Finland ITRF96 and Sweden ITRF97). The ETRS89 realizations have been adopted as the basis for geospatial data. The Finnish ETRS89 realization, called EUREF-FIN, was measured in 1996–1997 and realized through ITRF96(1997.0) with official

formulas provided by EUREF (Boucher and Altamimi 2008), yielding ETRF96 coordinates in 1997.0. The formulas correct the rigid plate motion to epoch 1989.0, but for intraplate deformations the epoch remains in 1997.0, which can be considered the reference epoch for EUREF-FIN. Since the most accurate GNSS processing has to be done in adequate ITRFyy, accurate transformations between ITRF and national ETRS89 realizations are needed. The connection between different threedimensional reference frames is generally presented with a linear 7-parameter similarity transformation. EUREF provides a 14-parameter similarity transformation approach (including time evolution of the parameters) from ITRS to ETRS89 that takes into account the rigid plate motion of the Eurasian tectonic plate (Boucher and Altamimi 2008). In the context of a GNSS campaign, EUREF does not recommend using velocity models. However, any similarity transformation alone cannot depict the influence of PGR since it deforms the crust unequally. Therefore, the EUREF transformation does not describe discontinuities and intraplate deformations; they remain as transformation residuals. In the Fennoscandian area, the EUREF approach is therefore not cm-level accurate when transforming from present ITRF to national ETRS89 realizations. The residuals are already up to 10 cm within a 10-year time span.

340

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Fig. 10.1 Horizontal velocity field in ETRF2000 (EPN 2010)

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© FGI/PH 2010

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence

ITRF2000 to ETRF96 is done in two steps. The first step transforms ITRF2000(tc) coordinates to ITRF96(tc), keeping the epoch of coordinates untouched. The transformation parameters are given at reference epoch t0 (for ITRF2000 t0 ¼ 1997.0) and the parameters have to be converted to the epoch of observations (tc) with the change rates of the parameters. In the second step, ITRF96(tc) coordinates are transformed to ETRF96(tc). Since no intraplate or epoch correction is recommended, the epoch of coordinates remains at tc.

To overcome this problem, the Nordic Geodetic Commission (NKG) has created a 3-D velocity model, NKG_RF03vel, and a procedure that also takes into account intraplate deformations. This study evaluates the NKG approach and compares it to the transformation recommended by EUREF. Transformations were evaluated using a 100-point network that is defining the Finnish ETRS89 realization, EUREF-FIN. EUREF-FIN was originally measured in 1996–1997 and the same network was re-measured in 2006. The re-measurement campaign was processed with Bernese 5.0 and the coordinates for fiducial stations at the central epoch of observations, tc ¼ 2006.5, were obtained from the official IERS ITRF2000 solution (ITRF 2010a). The resulting campaign coordinates were transformed using EUREF and NKG approaches and compared to the official EUREF-FIN coordinates.

10.2

10.2.2 Official NKG Transformation The NKG transformation was done according to the recommendations given by the NKG working group for positioning and reference frames (NKG WG). The NKG transformation takes intraplate deformations into account using velocity model NKG_RF03vel. The model corrects both horizontal and vertical intraplate deformations. The horizontal part originates from the GIA model by Milne et al. (2001) that was rotated to the GPS-derived velocities by Lidberg et al. (2007) (Fig. 10.3). The vertical part, NKG2005LU (ABS) model by NKG working group for height determination, is constructed from tide gauge, levelling and

Transformation Approaches

10.2.1 Official EUREF Transformation The EUREF transformation was made according to the recommendation and parameters given by Boucher and Altamimi (2008). The transformation from 340

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Fig. 10.2 Vertical velocity field in ETRF2000 (EPN 2010)

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P. H€akli and H. Koivula

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Fig. 10.3 NKG_RF03vel model for horizontal intraplate deformations in Fennoscandia (Nørbech et al. 2006)

permanent GPS data (Fig. 10.4) (Nørbech et al. 2006; Lidberg 2008). The Nordic ETRS89 realizations are based on different ITRF solutions and reference epochs leading up to discrepancies of a few cm. In order to take into account these discrepancies, a common reference frame was needed. The common Nordic frame, NKG_RF03, was realized with a NKG2003 GPS campaign and computed in ITRF2000 at epoch 2003.75. The complete description of NKG_RF03 is given by Jivall et al. (2005). The transformation is determined through the NKG_RF03 frame in order to determine the transformation for each Nordic country. For Finland, the transformation consists of three steps: 1. ITRF2000(tc) ! ITRF2000(2003.75) (NKG_ RF03) 2. ITRF2000(2003.75) ! ITRF2000(2003.75)1997.0 (¼intraplate corrected to 1997.0) 3. ITRF2000(2003.75)1997.0 ! EUREF-FIN The first step is to transform ITRF2000 coordinates at the epoch of observations to ITRF2000 at epoch 2003.75 in which the NKG_RF03 is expressed. The

rigid plate motion during the period is reduced with the ITRF2000 rotation pole for the Eurasian plate (given by Boucher and Altamimi 2008). In addition to the rigid plate motion, also intraplate deformations during the period have to be taken into account in order to simulate the ITRF2000 coordinates at epoch 2003.75 accurately. For this purpose the NKG_RF03vel intraplate velocity model is used. In the second step, the resulting coordinates in ITRF2000(2003.75) are further corrected for intraplate deformations to the reference epoch (tr) of the national ETRS89 realization, in Finland tr ¼ 1997.0. This step corrects the internal geometry of the network at the epoch 2003.75 to the one at the reference epoch of EUREF-FIN. In the third step these intraplate corrected ITRF2000(2003.75) coordinates (we are using notation ITRF2000(2003.75)1997.0) are transformed to EUREF-FIN with a 7-parameter similarity transformation. The parameters were determined between the intraplate corrected NKG_RF03(tr) coordinates and the Nordic ETRS89 realizations by the NKG WG.

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence 0˚ 8˚ 12 6˚ 4 ˚ 16˚ 20˚ 24˚ 28˚ 32˚ 3

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Fig. 10.4 NKG_RF03vel model for vertical intraplate deformations in Fennoscandia (Nørbech et al. 2006)

The standard error of unit weight for the parameter estimation with 12 fitting points in Finland was 1.6, 1.5 and 3.2 mm for North, East and the up component, respectively. The formulas, parameters and more detailed explanation are given by Nørbech et al. (2006).

10.3

Evaluation of the Transformations

10.3.1 EUREF Transformation Residuals of the official EUREF transformation for the vertical coordinates are up to 7 cm (Fig. 10.6) and show the neglected effect of the PGR between 1997.0 and 2006.5 (compare to the contours of the land uplift model NKG_RF03vel, Fig. 10.4). The horizontal residuals of the transformation (Fig. 10.5) have approximately the same direction, but there is a large difference in magnitude between Southern and

Northern Finland. The residuals vary from a couple of mms to 3 cm (see also Table 10.1). In order to understand this feature, and despite the fact that the EUREF does not recommend using intraplate velocities in the context of a GPS campaign, we subtracted the effect of PGR between 1997.0 and 2006.5 from the transformed coordinates. After removing the effect of PGR, horizontal residuals are more equal in size and direction throughout Finland. All residuals are approximately 15–20 mm and the direction is N-NW (and slightly rotating from the North to the West when moving from Southern to Northern Finland). This bias likely originates from different plate models used in ITRF2000 and ITRF96. Altamimi et al. (2002) report a significant disagreement between ITRF2000 velocities and those predicted by NNR-NUVEL-1A that were used in ITRF96. The difference between ITRF2000PMM (plate motion model) and NNR-NUVEL-1A, illustrated by Altamimi and Boucher (2002) and

P. H€akli and H. Koivula

82 Fig. 10.5 Horizontal residuals of official EUREF transformation

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Table 10.1 Statistics of transformation residuals from ITRF2000 to EUREF-FIN with different approaches (mm) n ¼ 95 Mean Std Rms Min Max

EUREF N 10.8 8.0 13.4 3.3 26.0

E 1.6 5.5 5.7 13.9 8.6

U 37.7 18.8 42.1 1.1 75.4

NKG-1 N 3.7 4.1 5.5 5.5 14.6

Altamimi et al. (2003), shows a strong agreement with our horizontal residuals after intraplate correction for the Fennoscandian area. Vertical residuals are reduced to a couple of cm when intraplate deformations were removed.

E 3.0 3.5 4.6 4.9 10.8

U 16.9 9.5 19.4 39.8 0.0

NKG-2 N 2.2 4.0 4.5 8.3 12.1

E 0.7 3.3 3.3 7.0 9.0

U 3.5 8.1 8.8 22.2 15.0

10.3.2 NKG Transformation The ITRF2000(2006.5) coordinates were transformed according to the official NKG transformation (solution labelled NKG-1). The solution includes intraplate

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence

Fig. 10.6 Vertical residuals of official EUREF transformation

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corrections of approx. 3–9 mm/year vertical and up to 2 mm/year horizontal velocities in Finland. The residuals that are at the level of a few cm are clearly smaller than those of the EUREF transformation without intraplate corrections. The horizontal residuals have a random nature, but the vertical residuals are systematically biased by an average of 16.9 mm (see Table 10.1). This is caused mainly by biased NKG_RF03 coordinates. The NKG2003 campaign was computed in ITRF2000(2003.75), but in Finland the vertical coordinates of NKG_RF03 differ approx. 13 cm from the ITRF2000 coordinates computed with positions and velocities published by the IERS (these were used in evaluation). The difference is of

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the same magnitude as the residuals (however, comparison was possible only at 4 Finnish EPN stations: METS, JOEN, VAAS and SODA). The NKG2003 campaign and resulting NKG_RF03 frame was computed using IGS00 cumulative solution. The NKG_RF03 was processed using GIPSY-OASISII, GAMIT/GLOBK and Bernese software and the final solution is a combination of these solutions. Constraining to ITRF2000 was done with GIPSY and GAMIT/GLOBK solutions, meaning that the NKG2003 campaign has a global alignment, through transformation parameters, to ITRF2000 (Jivall et al. 2005). The bias is partly caused by the cumulative IGS00 coordinates, but in Finland these are equal within a

P. H€akli and H. Koivula

84 Fig. 10.7 Horizontal residuals of NKG2 transformation

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few mm to the IERS-published ITRF2000 coordinates at epoch 2003.75. Instead, aligning the regional solution with transformation parameters to global ITRF2000 may lead to a biased solution since this method is sensitive to station selection (network effect), see e.g., Altamimi (2003) and Legrand and Bruyninx (2008). The ITRF positions and velocities published by the IERS are widely used, even if the positions usually need to be extrapolated outside of the temporal extent of the data used in the ITRF solution. Therefore, misaligned NKG_RF03 coordinates cannot be used for transformation, even if the solution itself is accurate. As a consequence, either new parameters from

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the IERS-published coordinates to national realization or an additional step from NKG_RF03 to ITRF2000 (2003.75) is needed. We solved for new national transformation parameters using the IERS-published positions and velocities. Figures 10.7 and 10.8 show the residuals for this solution (labelled NKG-2). This corrects most of the vertical bias in the NKG-1 solution. The accuracy (rms) of this transformation is better than 1 cm. The residuals of different transformations are summarized in Table 10.1. The NKG-2 transformation was also evaluated with ITRF2005, using only permanent FinnRef stations. The transformation has an additional step from ITRF2005(tc) to ITRF2000(tc) that was made

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence

Fig. 10.8 Vertical residuals of NKG-2 transformation

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Table 10.2 Comparison of ITRF2005/ITRF2000 transformations to EUREF-FIN (mm) n ¼ 12 Mean Stdev Rms Min Max

ITRF2000(2006.50) N E U 1.1 1.9 1.8 2.7 2.3 5.2 2.8 2.9 5.3 3.0 2.0 12.0 4.7 5.2 3.8

ITRF2005(2008.56) N E U 1.3 4.3 3.1 4.2 1.3 7.6 4.2 4.4 7.9 11.7 7.1 18.6 3.1 2.4 7.1

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plate motion with the ITRF2000 rotation pole and intraplate deformations with the NKG_RF03vel model. Table 10.2 summarizes the residuals for 12 permanent FinnRef stations. The accuracy of the ITRF2005 transformation was slightly worse, as expected. However, an accuracy of 1 cm (rms) can also be achieved from ITRF2005 to EUREF-FIN.

10.4 according to EUREF’s MEMO. The IERS-published ITRF2005 coordinates were used for the fiducial station METS (ITRF 2010b). The epoch of ITRF2005 coordinates was 2008.56, so the transformation includes an additional 2.06 year correction for rigid

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Conclusions and Discussion

We have studied several ways of transforming ITRF coordinates back to national ETRS89 realization in Finland, where the reference frame is under the influence of postglacial rebound (PGR). The official

86

EUREF transformation led to transformation residuals up to 7 cm in vertical and up to 3 cm in horizontal components. The pattern of vertical residuals is consistent with the effect of PGR and the magnitude is too large for accurate geodetic purposes. The NKG has created a common reference frame NKG_RF03, expressed in ITRF2000 at epoch 2003.75, and a transformation through it back to the Nordic ETRS89 realizations. The official NKG transformation includes a high-quality intraplate velocity model for Fennoscandia and corrects most of the PGR effect. However, a bias of 16.9 mm was found in vertical residuals and was identified as being the consequence of a network effect. NKG_RF03 was created using minimum constraints approach that is known to be sensitive for site selection. By using IERS-published official ITRF2000 coordinates instead of NKG_RF03, this bias was greatly reduced. This NKG-2 solution gives transformation residuals below 5 mm for horizontal and 8.8 mm (rms) for vertical coordinates, which is acceptable for most geodetic purposes. The main outcome of the study is that there are several influencing factors (e.g., plate rigid rotation, intraplate deformations and constraining approach to the ITRF) that need to be taken into account appropriately in order to get cm-level transformation accuracies back to national ETRS89 realizations. Our results verify that we need a velocity model to map coordinates with cm-level accuracy from the current ITRF to the Finnish realization of ETRS89. However, guidelines for using such models are missing or their use is not recommended. There are several different ways to implement such models. Therefore, guidelines for incorporating deformation models in a unified and standardized way or even a pan-European velocity model would be desirable.

References Altamimi Z (2003) Discussion on how to express a regional GPS solution in the ITRF, EUREF Publication No. 12. Verlag des Bundesamtes f€ ur Kartographie und Geod€asie, Frankfurt am Main, pp 162–167 Altamimi Z, Boucher C (2002) The ITRS and ETRS89 Relationship: New Results from ITRF2000. EUREF Publication No. 10. Verlag des Bundesamtes f€ ur Kartographie und Geod€asie, Frankfurt am Main, pp 49–52 Altamimi Z, Sillard P, Boucher C (2002) ITRF2000: a new release of the International Terrestrial Reference Frame for

P. H€akli and H. Koivula earth science applications. J Geophys Res 107(B10):2214. doi:10.1029/2001JB000561 Altamimi Z, Sillard P, Boucher C (2003) The impact of a nonet-rotation condition on ITRF2000. Geophys Res Lett 30 (2):1064. doi:10.1029/2002GL016279, 2003 Boucher C, Altamimi Z (1992) The EUREF Terrestrial Reference System and its First Realization. Report on the Symposium of the IAG Subcommission for the European Reference Frame (EUREF) held in Florence 28–31 May 1990. Ver€offentlichungen der Bayerischen Kommission Heft 52, M€unchen Boucher C, Altamimi Z (2008) Memo: specifications for reference frame fixing in the analysis of a EUREF GPS campaign. Version 7: 24-10-2008 EPN (2010) EPN cumulative solution GPS weeks 860–1540. ftp://epncb.oma.be/pub/station/coord/EPN/ EPN_A_ETRF2000_C1540.SSC. Accessed 16 Mar 2010 ITRF (2010a) Primary ITRF2000 solution. http://itrf.ensg. ign. fr/ITRF_solutions/2000/sol.php. Accessed 16 Mar 2010 ITRF (2010b) ITRF2005. http://itrf.ensg.ign.fr/ITRF_solutions/ 2005/ITRF2005.php. Accessed 16 Mar 2010 Jivall L, Lidberg M, Nørbech T, Weber M (2005) Processing of the NKG 2003 GPS Campaign. LMV-rapport 2005:7, Reports in Geodesy and Geographical Information Systems, G€avle 2005. Available at http://www.lantmateriet.se/templates/ LMV_Page.aspx?id¼2688. Accessed 16 Mar 2010 Johansson JM, Davis JL, Scherneck H-G, Milne GA, Vermeer M, Mitrovica JX, Bennett RA, Jonsson B, Elgered G, Elo´segui P, Koivula H, Poutanen M, R€onn€ang BO, Shapiro II (2002) Continuous GPS measurements of postglacial adjustment in Fennoscandia 1. Geodetic results. J Geophys Res 107(B8):2157. doi:10.1029/2001JB000400 Kenyeres A (2010) Categorization of permanent GNSS reference stations. Bolletino di Geodesia e Scienze Affini Legrand J, Bruyninx C (2008) EPN Reference Frame Alignment: Consistency of the Station Positions. EUREF 2008 Symposium, Brussels, Belgium, 18–21 June 2008 Lidberg M (2008) Geodetic Reference Frames in Presence of Crustal Deformations. Integrating Generations, FIG Working Week 2008, Stockholm, Sweden, 14–19 June 2008 Lidberg M, Johansson JM, Scherneck H-G, Davis JL (2007) An improved and extended GPS-derived 3D velocity field of the glacial isostatic adjustment (GIA) in Fennoscandia. J Geodesy 2007(81):213–230. doi:10.1007/s00190-006-0102-4 Milne GA, Davis JL, Mitrovica JX, Scherneck H-G, Johansson JM, Vermeer M, Koivula H (2001) Space-geodetic constraints on glacial isostatic adjustments in Fennoscandia. Science 291:2381–2385 Nocquet J-M, Calais E, Altamimi Z, Sillard P, Boucher C (2001) Intraplate deformation in western Europe deduced from an analysis of the ITRF97 velocity field. J Geophys Res 106 (B6):11239 Nocquet J-M, Calais E, Parsons B (2005) Geodetic constraints on glacial isostatic adjustment in Europe. Geophys Res Lett 32:L06308. doi:10.1029/2004GL022174, 2005 Nørbech T, Engsager K, Jivall L, Knudsen P, Koivula H, Lidberg M, Madsen B, Ollikainen M, Weber M (2006) Transformation from a Common Nordic Reference Frame to ETRS89 in Denmark, Finland, Norway, and Sweden – status report. Also Paper C in Lidberg M (2007) Geodetic Reference Frames in Presence of Crustal Deformations. Ph.D. Thesis, Chalmers University of Technology, G€oteborg, Sweden

Global Terrestrial Reference Frame Realization Within the GGOS-D Project

11

D. Angermann, H. Drewes, and M. Seitz

Abstract

The GGOS-D terrestrial reference frame has been computed in a common adjustment of station positions and velocities together with the Earth orientation parameters and the quasar coordinates (celestial reference frame). The data were processed as datum-free normal equations from homogeneously generated VLBI, SLR and GPS observation time series using identical standards for the modelling and parameterization. A major focus was on the analysis of the station position time series, investigations regarding seasonal variations in station motions and on the combination methodology for the terrestrial reference frame computation.

11.1

Introduction

The project GGOS-D was funded by the German Federal Ministry of Education and Research (BMBF) in the GEOTECHNOLIEN-Programme under the topic “Observation of System Earth from Space”. GGOS-D involved four German institutions: Deutsches GeoForschungsZentrum Potsdam (GFZ), Bundesamt f€ur Kartographie und Geod€asie (BKG), Institut f€ur Geod€asie und Geoinformation, Universit€at Bonn (IGG-B) and DGFI. An overview of the project is given by Rothacher et al. (2010). The major goals and challenges of this project were (1) the definition and implementation of common GGOS-D standards and a unique modelling and parameterization in the different software packages,

D. Angermann (*) H. Drewes M. Seitz Deutsches Geod€atisches Forschungsinstitut (DGFI), AlfonsGoppel-Strasse 11, 80539 M€ unchen, Germany e-mail: [email protected]

(2) the generation of homogeneously re-processed observation time series from the different space geodetic observation techniques, (3) the computation of the GGOS-D terrestrial and celestial reference frames, and (4) the generation of consistent, high-quality time series of geodetic-geophysical parameters describing the Earth System. In this paper, we focus on the computation of the terrestrial reference frame (TRF). The input data comprise observation time series from the space techniques Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR) and the Global Positioning System (GPS) based on identical standards for the modelling and parameterization. The data were processed as datum-free normal equations in two major steps (1) Analysis and accumulation of time series normal equations per technique, (2) Intertechnique combination and computation of the final TRF solution. A major focus was thereby on the investigation of seasonal variations in station positions and on the development of advanced combination methods.

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_11, # Springer-Verlag Berlin Heidelberg 2012

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11.2

D. Angermann et al.

Data Used for TRF Computation

The computation of the GGOS-D terrestrial reference frame is based on homogeneously processed VLBI, SLR and GPS observation time series.Table 11.1 gives an overview of the input data. The time series of the different techniques were provided as unconstrained datum-free normal equations. The GGOS–D input data also comprise the local tie information, which is available at the ITRS Centre at http://itrf.ensg.ign.fr/ local_survey.php. The common models used in the various software packages were not only implemented but also validated in detail by intensive comparisons. Based on the experiences gathered a refined set of standards was agreed upon the participating institutions and was subsequently implemented for the second project phase (Rothacher et al. 2007; Steigenberger et al. 2010). Most recent state-of-the models were taken into account, e.g., the use of 6-hourly ECMWF grids for computing the hydrostatic delay and the use of the hydrostatic mapping function VMF1 (B€ ohm et al. 2006) for VLBI and GPS, the correction for higherorder ionospheric terms for GPS, and the modelling of thermal deformation for the VLBI telescopes (Nothnagel 2008). In order to allow for a comparison and validation of the results, for each of the techniques two solutions were computed with different software packages. Based on the individual observation time series, intratechnique combinations were performed for VLBI and SLR. For GPS such a combination was not performed since the EPOS solution was not fully consistent

Table 11.1 GGOS-D data used for the global TRF computation. For each technique two series were computed using two different software packages. In case of GPS the second series computed with EPOS was used for comparisons only, since the time resolution of this solution is 1 week and there are also small differences w.r.t. the GGOS-D standards Institution Software Institution Software Time interval resolution No. of stations

GPS GFZ Bernese GFZ EPOS 1994–2007 1/7 days 240

SLR DGFI DOGS GFZ EPOS 1993–2007 7 days 71

VLBI DGFI OCCAM IGGB Calc/Solve 1984–2007 1 h sessions 49

with the Bernese solution. For the same reason, DORIS (Doppler Orbitography and Radiopositioning Integrated by Satellite) data were not included in the TRF computation, since the standards and models applied for the DORIS processing were not fully consistent with the GGOS-D standards.

11.3

Combination Methodology for the TRF Computation

The methodology applied in the GGOS-D project was based on combining datum-free normal equations of the VLBI, SLR and GPS observation time series. The tropospheric parameters and low-degree spherical harmonic coefficients, which were also included in the time series normal equations, were reduced and presently not considered for the TRF computations. The station positions and velocities of the observing stations (TRF) were combined in a common adjustment with the quaser coordinates (CRF), and the Earth Orientation Parameters (EOP). The TRF computation has been performed with the DGFI Software package DOGS. The general procedure of the combination methodology is given in Fig. 11.1.

Input: Datum-free normal equations (NEQ) Epoch 1

VLBI NEQ

SLR NEQ

GPS NEQ

Epoch 2

VLBI NEQ

SLR NEQ

GPS NEQ

Epoch n

VLBI NEQ

SLR NEQ

GPS NEQ

Accumulation of time series Multi-year NEQ‘s

VLBI NEQ

SLR NEQ

GPS NEQ

Inter-technique combination TRF (station positions and velocities), EOP, and quasar coordinates)

Fig. 11.1 Combination methodology for the terrestrial reference frame computation

11

Global Terrestrial Reference Frame Realization Within the GGOS-D Project

11.3.1 Accumulation of Time Series per Technique In the first part of the terrestrial reference frame computation, the time series of the daily/weekly normal equations were accumulated to a multi-year normal equation for each technique. The procedure comprises two major steps (1) The generation and analysis of station position time series to detect non-linear behaviours, and (2) the computation of the cumulative multi-year normal equations per technique. The time series analysis and accumulation was performed in an iterative procedure. At first, constant velocity parameters were set up in the epoch normal equations to represent linear station motions and the epoch station positions were transformed to positions at the reference epoch 2000.0. From the analysis of position time series discontinuities were identified for many stations, which are often caused by equipment changes or geophysical events (e.g., earthquakes). The discontinuities were parameterized by setting up new position and velocity parameters after the jump. The total number of discontinuities could significantly be reduced compared to ITRF2005, in particular for GPS stations. The ITRF2005 solution computed at DGFI (ITRF2005-D) comprises 221 discontinuities for 332 stations (67%), whereas 124 discontinuities for 240 stations (52%) were identified within GGOS-D. This improvement was mainly achieved by the homogeneuosly processed GGOS-D data sets and, in the case of GPS, by the implementation of absolute antenna phase center corrections. Figure 11.2 shows the position time series for the GPS station Yuzhno-Sakhalin (YSSK), Russia, located in a geodynamically very active region, the Sakhalin seismic belt. Two large earthquakes both with a magnitude of 8.3, at Hokkaido (25 September 2003) and at Kuril island (15 November 2006) caused discontinuities of about 1–2 cm in the position time series, in particular in the north and east component. The station velocity in the north component was changed by about 5 mm/year after the first earthquake, changing again to its nominal value after about 2 years. The precision of repeated station positions obtained from the accumulation of the time series solutions for each space technique was used to estimate scaling factors for the technique-specific normal equations. For stations with discontinuities separate positions

89

were estimated for each segment. Statistical tests were applied to decide whether the estimated velocities can be equated or not.

11.3.2 Computation of the TRF Solution Input for the combination of different techniques were the accumulated intra-technique normal equations for GPS, SLR and VLBI. The parameters include station positions, velocities, daily EOPs, and (for VLBI) also quasar coordinates (right ascension and declination). The connection of the different techniques’ observations is given by geodetic local tie measurements between the instruments’ reference points at colocation sites. The selection of suitable local ties is a critical issue because the number and the spatial distribution of “high quality” co-location sites is not optimal. Furthermore, there are discrepancies between the difference vectors derived from the space geodetic techniques and the local ties, as shown for example in Fig. 11.3 for VLBI and GPS co-locations. The results are given for the GGOS-D TRF computation in comparison with ITRF2005-D (Angermann et al. 2007, 2009), which is in excellent agreement with ITRF2005 (Altamimi et al. 2007). The agreement of the space geodetic solutions with the local ties is better for most stations of the GGOS-D computation, in particular for those located on the southern hemisphere. In the ideal case (without systematic errors) the EOP estimates must be (statistically) identical for all space techniques. Thus, their estimates are used as a criterion to validate the selected local ties and to stabilize the inter-technique combination as additional “global ties”. The selection and implementation of local ties for the inter-technique combination as well as the equating of station velocities at co-location sites was done in an iterative procedure (see Kr€ugel and Angermann 2007). The selection of local ties is performed on the basis of two criteria (1) the pole coordinates as common parameters provide an ideal basis to measure the consistency: A first combination of station coordinates is computed using a set of local ties, while the EOP are not combined. The mean offsets between the estimated coordinates of the pole are then used to quantify the consistency of the TRF. (2) The r.m.s. values of the 7-parameter similarity transformations between the combined TRF solution and the single-technique

90

D. Angermann et al. 2002

2001

2000

2004

2003

2005

2006

2007

North [mm]

0 –5 –10 –15 –20 25

East [mm]

20 15 10 5 0

Height [mm]

10 5 0 –5 –10 –15 1500

1000

500

0

2500

2000

JD2000 [Tage] Fig. 11.2 Position time series for the GPS station Yuzhno-Sakhalin (YSSK), Russia 25 ITRF 2005 GGOS D

20 15 10

HOB2

TIDB

CRO1

CONZ

SANT

FORT

MKEA

NLIB

PIE1

MDO1

ALGO

HRAO

TSKB

SHAO

WTZR

YEBE

MATE

NOTO

MEDI

NYA1

0

ONSA

5

Fig. 11.3 Comparison of the GGOS-D results with ITRF2005D. The 3D difference vectors (mm) between the VLBI and GPS solutions and the terrestrial difference vectors are given for 21

co-location sites. The stations located in the southern hemisphere are highlighted by yellow (grey) color

solutions are a measure of the deformation of the station networks (caused by the local tie implementation). The two above mentioned criteria were applied in various test computations by satisfying also that an optimal number of co-location sites with a good spatial distribution is selected.

To identify the best (most suitable) set of co-location sites, different solutions were computed, varying the co-locations and the assumed accuracy of the introduced local ties. As an example, the results for VLBI and SLR co-locations are given in Table 11.2. Shown are the mean pole offsets between the VLBI

11

Global Terrestrial Reference Frame Realization Within the GGOS-D Project

Table 11.2 Mean pole offsets and network deformation for VLBI and GPS co-locations obtained from the GGOS-D intertechnique combination in comparison with the DGFI solution for ITRF2005 (ITRF2005-D) Selected VLBI-GPS co-locations Mean pole difference (mas) Network deformation Epoch 2000.0 (mm)

GGOS-D 19

ITRF2005-D 13

0.035 0.3

0.041 1.0

and GPS estimates as a measure for the consistency. Furthermore, the r.m.s. position differences of a 7-parameter similarity transformation between the VLBI-only and the combined TRF solution (expressed at the reference epoch 2000.0) as a measure for the network deformation are shown. The GGOS-D results are compared with those obtained from ITRF2005-D. It shall be emphasized that the number of co-locations significantly increased for the GGOS-D computations, accompanied with a better agreement of the VLBI and GPS pole coordinates and a smaller network deformation. Major differences of the GGOS-D solution w.r.t. ITRF2005 are (1) an improved modelling of the individual space techniques, (2) a homogeneous re-processing by applying unified standards and models for all techniques, and (3) the inclusion of two more years of recent data until the end of 2007. Other tasks of the inter-technique combination include the weighting of the different techniques and the equating of station velocities of co-located instruments. The weighting was done by estimating variance factors for the normal equations based on the precision of repeated (daily/weekly) station positions. The station velocities of co-located instruments were estimated as separate parameters. The velocities were equated, if the differences are statistically not significant. To generate the TRF solution, minimum datum conditions were added to the combined normal equations, and the complete normal equation system was inverted. The origin was realized by SLR observations. As shown by R€ ulke et al. (2008), the network translation parameters derived from GPS are affected by orbital perturbances and cannot be used for the datum definition. The scale of the TRF solution was defined by SLR, VLBI and GPS observations. This was justified, since a comparison of the techniquespecific station networks w.r.t. the scale showed no significant differences. The orientation of the TRF was

91

defined by a No-Net-Rotation (NNR) condition w.r.t. ITRF2005. The kinematic datum of the TRF solution was given by an actual plate kinematic and crustal deformation model (APKIM) derived from observed station velocities (Drewes 2009). The GGOS-D combination results comprise the terrestrial reference frame (station positions and velocities), daily EOP and quasar coordinates, which were estimated in a common adjustment. The terrestrial reference frame results provide the basis for the generation of consistent, high-quality time series of geodetic-geophysical parameters describing the Earth System (Nothnagel et al. 2010).

11.4

Analysis of Seasonal Station Motions

In specific GGOS-D studies, the time series of co-located VLBI and GPS stations were analysed and compared (Tesmer et al. 2009). As an example, Fig. 11.4 shows the VLBI and GPS height time series for two co-location sites: Wettzell (Germany) and Ny-Alesund (Spitsbergen, Norway). The VLBI antenna of Wettzell has the most dense height series (two observations per week), which agrees rather well with GPS. In case of Ny-Alesund there are larger discrepancies between the VLBI and GPS height time series, which may be caused by solutionand/or technique-specific effects. The results given by Tesmer et al. (2009) show a rather good agreement of the height time series for most of the VLBI and GPS colocation sites. The strongest signals in the time series have mostly annually repeating patterns. The station height time series were compared with geophysical model results (Seitz and Kr€ugel 2009). It was found, that a large part of the observed annual signals can be explained by loading, which was computed from atmospheric, hydrospheric and non-tidal oceanic loading variations. Unfortunately, the geophysical models are not as accurate as necessary to reduce loading from the variations of the station positions. Thus, the reduction of loading effects from the original observations is problematic. However, these seasonal station motions will affect the terrestrial reference frame computations, if the temporal variations of station positions are described only by constant velocities, as it is done currently. The consequences are (1) Deviations of the station motions from a linear model (e.g., seasonal variations) will

92

D. Angermann et al. WTZR vs. WETTZELL: wmeans (each 7 days for −+35 days)

dR [cm]

1

0 VLBI GPS

−1 1997

1998

1999

2000

2001

2002

2003

2004

2005

time [year] NYAL vs. NYALES20: wmeans (each 7 days for −+35 days)

dR [cm]

1

0 VLBI GPS

−1 1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

time [year]

Fig. 11.4 Homogeneously re-processed VLBI (stars) and GPS (circles) height time series at two co-located sites: Wettzell (Germany) and Ny-Alesund (Spitsbergen, Norway). The figures illustrate 90-days moving weighted means computed each 7 days

produce errors in the combination results; (2) seasonal variations will affect the velocity estimations, in particular for stations with relatively short observation time spans (i.e., < P ðsn ð d zÞÞ2 Pn cos c þ kdz Dt for c þ kdz Dt p > = ; n¼0

> > : for c þ kdz Dt > p ; 0

(15.20)

where kdz is a conversion factor representing in the case of the time-varying SST the correlation time of the signal. This should be studied and determined in each region under study, since the characteristics of dz vary significantly for each area and in open or closed sea regions (Knudsen and Tscherning 2006). The degree variances sn ðd zÞ are determined as in (15.17) in order to fit the local characteristics of the time-varying SST. Since the time-varying SST is triggered by salinity and temperature variations, in-situ oceanographic data and climatology models can be used for the fit of the analytical model to empirical values. Given the analytical expressions for the covariance functions of all observations, it is possible to proceed to the simultaneous estimation of the deterministic parameters and of the signals z c, dz, T, hSLA nonsteric , etc., along with their prediction errors as:

1 ^x ¼ AT C1 A AT C1 yy yy y;

(15.21)

^zc ðPÞ ¼ Czs ðP; ÞC1 y Ax ^ ; yy

(15.22)

n o s^2c ðPÞ ¼ Czc zc ðP; PÞ Czc s C1 C1 AN1 AT C1 Cszc : yy yy yy z

(15.23)

15

On the Determination of Sea Level Changes by Combining Altimetric

^hSLA ðPÞ ¼ C SLA ðP; ÞC1 ðy A^ xÞ; hSTERIC s STERIC yy

(15.24)

SLA ðP; PÞ ChSLA s2^hSLA ðPÞ ¼ ChSLA STERIC hSTERIC STERIC s STERIC n o 1 1 T 1 C1 : yy Cyy AN A Cyy CshSLA STERIC

(15.25) The estimation of all other signals and prediction errors can be done according to (15.22)–(15.25). A short note on the estimation of variance components within this optimal combination scheme will be given. Given the general observation equation for LSC estimation with parameters and knowing that v ~ (0, Vy ¼ Cvv) we can write Cvv in a form as to depend on a unknown set ofPso-called variance components yi. Therefore, Vy ¼ yi Vi and the matrices Vi come from the original datai error CV matrices. Within the present combination scheme Vy would take the form " Vy ¼ yhalt Qhalt þ yhTG 1m mm

2

6 4

# yHISL TG q3

QhTG

0 QHISL TG

0

yhSLA 3 7 5

QhSLA

3q3q

þ yhSLA QhSLA þ yT QT þ yTrr QTrr GRACE GRACE 1k

kk

1p pp

1p pp

(15.26) The variance components can then be estimated by an iterative procedure such as the iterative almost unbiased estimation-IAUE (Rao and Kleffe 1988). According to IAUE, we first have to compute matrix W as T 1 1 T 1 1 W ¼ C1 A Cvv ; vv Cvv A A Cvv A

(15.27)

and then estimate the variance components ^y y ¼ Jþ k depending on according to ^y ¼ J1 k or ^ whether matrix J can be inverted. Matrices J and k can be analytical determined from the data error CV matrices Jij ¼ tr WCvi vi WCvj vj ;

(15.28)

1 ki ¼ ^ vT C1 v: vv Cvi vi Cvv ^

(15.29)

129

After the first variance components have been estimated, the iterations can be carried out according to ^y

a i

¼

^ya1 ^vT C1 Cv v C1 ^v i i i vv vv : trfWCvi vi g

(15.30)

where ^y a1 is the previous estimate and y^ ai is the i new one. When their difference is smaller than a a1 ^ ^ a certain threshold e, so that y y < e, then convergence is achieved, the variance components are estimated and the iterations can stop. Conclusions

A detailed combination scheme of satellite altimetry, tide-gauge, GRACE and GOCE data for the determination of various signals both static and variable has been presented. The latter, can be the disturbing potential, its second order derivatives, geoid heights, steric and non-steric sea level variations and the stationary and time-variable sea surface topography. It is clear that the versatility of LSC with parameters allows including other observations too, like gravity anomalies, salinity and temperature from oceanographic measurements, hydrological models, etc. Therefore, the proposed combination scheme can be extended easily as new data sources become available, so that better estimates will be derived. Another advantage of the combination strategy outlined is that rigorous signal estimation errors can be derived along with the possibility of variance component estimation so that the covariance matrices and errors models can be calibrated.

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130 altimeter, GRACE and geophysical models. J Geod 79:532–539 Garcia D, Chao BF, Rio JD, Vigo I (2006) On the steric and mass-induced contributions to the annual sea level variations in the Mediterranean Sea. J Geophys Res 111:C09030. doi:10.1029/2005JC002956 Garcia D, Ramillien G, Lombard A, Cazenave A (2007) Steric sea-level variations inferred from combined Topex/Poseidon altimetry and GRACE gradiometry. Pure Appl Geophys 164:721–731 Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman, San Fransisco, CA Knudsen P (1987) Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data. Bull Geod 61:45–160 Knudsen P (1991) Simultaneous estimation of the gravity field and sea surface topography from satellite altimeter data by least-squares collocation. Geophys J Inter 104(2):307–317 Knudsen P (1992) Estimation of sea surface topography in the Norwegian sea using gravimetry and Geosat altimetry. Bull Ge´od 66:27–40 Knudsen P (1993) Integration of gravity and altimeter data by optimal estimation techniques. In: Rummel R, Sanso` F (eds) Satellite altimetry for geodesy and oceanography, vol 50, Lecture Notes in Earth Sciences. Springer, Berlin, pp 453–466 Knudsen P, Tscherning CC (2006) Error Characteristics of dynamic topography models derived from altimetry and GOCE Gravimetry. In: Tregoning P, Rizos C (eds) Dynamic Planet 2005 – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools, IAG Symposia, vol 130. Springer, Berlin, pp 11–16

G.S. Vergos et al. Lombard A, Garcia D, Ramillien G, Cazenave A, Biancale R, Lemoine JM, Fletcher F, Schmidt R, Ishii M (2007) Estimation of steric sea level variations from combined GRACE and Jason-1 data. Earth Planet Sci Lett 254:194–202 Moritz H (1980) Advanced physical geodesy. Wichmann, Karlsruhe Rao CR, Kleffe J (1988) Estimation of variance components and applications. North-Holland Series in Statistics and Probability, vol 3 Rio MH, Hernandez F (2004) A mean dynamic topography computed over the world ocean from altimetry, in-situ measurements and a geoid model. J Geoph Res 109(12): C12032 Sanso` F, Sideris MG (1997) On the similarities and differences between systems theory and least-squares collocation in physical geodesy. Boll di Geod e Scie Aff 2:174–206 Swenson S, Wahr J (2002) Methods for inferring regional surface-mass anomalies from GRACE measurements of time-variable gravity. J Geophys Res 107(B9):2193 Tscherning CC and Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degreevariance models. Reports of the Department of Geodetic Science, 208, The Ohio State University, Columbus, OH Tziavos IN, Sideris MG, Forsberg R (1998) Combined satellite altimetry and shipborne gravimetry data processing. Mar Geod 21:299–317 Vergos GS, Tziavos IN and Andritsanos VD (2005) On the determination of marine geoid models by least-squares collocation and spectral methods using heterogeneous data. In: Sanso´ F (ed) IAG Symposia, A Window on the Future of Geodesy, vol 128. Springer, Berlin, pp 332–337

Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite and Submarine Measurements

16

J. Calvao, J. Rodrigues, and P. Wadhams

Abstract

The L3H phase of operation of ICESat’s laser in the winter of 2007 coincided for about two weeks with the cruise of the British submarine “Tireless” which was equipped with upward-looking sonars. This provided a rare opportunity for a simultaneous determination of the sea ice freeboard and draft in the Arctic Ocean.

16.1

Introduction

The determination of the thickness of the sea ice layer in the Arctic Ocean and surrounding seas is an essential component of the study of the climate of the Arctic. On the one side, the Arctic environment as a whole and, in particular, its sea ice cover, respond quickly to global climatic changes due to some amplification mechanisms that tend to be present in the polar regions, like the well-known albedo effect. As such, a decline in the volume of sea ice can be considered as one of the best indicators of the warming of our planet. On the other hand, because the sea ice acts as a regulator of heat and moisture transfer between the ocean and the atmosphere, its thickness is likely to affect significantly the climate of the Arctic and nearby regions. Basin-wide measurements of the thickness of the sea ice in the Arctic Ocean began in 1958 when the

J. Calvao (*) Lattex, IDL, Faculdade de Cieˆncias da Universidade de Lisboa, Lisboa, Portugal e-mail: [email protected] J. Rodrigues P. Wadhams Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK

U.S. submarine “Nautilus” reached the North Pole for the first time. Since then, regular Arctic cruises by U.S. submarines have collected a vast amount of sea ice draft data, mostly in the so-called SCICEX box, which roughly coincides with the portion of the Arctic Ocean outside international waters (but including the region north of Alaska). Most of these data sets are available through the National Snow and Ice Data Center (Boulder, CO) archive and have been extensively analysed by Rothrock et al. (1999, 2008). In the early 1970s British submarines started cruising in the Arctic Ocean. For more than three decades they have been taking ice thickness data in regions rarely visited by U.S. boats, such as Fram Strait and the waters north of Greenland. The latter are of special importance in the understanding of the large scale sea ice dynamics of the Arctic Ocean. Through Fram Strait passes most of the ice, fresh water and heat exchanged between the Arctic Ocean and the rest of the world oceans. The complex and variable sea ice conditions in Fram Strait partly reflect the diverse origins of that ice; some of it was transported all the way from the coasts of East Siberia by the Transpolar Drift, other parts may have been advected from the west or the east, and some is formed locally. On the other hand, the region north of Greenland and Ellesmere Island is known to have the thickest ice in

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the Arctic. The last data collected by British submarines suggest that, unlike the rest of the Arctic, there is no decline in ice thickness here in recent years. Data from earlier cruises have been processed at the University of Cambridge and the results published in several papers. Wadhams (1990) provides the first direct evidence of the thinning of the sea ice north of Greenland. Later, he and, independently, Rothrock, observed a significant overall thinning of the Arctic sea ice by comparing results from cruises in the mid 1970 s and in the mid 1990 s (Rothrock et al. 1999, Wadhams and Davis 2000). The Polar Oceans Physics group of the University of Cambridge is currently in the last stages of the statistical analysis of the data gathered during the last two Royal Navy submarine cruises, in the winters of 2004 and 2007. Data from the March 2007 cruise will be particularly relevant because they were taken in several regions of the Arctic with very different ice regimes, some of which would later become ice-free during the exceptional summer of 2007. In this paper we show some results from these two cruises. The quality and scientific importance of the submarine data are unquestionable. However, the use of different sonar equipment in different cruises, the frequent impossibility of an independent evaluation of the accuracy of the measurements, the scarsity of the voyages, the non-coincidence of the tracks, the varied times of the year of the cruises, and the difficulty in merging British and U.S. data in a single global sonic data bank on Arctic sea ice thickness, suggest that some caution is needed when they are used to derive long-term trends. The first use of satellite altimetry to retrieve sea ice freeboard (elevation of the ice floes above the sea level) and thickness is due to Laxon et al. (2003). They used data from the European Space Agency (ESA) satellites ERS-1 and ERS-2 in order to explore the correlation between ice thickness and the length of the ice season. More recently, the same group analysed other radar altimetry data, this time from ESA’s Envisat, obtained between the winters of 2002/2003 and 2007/2008 (Giles et al. 2008). They found a significant reduction in sea ice thickness in the region south of 81.5 N after the record minimum ice extent of September 2007 but no particular trend (in the winter season) between 2003 and 2007. The launch of NASA’s ICESat in January 2003, equipped with its high accuracy Geoscience Laser

J. Calvao et al.

Altimeter System (GLAS), allowed the first determinations of the central Arctic sea ice thickness from freeboard retrievals. Methods to apply laser altimetry to measurements of ice freeboard have been developed by Forsberg and Skourup (2005), Kwok et al. (2007) and Zwally et al. (2008), among others. In a thorough analysis of ten ICESat campaigns between 2003 and 2008, Kwok et al. (2009) derived the evolution of the Arctic sea ice thickness during this five year period, and were able to separate the contribution of first and multi-year ice. They concluded that the average winter ice thickness in the Arctic decreased from 3.3 m in February-March 2005 to less than 2.5 m in February-March 2008 and showed that this decline is essentially due to the disappearance of multi-year ice which, in the winter of 2008, was responsible for only one third of the total volume of ice in the Arctic Ocean. During a period of two weeks in March 2007 the L3H phase of operation of ICESat’s laser coincided with the cruise of the British submarine HMS “Tireless”. Large scale simultaneous observations of the Arctic Ocean’s sea ice thickness are very rare events. To our knowledge, this is only the second occurrence of this type. The first one was reported by Kwok et al. (2009), who used U.S. submarine data collected during a cruise in October/November 2005. The simultaneous measurement of the sea ice freeboard and draft offers the possibility of a cross-check (hopefully crossvalidation) between the results for Arctic sea ice thickness obtained independently by the complex satellite and submarine measurements. In this paper we present results for the sea ice thickness in the Arctic Ocean in the winter of 2007 (which preceded the extraordinary minimum ice extent of September 2007) and in the winter of 2004 obtained from ICESat and submarine data. In Sect. 16.2 we briefly describe the methods used in the retrieval of ice freeboard from ICESat and present overall results for the ice freeboard during the L2B and L3H phases. In Sect. 16.3 we give an overview of the techniques used to calculate ice draft from submarine observations and give average draft values for each section of the two cruises. In Sect. 16.4 we compare the ice thickness obtained by the two platforms and identify the main changes between the ice distributions in 2004 and 2007. We end up with a discussion on the possible causes of the observed discrepancies between ICESat and submarine results.

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Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite

16.2

ICES at Measurements

The laser altimeter system GLAS carried on board the ICESat satellite, especially dedicated to the observation of Polar Regions, generates profiles of the surface topography along the tracks with an accuracy of 15 cm. The beam width is 110 mrad and the pulse rate emission is 40 Hz, sampling the Earth surface with a footprint of 70 m diameter, spaced at 170 m intervals. Small-scale features inside the footprint modulate the amplitude and character of the waveforms, which can be used to determine sea-ice freeboard heights using a “lowest level” filtering scheme: specular laser returns from open water in leads between ice floes are used to define an ocean reference surface. The procedure applied to obtain the ice freeboard F ¼ h-N-MDT (where h is ICESat’s ellipsoidal height estimate of the surface, N is the geoid undulation and MDT is the ocean mean dynamic topography) for the whole Arctic basin (with the exception of points beyond 86 N) consisted of a high-pass filtering of the satellite data to remove low frequency effects due to the geoid and ocean dynamics (the geoid model ArcGP with sufficient accuracy to allow the computation of the freeboard was very recently made available (Forsberg et al. 2007). The original tide model was replaced by the tide model AOTIM5 and the tide loading model TPXO6.2. The inverse barometer correction was applied. As there are no MDT models with enough accuracy, the definition of the ocean surface was done through the identification of areas of open water (or thin ice, typically less than 30 cm thick) in each 20 km segment of the tracks to allow the interpolation of the ocean surface. Several solutions were tested to define the ocean surface and the computed freeboard values were interpolated on a 5 5 minute grid, where the submarine track was interpolated. Some waveform derived parameters in the ICESat data products were used to filter unreliable elevation estimates: surface reflectivity (i_reflctUncorr), detector gain (i_gain_rcv) and surface roughness (i_SeaIceVar). Figure 16.1a, b show the spatial distribution of the sea ice freeboard during the L2B (2004) and L3H (2007) phases of ICESat. Some comparisons between the ice freeboards in the two campaigns are now in order. The mean ice

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freeboard decreased between 2004 and 2007 in some regions of the central Arctic Ocean such as north of the Barents and Kara Seas in the Russian Sector, and between 120 W and 180 in the Canadian/Pacific sectors. Assuming that the depth of the snow on sea ice is similar in the two periods, this indicates a decrease in the sea ice thickness. The volume of ice in Fram Strait is much larger in 2007 than in 2004. There appears to be more thick ice in the vicinity of the north coasts of Greenland and Ellesmere Island in 2007 than in 2004. These observations are consistent with the strengthening of the Transpolar Drift in recent years reported by several authors, with large quantities of ice being transported across the central Arctic Ocean from the Chukchi and Beaufort Seas. Part of this ice is advected to the Atlantic through Fram Strait while the remaining piles up along the north coasts of Greenland and Ellesmere Island.

16.3

Submarine Measurements

In March 2007 the British submarine HMS “Tireless” carried out the route shown in Fig. 16.1b. After entering the Arctic Ocean through Fram Strait, it rounded northeast Greenland, thence cruised north of Greenland and Ellesmere Island roughly along the 85 N parallel until approximately 92 W, when it started heading SW towards the S Beaufort Sea. In the vicinity of (85 200 N, 64 080 W) the boat ran a set of parallel lines at low speed and depth with total length of 200 km under an area that was simultaneously used as a base for experiments of the EU DAMOCLES project. The outgoing part of the cruise ended at the location of the SEDNA ice camp (73 070 N, 143 440 W), where several other ice thickness measurements were performed (Hutchings et al. 2008). The submarine returned to the UK following almost exactly the same route as in the outbound journey and also collected draft data. However, because it was travelling at higher speed and depth, the quality of the data was somewhat inferior and will not be considered here. The April 2004 track, also shown in Fig. 16.1a, included a transect to and from the Pole and a diversion to (85 N, 62 W) in order to survey under a region which a month later was used for an ice camp experiment as part of the EU GreenICE project. Draft data were collected between the marginal ice zone in Fram Strait and the westernmost point of the

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Fig. 16.1 (a) L2B freeboard grid (m): max ¼ 0.97, mean ¼ 0.13, std ¼ 0.13. The track of the “Tireless” is shown in red. (b) L3H freeboard grid (m): max ¼ 0.84, mean ¼ 0.14, std ¼ 0.13. The track of the “Tireless” is shown in red

transect, including about 250 km under the GreenICE camp. Shortly after this survey the sonar stopped recording, except for a short period in the vicinity of the North Pole. In both cruises sea ice draft was measured with two upward-looking single-beam echosounders: the analogue system known as Admiralty Type 780 and the digital system known as Admiralty Type 2077. Because the latter malfunctioned during most of the 2007 voyage, in this paper we limit ourselves to the data obtained with AT780. Though the entire return from each ping is recorded in a paper chart, with the darkness of the trace a function of the intensity of the echo, we shall only take into account the first return from the insonified area (which is typically a circle with a diameter of a few metres for the 3 nominal beam width of the AT780 and ordinary cruise depths). This is a standard procedure in the analysis of submarine data: it is more reliable than to average over the entire pulse and it ensures compatibility with digital systems (where only the first return is recorded). Details of the techniques used to process this type of data can be found in Wadhams (1981).

Once the navigation data are included, the final product is the ice draft as a function of the position of the submarine or the along-track distance for the entire cruise. It is traditional to divide the full track into 50 km sections, as it has been shown that with such a length there is reliable statistics and a low probability of finding different ice regimes within one single section. Figure 16.2a, b show the mean ice draft obtained from the AT780 mounted in the “Tireless” for each of the sections of the 2004 and 2007 cruises. Notice that the 2004 cruise (which took place in April) did not coincide in time with ICESat’s winter campaign. However, because both measurements were made at the end of the winter, when the ice thickness is at or near its annual maximum, a comparison between them is, in our view, legitimate. This is also the reason why we used the freeboard values obtained throughout the whole ICESat winter campaign in 2007. A different approach, which the authors may follow in future work, would be to consider only the portion of the ICESat tracks that coincide in time and position with the track of the submarine.

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Fig. 16.2 (a) Mean ice drafts in the 2004 submarine cruise. (b) Mean ice drafts in the 2007 submarine cruise

16.4

Results

The sea ice thickness hi can be calculated from either the freeboard f (which includes the snow layer of depth hs because the laser pulse is reflected at the snow-air

interface if there is snow on top of the sea ice) or from the ice draft d through the relations hi ¼

rw r rs f w hs rw ri rw ri

(16.1)

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hi ¼

rw r d s hs ri ri

which follow from the assumption of isostatic equilibrium. Here rw ¼ 1024 kg=m3 , ri ¼ 920 kg=m3 and rs ¼ 360 kg=m3 are our assumed densities of the sea water, sea ice and snow, respectively. Values for the density of sea ice ranging between 850 and 925 kg/m3 have been reported (the later is used by Kwok et al. (2009)) while the density of the snow at the end of the winter may range from 300 to 360 kg/m3. Our choices have been widely used in the literature. While the snow term in the second equation can be safely neglected, this is not the case in the retrieval of the ice thickness from the ice (plus snow) freeboard measured with altimetry techniques, where the snow depth plays a crucial role. Unfortunately, the algorithms that attempt to extract snow depth from passive microwave satellite imagery (such as the NASA Team algorithm used to process raw AMSRE data) suffer from severe limitations. As such, in this paper we choose to work with values observed during recent winter field campaigns in the Beaufort and Lincoln Seas, which, typically, vary from zero to 30 cm. Here we use the upper value of 30 cm. In this case, it should be understood that the minima of the freeboard used to define a reference level as explained in Sect. 16.2 correspond not to zero elevation but to 30 cm elevation. Consequently, the value of the freeboard used to retrieve the sea ice thickness from (16.1) is the value that appears in the maps shown in Fig. 16.1a, b with 30 cm added. In fact, it is reasonable to assume that at the end of the winter the minima do not coincide with open water but with refrozen leads, which would have some snow on top. Tables 16.1a and 16.1b show the mean ice thicknesses obtained from satellite and submarine measurements for different regions of the Arctic visited by the submarine. Central (North) Fram Strait is the portion of the strait between 80 N and 82 N (82 N and 84 N); North Greenland 1 is the path of the submarine in 2004 along the 85 N parallel roughly between 20 W and 70 W, which coincides with the path of 2007; North Greenland 2 is the portion of the trajectory of the boat between 85 N and 86 N when it was heading northeastwards. Note that within each section, or region, the amount of invalid data differs for the satellite and the

Table 16.1a Mean sea ice thickness in the winter of 2004 Region

Sections

Central Fram Strait W Central Fram Strait E North Fram Strait Northeast Greenland North Greenland 1 North Greenland 2

01–18 30–33 34–37 38–43 44–57 59–62

Thickness (ICESat) 2.28 3.73 3.67 4.03 4.41 4.14

Thickness (Sub) 3.37 2.13 2.95 3.70 5.90 5.42

Table 16.1b Mean sea ice thickness in the winter of 2007 Region

Sections

Central Fram Strait North Fram Strait Northeast Greenland North Greenland N Ellesmere Island Canadian Basin Beaufort Sea

06–10 11–14 15–18 19–32 35–36 38–52 53–64

Thickness (ICESat) 2.85 2.88 2.78 4.65 4.65 2.90 2.27

Thickness (Sub) 4.43 3.93 3.77 5.95 7.00 5.42 3.73

submarine, which may explain some of the differences in the retrieved ice thickness. Figure 16.3a, b show the mean ice thickness for each of the 50nmjh sections into which the transects of the submarine were divided. ICESat values for each section are taken from a “square” box whose boundaries are the extreme values of the latitude and longitude of the submarine within that section.

16.5

Discussion

As it can be seen from Fig. 16.3a, b, although the results obtained from satellite and submarine data processing have good correlation concerning the shape of the curves representing sea ice thickness, there are visible discrepancies. The mean ice thickness computed from submarine data is consistently higher, with differences that can reach 1.5 m. Such differences occur mostly in the heavily ridged zone north of Greenland and Ellesmere Island (sections 44–57 in 2004 and 19–36 in 2007). There is reasonably good agreement in the Southern Beaufort Sea (last sections in 2007), where the submarine measured the thinnest ice of the whole transect.

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Fig. 16.3 (a) Mean ice thickness in the winter of 2004 for each section of the submarine cruise. (b) Mean ice thickness in the winter of 2007 for each section of the submarine cruise

Several reasons contribute to these results: (a) The comparison relates instantaneous data obtained from submarine draft data with gridded freeboard data computed from almost one month of satellite data, both converted into thickness, which means that this latter data set is a mean value for that period (moreover, the processing scheme produces some kind of attenuation of the freeboard grid). (b) The size of the footprints of the sonar and laser are quite different, which means that the integrated value and the resolution of the draft and freeboard are correspondingly different. (c) Beamwidth sonar correction were not performed, so the draft tends to be overestimated. Because of the comparatively narrow beam of the AT780 echosounder, such corrections have been considered

negligible in previous publications. However, a recent study by one of the authors suggests that this may not always be the case, namely in regions with a high density of pressure ridges. This might explain why the largest differences were found north of Greenland and Ellesmere Island. (d) A more sophisticated treatment of the snow may be required. We look forward to the release of version V12 of the NASA Team algorithm where, hopefully, some of the problems in the retrieval of the snow depth have been sorted. These should be considered as preliminary results. Clearly, more research is needed to fully understand the causes of the differences in the values of the sea ice thickness extracted from submarine sonars and satellite altimetry.

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References Forsberg R, Skourup H (2005) Arctic Ocean gravity, geoid and sea ice freeboard heights from ICESat and Grace. Geophys Res Lett 32:L215502. doi:10.1029/2005GL023711 Forsberg R, Skourup H, Andersen OB, Knudsen P, Laxon SW, Ridout A, Johannsen J, Siegismund F, Drange H, Tscherning CC, Arabelos D, Braun A, Renganathan V (2007) Combination of spaceborne, airborne and in-situ gravity measurements in support of Arctic Sea ice thickness mapping, Danish National Space Center, Technical report Nº 7, 2007 Giles KA, Laxon SW, Ridout AL (2008) Circumpolar thinning of Arctic sea ice following the 2007 record ice extent minimum. Geophys Res Lett 35:L22502. doi:10.1029/ 2008GL035710 Hutchings J et al (2008) Role of ice dynamics in the sea ice mass balance. Eos Trans AGU. 89(50). doi:10.1029/ 2008EO500003 Kwok R, Cunningham GF, Zwally HJ, Yi D (2007) Ice, Cloud, and land Elevation Satellite (ICESat) over Arctic sea ice: retrieval of freeboard. J Geophys Res 112:C12013. doi:10.1029/2006JC003978 Kwok R, Cunningham GF, Wensnahan M, Rigor I, Zwally HJ, Yi D (2009) Thinning and volume loss of the Arctic Ocean

J. Calvao et al. sea ice cover: 2003–2008. J Geophys Res 114:C07005. doi:10.1029/2009JC005312 Laxon S et al (2003) High interannual variability of sea ice thickness in the Arctic region. Nature 425:947–950 Rothrock DA, Yu Y, Maykut GA (1999) Thinning of the Arctic sea-ice cover. Geophys Res Lett 26(23):3469–3472. doi:10.1029/1999GL010863 Rothrock DA, Percival DB, Wensnahan M (2008) The decline in Arctic sea-ice thickness: Separating the spatial, annual, and interannual variability in a quarter century of submarine data. J Geophys Res 113:C05003. doi:10.1029/ 2007JC004252 Wadhams P (1981) Sea ice topography of the Arctic Ocean in the region 70 W to 25 E. Philos Trans Roy Soc Lond A302: 45–85 Wadhams P (1990) Evidence for thinning of the Arctic ice cover north of Greenland. Nature 345:795–797 Wadhams P, Davis N (2000) Further evidence of ice thinning in the Arctic Ocean. Geophys Res Lett 27:3973–3976 Zwally HJ, Yi D, Kwok R, Zhao Y (2008) ICESat measurements of sea ice freeboard and estimates of sea ice thickness in the Weddell Sea. J Geophys Res 113, C020S15. doi:10.1029/2007JC004284.

The Impact of Attitude Control on GRACE Accelerometry and Orbits

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U. Meyer, A. J€aggi, and G. Beutler

Abstract

Since March 2002 the two GRACE satellites orbit the Earth at relatively low altitude (500 km in 2002, still close to 460 km mid of 2009). GPS-receivers for orbit determination, star cameras and thrusters for attitude control, accelerometers to observe the surface forces, and a very precise microwave link (K-band) to measure the inter-satellite distance with micrometer accuracy are the principal instruments onboard the satellites. Determination of the gravity field of the Earth including its temporal variations from the satellites’ orbits and the inter-satellite measurements is the main goal of the mission. The accelerometers are needed to separate the gravitational acceleration from the surface forces acting on the satellites. They collect a wealth of information about the atmospheric density at satellite height as a by-product. These accelerations have not yet been analyzed thoroughly, because their interpretation is complicated due to numerous thruster spikes. We outline a method to model the thruster spikes and to clean the time series of the accelerations. The isolated effect of the modeled thruster pulses on the satellite orbits is studied and a first interpretation of the cleaned accelerations is given. A correlation between K-band residuals and regions of high atmospheric fluctuations was not observed, which is probably due to time variable signals of hydrological origin that dominate the residuals.

17.1

Introduction

The Gravity Recovery And Climate Experiment (GRACE, Tapley et al. 2004) satellites orbit the Earth at a height, where the atmospheric drag, in addition to solar radiation pressure and albedo effects,

U. Meyer (*) A. J€aggi G. Beutler Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland e-mail: [email protected]

still plays a major role in orbit dynamics. When estimating the Earth’s gravity field, these surface forces are considered as disturbing forces that either have to be modeled or preferably directly observed with an accelerometer onboard the satellite. In order to separate the surface forces from gravity, the accelerometer has to be located at the center of mass of the satellite. The position of the accelerometer relative to the center of mass is checked regularily by dedicated shaking maneuvers. If necessary, the center of mass is adjusted with movable masses inside the satellite.

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Direct solar pressure mainly depends on the satellite’s cross-section exposed to the sun and therefore varies twice per revolution. The influence of albedo is quite small in comparison to the other forces. As both satellites are identical and follow each other at a distance of 220 km or a timelag of 28 s on average in the same orbit, the influence of solar radiation pressure on the satellites will be greatly reduced in the difference of the accelerations between the two satellites. The highly variable part of atmospheric drag is the main signal remaining in the acceleration differences. Simultaneous acceleration differences are a measure for the spatial variability of the atmosphere’s density, at a distance of 220 km. If the timelag of the trailing satellite is taken into account when forming the difference, one obtains a measure for the temporal variability of the atmosphere’s density at one and the same location within 28 s. Both kinds of acceleration differences are studied subsequently. The accelerations originally measured in the satellite-fixed frames have to be transformed into a common reference frame before forming the differences. The transformation into a co-rotating frame with its axes pointing radially outward (R), normally to the orbital plane of the satellites (W), and orthogonally to these axes completing a right-handed frame (S, approximately in the direction of flight of the satellites) is performed using the attitude observations of the star cameras and the reduced dynamic satellite

orbits (J€aggi 2007). The interpretation of the resulting acceleration differences is complicated by numerous spikes (see Fig. 17.1), which usually may be attributed to one of the two satellites and which are highly correlated with thruster firings for attitude control, as recorded in the THR1B-files (Case et al. 2002). The thruster fire should, in principle, not be recorded by the accelerometers at all, because for each event two thrusters fire simultaneously and the corresponding thrusters are positioned in such a way, that their combined thrusts should only cause angular accelerations on the satellite. The observed linear accelerations may have two explanations: – The thrusters of a pair do not fire perfectly symmetrically and cause real linear accelerations. – The accelerometer is not positioned exactly in the center of mass of the satellite and therefore measures some artefacts due to thruster firings. As the thrusters fire frequently (500–800 times per day), it is necessary to develop a model for the thruster pulses to clean the observed accelerations.

17.2

Modeling Thruster Pulses

Six thruster pairs for attitude control and one thruster pair for orbit maneuvers are mounted on each satellite (the latter is not studied here). Time and duration of the thruster pulses are recorded in the THR1B-files.

R [μm/s2]

0.05 0 −0.05 240

260

280

300

320

340

360

380

400

420

260

280

300

320

340

360

380

400

420

260

280

300

320

340

360

380

400

420

S [μm/s2]

0.05 0 −0.05 240

Fig. 17.1 Accelerations of GRACE A (black) and differences GRACE B–A (gray), GRACE B shifted by 28 s

W [μm/s2]

0.05 0 −0.05 240

time [minutes] of day 90, 2007

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detail in Wu et al. (2006). Flury et al. (2008) discuss artefacts in the ACC1A data and their removal by the low-pass filter. The accelerometers do not sense the entire pulse. Therefore, we estimate scaling factors between the simulated and the actually measured accelerations for each thruster pair and each axis (R, S and W) of the reference frame. From the huge number of daily pulses we select only those which do not heavily overlap (which are separated by half the fit interval length of the low pass filter, i.e., by 70 s) and which have a certain minimal thrust duration (here 50 ms). The simulated accelerations by each of these pulses are fitted to the acceleration differences between the two satellites (the leading satellite shifted by 28 s) by estimating a bias, drift, and scale factor. The estimation of bias and drift parameters is necessary, because the signal of the thruster fire interferes with accelerations caused by atmospheric density variations. Because some of the pulses cannot be fitted very well, a screening step is needed. For all pulses of one and the same thruster pair a mean fit (RMS) is computed and badly fitted pulses are removed by a one sigma criterion. Mean scale factors are determined from the remaining pulses. Table 17.1 shows the results using data of the year 2003 and Table 17.2 for data of the year 2007, respectively. The thruster pairs 2 and 4 fire less frequently than the other thrusters and the estimation of scaling factors therefore is difficult (not shown). For all other thruster

The nominal thrust is 10 mN and the duration per thrust varies between 20 and several hundred milliseconds. To model that part of the thruster pulses, that is mapped to the measured accelerations, it is assumed that a constant scaling factor exists for each thruster pair and each axis of the reference frame. The processing procedure from the raw accelerations, measured onboard the satellites, to the ACC1B accelerations has to be understood first. An analog Butterworth filter of order 3 with a cut-off frequency of 3 Hz is applied onboard the satellites to the raw data to prevent aliasing. A shift by 0.14 s, called Butterworth-delay, is an unwanted artefact of this filter. Afterwards the data is sampled at a rate of 10 Hz, resulting in ACC1A-data. In the next steps the time scale is converted from On Board Data Handling (OBDH) to GPS time, the Butterworth-delay is removed and the data resampled at 1 Hz using a Charge Routing Network (CRN) class digital low-pass filter with a fit interval of 140.7 s and a target low-pass bandwidth of 0.035 Hz (Thomas 1999). This filter dampens the sharp spikes due to thruster firings and massively widens their shape. The effect of this transformation on orbit integration is not important, because the energy of the pulses is preserved. Together with the masses of the satellites, available in the MAS1B-files, we are now able to model the acceleration caused by a sharp 10 mN pulse on the satellites as in the ACC1A- or ACC1B-data (see Fig. 17.2). The filtering procedure is described in

0.4 20 0.3

Fig. 17.2 Left: original thruster pulse (black) and Butterworth-filtered pulse (gray), Butterworth-delay already removed. Right: ACC1A-pulse (gray) and ACC1B-pulse after low-pass filter (black), note the different scale

acc [μm/s2]

acc [μm/s2]

15

10

0.2

0.1

5 0 0

6386 6386.5 6387 6387.5

time [s]

−0.1

6350 6375 6400 6425

time [s]

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Table 17.1 Scaling factors of four thruster pairs, their error and the number of pulses considered, estimated from ACC1Baccelerations of 2003 2003 R Thruster 1 Thruster 3 Thruster 5 Thruster 6 S Thruster 1 Thruster 3 Thruster 5 Thruster 6

GRACE A Factor Sigma n

GRACE B Factor Sigma n

0.028 0.039 0.035 0.025

0.006 0.006 0.003 0.003

1,441 1,583 648 571

0.042 0.038 0.043 0.026

0.006 0.005 0.005 0.003

246 595 766 1,083

0.018 0.013 0.007 0.008

0.003 0.003 0.001 0.001

1,482 0.003 1,711 0.000 906 0.010 652 0.004

0.003 0.003 0.001 0.001

245 577 986 1,256

Table 17.2 Scaling factors of four thruster pairs, their error and the number of pulses considered, estimated from ACC1Baccelerations of 2007 2007 R Thruster 1 Thruster 3 Thruster 5 Thruster 6 S Thruster 1 Thruster 3 Thruster 5 Thruster 6

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pairs results of similar quality are obtained for 2003 and 2007. The scaling factors therefore can indeed be considered as constant over time (the change in sign in the S-direction from 2003 to 2007 is caused by the satellites’ switch of position in December 2005). All the pulses from the THR1B-files can now be modeled using the estimated scaling factors in the R, S, W-tripod. The results are promising (see Fig. 17.3), except for the W component. This poor quality in W is probably explained by the reduced sensitivity of the accelerometers in cross-track direction (Frommknecht et al. 2003). The accelerations in W are not further considered here, because they have the least influence on the inter-satellite distance and therefore on the gravity field estimation (J€aggi et al. 2011).

17.3

Effect on Satellite Orbits

We are now in a position to study the isolated impact of the thrusters on the satellite orbit and to compare it with the effect of the cleaned surface forces and with the combined effect of the original ACC1Baccelerations. Therefore, we determine orbits in four different ways: (a) Without ACC1B-DATA (b) With the complete, original ACC1B-data (c) Only with the modeled pulses of the thruster firings (d) With the accelerations reduced by the thrusterinduced effects The orbits were determined using the Celestial Mechanics Approach (CMA, Beutler 2005). Piecewise constant accelerations, set up for every 15 min interval, were constrained to zero with a sigma of 3.109 m/s2 and subsequently estimated together with all other parameters. This approach is very flexible, the pseudo-stochastic accelerations are able to absorb the missing non-gravitational accelerations to a large extent. To limit the model error by the static gravity field, AIUB-GRACE02S (J€aggi et al. 2011) was used as background model. AIUB-GRACE02S has been estimated from GRACE GPS and K-Band range-rate data of the years 2006 and 2007. Figure 17.4 shows the resulting K-band range-rate residuals. To separate the effects of interest from errors in the background models, the differences of the residuals from experiments (b) to (d) with respect to those of experiment (a) are shown (the model errors cancel out). The orbit perturbations caused by thruster firings and those caused by the cleaned surface forces are of comparable size. The simulated effect of the thruster fire on the orbit, obtained by direct numerical integration of the modeled accelerations and subsequent high-pass filtering (by subtraction of a moving mean over 3 min) is included in Fig. 17.4. The correlation with the K-band residuals is clearly visible. The differences are explained by the pseudostochastic accelerations. Therefore, we conclude that the thrusters significantly perturb the satellite orbits. Figure 17.5 shows the original range-rate residuals of experiments (b) and (d) and their difference due to the missing thruster accelerations in (d). The systematic effects are at least by a factor of three larger than the thrusters-induced effects. The thrusters (or the thruster

The Impact of Attitude Control on GRACE Accelerometry and Orbits

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mismodeling) are thus not responsible for these systematic effects. In a last orbit experiment we study the interaction between the linear thruster accelerations and the pseudo-stochastic pulses of the CMA using a Keplerian orbit. The Kepler ellipse, disturbed by the modeled thruster accelerations, is then introduced as “observed” orbit (3D-positions at 30 s intervals). These positions are used in an orbit determination, where pseudo-stochastic pulses are set up at 15-min

intervals. The resulting velocity residuals are compared with the range-rate residuals of the real orbit determination process in Fig. 17.6. The simulated and the real residuals are of the same order of magnitude. The jumps in the residuals every 15 min are due to the stochastic pulses. The Kepler orbit is a useful tool to assess the thruster-induced orbit perturbations in a qualitative way. We have not yet answered the question, whether the observed linear accelerations due to the thruster

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actuations are artefacts or real. GRACE orbits are determined according to the experiments (a) to (d) for 60 days of September and October 2007 to answer this question. Figure 17.7 shows daily RMSvalues of the range-rate residuals for all four cases. The results from Fig. 17.4 are confirmed, i.e., the thrusters do significantly perturb the orbits. It becomes clear by now, that their inclusion in the accelerometer files leads to a considerable improvement of the orbit fit. The linear thruster accelerations are thus mainly caused by non-symmetric firings of the thrusters.

17.4

Interpretation of Accelerations and Correlation with K-Band Residuals

Let us now use the cleaned accelerations to study the atmosphere. We limit our analysis to the acceleration differences in the direction of flight S of the satellites, which characterize the spatial or temporal variability of the atmosphere’s density. The state of the atmosphere mainly depends on the insolation, which is why sun-related coordinates are selected to represent the accelerations. The first coordinate is b, the angle

The Impact of Attitude Control on GRACE Accelerometry and Orbits

Fig. 17.7 Daily RMS of the range-rate residuals without using nongravitational accelerations (black), using the complete ACC1B-data (gray), considering only the modeled effect of the thrusters (black, dashed) or the cleaned effect of the surface forces (gray, dashed)

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between the W-axis of the co-rotating tripod and the direction to the sun. The second coordinate is u, the argument of latitude of the satellites on their orbit around the Earth. The Earth orbits the sun within a period of about 365 days and the orbital plane of the satellites rotates due to the precession of the nodes by about 40 per year. Hence the sun passes over the entire u, b-plane within 160 days and b varies by about 180 . The GRACE satellites orbit the Earth about 15 times per day and u therefore completes 15 full 360 cycles per day. Note that the selected coordinates are not strictly sun-fixed, the geocentric latitude of the sun varies between 23 during the year. Figure 17.8 shows acceleration differences for 160 days in the summer of 2007, referring to the same place, but separated 28 s in time. The center of the figure corresponds to the regions with the highest insolation. The heating of the atmosphere causes density fluctuations that result in the prominent signal visible in Fig. 17.8. Moreover, one sees the shadow entry and exit of the satellites, caused by a “discontinuity” in solar radiation pressure. White vertical lines mark arcs, where no orbits are available. Two horizontal lines currently cannot be explained. Figure 17.9, containing the simultaneous accelerations differences (thus corresponding to locations separated by 220 km in the north–south direction) shows much more variations than Fig. 17.8. The shadow entry

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We developed a method to model the linear accelerations on the satellites caused by the frequent thruster pulses. The impact of these pulses on the satellite orbits was studied and it was shown that the observed spikes represent real accelerations, which have to be taken into account for orbit determination. The method can be used to separate the surface forces from the thruster-induced accelerations for the radial and along-track direction. The analysis and interpretation of the cleaned accelerations, which mainly represent the atmospheric conditions at satellite height, is still at an early stage. To study the impact of atmospheric density fluctuations on orbit determination and on subsequent gravity field estimation, we plan to estimate and remove the strong seasonal variations of hydrological origin in the near future to be able to look for correlations between surface forces and K-band residuals. Acknowledgements The authors gratefully acknowledge the generous financial support provided by the Institute for Advanced Study (IAS) of the Technische Universit€at M€unchen.

β Fig. 17.9 Acceleration differences between the two GRACE satellites at the same time (separated 220 km in space), thruster fire removed

and exit are visible extremely well. Inclined, linear structures in the lower part of the figure are not yet explained. Strong atmospherical density variations can be observed close to the Earth’s north pole (around u ¼ +90 ), caused by interactions of the atmosphere with the magnetosphere. The corresponding variations close to the south pole (around u ¼ 90 ) are much less pronounced, because the southern atmosphere gets less insolation during the time period shown. The smoothness of the acceleration differences in Fig. 17.8 indicates that the signals visible in Fig. 17.9 really are atmospheric density fluctuations and not just noise. Currently, we cannot answer the question whether the atmosphere density variations shown in Fig. 17.9 have an impact on the orbit quality. The K-band rangerate residuals show a strong signal related to hydrology, which has to be removed before addressing the issue of the density variations.

References Beutler G (2005) Methods of celestial mechanics. Springer, Berlin. doi:10.1007/b138225 Case K, Kruizinga G, Wu S (2002) GRACE level 1B data product user handbook. D-22027. JPL, Pasadena, CA Flury J, Bettadpur S, Tapley B (2008) Precise accelerometry onboard the GRACE gravity field satellite mission. Adv Space Res 42:1414–1423. doi:10.1016/j.asr.2008.05.004 Frommknecht B, Oberndorfer H, Flechtner F, Schmidt R (2003) Integrated sensor analysis for GRACE – development and validation. Adv Geosci 2003(1):57–63 J€aggi A (2007) Pseudo-stochastic orbit modeling of low earth satellites using the global positioning system. Geod€atischgeophysikalische Arbeiten in der Schweiz, vol 73. doi:9783-908440-17-8 J€aggi A, Beutler G, Meyer U, Prange L, Dach R, Mervart L (2011) AIUB-GRACE02S – status of GRACE gravity field recovery with the celestial mechanics approach. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 30:5683. doi:10.1126/science.1099192 Thomas JB (1999) An analysis of gravity-field estimation based on Intersatellite Dual-1-way biased ranging. JPL, Pasadena, CA, pp 98–115 Wu SC, Kruizinga G, Bertiger W (2006) Algorithm theoretical basis document for GRACE level-1B data processing V1.2. D-27672. JPL, Pasadena, CA

Using Atmospheric Uncertainties for GRACE De-aliasing: First Results

18

L. Zenner, T. Gruber, G. Beutler, A. J€aggi, F. Flechtner, T. Schmidt, J. Wickert, E. Fagiolini, G. Schwarz, and T. Trautmann

Abstract

In standard gravity field processing, short-term mass variations in the atmosphere and the ocean are eliminated in the so-called de-aliasing step. Up to now the background models used for de-aliasing have been assumed to be error-free. As the accuracy assessed prior to launch could not yet be achieved in the analysis of real GRACE data, the de-aliasing process and related geophysical model uncertainties have to be considered as potential error sources in GRACE gravity field determination. The goal of this study is to identify the impact of atmospheric uncertainties on the de-aliasing products and on the resulting GRACE gravity field models. The paper summarizes the standard GRACE de-aliasing process and studies the effect of uncertainties in the atmospheric (temperature, surface pressure, specific humidity, geopotential) input parameters on the gravity field potential coefficients. Finally, the impact of alternative de-aliasing products (with and without atmospheric model errors) on a GRACE gravity field solution is investigated on the level of K-band range-rate residuals. The results indicate that atmospheric model uncertainties are small in terms of the associated spherical harmonic coefficients. The effect in terms of K-band observation residuals is negligible compared to other modeling errors.

18.1 L. Zenner (*) T. Gruber Institut f€ur Astronomische und Physikalische Geod€asie, Technische Universit€at M€ unchen, 80290, M€ unchen, Germany e-mail: [email protected] G. Beutler A. J€aggi Astronomisches Institut, Universit€at Bern, 3012, Bern, Switzerland F. Flechtner T. Schmidt J. Wickert GeoForschungsZentrum, 14473, Potsdam, Germany E. Fagiolini G. Schwarz T. Trautmann Deutsches Zentrum f€ ur Luft- und Raumfahrt, 82234, Weßling, Germany

Introduction

Mass redistributions inside, on, and above the Earth’s surface are responsible for time variable gravity field forces, which directly act on a satellite orbit. If these variations cannot be directly measured by repeated observations within short periods, they have to be removed during the gravity field determination in order to avoid aliasing due to undersampling. Shortterm mass variations in the atmosphere and in the oceans are part of the variations which cannot be measured with GRACE due to the inadequate spacetime sampling of the GRACE mission. To deal with this problem, the high frequency atmospheric and

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_18, # Springer-Verlag Berlin Heidelberg 2012

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oceanic signals are modeled and “removed” prior to or during the GRACE data processing. The “removal” of these high-frequency mass variations is called dealiasing (Flechtner 2007). As the GRACE results are still not meeting the expected pre-launch baseline accuracy, the de-aliasing process and related geophysical model uncertainties are considered here as a possible limiting factor for the GRACE gravity field determination. Therefore we explore ways to improve the de-aliasing process. In a first step, the de-aliasing process and its fundamental formulas, with the main focus on the processing sequence of the atmosphere, is reviewed (Sect. 18.2). As the standard processing scheme assumes error-free atmospheric and oceanic parameters and as it is well known that in areas with sparse observations the atmospheric models are degraded in quality (Salstein et al. 2008), there is reason to assume that one could improve the de-aliasing product and consequently the gravity field solution by taking into account uncertainties of the atmospheric parameters. This is why a mathematical model of error propagation of atmospheric model uncertainties into the gravityfield de-aliasing coefficients was developed. In order to study the impact of atmospheric model uncertainties on intermediate and final gravity field results, a test environment based on a real GRACE data set, data from the European Center for Medium Range Weather Forecast (ECMWF) atmospheric analysis (ECMWF 2009), and from the Ocean Model for Circulation and Tides (OMCT) (Dobslaw and Thomas 2007) was set up. The error scenarios for the atmosphere are defined in Sect. 18.2.2. The ocean is assumed as error-free in this study. For the time being, only atmospheric uncertainties provided by ECMWF were used to get a first insight into the effect of model errors on the de-aliasing coefficients (AOD) and GRACE. In Sect. 18.3.1 the impact of the error assumptions on the estimated de-aliasing coefficients and the geoid is studied. Differences between the error-scenarios are computed and compared to the GRACE error predictions. Finally, the newly computed de-aliasing coefficients are used for a GRACE gravity field determination. In order to identify their impact on the resulting gravity field, observation residuals are investigated in detail in Sect. 18.3.2. Sect. 18.4 summarizes the results and draws conclusions.

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18.2

Theory: The De-aliasing Process

18.2.1 Fundamental Formulas A short overview of the fundamental formulas of the current atmospheric and oceanic de-aliasing processing sequence is provided below. The following formulas outline the computation of the harmonic coefficients Cnm , Snm due to atmospheric and oceanic variations. Pkþ1=2 ¼ akþ1=2 þ bkþ1=2 Ps

(18.1)

Tv ¼ ð1 þ 0:608SÞT

(18.2)

Hkþ1=2 ¼ Hs þ

kmax X Pjþ1=2 RTv ln g Pj1=2 j¼kþ1

Z Z Cnm a2 ð1 þ kn Þ ¼ ½In þ PO ð2n þ 1ÞMg Snm Pnm ðcos yÞ Z0 In ¼ Ps

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(18.5)

Pk+1/2 ¼ pressure at half-levels ak+1/2, bk+1/2 ¼ model dependent coefficients Tv ¼ virtual temperature R ¼ gas constant for dry air Hk+1/2 ¼ geopotential height at half-levels g ¼ mean gravity acceleration a ¼ semi major axis of reference ellipsoid M ¼ earth mass kn ¼ loading love numbers Po ¼ ocean bottom pressure In ¼ vertically integrated atm. pressure Pnm ¼ assoc. norm. legendre polynomials N’ ¼ mean geoid height above sphere r ¼ a Four input parameters are needed for the determination of the atmospheric potential coefficients: Surface pressure Ps, surface geopotential height Hs, temperature T and specific humidity S at the 91 atmospheric model levels (full-level). After a few steps, the vertical integration of the atmospheric pressure In is performed in (18.5). The atmospheric pressure In is

18

Using Atmospheric Uncertainties for GRACE De-aliasing: First Results

then combined with the ocean bottom pressure Po in order to obtain the combined atmospheric and oceanic potential. The combined atmospheric and oceanic pressure is subsequently called PAo (18.6). In a final step (18.4), the numerical integration is performed and the atmosphere and ocean de-aliasing product (AOD) is obtained. A few comments should be made concerning the outlined procedure: Usually, a mean field is subtracted before-hand from the vertically integrated atmospheric pressure In and from the ocean bottom pressure Po, i.e., PAo contains the residual atmospheric and oceanic pressure. The mean fields are subtracted in order to analyze gravitational variations. Since the mean mass distribution of the atmosphere and ocean by definition refers to the static part of the gravity field, only the deviations from the mean value have to be taken into account. In the currently realized GRACE gravity field processing as well as in our investigations, mean fields obtained from the years 2001/2002 are used. For details we refer to Flechtner (2007).

18.2.2 Changing the Current De-aliasing Process: Two Error Scenarios Temperature, specific humidity, surface pressure, and surface geopotential height have so far been assumed to be error-free, although it is known that there are large uncertainties in the atmospheric input parameters, particularly in the surface pressure (Ponte and Dorandeau 2003). The goal of this study was to find out to what extent the atmospheric model uncertainties affect AOD and GRACE results. To answer this question, the atmospheric field errors are propagated into the vertically integrated atmospheric pressure In in a first step. As mentioned previously, for the time-being, the ocean is regarded as error-free. In a second step, in order to take the error of In into account and to further propagate it into the AOD coefficients Cnm , Snm , the present approach of numerical integration (18.4) was replaced by a least-squares adjustment (Gauß-Markov, model). Equation (18.4) was used as an observation equation: Mg X 2n þ 1 X Pnm ðcos yÞ a2 n 1 þ k n m ½Cnm cos ml þ Snm sin ml;

PAO ¼

(18.6)

149

where the combined residual atmospheric and oceanic pressure fields PAO are used as observations and the error of PAO as individual (inverse) weights. The error of PAO equates to the error of as the ocean In is assumed as error-free. In this way we obtain the AOD coefficients as a function of the errors of the input parameters, i.e., as a function of the weighting. In order to see whether the atmospheric model uncertainties significantly affect AOD and GRACE results, two so-called error scenarios were studied: “error-free AOD”: The uncertainties of the four atmospheric parameters are not taken into account. “full-error AOD”: The uncertainties of the four atmospheric parameters are taken into account. The “error-free” experiment thus assumes that all observations PAO have the same weight. The “fullerror” scenario assumes that the observations PAO are weighted individually using the inverse error of PAO on the diagonal of the weighting matrix. After performing the least-squares adjustment for these two scenarios, one obtains two sets of AOD products. By comparing them (Sect. 18.3), one gets insight into the effect of atmospheric uncertainties on the atmospheric and oceanic potential coefficients Cnm , Snm . The following results were achieved using data provided by the ECMWF (Operational archive, Atmospheric model, Analysis and Errors in analysis; ECMWF 2009).

18.3

Results: Impact of Atmospheric Uncertainties on AOD and GRACE

18.3.1 Effect of Model Uncertainties on AOD Figure 18.1 shows the difference between the AOD product for the “full-error” and the “error-free” experiment in terms of geoid heights. The atmospheric model uncertainties affect the geoid in the range of 0.8 to 0.3 mm. The largest differences are visible in regions where the weights for the observations are extreme (high or low). This result seems to be clear: The deviation between the two error-scenarios is only caused by the different weighting of the observations PAO. In the “error-free” case the combined atmospheric and oceanic pressure has the same weights,

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whereas in the “full-error” scenario the observations are weighted using the inverse error of PAO (which is equal to the vertically integrated atmosphere error since the ocean is assumed as error-free). Therefore, the error of the atmospheric pressure is reflected in the difference between the two error-scenarios. The effect of atmospheric uncertainties on AOD is in the sub-mm domain, which seems to be small. Figure 18.2 shows the difference between the two error-scenarios in terms of spherical harmonic degree variances. The difference between the two error-scenarios is presented in the dotted line marked with circles. Compared to the GRACE baseline (Kim 2000; lowest line in Fig. 18.2) the effect of atmospheric model uncertainties clearly is in the sensitivity range of GRACE. Therefore we conclude that an effect should be visible in the gravity field solution as well. This aspect will be studied in Sect. 18.3.2.

18.3.2 Effect of Model Uncertainties on K-Band-Residuals The K-band range-rate (KBRR) residuals are obtained from an extended orbit-determination problem, where one tries to minimize the differences between the computed K-band range-rate observations based on a priori models and the real observations made by GRACE by only adjusting orbit parameters. In theory, the resulting KBRRresiduals would become zero, if all force models (tides, static gravity field, AOD-model,. . .) as well

as the real observations were error-free. As it is known, this is not the case. Thus differences between the modeled and the real world occur in the KBRRresiduals. The residuals contain all model errors, measurement errors, and unmodeled forces. By applying the new de-aliasing coefficients, which take the atmospheric model uncertainties into account, we hope to reduce these residuals. This would imply that the new (“full-error”) AOD product is more realistic than the standard or “error-free” AOD. For this purpose the Bernese GPS Software (Dach et al. 2007) and the Celestial Mechanics Approach (J€aggi et al. 2011) is used to perform an orbit determination based on kinematic positions and K-band range-rate observations by using the gravity field model AIUB-GRACE02S (J€aggi et al. 2011) as known and using the different sets of AOD coefficients. By leaving all other settings unchanged, one gains insight into the effect of atmospheric model errors. For detailed parametrization information, we refer to J€aggi et al. (2011). In addition to the two error-scenarios described in Sect. 18.2.2, the overall effect of AOD by not applying AOD products for orbit determination was included as well. The impact of the three experiments on the KBRR residuals is shown in Fig. 18.3. Applying (dark grey “area” in Fig. 18.3) or not applying AOD (light grey “area”) clearly affects the KBRR-residuals. Removing the high frequent atmospheric and oceanic mass variations significantly reduces the RMS over 1 year of KBRR-residuals by about 10%. This implies that the inclusion of the short-term atmospheric and oceanic variations for gravity field solutions leads to more realistic results. Figure 18.3 shows that the amplitudes of the prominent peaks are reduced, but not removed, when applying AOD. One reason for that might be due to the fact that AIUB-GRACE02S is a static field where the time-variable gravity field signals are not yet modeled. Thus, the KBRRresiduals contain apart from other model errors mainly the unmodeled time-variable gravity forces (in particular hydrology). However, in general the signal in the residual series is reduced. Is the newly computed full-error AOD product able to further improve de-aliasing, i.e., to further reduce KBRR-residuals? Figure 18.3 (bold black line) shows that this is not the case. This is contrary to our

18

Using Atmospheric Uncertainties for GRACE De-aliasing: First Results

Fig. 18.2 Spherical harmonic degree variance differences between the “error-free” and “full-error” scenario in terms of geoid heights. 2007-08-01 00h. Unit (m)

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18.4

Conclusion and Discussion

Figure 18.4 is identical to Fig. 18.2, except that it also contains the achieved (as opposed to the expected prelaunch) accuracy of gravity field determination with GRACE (curve with asterisk). As the differences

“error-free” minus “full-error” are clearly below the curve “achieved GRACE baseline”, it is clear that the impact of the refined AOD model studied here is not visible in gravity field determination. It must be pointed out, however, that our results are based on atmospheric error fields taken from the operational analysis of the ECMWF. The results shown are therefore only true for the ECMWF error-fields. The effect of alternative uncertainty estimates from other sources [e.g., NCEP (National Centers for Environmental Prediction)] has not been investigated yet.

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Fig. 18.4 Degree variance differences of error-free AOD and full-error AOD in terms of geoid heights (m) for 2007-08-01 00h, and the predicted and current GRACE error estimates

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Also, one should take into account that the used errorfields might be too optimistic as they result from the assimilation model (ECMWF 2009). More work has to be carried out concerning the determination of representative error parameters (e.g., Schmidt et al. 2008). Previous investigations have shown that the surface pressure error has the most significant effect on the vertically integrated atmospheric pressure. It is therefore essential to determine reliable and reasonable surface pressure values and use them as input for the upcoming investigations. Furthermore, it must be emphasized that so far the ocean has been assumed to be error-free. To determine representative error-fields of ocean bottom pressure and to take them into account during AOD processing will be one of the next steps.The goal of this study was to improve the de-aliasing process by taking atmospheric model uncertainties into account. It was shown that atmospheric model uncertainties are not able to improve the de-aliasing process or GRACE orbit determination, because the actual error of GRACE is above the predicted baseline accuracy. It should be kept in mind, however, that model uncertainties will become more important if the achieved accuracy of GRACE can be further improved towards the assumed baseline. Acknowledgements This study was conducted as part of the IDEAL-GRACE project with the support of the German Research Foundation (Deutsche Forschungsgemeinschaft) and within the SPP1257 priority program “Mass transport and Mass Distribution in the System Earth”. The International Graduate School for Science and Engineering of the Technische Universit€at M€unchen also supported this work.

References Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS software version 5.0. Astronomical Institute, University of Bern, Bern Dobslaw H, Thomas M (2007) Simulation and observation of global ocean mass anomalies. J Geophys Res 112:C05040. doi:10.1029/2006JC004035 ECMWF (2009) MARS user guide. Technical Notes, p 5. http:// www.ecmwf.int/publications/manuals/mars/guide/MarsUser Guide.pdf. Accessed date 17 August 2011 Flechtner F (2007) AOD1B product description document. GRACE project documentation, JPL 327–750, Rev. 1.0. JPL, Pasadena, CA. 17 August 2011 J€aggi A, Beutler G, Meyer U, Prange L, Dach R, Mervart L (2011) AIUB-GRACE02S: status of GRACE gravity field recovery using the celestial mechanics approach. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg Kim J (2000) Simulation study of a low-low satellite-to-satellite tracking mission. Technical report, University of Texas at Austin, Austin, TX Ponte RM, Dorandeau J (2003) Uncertainties in ECMWF surface pressure fields over the ocean in relation to sea level analysis and modeling. J Atm Ocean Technol 20(2):301. doi:10.1175/1520-0426(2003)020 Salstein DA, Ponte RM, Cady-Pereira K (2008) Uncertainties in atmospheric surface pressure fields from global analyses. J Geophys Res 113:D14107. doi:10.1029/ 2007JD009531 Schmidt T, Wickert J, Heise J, Flechtner F, Fagiolini E, Schwarz G, Zenner L, Gruber T (2008) Comparison of ECMWF analyses with GPS radio occultations from CHAMP. Ann Geophys 26:3225–3234. http://www.ann-geophys.net/26/ 3225/2008/. Accessed date 17 August 2011

Challenges in Deriving Trends from GRACE

19

A. Eicker, T. Mayer-Guerr, and E. Kurtenbach

Abstract

The following contribution addresses some of the problems involved with the determination of long-term gravity field variations from GRACE satellite observations. First of all the choice of the time span plays a very important role, especially since it generally is a hard task to derive secular trends from only a few years of satellite data. Another issue, when one is interested in a single trend phenomenon, is the reduction of all other geophysical effects causing long-term gravity field variations. This paper uses the example of trends in continental hydrological water masses for the case of the High Plains aquifer to demonstrate some of the challenges implicated by trend analysis from GRACE.

19.1

Introduction

During the last years, the satellite mission GRACE (Tapley et al. (2004)) has provided significant improvement in the knowledge of the Earth’s gravity field, in the static as well as in the time variable component. The time variabilities derived from GRACE deliver information about short periodic variations such as the seasonal hydrological cycle (e.g. Schmidt et al. 2008), occasional incidents such as large earthquakes (Han et al. 2006), and also long-term trends. The investigations to be presented in this contribution are dedicated to the analysis of these long-term processes from GRACE observations. Examples are the melting of the ice sheets and glaciers in Greenland and Antarctica (Horwath and Dietrich 2009), rebound

A. Eicker (*) T. Mayer-Guerr E. Kurtenbach Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115 Bonn, Germany e-mail: [email protected]

effects due to glacial isostatic adjustment (GIA), see e.g. Steffen et al. (2008), or trends in the continental hydrological water masses due to water withdrawal or climatic changes, as outlined in Rodell et al. (2009). When trying to derive long-term trends from GRACE gravity field observations, one is faced with a number of challenges. First of all, the time span used in the analysis is very important because it is very difficult to derive long-term gravity field variations from only 7 years of GRACE data or even less. The other important aspect that has to be taken into account is that when the focus is on one single phenomenon, all other effects causing long-term gravity field variations have to be removed. This is generally performed by the application of geophysical models, with the drawback that inaccuracies or uncertainties of these models are introduced into the trend analysis. Some confusion might be caused by the fact that these models are not only used for signal separation but also, with a different intention, in the GRACE data analysis (linearization, de-aliasing). The latter point is discussed in

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Sect. 19.2 while Sect. 19.4 is dedicated to the problem of signal separation. The High Plains aquifer (marked in white in Fig. 19.2), located in the central United States with an area of 450,000 km2, serves as an example in this article to demonstrate some of the issues that have to be addressed when deriving long-term gravity field variations from GRACE. This investigation area was chosen because of an assumed mass loss trend due to ground water depletion caused by large-scale irrigation, see Strassberg et al. (2009). It is subject to discussion, whether GRACE is able to detect such human ground water withdrawals, see Rodell (2002) and Strassberg et al. (2009). However, this paper does not intend to give quantifications of potential mass loss or to draw qualitative conclusions whether such investigations are indeed possible with GRACE. The main intention is to point out some of the issues that have be considered when such investigations shall be performed. They are not specific for the example area but are valid for every investigation dealing with the interpretation of GRACE data.

19.2

Gravity Field Processing

In order to interpret the GRACE solutions and to derive, for example, trend information out of the GRACE products it is necessary to understand some details about the processing procedures. During the processing of the original GRACE data (level 1B), the observations y are expressed as non-linear functionals y ¼ f (x) of the unknown gravity field parameters x. In the examples described below, these parameters are given as a set of spherical harmonics coefficients for each monthly solution. After a linearization process the coefficients can be estimated by means of a least-squares adjustment. The linearization is given by: y ¼ f ðx0 Þ þ

@f ðx x0 Þ: @x x0

(19.1)

Here x0 describes the approximate solution, functionals of which are reduced from the original observations. This means that the least squares adjustment delivers a solution for Dx ¼ x x0. In order to keep the linearization error small, the approximate

solution should be as close as possible to the real gravity field signal in the corresponding month as possible. Therefore, the following models are taken into account by the processing centers: a static gravity field solution, a trend for the lower spherical harmonics, tides (direct tides, Earth and ocean tides), periodic changes of the centrifugal potential due to polar motion (pole tide and ocean pole tide), and mass variations of the atmosphere and the ocean. Apart from the linearization, these models serve also for the reduction of short-term mass variations that cannot be modelled by monthly representations (dealiasing). To obtain the complete gravity field signal for a monthly solution, the components removed by these models should be restored according to: x ¼ x0 þ Dx:

(19.2)

But in case of the official GRACE products, this is not carried out in a consistent manner. Only the static solution is re-added, but the trend and all other models are not present in the solutions. The atmosphere and ocean model, however, is delivered as an additional product in terms of monthly mean values. The consideration of these background models differs slightly from one analysis center to the next, for example by a different treatment of the permanent tides and different trend models for the lower harmonic coefficients. Details can be found in Flechtner (2007), Bettadpur (2007), and Watkins and Yuan (2007). Another aspect that has to be taken into account when dealing with GRACE solutions is the fact that the coefficients of degree n ¼ 1 cannot be determined from GRACE observations, and are therefore set to zero. This corresponds to the realization of a coordinate system in the center of mass (CM) of the complete Earth system (solid Earth including ocean and atmosphere and all other subsystems). The CM varies in comparison to the center of the solid Earth (CE) or the center of figure (CF). A good discussion of this topic can be found in Blewitt (2003). While there exist models to account for the seasonal variations of these changes, the longterm variations are still under investigation (Klemann and Martinec 2009). It is commonly known that the accuracy of the GRACE monthly solutions decreases with increasing spherical harmonic degree, i.e. with increasing resolution. Therefore, the solutions have to be filtered in order to allow a meaningful interpretation of the

19

Challenges in Deriving Trends from GRACE

results. The choice of the filtering technique has a significant influence of the solution. As there have been a number of investigations on this topic in the past, see for example Steffen et al. (2009) and Werth et al. (2009), we will not go into further detail at this point.

19.3

Time Span

Some of the geophysical phenomena causing longterm gravity field variations have been going on for time spans of decades to millenia and even longer. In contrast to this, the measurement period of GRACE with only a few years of observations is very short. Taking into account the strong inter-annual gravity field variations, it is very difficult to derive representative trends from such a short time series. It has been shown, that especially when only parts of the GRACE period are evaluated, the estimated trend changes significantly, an example given by Horwath and Dietrich (2009) for the case of ice mass loss in Western Antarctica. In the study at hand, we perform a similar analysis for the temporal evolution in the High Plains aquifer, using the GFZ-RL04 monthly gravity field series (Flechtner et al. 2009) for the time span January 2004 to December 2008. The monthly gravity field solutions provided by the CSR (Bettadpur 2007), the JPL (Watkins and Yuan 2007) and the ITG-Grace03s (Mayer-G€urr et al. 2007) time series were evaluated as well, all providing similar results to those presented below. But for reasons of clarity and because the ITGGrace solutions are so far only available until April 2007, only the results for the GFZ solutions will be displayed. The monthly gravity field models are smoothed by a 500 km Gaussian filter and integrated over the area of the High Plains aquifer. Figure 19.1 displays the mass variations of the GFZ-RL04 monthly gravity field solutions in terms of geoid heights. If only the time span until the end of December 2006 is considered, the linear trend (black line on the left) shows a strongly negative behavior suggesting an interpretation as mass loss in the area. This is the time span under investigation in Strassberg et al. (2009), but the examination of the complete time series up to today shows that statements about such trends are critical from only a few years. When introducing the next 2 years of GRACE data,

155

the trend for 2007–2008 (black line on the right) shows a different behavior by being slightly positive. The overall linear trend for the whole time span (indicated by the gray line) is therefore slightly positive as well. Of course the results raise the question, whether a linear function is the correct representation of the long-term variability in this specific area. Alternatively, one could think about the use of basis functions changing slowly in time or applying low pass filters to remove short-term variations.

19.4

Separation of Trend Signals

GRACE can only observe the integrated mass signal without being able to distinguish between different mass sources. In order to separate different mass signals external information is required, which is commonly obtained from geophysical models. More precisely this implies that if one is interested in only one specific geophysical phenomenon (e.g. continental water variations), the GRACE solutions must be reduced by models of all other mass variations. At this point it has to be emphasized that this task of signal separation is in principle independent of the linearization and dealiasing process described above. This is especially important when considering the different error sources. During the linearization process, the introduction of approximate values does not lead to any additional model errors, apart from linearization errors which are negligible in case of GRACE. Therefore, after readding all models that are used in the processing, the result is the complete mass signal which is only influenced by the observation errors of GRACE. This solution should be independent of the inaccuracies in the models as they are only used as approximation values. In contrast to this, the model errors play a very important role for the task of signal separation, as the quality of the separation can only be as good as the accuracy of the models. In case of GRACE it is difficult to distinguish between signal separation and linearization as the restore step is not carried out in a consistent manner, as described in Sect. 19.2. In the context of our example of mass changes in the High Plains aquifer, the main focus is on the determination of changes in the continental water budget. Therefore, all other effects have to be removed from the GRACE signal. There is a wide variety of different mass variations that have to be considered.

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Fig. 19.1 Monthly solutions as provided by the GFZ-RL04 time series averaged over the High Plains aquifer region. One overall trend has been calculated for the entire time span (gray

line) and individual trends for the years 2004–2006 and 2007–2008 (black lines)

The first ones are the variabilities of the atmospheric masses and the effects on the ocean caused by these variabilities. These short- and long-periodic changes are included in the so-called atmosphere and ocean dealiasing product (AOD1B; see Flechtner 2005) which is reduced during the analysis process especially to prevent the solutions from aliasing errors caused by short periodic signals and is not restored afterwards. This means that the separation of the atmosphere masses has already been performed during the data processing step. Nevertheless, if a more suitable atmospheric model (e.g. a local model) is available, it might be a good choice to re-add the global model and remove a more accurate local model which is better tailored to the specific area. Another source for long-term variations might be the long-periodic part of the tidal forces (e.g. the 18.6year cycle), of the Earth tides and the ocean tides which appear as trends when only a few years of data are analyzed. However, those tidal effects are also taken into account by models during the GRACE gravity field analysis process from all of the different analysis centers and can therefore be regarded as nonexistent in the GRACE monthly solutions. These assumptions are only valid, of course, to the extent

of the accuracy of the given models, as errors in the modeling of these long-term variations might be wrongly interpreted as originating from a different source. It has to be emphasized that the AOD1B and tide models are used twice, each time with a different objective: for linearization but also for signal separation. A different issue is the handling of the trend which is removed during the GRACE processing. Most analysis centers use rates given for the lower spherical harmonic coefficients (dot coefficients) as proposed by the IERS conventions (McCarthy and Petit 2004). While these rates might be sufficient as approximation values, they are not suitable for signal separation. They were derived during the development of the EGM96 gravity field model by a combination of models and observations. The intention of these rates was to account for the integrated mass trend. It is not possible to distinguish which subsystems contribute to the dot coefficients. There is certainly some part of the glacial isostatic adjustment effect present, but a complete GIA model cannot be described by only a few coefficients. Furthermore, we do not know for certain whether there are also some hydrological effects comprised in those coefficients. Therefore, we

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Challenges in Deriving Trends from GRACE

157

Fig. 19.2 Spatial distribution of different phenomena that cause long-term trends in GRACE data, displayed in terms of geoid heights per year in mm. Top: dot coefficients, middle: glacial isostatic adjustment, bottom: secular variations in pole tide model. The region of the High Plains aquifer in the Central United States is illustrated in white

−0.30 −0.24 −0.18 −0.12 −0.06

0.00

0.06

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0.0 0.2 [mm/year]

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believe that it is preferable to restore these rates and to subtract the different models according to present knowledge afterwards. In order to give an impression of the order of magnitude and the spatial pattern,

−0.6

−0.4

the rates given by the IERS are displayed in the top part of Fig. 19.2 as changes in geoid heights in mm per year. A considerably large negative rate is visible for our example of the area containing the High Plains aquifer.

158

After re-adding the mass trend proposed by the IERS dot coefficients, the next step is to remove the known long-term mass change phenomena. As a first aspect in this context the glacial isostatic adjustment has to be mentioned, causing a mass redistribution due to the uplift of the land mass as a reaction to the deglaciation after the last ice age. This phenomenon can be observed especially in the Scandinavian and Northern Canadian area, but the effects can be witnessed further south as well. The spatial distribution of the GIA is displayed in the middle part of Fig. 19.2 as a rate in terms of millimeter of geoid height per year, the corresponding model was provided by Klemann and Martinec (2009). In the area of the High Plains aquifer positive as well as negative rates can be found in the GIA model, with the positive trend prevailing. The second aspect that has to be considered is the effect of the change in the centrifugal potential caused by polar motion (pole tide). While all of the analysis centers reduce a pole tide model during their gravity field estimation process, this model only contains the short-term variations of the pole tides, but not the secular mass changes. The spatial distribution of these secular variations, calculated according to the IERS conventions, can be found in the bottom part of Fig. 19.2. This effect is rather small globally, but in the investigation area in the central United States a comparably large positive trend can be observed nevertheless. The influence that the effects described above have on the estimation of the trend in the investigation area can be observed in Fig. 19.3. The original GFZ-RL04 time series is plotted in gray, where only the effects given by the de-aliasing product and the different tidal models have been removed. The remaining effects have been integrated over the High Plains region and the corresponding rates are shown as well. The effect caused by GIA is indicated by the orange line, here the positive trend can be noticed. The pole tide trend is given in green, even though this effect is quite small globally, in the region at interest a significant positive trend is present. The IERS rates are plotted in the black line and as already concluded from the spatial distribution, a large negative trend is visible when integrating over the High Plains region. Re-adding these rates and subtracting the GIA and pole tide model results in the reduced time series illustrated in red with the corresponding linear trend. In contrast to the original time series, the overall trend

A. Eicker et al.

is now negative. After having taken into account the variations of atmosphere and ocean, of tides and solid Earth mass transports, the remaining variations can be interpreted as changes in the hydrological budget. Here it has to be kept in mind that the changes in the gravitational potential (expressed in terms of geoid heights) does not only contain the changes in the water masses but also the reaction of the Earth’s crust due to loading. These effects can be separated using a loading theory. Specifically this implies the multiplication of the spherical harmonic coefficients with the load love numbers, see Wahr et al. (1998). Within this step it is reasonable to convert the resulting trend signal from potential to the generating water masses in terms of equivalent water heights. This conversion would have been mathematically possible at an earlier stage as well, but it is difficult to interpret effects such as GIA in terms of water heights. The resulting trend in terms of equivalent water heights of the original and the reduced time series is shown in Fig. 19.4. In contrast to the analysis in terms of geoid height, the trend is slightly positive. It is doubtful whether this trend can be regarded as significant. It is extremely difficult to make a reliable statement in this matter, as not only the currently achievable accuracy of GRACE has to be taken into account, but also errors in the background models used for signal separation. Especially for the geophysical models plausible error estimations are hardly available.

19.5

Conclusions

This article shows some of the pitfalls occurring when analyzing and interpreting long-term trends derived from GRACE data. The first important conclusion is the need of long time series to derive trends especially in areas with strong inter-annual variability. In order to give reliable statements concerning secular mass changes longer time series are inevitable, which emphasizes the importance of a GRACE follow-on mission without any gap. The second issue deals with the separation of signals originating from different geophysical processes. For example, if one is interested in changes in the hydrological water masses, the contributions of atmosphere, oceans, tides, and solid Earth have to be taken into account. These effects are usually removed by models. It has to be

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Challenges in Deriving Trends from GRACE

Fig. 19.3 Influence of the different background models on the trend in the High Plains aquifer: original GFZ-RL04 time series (gray), GIA model (orange), pole tide trend (green), rates

159

proposed by the IERS (black). Reduced time series with corresponding linear trend: red

Fig. 19.4 The original (gray) and the reduced (red) time series expressed in terms of water heights in mm

kept in mind that in case of GRACE some part of the separation step has already taken place during the data analysis, while further models have to be reduced by the user. Other models are only used for linearization or de-aliasing purposes and are not suitable for signal

separation. This article summarizes the correct use of the different models and demonstrates their importance for the interpretation of trend signals from GRACE data using the example of the High Plains aquifer region.

160 Acknowledgments The authors would like to thank Volker Klemann from the GFZ for providing the GIA model. The support by the DFG (Deutsche Forschungsgemeinschaft) within the frame of the special priority program SPP1257 “Mass transport and mass distribution in the Earth system” is gratefully acknowledged.

References Bettadpur S (2007) UTCSR level-2 processing standards document for level-2 product release 0004. GR-03-03. CSR, Austin, TX Blewitt G (2003) Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth. J Geophys Res 108(B2):2103. doi:10.1029/2002JB002082 Flechtner F (2005) GRACE AOD1B product description document (Rev. 2.1) Flechtner F (2007) GFZ level-2 processing standards document for level-2 product release 0004. GR-GFZ-STD-001. GFZ, Potsdam Flechtner F, Dahle Ch, Neumayer KH, Koenig R, Foerste Ch (2009) The release 04 CHAMP and GRACE EIGEN gravity field models. In: Flechtner F, Gruber T, Guentner A, Mandea M, Rothacher M, Wickert J (eds) Satellite geodesy and earth system science – observation of the earth from space. Springer, Berlin (in preparation) Han S-C, Shum CK, Bevis M, Ji C, Kuo C-Y (2006) Crustal dilatation observed by GRACE After the 2004 SumatraAndaman Earthquake. Science 313:658–662. doi:10.1126/ science.1128661 Horwath M, Dietrich R (2009) Signal and error in mass change inferences from GRACE: the case of Antarctica. Geophys J Int 177(3):849–864. doi:10.1111/j.1365-246X.2009.04139.x Klemann V, Martinec Z (2009) Contribution of glacial-isostatic adjustment to the geocenter motion. Tectonophysics. doi:10.1016/j.tecto.2009.08.031 Mayer-G€urr T, Eicker A, Ilk KH (2007) ITG-Grace03 gravity field model. http://www.geod.uni-bonn.de/itg-grace03.html

A. Eicker et al. McCarthy DD, Petit G (2004) IERS conventions 2003. IERS technical notes, 32 Verlag des Bundesamts fuer Kartographie und Geod€asie, Frankfurt am Main Rodell M (2002) The potential for satellite-based monitoring of groundwater storage changes using GRACE: the High Plains aquifer, Central US. J Hydrol 63:245–256. doi:10.1016/ S0022-1694(02)00060-4 Rodell M, Velicogna I, Famiglietti JS (2009) Satellite-based estimates of groundwater depletion in India. Nature 460: 999–1002. doi:10.1038/nature08238 Schmidt R, Petrovic S, G€untner A, Barthelmes F, W€unsch J, Kusche J (2008) Periodic components of water storage changes from GRACE and global hydrology models. J Geophys Res 113:B08419. doi:10.1029/2007JB005363 Steffen H, Denker H, M€uller J (2008) Glacial isostatic adjustment in Fennoscandia from GRACE data and comparison with geodynamic models. J Geodyn 46(3–5):155–164. doi:10.1016/j.jog.2008.03.002 Steffen H, Petrovic S, M€uller J, Schmidt R, W€unsch J, Barthelmes F, Kusche J (2009) Significance of secular trends of mass variations determined from GRACE solution. J Geodyn 48(3–5):157–165. doi:10.1016/j.jog.2009.09.029 Strassberg G, Scanlon BR, Chambers D (2009) Evaluation of groundwater storage monitoring with the GRACE satellite: case study of the High Plains aquifer, central United States. Water Res Int 45:W05410. doi:10.1029/2008WR006892 Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L09607 Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103(B12):30, 20530, 229 Watkins M, Yuan D-N (2007) JPL level-2 processing standards document for level-2 product release 04. ftp://podaac.jpl. nasa.gov/pub/grace/doc/ Werth S, G€untner A, Schmidt R, Kusche J (2009) Evaluation of GRACE filter tools from a hydrological perspective. Geophys J Int 179(3):14991515

AIUB-GRACE02S: Status of GRACE Gravity Field Recovery Using the Celestial Mechanics Approach

20

A. J€aggi, G. Beutler, U. Meyer, L. Prange, R. Dach, and L. Mervart

Abstract

The gravity field model AIUB-GRACE02S is the second release of a model generated with the Celestial Mechanics Approach using GRACE data. Intersatellite K-band range-rate measurements and GPS-derived kinematic positions serve as observations to solve for the Earth’s static gravity field in a generalized orbit determination problem. Apart from the normalized spherical harmonic coefficients up to degree 150, arc-specific parameters like initial conditions and pseudo-stochastic parameters are solved for in a rigorous least-squares adjustment based on both types of observations. The quality of AIUB-GRACE02S has significantly improved with respect to the earlier release 01 due to a refined orbit parametrization and the implementation of all relevant background models. AIUB-GRACE02S is based on 2 years of data and was derived in one iteration step from EGM96, which served as a priori gravity field model. Comparisons with levelling data and models from other groups are used to assess the suitability of the Celestial Mechanics Approach for GRACE gravity field determination.

20.1

Introduction

The Gravity Recovery And Climate Experiment (GRACE) mission, launched on March 17, 2002, has significantly improved our knowledge of the Earth’s gravity field in terms of accuracy, spatial and temporal resolution. The time-varying part of the Earth’s gravity field has been inferred with unprecedented accuracy from space by analyzing GPS, accelerometer, and

A. J€aggi (*) G. Beutler U. Meyer L. Prange R. Dach Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland e-mail: [email protected] L. Mervart Institute of Advanced Geodesy, Czech Technical University, Thakurova 7, 16629, Prague, Czech Republic

inter-satellite K-band observations (Tapley et al. 2004). For the static part GRACE models report an accuracy of about 3 cm in terms of geoid heights at a spatial resolution of 200 km (F€orste et al. 2008a), which is an improvement by about one order of magnitude compared to the high-resolution model EGM96 (Lemoine et al. 1997). Apart from the official GRACE models derived by the Center for Space Research (CSR) of the University of Texas at Austin, the Jet Propulsion Laboratories (JPL), and the GeoForschungsZentrum Potsdam/Groupe de Recherche de Ge´ode´sie Spatiale (GFZ /GRGS), alternative state-of-the-art GRACE gravity field models have been computed, e.g., at the Institut f€ur Geod€asie und Geoinformation of the University of Bonn (Mayer-G€urr 2008).

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_20, # Springer-Verlag Berlin Heidelberg 2012

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Gravity field recovery from satellite data has also been initiated at the Astronomical Institute of the University of Bern (AIUB). The so-called Celestial Mechanics Approach (CMA) has been applied to both the high-low satellite-to-satellite tracking (hlSST) data of CHAMP and the low-low (ll) SST data of GRACE. Consolidated results for CHAMP and first results for GRACE may be found in Prange et al. (2009) and J€aggi et al. (2010), respectively. This article focuses on updates of the CMA, which were implemented to improve the combined processing of GRACE hl-SST and ll-SST data of the K-band ranging system (Sect. 20.2). Various gravity field recovery experiments illustrate the sensitivity of the CMA on, e.g., the orbit parametrization or the background modeling (Sect. 20.3). A significantly improved gravity field model AIUB-GRACE02S has been derived from GRACE data covering the years 2006 and 2007 (Sect. 20.4). The quality of this 2-year solution is assessed and perspectives for future developments are presented.

20.2

Celestial Mechanics Approach

GRACE gravity field determination using the CMA is based on the analysis of Level 1B inter-satellite Kband measurements (Case et al. 2002) and GPSderived kinematic positions (J€aggi et al. 2009). The static part of the gravity field is modeled with a series of normalized spherical harmonic (SH) coefficients (Heiskanen and Moritz 1967) from degree 2 up to a maximum degree of 150. The time variable part is currently not yet modeled. Based on a priori orbits derived from the kinematic positions of both GRACE satellites and the inter-satellite measurements (Sect. 20.1) normal equations for both types of observations are set up on a daily basis for the unknown gravity field coefficients and for additional arc-specific parameters, i.e., two normal equation systems based on the kinematic positions of GRACE A and B (Sect. 20.2), and one normal equation system based on the ll-SST data of the K-band ranging system (Sect. 20.3). The resulting daily normal equations are then combined into one system for each daily arc. Finally, arc-specific parameters are pre-eliminated and the combined daily normal equations are accumulated into monthly, annual, and multi-annual systems, which are eventually inverted to solve for the SH coefficients

without applying any regularization. Details about the general procedure may be found in (J€aggi et al. 2010).

20.2.1 A Priori Orbit Generation As opposed to the processing scheme described by J€aggi et al. (2010), improved a priori orbits are used here for gravity field recovery: Kinematic positions and K-band measurements are already taken into account for the a priori orbit generation. Using a specific force model (a priori gravity field model, accelerometer data, ocean tide model,. . .), only the weighted kinematic positions of both GRACE satellites are fitted in a first step by numerically integrating the corresponding equations of motion (Beutler 2005) and by adjusting the arc-specific orbit parameters. Efficient numerical integration techniques are applied to solve the so-called variational equations (Beutler 2005) in order to obtain the needed partial derivatives with respect to the orbit parameters. The normal equations of this first step are stored as they are subsequently needed for the generation of the “final” a priori orbits. Apart from the initial osculating elements constant and once-per-revolution empirical accelerations are set up, which act over the entire arc in the radial, along-track, and cross-track directions, and constrained piecewise constant accelerations acting over 15 min intervals in the same three directions. Additional polynomial coefficients up to degree 3 are set up per arc in the along-track direction in order to compensate for instrument drifts when using accelerometer data. For numerical reasons the arc-specific parameters are not set up separately for the two GRACE satellites. Transformed parameters representing the mean values of the corresponding parameters of both GRACE satellites, which are mainly determined by the kinematic positions, and half of the differences, which are mainly determined by the K-band observations are used. Based on the orbits of GRACE A and B from the first step, K-band measurements are used to set up the K-band normal equations for the same arc-specific parameters. The three normal equation contributions are then combined into one system for each daily arc, which is eventually inverted to solve for the arcspecific orbit parameters. The a priori orbits used for gravity field recovery are finally obtained by propagating the improved state vectors by numerical integration. As these orbits represent the K-band

20

AIUB-GRACE02S: Status of GRACE Gravity Field Recovery

observations already at a rather comfortable level even when using an a priori gravity field model from the pre-GRACE era, the gravity field solution should not be harmed by linearization effects (see Sect. 20.3).

20.2.2 Daily Normal Equations from Positions Based on the a priori orbits of GRACE A and B gravity field recovery from orbit positions is set up as a generalized orbit improvement process for each GRACE satellite as described by J€aggi et al. (2010). The actual orbit of one satellite is expressed as a truncated Taylor series with respect to the unknown arc-specific orbit parameters and the unknown SH coefficients about the a priori orbit. In analogy to Sect. 20.2.1 the partial derivatives of the a priori orbit are computed with respect to all parameters. These partials eventually allow it to set up the daily normal equations based on kinematic positions for all parameters according to a standard least-squares adjustment. In analogy to Sect. 20.2.1 the arc-specific parameters are set up as the sum and the difference of the original parameters in order to avoid numerical problems.

20.2.3 Daily Normal Equations from II-SST Based on the a priori orbits of GRACE A and B gravity field recovery from K-band measurements is set up as a differential orbit improvement process as described by J€aggi et al. (2010). The actual orbit difference is expressed as a truncated Taylor series with respect to the unknown parameters about the a priori orbit difference. The same techniques as in Sect. 20.2.2 are applied to solve the variational equations separately for both GRACE satellites. As K-band observations only contain information about the line-of-sight orbit difference, a projection of the partial derivatives on the line-of-sight direction is required. These projected partials eventually allow it to set up the daily normal equations based on K-band measurements for all parameters. The transformed arc-specific parameters are set up for both GRACE satellites, which implies that the daily normal equation matrices would be singular if only K-band data were used.

163

20.3

Gravity Field Recovery Studies

Gravity field models from GRACE kinematic positions and K-band range-rate data of the year 2007 were derived using the methodology outlined in Sect. 20.2. Due to the computational burden of gravity field recovery, only static fields up to degree 120 were estimated. EGM96 served as a priori model up to the same degree. A spacing of 15 min for the piecewise constant accelerations turned out to be sufficient to account for various model shortcomings. In order to illustrate the sensitivity of the CMA on orbit parametrization (Sect. 20.3.1), data weighting (Sect. 20.3.2), linearization effects (Sect. 20.3.3), and background models (Sect. 20.3.4), different solutions were made, which are subsequently compared with the gravity field model ITG-GRACE03S (Mayer-G€urr 2008) in terms of the square-roots of the degree difference variances up to degree 120.

20.3.1 Orbit Parametrization As the two GRACE satellites are only separated by about 30 s on the same orbital trajectory, they experience almost the same perturbations. An orbit and gravity field determination tailored to the GRACE configuration may therefore use this information and introduce additional, relative constraints between the estimated piecewise constant accelerations of both GRACE satellites. A series of gravity field models was thus generated to study the impact of relative constraints, which have been chosen to be 100 times tighter than the absolute constraints imposed on the piecewise constant accelerations. Figure 20.1 shows the impact of such constraints. The largest part of the improvement w.r.t. ITG-GRACE03S is already obtained by applying relative constraints only for the along-track direction. Additional constraints for the radial direction have a small positive impact, as well, whereas relative constraints for the cross-track direction do not significantly change the solution. The agreement (consistency) with the reference model for degree 2 is, however, degraded when relative constraints are applied for the radial and the along-track directions. Comparisons with EIGEN-GL04C (F€orste et al. 2008a) even revealed a degradation when relative constraints are only applied for the along-track

A. J€aggi et al.

164 Fig. 20.1 Square-roots of degree difference variances of annual recoveries when using different constraining options

101

Geoid heights (m)

100

10−1

10−2

10−3

ITG−GRACE03S no rel. constraints along−track only all directions

10−4

0

20

40

60

80

100

120

Degree of spherical harmonics Fig. 20.2 Square-roots of degree difference variances of annual recoveries when using additional iteration steps

101 100

Geoid heights (m)

10−1 10−2 10−3 10−4 ITG−GRACE03S Iteration 1 Iteration 2 Difference

10−5 10−6

0

20

40

60

80

100

120

Degree of spherical harmonics

direction (not shown). Further investigations are necessary to better understand this mechanism.

20.3.2 Data Weighting According to J€aggi et al. (2010) a nominal weighting ratio of 1:108 between the GPS (L1) carrier phase

observations (error propagation taken into account by epoch-wise covariance information) and the K-band range-rate observations has been used for the determination of the gravity field solutions shown in Fig. 20.1. It turned out, however, that the agreement with ITG-GRACE03S may be slightly improved if stronger ratios are used, e.g., 1:109 or even 1:1010 (not shown). Despite the fact that a weighting ratio of 1:1010 is

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AIUB-GRACE02S: Status of GRACE Gravity Field Recovery

165

unrealistic in view of the a posteriori RMS errors, which are about 1 mm for the (L1) GPS carrier phase observations, and about 0.23 mm/s for the K-band range-rate observations (see Fig. 20.4), the latter was chosen for all subsequent solutions.

piecewise constant accelerations are able to compensate for the unmodeled non-gravitational accelerations to a great extent. Note that the cross-track component of the accelerometer data has not been taken into account for the solutions presented in this article, as almost no impact could be detected. The differences between the nominal and the experimental solutions are visible only up to about degree 90 because of an inadequate implementation of the ocean tide model used in the CMA at that time (see Sect. 20.3.4.3), and because of the omission errors due to the limited maximum degree of 120.

20.3.3 Linearization Gravity field recovery may depend on the a priori gravity field model used to set up the (linearized) observation equations. In order to rule out such dependencies for the CMA, EGM96 served as a priori gravity field model for all solutions presented in this article. Figure 20.2 shows the agreement of this baseline solution (iteration 1) with ITG-GRACE03S. A full iteration cycle was performed in order to assess the magnitude of linearization effects. For that purpose the previously obtained gravity field model (iteration 1) was used as a priori model for an additional iteration step (iteration 2). Figure 20.2 confirms that the agreement with ITG-GRACE03S is almost identical for both iteration steps. For most degrees the differences between the two iterations are about one order of magnitude smaller than the differences w.r.t. ITGGRACE03S. Gravity field recovery with the CMA is thus considered as highly independent of the a priori gravity field.

20.3.4 Background Modeling Since the results presented by J€aggi et al. (2010) all relevant background models are now implemented in the CMA. Their impact on static GRACE gravity field solutions is studied in the following subsections.

20.3.4.1 Non-gravitational Accelerations Accelerometer data are taken into account for GRACE gravity field recovery to separate the non-gravitational accelerations from the gravitational signal. Figure 20.3 compares the agreement of this nominal solution and of an experimental solution obtained without accelerometer data with ITG-GRACE03S. As expected, GRACE gravity field recovery clearly benefits from using accelerometer data. It is, however, remarkable that solutions of quite good quality may be obtained with the CMA even if accelerometer data are left out from the processing. Obviously, the estimated

20.3.4.2 Atmospheric and Oceanic Dealiasing Figure 20.3 also shows the impact of the non-tidal atmosphere and ocean short-term mass variations (Flechtner et al. 2006) on a static gravity field solution. Although the atmospheric and oceanic dealiasing (AOD) products are important for the generation of GRACE monthly solutions and clearly improve the K-band range-rate residuals (see Fig. 20.4 or Zenner et al. (2011) for more details), only a relatively small effect is visible in a static 1-year solution. 20.3.4.3 Ocean Tides The implementation of the ocean tide model in the CMA has been revised. Figure 20.3 shows that the improved implementation mainly had a positive effect on the high degrees above 90. The analysis of ocean tides also included the implementation of the models FES2004 (Lyard et al. 2006) and EOT08a (Savcenko and Bosch 2008) in the CMA. As no large differences on a static one-year solution have been found when using FES2004 (not shown) or EOT08A, the more recent EOT08a ocean tide model has been selected for the generation of the AIUB-GRACE02S model.

20.4

AIUB-GRACE02S

The AIUB-GRACE02S model was derived from kinematic positions and K-band range-rate data covering the years 2006 and 2007 using the methods from Sects. 20.2 to 20.3. As opposed to Sect. 20.3, the maximum degree was increased from 120 to 150. EGM96 thus also served as a priori model up to degree 150.

A. J€aggi et al.

166 Fig. 20.3 Square-roots of degree difference variances of annual recoveries when using different processing options

101

100

Geoid heights (m)

10−1

10−2

−3

10

ITG−GRACE03S no ACC data nominal solution no AOD data OT improved

10−4

0

20

40

60

80

100

120

Degree of spherical harmonics

Fig. 20.4 Daily RMS values of K-band range-rate residuals of annual and monthly gravity field solutions

0.45 no AOD data for annual solution nominal annual solution nominal monthly solutions

K−band range−rate RMS (µm/s)

0.4

0.35

0.3

0.25

0.2

0.15

0

20.4.1 Comparison with Other Models Figure 20.5 shows the agreement of the two solutions obtained from the 2006 and 2007 data sets and of the combined 2-year solution AIUB-GRACE02S w.r.t. ITG-GRACE03S. Compared to the figures from the previous section, clear improvement results for the higher degrees, but also the medium to high degrees

50

100

150 200 Day of year 2007

250

300

350

show a better agreement with ITG-GRACE03 due to the increased maximum degree. Figure 20.5 even suggests that a maximum degree larger than 150 might have been chosen for the 2-year solution. The quality of the lower degrees improved slightly as well. Figure 20.6 shows the agreement of the three satellite-only models AIUB-GRACE02S, EIGEN-5S (F€orste et al. 2008b), and GGM03S (Tapley et al.

20

AIUB-GRACE02S: Status of GRACE Gravity Field Recovery

Fig. 20.5 Square-roots of degree difference variances of annual recoveries and the combined 2-year solution

167

101

Geoid heights (m)

100

10−1

10−2

10−3

ITG−GRACE03S 2006 2007 AIUB−GRACE02S

10−4

0

Fig. 20.6 Square-roots of degree difference variances of different satellite-only models

50 100 Degree of spherical harmonics

150

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Geoid heights (m)

100

10−1

10

−2

10−3

10−4

0

2007) with ITG-GRACE03S. Although AIUBGRACE02S is based on only about half of the data volume of the other models, the agreement with ITGGRACE03S is comparable to that of EIGEN-5S and GGM03S for most degrees. Deficiencies in the AIUB-GRACE02S solution are only visible for degrees below 20. Longer data spans and, in particular, the estimation of time variable signals will further improve the low degree SH coefficients. Figure 20.4

ITG−GRACE03S GGM03S EIGEN−5S AIUB−GRACE02S

50 100 Degree of spherical harmonics

150

indicates that the time variability of the Earth’s gravity field has a clear impact on the residuals of gravity field recovery.

20.4.2 Validation with External Data T. Gruber from the Institut f€ur Astronomische und Physikalische Geod€asie of the Technische Universit€at

A. J€aggi et al.

168 Table 20.1 RMS of differences (cm) between levelling and model geoid heights up to a maximum degree of 120 Levelling data EUREF GPS BRD EUVN BRD GPS Canada GPS 1998 Canada GPS 2007 Australia GPS Japan GPS USA GPS

2007 Solution 22.2 04.4 05.0 20.9 15.3 25.0 11.2 34.2

AIUB-GRACE02S 22.2 05.3 05.7 20.2 14.5 24.6 10.8 33.6

M€unchen compared the geoid heights derived from AIUB-GRACE02S, ITG-GRACE03S, EIGEN-5S, and GGM03S with geoid heights derived from different sets of recent levelling data using the method described in (Gruber 2004). Table 20.1 shows the RMS values of the differences around the mean values between the filtered levelling geoid heights and geoid heights from the different models up to degree 120. The solution AIUB-GRACE02S is of the same quality as the other satellite-only models. More pronounced differences between the models are only visible for some of the high-quality data sets, e.g., the BRD EUVN or BRD GPS data sets. Compared to the solution obtained from the GRACE data of the year 2007, AIUB-GRACE02S has improved for all data sets except for BRD EUVN and BRD GPS. The reason is yet unclear.

20.5

Conclusions

We used the CMA for gravity field determination with GPS-derived kinematic GRACE positions and Level 1B K-band range-rate data to generate the static gravity field model AIUB-GRACE02S with data covering the years 2006 and 2007. We showed that it is possible to start the recovery with EGM96 as a priori model and that the solution may be obtained in one iteration step. Accelerometer data are important for the quality of the resulting gravity field model, but models of remarkable quality may also be obtained with the CMA without accelerometer data. The overall quality of AIUB-GRACE02S was found to be comparable to that of other well-known satellite-only gravity fields, which is also confirmed by the validation with terrestrial measurements.

ITG-GRACE03S 21.9 03.7 04.4 19.6 15.0 24.3 10.3 33.4

EIGEN-5S 22.6 05.9 06.3 19.9 15.5 24.5 11.8 33.3

GGM03S 22.5 05.4 06.2 19.7 14.8 24.2 10.6 33.3

The quality of the very low degrees of AIUBGRACE02S is not yet optimal and should be improved in a future release. Part of the degradation might be due to too strong relative constraints applied to the piecewise constant accelerations. This aspect has to be further studied. The unmodeled time variability of the Earth’s gravity field, which clearly shows up in the a posteriori K-band range-rate residuals, could also contribute to the not yet optimally estimated low degree SH coefficients. Further refinements of the CMA will include the estimation of time variable signals for the low degree SH coefficients. This and the use of a longer data span will further improve GRACE gravity field recovery with the CMA. Acknowledgements The authors gratefully acknowledge the generous financial support provided by the Swiss National Science Foundation and the Institute for Advanced Study (IAS) of the Technische Universit€at M€unchen.

References Beutler G (2005) Methods of celestial mechanics. Springer, Berlin Case K, Kruizinga G, Wu S (2002) GRACE level 1B data product user handbook. D-22027. JPL, Pasadena, CA Flechtner F, Schmidt R, Meyer U (2006) De-aliasing of shortterm atmospheric and oceanic mass variations for GRACE. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the earth system from space. Springer, Heidelberg, pp 83–97 F€orste C, Schmidt R, Stubenvoll R, Flechtner F, Meyer U, K€onig R, Neumayer H, Biancale R, Lemoine JM, Bruinsma S, Loyer S, Barthelmes F, Esselborn S (2008a) The GeoForschungsZentrum Potsdam/Groupe de Recherche de Ge´ode´sie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C. J Geod 82:331–346 F€orste C, Flechtner F, Schmidt R, Stubenvoll R, Rothacher M, Kusche J, Neumayer KH, Biancale R, Lemoine JM,

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Barthelmes F, Bruinsma S, K€ onig R, Meyer U (2008b) EIGEN-GL05C – a new global combined high-resolution GRACE-based gravity field model of the GFZ-GRGS cooperation. Geophysical Research Abstracts, vol. 10. EGU, Vienna Gruber T (2004) Validation concepts for gravity field models from satellite missions. In: Proceedings of second international GOCE user workshop “GOCE, The Geoid and Oceanography”, ESA-ESRIN, Frascati Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco, CA J€aggi A, Dach R, Montenbruck O, Hugentobler U, Bock H, Beutler G (2009) Phase center modeling for LEO GPS receiver antennas and its impact on precise orbit determination. J Geod 83:1145–1162 J€aggi A, Beutler G, Mervart L (2010) GRACE gravity field determination using the celestial mechanics approach – first results. In: Mertikas S (ed) Gravity, geoid and earth observation. Springer, Berlin, pp 177–184 Lemoine FG, Smith DE, Kunz L, Smith R, Pavlis EC, Pavlis NK, Klosko SM, Chinn DS, Torrence MH, Williamson RG, Cox CM, Rachlin KE, Wang YM, Kenyon SC, Salman R, Trimmer R, Rapp RH, Nerem RS (1997) The development of the NASA GSFC and NIMA joint geopotential model. In: Segawa J, Fujimoto H, Okubo S (eds) IAG symposia: gravity, geoid and marine geodesy. Springer, Berlin, pp 461–469

Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: insight from FES2004. Ocean Dyn 56:394–415 Mayer-G€urr T (2008) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb€ogen am Beispiel der Satellitenmissionen CHAMP und GRACE. Schriftenreihe 9, Institut f€ur Geod€asie und Geoinformation, University of Bonn, Bonn Prange L, J€aggi A, Dach R, Bock H, Beutler G, Mervart L (2009) AIUB-CHAMP02S: the influence of GNSS model changes on gravity field recovery using spaceborne GPS. Adv Space Res 45:215–224 Savcenko R, Bosch W (2008) EOT08a – empirical ocean tide model from multi-mission satellite altimetry. DGFI report 81, Deutsches Geod€atisches Forschungsinstitut, Munich Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 305(5683):503–505 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Poole S (2007) The GGM03 mean earth gravity model from GRACE. Eos Trans AGU 88(52) Zenner L, Gruber T, Beutler G, J€aggi A, Flechtner F, Schmidt T, Wickert J, Fagiolini E, Schwarz G, Trautmann T (2011) Using atmospheric uncertainties for GRACE de-aliasing – first results. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg

.

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes

21

B.C. Gunter, T. Wittwer, W. Stolk, R. Klees, and P. Ditmar

Abstract

In this study we address the question of whether regional gravity field modeling techniques of GRACE data can offer improved resolution over traditional global spherical harmonic solutions. Earlier studies into large, equatorial river basins such as the Amazon, Zambezi and others showed no obvious distinction between regional and global techniques, but this may have been limited by the fact that these equatorial regions are at the latitudes where GRACE errors are known to be largest (due to the sparse groundtrack coverage). This study will focus on regions of higher latitude, specifically Greenland and Antarctica, where the density of GRACE measurements is much higher. The regional modeling technique employed made use of spherical radial basis functions (SRBF), complete with an optimal filtering algorithm. Comparisons of these regional solutions were made to a range of other publicly available global spherical harmonic solutions, and validated using ICESat laser altimetry. The timeframe considered was a 3 year period spanning from October 2003 to September 2006.

21.1

Introduction

The launch of the Gravity Recovery and Climate Experiment (GRACE) in 2002 started a new wave of research into the Earth’s mass transport processes. The measurements from the mission’s twin satellites have enabled the multi-year tracking of many large scale processes, such as continental hydrology and ice mass changes in the cryosphere. While these first studies have produced some truly excellent results, there is always the desire to push the boundaries of what

B.C. Gunter (*) T. Wittwer W. Stolk R. Klees P. Ditmar Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands e-mail: [email protected]

GRACE can observe, in terms of spatial and temporal resolution. Previous studies have demonstrated that the current processing standards of GRACE data provide mass change accuracies on the order of 2 cm of equivalent water height (EWH) over spatial scales of 400 km and time intervals of 1 month (Klees et al. 2008a). This analysis was done by comparing the performance of a range of different GRACE processing strategies, including both regional and global methods, over selected river basins and other “dry” regions where little to no hydrological signal is expected. The global methods tested primarily involved traditional spherical harmonic solutions from various processing centers (CSR, GFZ, JPL, CNES, DEOS), but with various spatial filters applied. The regional solutions examined included the “mascon” approach (Luthcke et al. 2006) as well as

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_21, # Springer-Verlag Berlin Heidelberg 2012

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solutions computed using spherical radial basis functions. In short, the overall conclusion of this earlier study was that there was no clear advantage to using regional techniques over global methods for the river basins studied. In fact, it turned out that the choice of spatial filter was the most important aspect in the comparisons; however, one of the limitations of this particular study was that most of the regions studied were at relatively low latitudes, where the density of GRACE measurements is the lowest. For higher latitude regions, it is possible that the increased data density might offer a higher signal-to-noise ratio that regional techniques might be able to better exploit. As a result, a follow-on study was conducted (Stolk 2009) to perform a similar analysis over regions such as Greenland and Antarctica, to see if the conclusions would be the same. This paper will provide an overview of the methods and conclusions of this follow-on study.

21.2

Spherical Radial Basis Functions

The focus of the regional techniques for the highlatitude regions involved the application of spherical radial basis functions (SRBF). The general concept behind this approach is to use a distribution of spacelocalizing functions to represent any complex spherical shape, such as the Earth’s gravity field. The functions can be constructed using a number of different methods, although the kernel adopted for the current study makes use of Poisson wavelets of order three (Holschneider et al. 2003; Wittwer 2009). The shape and spatial distribution of the SRBFs are determined by the depth (i.e., bandwidth) and the level (i.e., spacing on a Reuter grid) assigned to each SRBF, as illustrated in Fig. 21.1. As with spherical harmonic solutions, SRBF solutions suffer from north-south error patterns (i.e., “stripes”), which require the application of a suitable filter. The anisotropic, non-symmetric (ANS) filter developed by Klees et al. (2008b) offers a number of benefits over other traditional filtering techniques, such as destriping or Gaussian smoothing, primarily because use is made of the full statistical information of the solution (i.e., signal and noise variance-covariance matrices are used). For example, if spherical harmonics are used to parameterize the time-variable gravity field, and we let N^ x ¼ b represent the normal

equations for a monthly GRACE solution, the ANS filter W can be applied as follows: ^xw ¼ W^x ¼ ðN þ D1 Þ1 b

(21.1)

where N is the normal matrix, D the signal covariance matrix (i.e., the auto-covariance matrix of the vector ^x), ^x the estimated parameter vector, and b the rightside vector. The matrix N is determined from the partial derivatives of the system dynamics; however, the auto-covariance matrix, D, must be determined empirically. This is done through an iterative process whereby the (time-independent) variances of the signal from the actual time series of monthly solutions (e.g., 36 months for this study) are computed at the nodes of an equal-angular grid and then transformed back to the spherical harmonic domain to form D. This signal covariance information has the effect of suppressing spurious noise in regions that typically do not have much mass variations (e.g., oceans and deserts), while also allowing the solution to adjust more freely in areas where the mass change signal has larger variations (e.g., river basins). Since this signal covariance matrix is computed from the time series of GRACE solutions, it is particular to the solution technique. A straightforward generalization of this concept to a SRBF parameterization is obtained when the relationship between spherical harmonic coefficients, x, and SRBF coefficients, a, is exploited. This relationship can be written as x ¼ Qa

(21.2)

Hence, given the auto-covariance matrix in the spherical harmonic domain, D, we can obtain the corresponding auto-covariance matrix in the SRBF domain according to a ¼ Qþ x ) Da ¼ Qþ DðQþ ÞT

(21.3)

where Q+ ¼ (QTQ)1QT is the pseudo-inverse of the Q matrix. This approach, however, fails because the spectrum of a given SRBF parameterization comprises spherical harmonic degrees, which may exceed the maximum degree of a given GRACE monthly solution (the number of harmonic coefficients in x is often much larger than the number of SRBF coefficients in a).

21

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes

173

Fig. 21.1 Example spherical radial basis functions

Therefore, the optimal filter needs to be designed directly in the SRBF domain. If f is the time-variable gravity signal in terms of equivalent water heights, and a comprises the SRBF coefficients, we write the SRBF synthesis as f ¼ Ba

(21.4)

Using the pseudo-inverse of B, B+ ¼ (BTB)1BT, we write a ¼ Bþ f

(21.5)

and obtain the auto-covariance matrix in the SRBF domain, Da, as Da ¼ Bþ D ðBþ ÞT

(21.6)

Hence, if Na a ¼ ba is the system of normal equations in terms of SRBFs, the equivalent expression of (21.1) is aW ¼ Wa a ¼ ðNa þ Da Þ1 ba

(21.7)

With these relationships, the signal covariance matrix now can be computed, and the ANS filter applied to the SRBF coefficients. Note that since the computation of the signal covariance matrix is done iteratively, an initial set of values must first be chosen. The standard deviations chosen for this initial signal variance covariance matrix are essentially arbitrary, although proper choices might reduce the number of iterations needed. For the current study, the initial standard deviations were set to 50 mm globally. This initial standard deviation is propagated from the spatial domain to the frequency domain using (21.6), then a new signal variance matrix is created from the

filtered solution. Iteration is halted when the difference in equivalent water height between two consecutive iterations for each grid point is less than 35 mm (chosen experimentally to balance convergence speed and the determination of accurate signal variability). The determination of the optimal values for the level and depth of the SRBF solutions depends on the spatial variations, and noise content, of the data involved. Placing a dense grid of functions at a relatively shallow depth (i.e., small bandwidth) may result in noisy solutions, especially for GRACE data. The general approach used here was to employ a level high enough to represent what was believed to be the signal content in the data, and to place these functions as deep as possible in an attempt to smooth out the noise in the data. Many combinations of level and depth were evaluated, with the determination that a level 90 (i.e., ~220 km Reuter grid spacing), depth 900 km parameterization offers the highest quality solutions for Greenland and Antarctica.

21.3

Comparisons

Having finalized the optimal parameterization and filtering of the SRBF solutions, the next step was to compare the results of the mass change estimates derived from these solutions to those derived from other techniques, over Greenland and Antarctica. Since the goal of the study was simply to compare global versus regional techniques, only a limited set of spherical harmonic solutions were involved, and included those from the Center for Space Research (CSR) and the Delft Institute of Earth Observation and Space Systems (DEOS), who now produce a set of publically available monthly gravity solutions called the DEOS Mass Transport (DMT-1) models

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Table 21.1 Descriptions of the various global and regional solutions used for comparison Model name CSR DS400

Solution type Global

CSR DS0 DMT-1

Global Global

SRBF global

Regional

SRBF regional

Regional

Description Spherical harmonic solution to 60x60 derived from CSR RL04 data (see http://podaac.jpl.nasa.gov/ grace) ; destriped; 400km Gaussian smoothing applied; SLR C20 values; degree 1 coefficients taken from Swenson et al. (2008) Similar to above, except without Gaussian smoothing applied DEOS Mass Transport models (see http://www.lr.tudelft.nl/psg/grace); spherical harmonic solutions to 120x120 generated from KBR L1B data using the range-combination approach (Liu et al. 2009); anisotropic non-symmetric (ANS) filter applied (Klees et al. 2008b) Spherical radial basis function approach using Poisson wavelets, in which a global distribution of nodes with a Reuter grid spacing of level 90 and depth of 900km is used. Low level data derived from the same KBR L1B data as the DMT-1 solution, with a similar anisotropic non-symmetric filter applied (adapted for use with SRBF’s) Similar to the SRBF global approach, but using only regional data (i.e., within a 30 extended boundary from the target region)

(Liu et al. 2009). A summary of the solutions used in the comparisons is provided in Table 21.1. In short, the CSR solutions used both un-filtered and Gaussian filtered solutions, with an additional destriping filter applied similar to that of Swenson et al. (2008) (see Gunter et al. (2009) for further details of the CSR solution processing). The DMT-1 solutions are global spherical harmonic solutions computed using the acceleration approach (Ditmar and van Eck van der Sluijs 2004; Liu 2008), and with the ANS filter applied. Two types of SRBF solutions were tested, one using a global distribution of functions (SRBF global), and one using a more regional distribution (SRBF regional) in which a latitudinal buffer of 30 was used to reduce edge effects. For each solution, both the long-term linear trends (with bias and annual/ semi-annual terms included) and monthly variations in the signals were examined. The timeframe considered was a 3 year period spanning from October 2003 to September 2006. Some selected results from the comparisons are shown in Figs. 21.2 and 21.3. Figure 21.2 shows the geographical plot of the linear trends for the CSR400, DMT-1 and SRBF regional solutions over Greenland and Antarctica. Figure 21.3 is a plot of the maximum amplitude of the annual signal variation which is coestimated along with the linear trend parameter. This is useful to visualize where the largest fluctuations in mass change exist. The first observation that can be made from looking at these two figures is that the resolution for the ANS filtered solutions is much higher than those of the CSR DS400 solution, particularly for Greenland. The

DMT-1 and SRBF solutions are quite similar, but differences do exist. It is also interesting to note that the amplitude plots for the DMT-1 and SRBF regional solutions show subtle difference as well. For example, the SRBF solution shows a noticeable variation at the tip of the Antarctic Peninsula, where the DMT-1 solution does not. Similarly, in the Amundsen Sea sector (SW Antarctica), the SRBF solutions show two distinctive peaks, where the DMT-1 solutions show only one.

21.4

Validation

The determination of whether the differences seen in Figs. 21.2 and 21.3 represent genuine improvements in the signal recovered by the SRBF solutions is a difficult question to answer, and is a topic of current and future research efforts. One attempt made in this study to do this utilized surface elevation change data from the Ice, Cloud and Land Elevation Satellite (ICESat), a laser altimetry mission launched in 2003. ICESat observes the volume changes due to ice mass changes, which are naturally correlated to the mass changes observed by GRACE. The spatial resolution of ICESat is also much higher than that of GRACE, so a test was developed whereby the ICESat data was smoothed using a full-width Gaussian filter [as opposed to the traditional half-width filter normally used in geodesy, e.g., Jekeli (1981)] at intervals ranging from 0 to 2,500 km. Trend maps over the 3-year time period for both GRACE and ICESat were computed and each map was individually normalized. The normalization was needed because the ICESat

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes DMT–1

CSR DS400 0°

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Fig. 21.2 Geographical plot of the 3-year trend computed from selected global and regional solutions, in units of equivalent water height CSR DS400

DMT–1

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Fig. 21.3 Geographical plot of the estimated annual amplitude variations computed from selected global and regional solutions, in units of equivalent water height

176

B.C. Gunter et al.

map represents physical height changes (dh/dt, in cm/ yr), whereas GRACE maps represent annual changes in EWH (also in cm/yr). As these are not the same quantities, the normalization allows a more direct comparison of the two data types under the assumption that a strong change in volume directly corresponds to a strong change in mass (and vice-versa). For each smoothing increment, correlations were computed between the smoothed ICESat map and the corresponding GRACE map. A peak in the resulting correlation curve would give an indication of the spatial resolution of the GRACE solution tested. The results of this test for all of the GRACE solutions mentioned in Table 21.1 are provided in Fig. 21.4. For Greenland, the correlations with ICESat for the ANS filtered solutions (DMT-1 and the SRBF solutions) peak at around 1,300 km (full-width) 1

Correlation

0.8

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1500

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0.8 0.6 CSR DS400 CSR DS0 DMT–1 SRBF global SRBF regional

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ICESat Gaussian smoothing full width [km] Fig. 21.4 Results of the spatial correlation test to ICESat data for Greenland (top) and Antarctica (bottom), for the various GRACE solutions

Gaussian smoothing, where the CSR solutions peak in the 2,200–2,500 km range. This implies that the ANS filter is the driving force for the accuracy levels in Greenland, and not necessarily the solution technique. For Antarctica, the situation is slightly different. Here, the correlation peak of the SRBF regional solution is approximately 5–10% higher than the SRBF global solution and the unfiltered CSR solution. This would suggest that the SRBF regional approach is achieving slightly better spatial resolution than the other global approaches. Conclusions

The results of the analysis for this study supports the earlier conclusions by Klees et al. (2008a) that the choice of the spatial filter used in the GRACE processing has the largest impact on the comparisons. When compared to the standard destriping and Gaussian filter approach (i.e., DS400), the anisotropic, non-symmetric (ANS) filter offers many benefits in terms of improved spatial resolution. That said, there were other indications that the choice of solution method may also offer some improvements, although to a much smaller degree. For Antarctica, the SRBF regional solution had the best spatial correlation when compared to the corresponding height change data from ICESat (Fig. 21.4), and was the only solution to observe annual variations in the Antarctic Peninsula (Fig. 21.3). For Greenland, all ANS filtered solutions (global and regional) performed essentially the same, with all of them offering substantial improvements over the corresponding CSR fields (DS400 and DS0). This is primarily due to the fact that the CSR fields have inherently lower resolution (with maximum degree and order 60), and because a Gaussian filter was applied (equivalent ANS filtered CSR solutions were not possible since the monthly noise covariance matrices are not publicly available). Regardless, the results suggest that, at a minimum, the regional SRBF techniques are equivalent to other global spherical harmonic solutions (i. e., DMT-1), but that there is also the possibility that a 5–10% improvement might be gained, depending on the region. Future studies will attempt to verify these results with more extensive comparisons with independent data sets, such as in-situ glaciological measurements or other satellite measurements.

21

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes

Acknowledgements The authors would like to thank Tim Urban at the UT-Austin Center for Space Research for providing the ICESat crossover height change data.

References Ditmar P, van Eck van der Sluijs AA (2004) A technique for modeling the Earth’s gravity field on the basis of satellite accelerations. J Geodesy 78:12–33 Gunter BC, Urban T, Riva REM, Helsen M, Harpold R, Poole S, Nagel P, Schutz B, Tapley B (2009) A comparison of coincident GRACE and ICESat data over Antarctica. J Geodesy 83(11):1051–1060. doi:10.1007/s00190-009-0323-4 Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys Earth Planet Inter 135:107–124 Jekeli C (1981) Alternative methods to smooth the Earth’s gravity field. Technical Report 327, Ohio State University, Department of Geodetic Science and Surveying, December 1981 Klees R, Liu X, Wittwer T, Gunter BC, Revtova EA, Tenzer R, Ditmar P, Winsemius HC, Savenije HHG (2008a) A comparison of global and regional GRACE models for land hydrology. Surv Geophys 29(4–5):335–359

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Klees R, Revtova EA, Gunter BC, Ditmar P, Oudman E, Winsemius HC, Savenije HHG (2008b) The design of an optimal filter for monthly GRACE gravity models. Geophys J Int 175:417–432 Liu X (2008) Global gravity field recovery from satelliteto-satellite tracking data with the acceleration approach. Ph.D. Thesis, Netherlands Geodetic Commision, Publication on Geodesy 68, Delft, The Netherlands Liu X, Ditmar P, Siemes C, Slobbe DC, Revtova EA, Klees R, Riva R, Zhao Q (2009) DEOS Mass Transport model (DMT1) based on GRACE satellite data: methodology and validation. Geophys J Int 181(2):769–788 Luthcke S, Rowlands D, Lemoine F, Klosko S, Chinn D, McCarthy J (2006) Monthly spherical harmonic gravity field solutions determined from GRACE inter-satellite range-rate data alone. Geophys Res Lett 33:L02402 Stolk W (2009) An evaluation of the use of radial basis functions for mass change estimates at high latitudes. Msc Thesis, Delft University of Technology, The Netherlands Swenson S, Chambers D, Wahr J (2008) Estimating geocenter variations from a combination of GRACE and ocean model output. J Geophys Res 113, B08410, doi:10.1029/ 2007JB005338 Wittwer T (2009) Regional gravity field modeling with radial basis functions. Ph.D. Thesis, Netherlands Geodetic Commision, Publication on Geodesy 72, Delft, The Netherlands

.

A New Approach for Pure Kinematical and Reduced-Kinematical Determination of LEO Orbit Based on GNSS Observations

22

A. Shabanloui and K.H. Ilk

Abstract

The geometrical point-wise satellite positions of a Low Earth Orbiter (LEO) equipped with a Global Navigation Satellite System (GNSS) receiver can be derived by GNSS analysis techniques based on hl-SST (high-low Satellite to Satellite Tracking) observations. In the geometrically determined LEO orbit, there is no connection between subsequent positions, and consequently, no information about the velocity and the acceleration or in general kinematical information of the satellite is available. If the kinematical parameters which consistently connects positions, velocities and accelerations are determined by a best fitting process based on the observations, we perform a pure Kinematical Precise Orbit Determination (KPOD). In addition, the proposed approach has a capability to use certain dynamical constraints based on the dynamical force function model. In this case, we introduce a Reduced-Kinematical Precise Orbit Determination (RKPOD) of a specific level depending on the strength of the “dynamical constraints”. The various possibilities and the corresponding results of CHAMP orbits based on GNSS observations are presented.

22.1

Introduction

Estimated positions based on GNSS techniques are purely geometric and there is no connection between subsequent positions, consequently no information about the velocity and the acceleration of the LEO is available. Therefore, to describe the time dependency of the satellite’s motion, it is necessary to provide a properly constructed function which consistently connects positions, velocities and accelerations. Such

A. Shabanloui (*) K.H. Ilk Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115, Bonn, Germany e-mail: [email protected]

a continuous function can be provided by a least squares adjustment process based on a table of geometrical positions. It should be pointed out that the orbit determination in this investigation is restricted to short arcs. In order to keep the accumulated effects of the disturbing forces on LEO as small as possible, satellite arcs can be divided into short arcs. Especially, in case of the measurement of LEO’s surface forces (e.g. CHAMP or twin-GRACE) or its compensation (e.g. GOCE), it is preferable to reduce the residual effects of an incomplete compensation by selecting short arcs. In this paper, the kinematical orbit is represented by a sufficient number of approximated parameters, including the boundary values of the short arc. These parameters are determined in such a way that

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_22, # Springer-Verlag Berlin Heidelberg 2012

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180

the hl-SST observations (or geometrical positions as pseudo-observations) are approximated in the best possible way w.r.t. a properly selected norm. The approach is based on approximated parameters, which have also a clearly defined relation to the dynamical model of the satellite’s motion. If the kinematical representation parameters are determined by a best fitting process based on the hl-SST observations or geometrical positions, then a pure kinematical POD is realized. If instead all orbit representation parameters are determined by a model of the forces acting on the LEOs, then a Dynamical Precise Orbit Determination (DPOD) is introduced. In addition, there is also a possibility to use certain dynamical constraints based on the dynamical force function models. Therefore, a reduced-kinematical precise orbit determination of a specific level depending on the strength and contribution of the “dynamical restrictions” can be realized. It should be pointed out that this formulation of a POD problem allows a smooth transition from pure KPOD to RKPOD and finally fully DPOD of a LEO (Shabanloui 2008). In addition to the kinematic and the dynamic orbit determination technique, J€aggi (2007) also introduced a further type of methods called reduced dynamic methods which is located somewhere in between the kinematic and purely dynamic methods. This group of methods is characterized by the fact that additionally to the parameters of the dynamic methods, stochastic parameters of various types are introduced at intervals of a few minutes along the orbit. In this way the total orbit is divided into short arcs with different continuity properties at the arc boundaries depending on the type of the stochastic parameters. There are various articles which demonstrate the advantages of this method (e.g. J€aggi 2007). On the other hand, Sˇvehla and Rothacher (2003) proposed a reduced-kinematic POD which is here defined as a GPOD with a-priori low accuracy dynamical information between successive epochs in radial, cross and along-track directions. Their proposed reduced-kinematic method is only used to improve the characteristics of the purely GPOD by a considerable reduction of spikes and jumps. It should be mentioned that the proposed reduced-kinematical (geometrical) POD by Sˇvehla and Rothacher (2003) reduces geometrical information, while our new proposed reduced-kinematical POD in this investigation

A. Shabanloui and K.H. Ilk

reduces kinematical information in the LEO orbit (Shabanloui 2008).

22.2

Methodology

The precise orbit determination procedure is formulated as a Boundary Value Problem (BVP) of Newton-Euler’s equation of motion in the form of an integral equation of Fredholm type (Sect. 22.2.1). The solution of a Fredholm integral can be formulated in a semi-analytical way, either as a series of Fourier coefficients (Sect. 22.2.2), or as a series of Euler and Bernoulli polynomials (Sect. 22.2.3). The kinematical LEO orbit can be represented based on the combination of Euler–Bernoulli polynomials and Fourier series (Sect. 22.2.5). With introducing “dynamical restrictions” to the Fourier coefficients, the reduced kinematical POD is realized (Sect. 22.2.6).

22.2.1 Equation of Motion Based on the Linear Extended Newton Operator The basic idea was proposed as a general method for artificial satellite orbit determination by Schneider in 1968 (Schneider 1968). Before applying this method in the orbit determination, it was modified for the Earth gravity field determination by Reigber (1969). This orbit determination technique has been further developed and modified to various applications in satellite geodesy, especially to recover the Earth’s gravity field based on POD methods (Ilk 1977). A technique for the numerical solution of two BVP of Newton–Euler’s equation of motion as well as of Lagrange’s equation of motion based on Schneider’s proposed method was developed in the same publication. Later, this technique has been successfully applied to recover the Earth gravity field based on high–low SST and low–low SST observations (Mayer-G€urr 2006), but was not applied to produce the kinematical precise orbit of a LEO. At that time, orbit observations were sparse and not suited for precise kinematical orbit computation. The availability of GNSS changed the situation dramatically. The mathematical–physical model for densely tracked LEO orbits is based on the formulation of the equation of motion as (Ilk 1977)

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination

€rðtÞ ¼ aðt; r; r_ Þ

(22.1)

where the vectors r, r_ and €r denote the position, velocity and acceleration of the LEO, respectively. The function a indicates the specific (mass-related) force function acting on the LEO. The solution of Newtons equation of motion formulated as a BVP with the boundary values rA :¼ rðtA Þ;

rB :¼ rðtB Þ

ð1 rðtÞ ¼ rðtÞ T

K II ðt; t0 ÞaII ðt0 ; r; r_ Þdt0 (22.2)

t0 ¼0

with the starting time tA, the end time tB, the normalized time variable t and the arc length T t tA t :¼ ; T

with t 2 ½tA ; tB ;

T :¼ tB tA

K II ðt; t0 Þ ¼

0 1 sinmð1tÞsinmt ; sinm m 0 1 sinmð1t Þsinmt ; sinm m

sinðupt0 ÞaII ðt0 ; r; r_ Þdt0 :

t0 ¼0

(22.6)

It can be shown (Ilk (1977)), that the solution series (22.5) contains a generalized Fourier series of the difference function dðtÞ : ¼ rðtÞ rðtÞ 1 X ¼ du sinðuptÞ :¼ d1 F ð tÞ

(22.7)

u¼1

du ¼ 2

0

tt:

sinmð1 tÞ sinmt (22.3) rA þ rB sinm sinm qﬃﬃﬃﬃﬃﬃ with the mean anamoly m ¼ GM a3 T, where a is the semi-major axis of the LEO orbit and GM denotes the standard Earth’s gravitational constant. The reduced force function acting on the satellite reads m2 rðtÞ: T2

(22.4)

The solution of the BVP (Fredholm’s integral equation in (22.2)) reads rðtÞ ¼ rðtÞ þ dðtÞ 1 X du sinðuptÞ ¼ r ð tÞ þ

dðt0 Þsinðupt0 Þdt0 :

(22.8)

t0 ¼0

r ð tÞ ¼

u¼1

ð1

ð1

t0 t

The Keplerian orbit as the reference motion reads

aII ðt; r; r_ Þ ¼ aðt; r; r_ Þ

2T 2 du ¼ 2 2 u p m2

with the Fourier coefficient

as well as the the integral kernel (

with the sine coefficient

22.2.2 Interpretation of the Solution of Fredholm’s Integral Equation as Fourier Series

reads according to Schneider (1968)

2

181

(22.5)

If the difference function d (t) is continued to an odd periodic function with the period 2T or to the normalized interval [–1, 1], then we get because of property of sine function dðtÞ ¼ dðtÞ

(22.9)

and all cosine terms vanish (i.e. special case of Fourier series). Therefore, all properties of the Fourier series hold for the solution of the differential function. If we restrict the upper summation indices of the Fourier series to finite number n then the Fourier series in (22.7) taking the Fourier remainder function (RF (t)) into account reads n d1 F ðtÞ ¼ dF ðtÞ þ RF ðtÞ n 1 X X ¼ du sinðuptÞ þ du sinðuptÞ: u¼1

u¼nþ1

(22.10)

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When neglecting the Fourier remaining term RF (t) in (22.10), then Fourier coefficients from discrete not necessary equidistant positions can be estimated by a least square adjustment (Shabanloui 2008).

J X 2 ð1Þjþ1

du ¼

2jþ1

ðupÞ

j¼1

ð1Þu d½2j ð1Þ d½2j ð0Þ

þ zðb; J; uÞ: (22.13)

22.2.3 From Fourier Series to Series of Euler–Bernoulli Polynomials

Introducing (22.13) into (22.7) results in

If (22.8) is partially integrated, then the first integration by parts reads

d(tÞ ¼

ð1

2jþ1

ð1Þu dj2jj ð1Þ

ð u pÞ j2jj d ð0Þ þBðb; J; uÞ sin ðuptÞ: u¼1

j¼1

(22.14)

dðt0 Þsinðupt0 Þdt0

du ¼ 2 t0 ¼0

¼ 2dðt0 Þ þ2

1 J X X 2ð1Þjþ1

1 up

1 1 cosðupt0 Þj0 up ð1 d½1 ðt0 Þcosðupt0 Þdt0 :

(22.11)

dðtÞ ¼

t0 ¼0

1 X

du sinðuptÞ

u¼1

By inserting the limits and considering that the difference function at the boundaries is zero, then we get 1 du ¼ 2 up

If we separate the inner sum of (22.14) in terms of even and odd indices u we get,

¼

½1

0

0

2jþ1

j¼1

þ

ð1

J X 2 ð1Þjþ1

J X 2 ð1Þj j¼1

0

d ðt Þcosðupt Þdt : þ

t0 ¼0

ð2pÞ

1 X

ðpÞ2jþ1

d½2j ð1Þ d½2j ð0Þ

d½2j ð1Þ þ d½2j ð0Þ

1 X sinð2uptÞ u2jþ1 u¼1

1 X sinð2u 1Þpt u¼1

ð2u 1Þ2jþ1

zðb; J; uÞsinðuptÞ:

u¼1

(22.15)

After 2J + 2 integrations by parts (Klose 1985), the general expression reads du ¼

J X

d½2j ðt0 Þ

j¼1

þb

The terms in (22.15) can be replaced by the absolutely and uniformly continuous series expansions of the Euler polynomials (Abramowitz and Stegun 1972)

2 ð1Þjþ1

1 cosðupt0 Þj0 2jþ1

ðupÞ ð1

2 ðupÞ2Jþ2

d½2Jþ2 ðt0 Þsinðupt0 Þdt0

t0 ¼0

E2j ðtÞ ¼

1 4 ð1Þj ð2jÞ! X sinð2u 1Þpt 2jþ1 2jþ1 p u¼1 ð2u 1Þ

(22.16)

(22.12) and the Bernoulli polynomials

with b¼

þ1 1

J ¼ 1; 4; 5; 8; 9; 12; . . . ; J ¼ 2; 3; 6; 7; 10; 11; . . . :

Evaluating the integration limits in the first term of (22.12) and denoting the remaining terms as x (b, J, u) we get

B2jþ1 ðtÞ ¼

1 2 ð1Þjþ1 ð2j þ 1Þ! X sinð2uptÞ : (22.17) u2jþ1 ð2pÞ2jþ1 u¼1

If (22.16) and (22.17) are inserted into (22.15), then we get:

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination

d ð tÞ ¼

1 X u¼1

¼

J X j¼1

þ

22.2.5 Kinematical Precise Orbit Determination

du sinðuptÞ ¼ d1 F ð tÞ e2j E2j ðtÞ þ

J X

b2jþ1 B2jþ1 ðtÞ

j¼1

1 X u¼1

zðb; J; uÞsinðuptÞ ¼ dJP ðtÞ þ RP ðtÞ (22.18)

with the term RP (t) as the Euler–Bernoulli truncation error term from degree J + 1 to infinity and the coefficients of the Euler polynomials (Shabanloui 2008) 1 ½2j d ð1Þ þ d½2j ð0Þ 2ð2jÞ!

e2j ¼

(22.19)

and the coefficients of the Bernoulli polynomials b2jþ1 ¼

1 d½2j ð1Þ d½2j ð0Þ : ð2j þ 1Þ!

183

(22.20)

In Shabanloui (2008), it was demonstrated that the upper degree J of the series in terms of Euler– Bernoulli polynomials should be very high to achieve an approximation accuracy at the sub-millimeter level. Also, it was shown that a fit of a series in terms of Euler–Bernoulli polynomials to the geometrically determined LEO’s short arc at the reasonable upper degree Jmax ¼ 4 should guarantee a sufficient determination of the Euler–Bernoulli coefficients corresponding to sufficiently precise arc derivatives at the boundaries (refer to (22.19) and (22.20)). Therefore, the resulting residual sine series after removing the Euler–Bernoulli terms should show a fast convergence and small residuals when compared to the true ephemerides (Shabanloui 2008). The satellite arc is presented kinematically by the selected reference motion, the Euler–Bernoulli polynomial up to degree Jmax and the residual Fourier series up to index n as rðtÞ ¼ rðtÞ þ

Jmax X

e2j E2j ðtÞ

j¼1

22.2.4 Determination of the Euler–Bernoulli Polynomials Coefficients If we restrict the upper summation indices of the Euler–Bernoulli polynomials to finite number J, then the LEO short arc can be represented based on GNSS observations according to (22.18) by omitting RP (t) as dðtÞ ’

J X j¼1

e2j E2j ðtÞ þ

J X

b2jþ1 B2jþ1 ðtÞ: (22.21)

j¼1

It should be mentioned that an Euler polynomial of degree 2j and a Bernoulli polynomial of degree 2j + 1 belong together as a pair. Based on the selected reference motion (refer to (22.3)), the LEO’s short arc can be approximated with the Euler–Bernoulli polynomial of degree Jmax in the space or spectral domain. We may note that the approximation quality increases with increasing upper degree J and the convergence of the Euler–Bernoulli polynomials requires a large upper index J (Shabanloui 2008).

þ

Jmax X j¼1

b2jþ1 B2jþ1 ðtÞ þ

n X

u sinðuptÞ: d

u¼1

(22.22) u denotes the residual Fourier coefwhere the vector d ficient. Based on the (22.22), in the first step, Euler–Bernoulli coefficients up to degree Jmax ¼ 4 are estimated from discrete GNSS observations. In the second step, the short arc boundary values as well as the residual Fourier coefficients are determined based on GNSS observations which are reduced w.r.t. the Euler–Bernoulli term up to degree Jmax ¼ 4.

22.2.6 Reduced-Kinematical Precise Orbit Determination The kinematical orbit parameters contain no dynamical information of the force function model; they are only based on observations taken at discrete epochs of the satellite motion along the orbit. In the following, we will extend the hybrid case of the kinematical orbit determination procedures as treated in Sect. 22.2.5 in

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such a way that dynamical restrictions can be introduced in the orbit determination procedure. This dynamical ~u information is contained in the orbit coefficients d which are related to the force function according to ~j (i and j as ~ i to d (22.6). If the dynamical quantities d start and end indices) from (22.6) are considered as a-priori dynamical information with the corresponding ~i Þ to Cðd ~j Þ, then the observavariance-covariance Cðd tion equation taking dynamical restrictions into account reads as ¼

A1 0

A2 I

x1 x2

(22.23)

with (pseudo) observations,

Diff. (m)

l1 ¼ ð rðt1 Þ . . . rðtK Þ ÞT ~ T ~ l2 ¼ d . . . dj i where l1 and l2 denote geometrically determined LEO ~i to d ~j which positions and dynamical restrictions d are derived from (22.6), respectively. The terms A1 and A2 are design matrices w.r.t. the boundary values and the Fourier series coefficients which are computed based on (22.3) and (22.5), and x1 and x2 are the unknown boundary values and the Fourier series coefficients, respectively. The a-priori variancecovariance information reads C¼

C1

0

d

x

dy

10

dz

20

30

20

30

20

30

Minute 0.0004 0 -0.0004

0

10 Minute

0 C2 C1 ¼ diagðCðt1 Þ CðtK ÞÞ ~ ~ C2 ¼ diag Cðd i Þ Cðdj Þ

0.02 0.01 0 -0.01 -0.02 0

Diff. (m/s)

Numerical Tests

The orbit determination approach is tested based on simulated GNSS observations for a 30 min arc of a simulated CHAMP orbit with a sampling rate of 30 s. The geometrical positions of CHAMP are estimated based on the simulated hl-SST carrier phase observations which are contaminated with a white noise of 2 cm.

(22.24)

where C1 and C2 denote LEO position variancecovariance matrix derived from GPOD procedure and variance-covariance of dynamical restriction, respectively. Based on the estimated boundary values and the Fourier coefficients from (22.23), the Euler–Bernoulli polynomials up to degree Jmax ¼ 4 by a least squares adjustment are estimated (for more details refer to Shabanloui 2008). The coefficients of the residual Fourier series up to degree n from (22.22) are determined based on the estimated boundary

Diff. (m/s2)

l1 l2

22.3

2E-005 0 -2E-005

0

10 Minute

Coef. (m)

values, the Euler–Bernoulli coefficients and the Fourier coefficients. Therefore, the orbit parameters are determined in such a way that the kinematical parameters are constrained according to the specified level of “dynamical restrictions”.

0.08 0.04 0 -0.04 -0.08 -0.12

dv,x

dv,y

20

dv,z

40

60

Index

Fig. 22.1 Position differences, velocity differences, acceleration differences and the coefficients of the residual Fourier series for the kinematical POD, respectively

Diff. (m)

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination 0.02 0.01 0 -0.01 -0.02

d

0

x

dy

10

dz

20

30

20

30

20

30

Diff. (m/s)

Minute 0.0008 0.0004 0 -0.0004 -0.0008

0

10

Diff. (m/s2)

Minute 4E-005 2E-005 0 -2E-005 -4E-005

0

10

Coef. (m)

Minute dv,x

0.08

dv,y

dv,z

185

To verify results, kinematical ephemerides are determined based on estimated orbit representation parameters at the interval of 10 s and compared with the true ephemerides. The kinematical position, velocity and acceleration differences as well as the Fourier coefficients are shown in Fig. 22.1. It should be pointed out that the kinematical position differences are in range of the given white noise (2 cm). To test the proposed RKPOD procedure, the same short arc of CHAMP as used in the kinematical case, ~ 1 to d ~5 has been selected. The Fourier coefficients d are determined based on EGM96 and have been introduced to the observation equations as dynamical restrictions. The reduced-kinematical position, velocity and acceleration differences at the interval 10 s as well as the residual Fourier coefficients for the reduced-kinematical case are shown in Fig. 22.2. The comparison of the RKPOD with the KPOD shows some improvements in the results, because of introducing dynamical restrictions to estimation procedure (refer to Table 22.1).

0 -0.08

Conclusions

Fig. 22.2 Position differences, velocity differences, acceleration differences and the coefficients of the residual Fourier series for the reduced-kinematical POD case, respectively

The new proposed kinematical and reducedkinematical POD procedures open a wide window to represent low-flying orbits. The method is very flexible with the possibility of smooth transition from pure kinematical to fully dynamical POD.

Table 22.1 3D RMS of positions, velocities and accelerations for kinematical and reduced-kinematical POD numerical tests

Acknowledgment We gratefully acknowledge the financial support of the BMBF under the project ”REAL-GOCE”.

20

40

60

Index

Method KPOD RKPOD

Pos.(m) 0.011339 0.008873

Vel.(m/s) 0.000351 0.000337

Acc.(m/s2) 0.000015 0.000016

References The geometrically estimated positions are used as pseudo-observations in case of the kinematical and reduced kinematical orbit determination procedures. To demonstrate the kinematical POD procedure, the Keplerian orbit is used as reference motion. A series of Euler–Bernoulli polynomials up to degree Jmax ¼ 4 has been fitted to the difference function d(t). Based on geometrical determined LEO positions, the Keplerian orbit and the estimated Euler–Bernoulli polynomials coefficients, the residual Fourier coefficients up to the index n ¼ 30 are estimated.

Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York Klose U (1985). Beitr€age zur L€osung einer Integralgleichung vom HAMMERSTEINschen Typ. Diploma Thesis, TUM, Germany Ilk KH (1977) Berechnung von Referenzbahnen durch L€osung selbstadjungierter Randwertaufgaben, DGK, Reihe C, Heft 228, Munich, Germany J€aggi A (2007) Pseudo-stochastic orbit modeling of low earth satellites using the GPS. Ph.D. Thesis, University of Bern, Switzerland Mayer-G€urr T (2006) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb€ogen am Beispiel der Satellitenmission

186 CHAMP und GRACE. Ph.D. Thesis, IGG, University of Bonn, Germany Reigber C (1969) Zur Bestimmung des Gravitationsfeldes der Erde aus Satellitenbeobachtungen, DGK, Reihe C, Heft Nr. 137, Munich, Germany Schneider M (1968) A general method of orbit determination. Ph.D. Thesis, Ministry of Technology, Farnborough, England

A. Shabanloui and K.H. Ilk Shabanloui A (2008) A new approach for a kinematic-dynamic determination of low satellite orbits based on GNSS observations. Ph.D. Thesis, IGG, University of Bonn, Germany Sˇvehla D, Rothacher M (2003) Kinematic, reduced-kinematic, dynamic and reduced-dynamic precise orbit determination in the LEO orbit, 2nd CHAMP Science Meeting, Potsdam, Germany

Pure Geometrical Precise Orbit Determination of a LEO Based on GNSS Carrier Phase Observations

23

A. Shabanloui and K.H. Ilk

Abstract

The interest in a precise orbit determination of Low Earth Orbiters (LEOs) especially in pure geometrical mode using Global Navigation Satellite System (GNSS) observations has been rapidly grown. Conventional GNSS-based strategies rely on the GNSS observations from a terrestrial network of ground receivers (IGS network) as well as the GNSS receiver on-board LEO in double difference (DD) or in triple difference (TD) data processing modes. With the advent of precise orbit and clock products at centimeter level accuracy provided by the IGS centers, the two errors associated with broadcast orbits and clocks can be significantly reduced. Therefore, higher positioning accuracy can be expected even when only a single GNSS receiver is used in a zero difference (ZD) procedure. Along with improvements in the International GNSS Services (IGS) products in terms of Global Position System (GPS) satellite orbits and clock offsets, the Precise Point Positioning (PPP) technique based on zero (un-) differenced carrier phase observations has been developed in recent years. In this paper, the zero difference procedure has been applied to the CHAllenging Minisatellite Payload (CHAMP) high–low GPS Satellite to Satellite Tracking (hl-SST) observations, then the solution has been denoted as Geometrical Precise Orbit Determination (GPOD). The estimated GPOD CHAMP results are comparable with results of other groups e.g. Sˇvehla at TUM (Sˇvehla D, Rothacher M (Sˇvehla and Rothacher 2002) and Bock at Bern (Bock 2003) but because of different outliers detection and data processing strategies, the GPOD results presented here are more or less different than the other groups’ results. The estimated geometrical orbit of CHAMP is point-wise and its accuracy relies on the geometrical status of the GNSS satellites and on the number of the tracked GNSS satellites as well as on the GNSS measurement accuracy in the data

A. Shabanloui (*) K.H. Ilk Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115, Bonn, Germany e-mail: [email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_23, # Springer-Verlag Berlin Heidelberg 2012

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processing. The position accuracy of 2–5 cm of CHAMP based on high–low GPS carrier phase observations with zero difference procedure has been achieved. These point-wise absolute positions can be used to estimate kinematical orbit of the LEOs.

23.1

Introduction

Among many possibilities to determine precise orbit of LEOs based on GNSS observations, zero differenced of high–low GPS-SST observations makes possible to be independent of the ground GNSS stations. In other words, geometrical LEO orbit can be estimated point-wise at the tracked epochs with only high–low observations which connect the GPS satellites to LEO with high precision. With the proposed method, only table of absolute positions at the desired epochs are estimated and subsequently no velocities and no accelerations and other kinematical parameters can be directly estimated in this procedure. In order to estimate kinematical parameters (e.g. velocities) specified functions or polynomials have to be fitted to the determined geometrical absolute positions of a LEO based on high–low GPS-SST observations. In this paper, methodology, numerical tests for the CHAMP satellite and some conclusions for the proposed geometrical precise orbit determination are presented.

the outliers in the high–low GPS-SST observations have to be flagged in the pre-processing process and excluded from the main GPS data processing. The detection methods are described in Sect. 23.2.2 in more detail.

23.2.1 Zero Differenced High–Low GPS-SST Observations In the zero (un-)differenced observations (refer to Fig. 23.1), always only the geometrical connection between GPS satellites (as sender) and GPS receiver on-board LEO (as receiver) is used to estimate the LEO geometrical orbit (Shabanloui 2008). The zero differenced carrier phase GPS-SST observations between the GPS satellite s and the LEO satellite r at frequency i with respect to the ambiguity parameter and all the error terms can be written as (Shabanloui 2008) si

23.2

E e3

Methodology

Zero differenced procedure of GPS-SST observations is used to estimate the geometrical absolute positions and the clock offset of a LEO. The approximated LEO absolute positions and the clock offsets can be determined epoch-wise based on code pseudo-range GPS-SST observations. The geometrical LEO absolute positions are improved with the accurate screened carrier phase observations. It has to be mentioned that the estimator, based on the least squares assuming standard normal distribution is not robust. Therefore, any outliers in the high–low GPS-SST observations, prior variances or other kinds of random parameters could dramatically affect the estimates of the unknown parameters and the variance components. Therefore,

Geocenter

rsi(t – –r tsi)

r

r(t)

E e2 E e1

Fig. 23.1 Zero differenced geometrical orbit determination

23

Pure Geometrical Precise Orbit Determination of a LEO s Fsr;i ðtÞ ¼ li fsr;i ðtÞ ¼ rsr ðtÞ cdtr ðtÞ þ li Nr;i

þ xsr;Fi ðtÞ þ eFi ðtÞ

(23.1)

s s where at epoch t, Fr,j and fr,j are the observed carrier phase between the GPS satellite s and the LEO GPS receiver r at frequency of i in unit of length and cycle respectively, rr,s is the true geometrical distance between the corresponding GPS and LEO, cdtr is the s LEO GPS receiver clock offset, Nr,j is the ambiguity parameter, li is the wavelength of the given GPS signal at frequency of i. The terms xsr,Фi and eФi represent the summation of all error effects on the LEO and the GPS satellites and the remaining error that cannot be modeled in the carrier phase observations, respectively (Shabanloui 2008). In the zero differenced principal, the error terms either have to be modeled with accurate specified models or have to be eliminated in the data processing procedure. For example, to eliminate the largest part of ionospheric effect on the carrier phase observations, the linear combination (L3) of the carrier phase observation at two frequencies (L1) and (L2) has been used. Consequently, the ionosphere free carrier phase observations at the epochs t can be written as (Shabanloui 2008):

s þ xsr;F3 ðtÞ þ eF3 ðtÞ Fsr;3 ðtÞ ¼ rsr ðtÞ cdtr ðtÞ þ l3 Nr;3

(23.2) with considering the GPS signal sending time and the Sagnac effect on the carrier phase observation, the observation equation can be rewritten as (Shabanloui 2008), Fsr;3 ðtÞ ¼ RZ ðoe tsr Þrs ðt tsr Þ rr ðtÞ s þ l3 Nr;3 cdtr ðtÞ þ xsr;F3 ðtÞ þ eF3 ðtÞ

(23.3) or the observation equation reads as (Leick 1995), s Fsr;3 ðtÞ ¼ rsm ðt tsr Þ rr ðtÞ þ l3 Nr;3 cdtr ðtÞ þ xsr;F3 ðtÞ þ eF3 ðtÞ

(23.4)

where oe, tsr and Rz(oetsr) represent rotation rate of the Earth, GPS signal travel time between sender and receiver and rotation matrix of the ITRF around the Z axis by the angle oetsr, respectively. The term rs(ttsr) represents the absolute position of the GPS

189 s satellite s at the sending time, rm (ttrs) is the GPS satellite position after applying the GPS signal travel time and the Sagnac effect and finally rr(t) is LEO receiver r at the receiving time, respectively.

23.2.2 Data Pre-processing For all applications of GPS, an efficient pre-processing and data screening of GNSS observations are necessary. It is particularly an important issue for the processing of the high–low GPS-SST observations of space-borne GNSS receivers on-board LEOs to determine precise orbit of the LEOs. In other words, any orbit determination procedure based on high–low GPS-SST observations depends crucially on the ability to remove invalid or degraded observations from the estimation procedures. It is clear that code pseudo-range GPS-SST observations are only used to determine initial geometrical absolute positions of a LEO. The procedure applied to detect outliers in the code observations is based on the “Majority voting”, which used the estimated LEO clock offset at every epoch (Bock 2003). In other words, the Majority voting is based on an epoch-wise processing of the GPS-SST observations. Therefore, at the first step, the receiver clock is synchronized with the GPS time, then based on the Majority voting algorithm, the bad observations are flagged to be excluded from GPS data processing. Based on the iterative least squares data processing, these flagged observations may be detected and subsequently be excluded from the main GPS data processing. The same Majority algorithm can be applied to the subsequent time differenced carrier phase observations between two epochs to detect outliers in the high–low carrier phase observations (Bock 2003). But the key factor limiting the performance of the pre-screening procedure of the carrier phase observations is the quality of the a-priori LEO absolute positions. Therefore, the rejection threshold should be set not too small. Consequently, a second screening procedure has to be applied to the carrier phase observations to find all bad observations which deteriorate the solution. The second issue, in case of carrier phase observation, is the cycle slips in the high–low GPS-SST observations. In order to obtain high precision orbit based on the carrier phase observations, cycle slips in the carrier phase observations have to be detected,

190

A. Shabanloui and K.H. Ilk

identified and repaired at the pre-processing stage. A slip of only few cycles can influence the measurements such that a geometrical precise orbit cannot be achieved. Therefore, the detection of the cycle slips in the carrier observations is the major issue in the geometrical precise orbit determination procedure. Repairing process in this case needs to determine the size of cycle slips at the every carrier phase frequency, but sometimes it is very difficult to identify the size of an every frequency part. Therefore, after identification of the cycle slips, to avoid the repairing of the carrier phase, as an alternative, the new ambiguity parameters can be introduced to detected carrier phases in the GPS main processing. In order to detect the cycle slips in carrier phase observation, the observations or combination of observations have to be used sensitively to the cycle slips. Therefore, the detection of cycle slips in this case can be performed based on the quantities which based on the geometrical free combination of carrier phase observation at Li frequencies (i.e. fi, i ¼ 1,2) and code pseudo-range observations. The geometrical free combination (frs, GF(t) of carrier phase observation can be formulated as (Hofmann-Wellenhof et al. 2001):

GPS ambiguity terms, but is non-linear with respect to LEO absolute position. Therefore, to estimate geometrical absolute position, the observation equation has to be linearized. The linearized ionosphere-free GPSSST carrier phase observation reads (Xu 2007),

f1 s f ðtÞ f2 r;2 f1 s bion ðtÞ f2 s ¼ Nr;1 Nr;2 ð1 12 Þ þ ef ðtÞ f2 f1 f2 (23.5)

which xr(t), yr(t) and zr(t) represents the coordinates of the LEO absolute position at time t. The linearized observation can be rewritten as (Shabanloui 2008),

fsr;GF ðtÞ ¼ fsr;1 ðtÞ

the left side of the geometrical free carrier phase observations at two frequencies shows that only time-varying quantity is the ionosphere term bion. This equation shows that the ionosphere effect has been reduced with respect to the original carrier phase observations (Shabanloui 2008). Now, if there are no cycle slips, the temporal variations of the geometrical free of carrier phase observations would be small for normal ionosphere conditions. The geometrical free observations have disadvantages to determine the cycle slips on L1 and L2 or both, but with introducing new ambiguities, this problem can be solved.

23.2.3 Solution Obviously, the carrier phase GPS-SST observation is linear with respect to the LEO clock offset and the

Fsr;3 ðtÞ ¼ Fsr;3;0 ðtÞ þ

@Fsr;3 ðtÞ ðx x0 Þ @x

(23.6)

where x is a vector of the unknown parameters which contains the LEO absolute position, the LEO clock offset and GPS ambiguity terms at the time t respectively; and x0 is a vector of the initial values of the unknown parameters as follows, 0

1 xr ðtÞ B yr ðtÞ C B C B zr ðtÞ C B C B C x :¼ B cdtr ðtÞ s1 C; B l3 Nr;3 C B C B .. C @ . A sn l3 Nr;3

x0 :¼ xj0

(23.7)

DFsr;3 ðtÞ ¼ asr ðtÞDxðtÞ; wsr ðtÞ ¼

s20 cosðzsr ðtÞÞ: s2Fs

(23.8)

r;3

The terms wrs(t), s2Fs and zrs(t) represent the obserr;3 vation weight, carrier phase observation variance and zenith distance of the GPS satellite s from the GPS receiver r, respectively. If we assume that, a number n of GPS satellites si, i¼1, . . ., n are available at the time t. The Gauss-Markov model for all observed carrier phase observations corresponding to weight matrix with considering the unit vector between the GPS satellite s and LEO on-board receiver r reads (Shabanloui 2008), rsm ðt tsr Þ rr ðtÞ esri ðtÞ ¼ rs ðt ts Þ rr ðtÞ m r i i esr;y esr;zi ; ¼ esr;x

(23.9)

23

Pure Geometrical Precise Orbit Determination of a LEO

1 1 DFsr;3 ðtÞ C B .. A @ . n ðtÞ DFsr;3 1 esr;y ðtÞ esr;z1 ðtÞ 1 .. .. .. . . . n esr;y ðtÞ esr;zn ðtÞ 1

23.3

0

0

1 esr;x ðtÞ B .. ¼@ . n ðtÞ esr;x

191

1 .. . 0

1 0 0 .. .. C: . .A 0 1

0

1 Dxr ðtÞ B Dyr ðtÞ C B C B Dzr ðtÞ C B C B C :B DcdtrsðtÞ C þ «ðtÞ; 1 B l3 Nr;3 C B C B C .. @ A . sn l3 Nr;3 Wl ðtÞ ¼ diagðwsr1 ðtÞ wsrn ðtÞÞ:

(23.10)

Obviously, at the first epoch, the number of observations are smaller than the number of the unknowns (under determined problem). Therefore, the carrier phase observations of the other epochs have to be summed up and the observation equation has to be solved in a batch processing for all desired epochs. The observation equations for all observed epochs can be written as, « þ Dl ¼ ADx; Wl

(23.11)

by a least square adjustment, the unknowns read 1 T D^x ¼ AT Wl A A Wl Dl:

Numerical Tests

To verify the proposed geometrical orbit determination procedure, four 30 min arcs (short arcs) of CHAMP have been selected. As a pre-processing step, the outliers and cycle slips in carrier phase observations are removed with the “Majority voting” technique and different combination form of the observations. Finally, to minimize or mitigate the effect of multi-path effect nearby weighting the carrier phase observations with respect to the GPS zenith distance, a 15 cut-off angle has been applied to the observations. The ground tracks of four CHAMP short arcs are shown in Fig. 23.2. Because of the fact that the accuracy of GPS-SST carrier phase observations are in the range of some millimeters, one can expect that the millimeter level of accuracy for the GPS receiver is achievable. In case of GPS receivers at the ground stations, the problematic modeling of the atmosphere parameters can reduce the accuracy of GPS estimated positions. The situation is less critical for GPS receiver on-board LEO. There is no tropospheric effect in the altitude of LEOs (e.g. 430 km); but at this altitude, the ionosphere can dramatically affect the GPS-SST observations. The multipath effect is another major problem in the data processing of LEOs. To validate the estimated geometrical CHAMP orbit externally, the CHAMP PSO dynamical orbits provided by GFZ-Potsdam have been used. In Fig. 23.3, the differences between estimated CHAMP absolute positions and PSO dynamical orbit for the four short arcs (cases a–d) are shown. Figure 23.4 represents the carrier phase GPS-SST observations for four short arcs. It should be pointed

(23.12)

The estimated unknowns are the corrections to the LEO absolute positions, the LEO clock offsets and the GPS ambiguity terms. Because of the non-linearity of the observations, the convergence is achieved after few iterations e.g. after i iterations; subsequently the unknowns vector and variance-covariance read,

90°N

d

60°N 30°N

c

0° 30°S

b

a

60°S

^xi ¼ ^xi1 þD ^ xi ;

1 C^xi ¼ AT Wl A

(23.13)

the estimated unknowns are the LEO absolute positions, the LEO clock offsets and the float GPS ambiguity terms at all desired epochs.

90°S 180°W

120°W

60°W

0°

60°E

120°E

180°

Fig. 23.2 The ground tracks of four 30 min short arcs for the time: (a) 2002 03 21 13h 30m 0.0s – 14h 00m 0.0s, (b) 2002 07 20 12h 48m 0.0s – 13h 18m 0.0s, (c) 2003 03 21 17h 20m 0.0s – 17h 50m 0.0s, (d) 2003 03 31 17h 00m 0.0s – 17h 30m 0.0s

a

A. Shabanloui and K.H. Ilk 0.12

dx

dy

dz

Diff. (m)

0.08 0.04 0 -0.04

a

0.02

Residuals (m)

192

0.01 0 -0.01 -0.02

-0.08

52354.565

52354.565 52354.57 52354.575 52354.58 MJD (days) 0.08

dx

dy

dz

Diff. (m)

0.04 0

52354.575

52354.58

MJD (days)

b

0.02

Residuals (m)

b

52354.57

0.01 0 -0.01

-0.04 -0.02

-0.08

52475.535

MJD (days)

c

0.2

dx

dy

dz

Diff. (m)

0.1 0

c

0.012

Residuals (m)

52475.535 52475.54 52475.545 52475.55 52475.555

0.008

-0.1

0 -0.004 -0.008 52719.725 52719.73 52719.735 52719.74 MJD (days)

MJD (days) dx

dy

dz

Diff. (m)

0.1 0

d

0.012

Residuals (m)

52719.725 52719.73 52719.735 52719.74

d

52475.55

0.004

-0.2

0.2

52475.54 52475.545 MJD (days)

0.008 0.004 0 -0.004 -0.008 -0.012

-0.1

52729.71 52729.715 52729.72 52729.725 52729.73 MJD (days)

-0.2 52729.71 52729.715 52729.72 52729.725 52729.73 MJD (days)

Fig. 23.3 Absolute position differences between estimated geometrically orbit and PSO dynamical CHAMP orbit for the case (a), case (b), case (c) and case (d)

out that the residuals for all four short arcs are in the range of 2 cm. The geometrical absolute positions of CHAMP can be estimated with an accuracy of 2–5 cm. In the cases (c) and (d), the residuals in some epochs show zero values which means availability of minimum four GPS satellites at these epochs. To show real precision of estimated CHAMP orbits, polynomial of degree five seems to be a proper degree to fit to the differenced ephemeris. Therefore, a polynomial of degree five is fitted to the differences between the

Fig. 23.4 Carrier phase GPS-SST observation residuals for case (a), case (b), case (c) and case (d)

estimated CHAMP orbit and the PSO dynamical orbit. After removing trend with the polynomial of degree five, RMS values of orbits is shown in Table 23.1. Table 23.1 RMS values of the estimated CHAMP orbits w.r.t CHAMP PSO dynamical orbit after removing trend with polynomial degree five Case a b c d

X(m) 0.0047 0.0024 0.0140 0.0087

Y(m) 0.0096 0.0062 0.0171 0.0073

Z(m) 0.0122 0.0075 0.0085 0.0105

23

Pure Geometrical Precise Orbit Determination of a LEO

23.4

Conclusions

Conventional GNSS-based POD strategies rely on the GNSS observations from a terrestrial network of ground receivers (IGS network) as well as the GNSS receiver on-board LEO in different difference data processing modes. Advent of GNSS orbits and clock offset, the geometrical precise orbit of LEO has been realized based on high–low GPS-SST observations. Zero differenced procedure provides an efficient possibility to estimate geometrical orbit of a LEO. The proposed geometrical LEO POD strategy could be characterized as pure geometrical type, point-wise and three-dimensional. In other words, only geometrical parameters (e.g. absolute positions) can be estimated based on the geometrical observations. An efficient data pre-processing and data screening procedure are very important to achieve precise geometrical LEO orbits based on GPS-SST observations. Specially, the detection, identification and removing (repairing) procedure are the most important issues in the main data processing. In this paper, real case results demonstrate that an accuracy of centimeter has been achieved for the geometrical orbit of CHAMP. It has to be mentioned that the estimated geometric precise LEO orbits can be used for further investigations e.g. determination of kinematical (continuous) LEO orbit for space borne recovery of the Earth gravity field based on GPS-SST observations. It is clear that,

193

geometrical or kinematical POD of a LEO have not been affected by the dynamical information from the Earth gravity field. Acknowledgment We gratefully acknowledge the financial support of the “German Federal Ministry for Education and Research” (BMBF) under the project “LOTSE-CHAMP/ GRACE”.

References Bock H (2003) Efficient methods for determining precise orbits for low earth orbiters using the global positioning system (GPS). PhD thesis, Astronomical Institute, University of Bern, Switzerland Hofmann-Wellenhof B, Lichtenegger H, Collins J (2001) GPS, theory and practice. Springer, New York Leick A (1995) GPS satellite surveying, 3rd edn. Wiley, New York Shabanloui A (2008) A new approach for a kinematic-dynamic determination of low satellite orbits based on GNSS observations. PhD thesis, Department of Astronomical, Physical and Mathematical Geodesy, Institute of Geodesy and Geo-information, University of Bonn, Germany Sˇvehla D, Rothacher M (2002) Kinematic orbit determination of LEOs based on zero or double difference algorithms using simulated and real SST data. In: Adam J, Schwarz K-P (eds) Vistas for geodesy in the new millennium, vol 125. Springer, Berlin, pp 322–328 Xu G (2007) GPS, theory, algorithms and applications, 2nd edn. Springer, Berlin

.

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field: Boundary Problems and a Target Function

24

P. Holota and O. Nesvadba

Abstract

The purpose of the presentation is to discuss the combination of terrestrial and satellite gravity field data and to show its spectral and space domain interpretation. Potential theory is of key importance in this field. However, the problems discussed are overdetermined by nature. Therefore, methods typical for the solution of boundary-value problems are used together with an optimization concept, target functions and regularization techniques. Two cases are treated. They are motivated by the use of a satellite-only model of the gravity field of the Earth or by data coming from satellite missions (especially GOCE) in common with terrestrial gravity measurements. For the results reached in the spectral domain summation techniques are applied in order to find the interpretation of the results in terms of kernel (Green’s) functions related to the particular combination scheme. This enables to show the tie between the global and the local modelling of the gravity field. In order to demonstrate the efficiency of the procedure results of numerical simulations are added. Differences of the closed loop simulation are very promising.

24.1

Introduction

In this paper a concept is discussed enabling combination of terrestrial gravity measurements with a satellite-only model of the Earth’s gravity field or with data coming from satellite missions, in particular GOCE [treated in terms of the space-wise approach, see e.g.

P. Holota (*) Research Institute of Geodesy, Topography and Cartography, 250 66 Zdiby 98, Praha-vy´chod, Zdiby, Czech Republic e-mail: [email protected] O. Nesvadba Land Survey Office, Pod Sı´dlisˇteˇm 9, 182 11, Praha 8, Czech Republic e-mail: [email protected]

Migliaccio et al. (2004)]. The paper ties to Holota (2007) and Holota and Nesvadba (2006, 2007, 2009). In the sequel OTOPO means a domain bounded by two surfaces, from above by a geocentric sphere of radius Re and from below by a surface GTOPO , which is star-shaped at the origin and about as irregular as the surface of the Earth. Moreover, let h be the height of the boundary point x 2 GTOPO above a geocentric sphere of radius Ri . We will consider the following two problems D T ¼ 0 in jxj

@T þ 2T ¼ jxjDg @jxj

O TOPO for

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_24, # Springer-Verlag Berlin Heidelberg 2012

x 2 GTOPO

(24.1) (24.2)

195

196

P. Holota and O. Nesvadba

T ¼ t for

jxj ¼ Re

(24.3)

where t means the input from an available satelliteonly model of the Earth’s gravity field and D T ¼ 0 in jxj

@T þ 2T ¼ jxjDg @jxj @2 T @ j x j2

¼G

O TOPO for

for

x 2 GTOPO jxj ¼ Re

(24.4) (24.5)

(24.6)

where G represents the input from satellite gradiometry. Note also that DT means Laplace’s operator applied on T, Dg is the gravity anomaly, x ðx1 ; x2 ; x3 Þ, jxj ¼ ðS3i¼1 x2i Þ1=2 and x1 ; x2 ; x3 are the usual rectangular Cartesian co-ordinates in threedimensional Euclidean space R3 . Our starting point will be the transform of the domain O TOPO given by x1 ¼ ½ r þ oðrÞ hð’; lÞ cos ’ cos l

(24.7)

x2 ¼ ½ r þ oðrÞ hð’; lÞ cos ’ sin l

(24.8)

x3 ¼ ½ r þ oðrÞ hð’; lÞ sin ’

(24.9)

where ’ and l are the spherical coordinates of the point x 2 O TOPO (geocentric latitude and longitude, respectively), r is defined by jxj ¼ r þ oðrÞ hð’; lÞ and oðrÞ is a suitably chosen and twice differentiable auxiliary function of r 2 ½Ri ; Re . The transformation, as defined in Holota and Nesvadba (2009), is a oneto-one mapping between O TOPO and a domain O, O f x; x 2 R3 ; Ri < jxj Re is not regular at infinity, for r ! 1 it does not decrease as c=r (c is a constant) or faster. This is

198

P. Holota and O. Nesvadba

caused by measurement errors. The data given for r ¼ Ri are sufficient to determine a harmonic function in Oext fx 2 R3 ; r>Ri g and thus in O Oext . The data for r ¼ Re are excess data and give rise to “interðeÞ nal” terms ðr=Re Þn T~n that are not regular at infinity. In solving this overdetermined problem, we will look for a harmonic function f , which is regular at infinity and minimizes the functional ð Fðf Þ ¼

ð ðf ; gÞ1 ¼

hgrad f ; grad gi dx

(24.36)

Oext ð1Þ

We look for a function f 2 H2 ðOext Þ that minimizes the functional ð Cðf Þ ¼ jgradðf T~ Þj2 dx

(24.37)

O 2

ðf T~ Þ dx

(24.31)

O

In particular we suppose that f 2 H2 ðOext Þ, where H2 ðOext Þ is a space of harmonic functions with inner product ð ðf ; gÞ

In this case the function f is defined by the following integral identity ð

ð

hgrad f ; gradvidx ¼ hgrad T~ ; gradvidx (24.38)

O

O ð1Þ

r 2 fg dx

(24.32)

valid for all v 2 H2 ðOext Þ. In consequence f is given by (24.34), but now with an ¼ 0 for all n.

Oext

Hence, assuming F has its minimum at a point f 2 H2 ðOext Þ, we know that Gaˆteaux’ differentials of F equals zero at the point f . This yields ð

ð fv dx ¼ T~ v dx for all v 2 H2 ðOext Þ

O

(24.33)

O

24.6

ð1Þ

Optimization in H2 -Traces ð1Þ

Considering the fact that functions from H2 ðOext Þ have precisely defined traces on the boundary @O of ð1Þ

O, we can also look for f 2 H2 ðOext Þ that minimizes the functional ð

and we easily obtain that

Yðf Þ ¼

1 nþ1 h i X Ri T~nðiÞ þ an T~nðeÞ f ¼ r n¼0

2 ð f T~ Þ dS

(24.34)

Again, Y attains its minimum and ð

with

ð fv dS ¼

an ¼

ð2n 1Þð 1 q2 Þ n2 q 2ð 1 q2n1 Þ

@O

(24.35)

see Fig. 24.1. In particular a0 ¼ ð1 þ qÞ=2q and lim an ¼ 0 as n ! 1, see Holota and Nesvadba (2006, 2007).

Optimization in ð1Þ

ð1Þ H2

Let H2 ðOext Þ be the space of harmonic functions on Oext equipped with inner product

T~ v dS

(24.40)

@O

ð1Þ

valid for all v 2 H2 ðOext Þ is the respective condition for Y to have a minimum at the point f . Hence, the function f can be represented by (24.34) too, but with an ¼ bn , n ¼ 1; 2; . . . 1 and bn ¼

24.5

(24.39)

@O

1 þ q n1 q 1 þ qn1

(24.41)

see Fig. 24.1. One can show that b0 ¼ 1 and that lim bn ¼ 0 as n ! 1.

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field

Fig. 24.1 Values of an and bn for Ri ¼ 6; 378 km and two cases of Re : Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km, i.e., for q ¼ 0:96228 and q ¼ 0:94099, respectively

199

1,2

· a n for q = 0, 96228

1,0

a n for q = 0, 94099

0,8

+ b n for q = 0, 96228

0,6

× b n for q = 0, 94099

0,4 0,2 degree n 0,0 0

24.7

AðiÞ n ¼

Optimized Solution: Target Function Fðf Þ

In order to show more clearly how the optimization works we have to return to the structure of the ðiÞ ðeÞ harmonics T~n and T~n . For the combination of gravimetry and a satellite-only model they are given by (24.25) and (24.26). Thus the optimization leads to f ¼

1 nþ1 X Ri n¼0

r

AðiÞ n

Ri f Dg þ AðeÞ n tn n1 n

25

(24.42)

with nþ1 Þ=Dn AðiÞ n ¼ ðn 1Þð 1 an q

(24.43)

n AðeÞ n ¼ ½ðn þ 2Þq þ an ðn 1Þ =Dn

(24.44)

f¼

100

125

AðiÞ n

r

Ri f R2e Dgn þ þAðeÞ Gn n n1 ðn þ 1Þðn þ 2Þ

AðeÞ n ¼

ðn þ 1Þðn þ 2Þ ½ðn þ 2Þqn þ an ðn 1Þ Dgn (24.47)

illustrated in (comparative) Fig. 24.3. For a greater range of degree n the plots shown in Figs. 24.2 and 24.3 can be seen in Holota and Nesvadba (2006, 2007).

24.8

(24.45)

150

n1 nðn 1Þ an ðn þ 1Þðn þ 2Þqnþ1 g Dn (24.46)

Optimization – Comparizon: Target Functions Fðf Þ, Cðf Þ and Yðf Þ

Similarly as for the optimization based on the target function Fðf Þ we can proceed in case of target functions Cðf Þ and Yðf Þ. The optimized solution is again given by (24.42) for the first problem and ðiÞ ðeÞ (24.45) for the second problem, but with An and An illustrated in Figs. 24.2 and 24.3.

1 nþ1 X Ri n¼0

with

75

and

and

illustrated in (comparative) Fig. 24.2. For the combination of gravimetry and satellite ðiÞ ðeÞ gradiometry T~n and T~n are given by (24.29) and (24.30). Thus the optimization leads to

50

24.9

Space Domain Interpretation: Terrestrial Term

Our aim is now to sum the series

200

P. Holota and O. Nesvadba

Φ(f)

Ψ (f)

Θ (f )

2,0

2,0

2,0

• q = 0, 96228

q = 0, 94099 1,5

1,5

1,5 (i)

(i)

1,0

(i)

An

An

0,5

An

1,0

1,0

0,5

0,5

(e)

An

0,0

0,0

0,0 (e)

(e)

An

An

degree n -0,5

-0,5 0

2

-0,5 0

50

25

50

0

25

50

ðeÞ Fig. 24.2 Combination of gravimetry and a satellite-only model: Coefficients AðiÞ n and An for target functions Fðf Þ, Cðf Þ, Yðf Þ and for q ¼ 0:96228 and q ¼ 0:94099, i.e., for Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km

Φ( f )

Ψ (f)

2,0

Θ (f)

2,0

2,0

1,5

1,5

• q = 0, 96228

q = 0, 94099 1,5 (i)

1,0

0,5

(i)

(i)

An

An

An 1,0

1,0

0,5

0,5

(e)

An

0,0

0,0

0,0 (e)

-0,5

-0,5 0

25

(e)

An

degree n 50

An -0,5

0

25

50

0

25

50

ðeÞ Fig. 24.3 Combination of gravimetry and satellite gradiometry: Coefficients AðiÞ n and An for target functions Fðf Þ, Cðf Þ, Yðf Þ and for q ¼ 0:96228 and q ¼ 0:94099, i.e., for Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km

fterr ¼

1 nþ1 X Ri n¼0

r

AðiÞ n

Ri f Dg n1 n

(24.48)

representing the terrestrial term in (24.42) or (24.45) for the target function Fðf Þ. To our opinion the function combines the effect of boundary data and the smoothness in constructing the optimized solution optimally. For the first problem we obtain

R2i 1q f R3 2 ð1 þ qÞq f Dg1 Dg0 þ 2i r 2ð2q 1Þ r 6q2 ð Ri f ds þ S ðr; cÞ Dg 4p s (24.49) 1 X 2n þ 1 Ri nþ1 AðiÞ Pn ðcos cÞ S ðr; cÞ ¼ n r n1 n¼2

fterr ¼

(24.50)

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field

201

15 Classical Stokes' function S(psi) = S(1,psi) 12 S* Kernel, terrestrial data and a satellite-only model, q = 0.94099 9

S* Kernel, terrestrial data and a satellite-only model, q = 0.96228 S* Kernel, gravimetry and satellite gradiometry, q = 0,94099

6 S* Kernel, gravimetry and satellite gradiometry, q = 0.96228 3 0 -3 -6 psi ® -9 0°

30°

60°

90°

120°

150°

180°

Fig. 24.4 Kernel functions in the integral representation of the terrestrial term (Stokes’ function drawn for comparison) ðiÞ

with An given by (24.43). Note that c is the angle between the computation and the running point of the integration. Similarly, for the second problem we have fterr ¼

R2i 1 þ q f R3 1 þ q f Dg0 2i Dg1 r 4q r 6q2 ð Ri f ds þ S ðr; cÞ Dg 4p s

ð1Þ

fsat ¼

1 nþ1 X Ri n¼0

r

AðeÞ n tn

(24.52)

and (24.51)

where the kernel S ðr; cÞ appears again, but with ðiÞ coefficients An given by (24.46). It is clear that for the space domain interpretation of the terrestrial term the dependence of S ðr; cÞ on the angle c is of considerable instructive value. For r ¼ Ri the diagrams were derived (indirectly) from summation of about 300–450 first terms of the difference between the series expression of Stokes’ function and the series in (24.50), see Fig. 24.4. The figure shows that in fterr the influence of distant zones is considerably suppressed.

24.10 Space Domain Interpretation: Satellite Term Of course, our aim also is to sum the satellite term

ð2Þ fsat

¼

1 nþ1 X Ri n¼0

r

AðeÞ n

R2e Gn ðn þ 1Þðn þ 2Þ

(24.53)

in (24.42) and (24.45), respectively. After some algebra we have ( ) ð ð1Þ ð1Þ S ðr; cÞ 0 t 1 fsat ¼ ds ð2Þ ð2Þ S ðr; cÞ G 4p s 0 fsat (24.54) 1 X Ri nþ1 AðeÞ Pn ðcos cÞ Sð1Þ ðr; cÞ ¼ n ð2n þ 1Þ r n¼0 (24.55) ðeÞ

with An given by (24.44) and Sð2Þ ðr; cÞ ¼

1 X n¼0

ðeÞ

AðeÞ n

nþ1 2n þ 1 Ri Pn ðcos cÞ ðn þ 1Þðn þ 2Þ r

with An given by (24.47). The kernels Sð1Þ and Sð2Þ are illustrated in the following figure (Fig. 24.5).

202 Fig. 24.5 Kernel functions in the integral representation of the satellite terms

P. Holota and O. Nesvadba 8000

10

7000

9

S^(1) Kernel: gravimetry and a satellite only model, Re = Ri + 400km

6000 5000

8 7

S^(1) Kernel: gravimetry and a satellite only model, Re = Ri + 250km

4000

S^(2) Kernel: gravimetry and satellite gradiometry, Re = Ri + 400km S^(2) Kernel: gravimetry and satellite gradiometry, Re = Ri + 250km

6 5

3000 4 2000

3

1000

2

0

1

psi →

-1000

psi →

0 0°

30°

60°

90°

0°

30°

60°

90°

Fig. 24.6 (Above) fterr and (below) fsat for r ¼ Ri in case of the combination of gravimetry and a satellite-only model (r ¼ 6; 378 km and Re ¼ Ri þ 250 km, i.e. q ¼ 0:96228)

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field

203

Fig. 24.7 (Above) fterr and (below) fsat for r ¼ Ri in case of the combination of gravimetry and a satellite gradiometry (r ¼ 6; 378 km and Re ¼ Ri þ 250 km, i.e. q ¼ 0:96228)

24.11 Numerical Simulations and Final Comments The procedure discussed here was tested by means of simulated data. For W we took the EGM08 potential from Pavlis et al. (2008) and U was the potential of the Somigliana-Pizzetti normal gravity field with parameters of the GRS1980 system. f was simulated at more than The anomaly Dg 10,000,000 points of a “regular” grid on the sphere of radius Ri ¼ 6; 378 km given by the tenth level of the icosahedron refinement. The hierarchically created grids were also exploited for Romberg’s integration method in calculating the integrals in (24.49), (24.51) and (24.54), see Nesvadba et al. (2007).

Similarly, also the data t and G were simulated at more than 10,000,000 points of a “regular” grid, but on the sphere of radius Re ¼ Ri þ 250 km. The restrictions of T ¼ W U and @ 2 [email protected] 2 were used for this purpose. The case of gravimetry and a satellite-only model was treated first. The terrestrial term represented by (24.49) and the satellite term given by the first of (24.54) were computed for r ¼ Ri and Re ¼ Ri þ 250 km. They are shown in Fig. 24.6. Figure 24.7 concerns the combination of gravimetry and satellite gradiometry. The fterr term is given by (24.51) and fsat by the second of (24.54). The computation was done for r ¼ Ri and Re as above. In both the cases the composition (optimized solution) f ¼ fterr þ fsat was compared with T as obtained

204

directly from W and U, and restricted to r ¼ Ri . The results of the comparison are nearly the same in both the cases considered. Globally the RMS of the differences expressed in GeoPotential Units (1 GPU 1 m2 s2 ) does not exceed 0.06 GPU (max. difference smaller than 0.9 GPU) and excellently conforms to an apriori estimate of the error obtained from Romberg’s integration method. The results well confirm our earlier conclusions derived from numerical simulations based on the gravity field model EGM96, see Holota and Nesvadba (2006, 2007). In addition the dependence of the kernels Sð1Þ and Sð2Þ on the angle c, shown in Fig. 24.5, offers a suitable springboard for investigations on polar cap problem in using satellite data. Acknowledgements The work on this paper was done within ESA GOCE Project ID: 4311. The presentation of the paper at the IAG 2009 Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, August 31 to September 4, 2009, was sponsored by the Ministry of Education, Youth and Sports of the Czech Republic through Projects No. LC506. All this support is gratefully acknowledged.

References Holota P (2007) On the combination of terrestrial gravity data with satellite gradiometry and airborne gravimetry treated in terms of boundary-value problems. In: Tregoning P and

P. Holota and O. Nesvadba Rizos C (eds) Dynamic Planet. IAG Symp., Cairns, Australia, 22–26 Aug 2005. IAG Symposia, vol 130, Springer, Berlin etc., Chap 53, pp 362–369 Holota P and Nesvadba O (2006) Optimized solution and a numerical treatment of two-boundary problems in combining terrestrial and satellite data. Proc. 1st Intl. Symp. of the IGFS, 28 Aug-1 Sept, 2006, Istanbul, Turkey. Spec. Issue: 18, Genl. Command of Mapping, Ankara, 2007, pp 25–30 Holota P and Nesvadba O (2007) A regularized solution of boundary problems in combining terrestrial and satellite gravity field data [CD-ROM]. Proc. ‘The 3rd International GOCE User Workshop’, ESA-ESRIN Frascati, Italy, 6–8 Nov 2006 (ESA SP-627, Jan 2007), pp 121–126 Holota P and Nesvadba O (2009) Domain transformation, boundary problems and optimization concepts in the combination of terrestrial and satellite gravity field data. In: Sideris M (ed) Observing our changing earth, Proc. 2007 IAG Gen. Assembly, Perugia, Italy, July 2–13, 2007. IAG Symposia, vol 133, Springer, Berlin etc. pp 219–228 Migliaccio F, Raguzzoni M, Sanso` F (2004) Space-wise approach to satellite gravity field determination in the presence of coloured noise. J Geod 78:304–313 Nesvadba O, Holota P and Klees R (2007) A direct method and its numerical interpretation in the determination of the Earth’s gravity field from terrestrial data. In: Tregoning P and Rizos C (eds) Dynamic Planet. IAG Symp., Cairns, Australia, 22–26 Aug 2005. IAG Symposia, vol 130, Springer, Berlin etc., Chap. 54, pp 370–376 Pavlis NK, Holmes SA, Kenyon SC and Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008. Presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008 Sokolnikoff IS (1971) Tensor analysis, theory and applications to geometry and mechanics of continua. Nauka Publishers, Moscow (in Russian)

Moho Estimation Using GOCE Data: A Numerical Simulation

25

Mirko Reguzzoni and Daniele Sampietro

Abstract

The GOCE mission, exploiting for the first time the concept of satellite gradiometry, promises to estimate the Earth’s gravitational field from space with unprecedented accuracy and spatial resolution. Also inverse gravimetric problems can get benefit from GOCE observations. In this work the general problem of estimating the discontinuity surface between two layers of different density is investigated. A possible solution based on a local Fourier analysis and Wiener deconvolution of satellite data (such as gravitational potential and its second radial derivative) is proposed. Moreover a numerical method to combine in an efficient way gridded satellite data with sparse ground data, like gravity anomalies, has been implemented. Numerical simulations on different synthetic Moho profiles have been carried out. Finally a two-dimensional simulation on realistic data over the Alps has been set up. The results confirm that GOCE data can significantly contribute to the detection of geophysical structures, leading to a much better determination of the signal long wavelengths (up to about 200 km). The use of local ground data improves the satellite-only estimate, making possible the recovery of higher resolution details.

25.1

Introduction

The crust-mantle boundary, the so called Moho discontinuity, has been studied with profitable results by means of seismic profiles and ground gravity anomalies

M. Reguzzoni Department of Geophysics of the Lithosphere, OGS, c/o Politecnico di Milano, Polo Regionale di Como, Via Valleggio 11, 22100 Como, Italy e-mail: [email protected] D. Sampietro (*) DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio 11, 22100 Como, Italy e-mail: [email protected]

(e.g. Grad and Tiira 2009). However there are areas where, due to the lack of observations, the crustal structure has not been yet determined by geophysical sounding. In such areas the Moho surface can be estimated using isostatic theories, but especially young orogenic regions are not necessarily in isostatic equilibrium. Since satellite data are collected worldwide, they can be profitably used to overstep these problems and correct isostatic models even where ground observations are not available. Studies on this subject have been done using GRACE data (see e.g. Shin et al. 2007) or simulated GOCE gradiometric data (see e.g. Benedek and Papp 2009). In our solution, the gravitational potential and its second radial derivative at

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_25, # Springer-Verlag Berlin Heidelberg 2012

205

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M. Reguzzoni and D. Sampietro

satellite altitude, as observed from a GOCE like mission, are combined with point-wise observations at ground level (e.g. gravity anomalies) to model the Moho surface. This inverse gravimetric problem is in general ill-posed: in fact the solution is not unique (i.e. the space of all solutions corresponding to a fixed potential is infinite-dimensional) and not stable because the Newton operator is a smoother and its inverse is unbounded (Sanso´ 1980). In this work the non-uniqueness has been treated by taking peculiar hypotheses on the shape of the density discontinuity. In fact it has been proved that to guarantee the uniqueness of the solution it is possible to consider a two layer density discontinuity, in planar or spherical approximation, where the two different densities are known (see e.g. Sampietro and Sanso´ 2009 and the references therein). Note that for a lot of geophysical problems, included the one considered in this work, these kinds of approximations are commonly adopted and in general well satisfied (see e.g. Gangui 1998). In the first part of the paper the proposed algorithm is theoretically illustrated (Sect. 25.2) and numerically assessed (Sect. 25.3) on the basis of one-dimensional Moho profiles in planar approximation. Then the algorithm is generalized to the two-dimensional case and tested on a realistic simulation in the Alpine area (Sect. 25.4). The results obtained in both cases are presented in the paper, drawing conclusions and discussing future perspectives.

ð ð ð z0 þDðxÞ

TðxÞ ¼

Tz ðxÞ ¼

z0 hðxÞ

@ @z

ð ð ð z0 þDðxÞ z0 hðxÞ

Methodology

The analysis is developed accordingly to the inversion algorithm described in Reguzzoni and Sampietro 2008 and to the theory of partitioned (or stepwise) collocation solution (see Tscherning 1974). Let us recall here the main concepts. We consider a Cartesian coordinate system, where the (x, y) plane is at satellite level, x is oriented along track and z is pointing downward. The topography profile h(x) and the Moho profile D(x) are both defined with respect to the ground level z ¼ z0 ¼ 250 km. In order to solve the gravimetric inverse problem the following deterministic models have to be inverted (Heiskanen and Moritz 1967):

(25.1)

G r dxd dz qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 ðx xÞ þ þ ðz zÞ

z¼z0

(25.2)

Tzz ðxÞ ¼

@ @z2 2

ð ð ð z0 þDðxÞ z0 hðxÞ

G r dx d dz qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðx xÞ2 þ 2 þ ðz zÞ2

z¼0

(25.3) where T is the anomalous potential, G is the gravitational constant, r the density contrast and x, , z are integration variables in the x, y, z directions respectively (planar approximation). Modelling the Moho as DðxÞ ¼ D þ dDðxÞ, linearizing (25.1)–(25.3) around D in the z direction and finally integrating with respect to over a bounded interval, we get:

dTðxÞ ¼

dTz ðxÞ ¼

25.2

G r dx d dz qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðx xÞ2 þ 2 þ z2

dTzz ðxÞ ¼

ð þ1 1

ð þ1 1

ð þ1 1

kT ðx xÞ dDðxÞ dx

(25.4)

kTz ðx xÞ dDðxÞ dx

(25.5)

kTzz ðx xÞdDðxÞ dx

(25.6)

where dT(x), dTz(x) and dTzz(x) are the residual gravitational effects, i.e. the actual effect minus the effects due to the topography and to the mean Moho D, while kT, kTz and kTzz are convolution kernels (for details on the form of these kernels we refer again to Reguzzoni and Sampietro 2008). To estimate the Moho we have to solve the system obtained by inverting (25.4)–(25.6) degraded with the corresponding observation noise. As for the noise in (25.4) and (25.6), none of the two quantities is a direct observation of the GOCE

25

Moho Estimation Using GOCE Data: A Numerical Simulation

mission: in fact the potential T is derived from GPS tracking data, for example by applying the so called energy integral approach (Jekeli 1999; Visser et al. 2003), while the second radial derivatives Tzz are obtained by preprocessing the gradiometer observations taken in the instrumental reference frame (Cesare 2002; Pail 2005). However the resulting potential is known to have an almost white error, while the second radial derivatives have a time-correlated error with spectral characteristics almost identical to the original observations (Migliaccio et al. 2004a). As for the ground gravity anomalies Tz, a hypothesis of white noise is generally adopted. At this point a remark is due: the three different observations have complementary spectral characteristics and spatial distribution. In particular T and Tzz contain the low and medium frequencies of the signal because they are observed at satellite altitude; they are regularly sampled along the orbit and therefore are given on a regular grid when inverting a Moho profile; note that, in the two-dimensional case, T and Tzz can be anyway predicted on a grid (Migliaccio et al. 2007) thanks to the good spatial coverage of the GOCE data. On the other hand, Tz contains the highest frequencies of the gravimetric signal because it is observed at ground level; Tz observations are generally available at sparse points and cannot be reasonably gridded due to their typical dishomogeneous spatial distribution, e.g. in mountain areas. For these reasons it is important to combine, in an optimal way, the point-wise gravity data with the gridded satellite data. This can be achieved by means of collocation, treating dD as a random signal: dD^ ¼ CTy;dD C1 y;y y

(25.7)

207

the covariance matrix between a and b plus the error covariance matrix Cna ;nb . Since we consider the noise of the three observations uncorrelated, the error covariances are present only on the diagonal blocks of the matrix Cy,y. It can be noticed that all the needed covariance matrices, with the exception of the error covariance matrices, can be computed by propagating CdD,dD through the corresponding convolution kernels. In this way the resulting covariance matrices describe the different spectral contents of the observables. The solution of (25.7) can be obtained by using a partitioned method separating the gridded observations from the sparse ones:

dD^ ¼ mT

dTzz þ lT dTz dT

(25.8)

where: 2 6 m¼4

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T

31 02 B6 @4

7 5

3 CTzz ;dD

7 6 54

CT;dD

2 3T 2 C C 6 Tzz ;Tz 7 6 Tzz ;Tzz B 5 4 l ¼ @CTz ;Tz 4 CT;Tz CT;Tzz 2

6 B @CTz ;dD 4

CTzz ;Tz

3T 2 7 6 5 4

CT;Tz

CTzz ;Tz

7 C 5lA;

CT;Tz

0

0

3 1

2

CTzz ;T

31 2

311 C 6 Tzz ;Tz 7C 4 5A CT;Tz

7 5

CT;T

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T

31 2 6 4

7 5

31 CTzz ;dD

7C 5A:

CT;dD

The solution can be rearranged in the following way: T T dTzz T dTzz ^ þ l dTz m2 dD ¼ m 1 dT dT

(25.9)

with: where: 2

3

2

3

CTzz ;dD CTzz ;Tzz CTzz ;T CTzz ;Tz 7 7 6 6 Cy;dD ¼ 4 CT;dD 5; Cy;y ¼ 4 CT;Tzz CT;T CT;Tz 5; CTz ;dD CTz ;Tzz CTz ;T CTz ;Tz 3 2 dTzz 7 6 y ¼ 4 dT 5; dTz dD is the unknown Moho depth, y are the observations, Ca,b is the covariance matrix between a and b and Ca;b

2 m1 ¼ 4 2 m2 ¼ 4 0

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T 2

B l ¼ @CTz ;Tz 4

31 2 5

31 2 5

CTzz ;Tz CT;Tz

4

3T

4

CTzz ;dD CT;dD CTzz ;Tz

3 5;

3

5; CT;Tz 11 0

5 m C 2A

2

B @CTz ;dD 4

CTzz ;Tz CT;Tz

3T

1

5 m C 1 A:

The matrices m1 and m2 are the two heaviest terms to be computed, because there are in general much

208

more satellite data than ground observation points and, therefore, the covariance matrix of the satellite data is much larger than the one of the ground data. In other words, after computing m1 and m2, the determination of l is easy from the numerical point of view. Note that the first term of the sum in (25.9) is exactly the collocation estimate of the Moho when considering as observations T and Tzz only. Since both these observations are regularly sampled, and the Moho is estimated on the same gridded points, the matrix m1 can be efficiently computed in terms of Multiple Input Single Output (MISO) Wiener filter in the frequency domain (Papoulis 1977; Sideris 1996). The other onerous term, namely m2, can be seen as the collocation transfer operator from satellite to ground data. Naturally, if the ground data were regularly sampled too, this term could be also efficiently computed in the frequency domain. In reality, as we explained before, ground data are generally sparse and cannot be conveniently gridded. Our idea is first to compute the term m2 as if the ground data were available on a regular grid, thus exploiting the properties of the Fourier transform; this leads to a discrete convolution kernel, which is interpolated by splines to derive a continuous model. By applying a principle of stationarity (invariance by translations), this continuous kernel is shifted to each ground observation point and then resampled over the satellite data grid to get the corresponding row of the matrix m2. Of course this

M. Reguzzoni and D. Sampietro

matrix is not Toeplitz even if it is computed according to the stationarity principle, because ground data are not regularly distributed. Note that the harmonicity of Tz predicted by m2 is in general not guaranteed when splines are used to model the continuous kernel; however this is not critical here, because all covariances and cross-covariances are anyway consistent with one another through the Newton integral and the use of a unique covariance of dD.

25.3

Tests on Synthetic Models

The proposed methodology has been assessed on different synthetic models. The aim of the first model is to illustrate the complementary behaviour of the three kinds of observations used. The reference Moho is taken as a simple profile with non-zero mean, a linear trend and a spike modelled by a Gaussian function, see Fig. 25.1. The Moho covariance function and the corresponding spectrum, which are computed disregarding the trend in the data, have a Gaussian shape as shown in Fig. 25.2. Even if it is quite simplistic, this model can be thought as the Moho counterpart of a mountain loading the crust. The depth of the root extends from 32 to 38 km (we fixed D ¼ 30 km) and the density contrast between the root and the underlying mantle is 0.6 g/cm3. The simulation is performed on nearly 150

Fig. 25.1 Estimated Moho using only T, only Tzz or combining T and Tzz. Reference Moho in black solid line

25

Moho Estimation Using GOCE Data: A Numerical Simulation

209

Fig. 25.2 Error spectra using only T, only Tzz or combining T and Tzz. Moho signal spectrum in black solid line. To convert timefrequency into space-frequency divide by the mean satellite velocity of about 7 km/s

Fig. 25.3 Error spectra comparing a single-track solution versus a ten-track solution. Error spectrum using only gridded ground Tz. Moho signal spectrum in black solid line. To convert time-frequency into space-frequency divide by the mean satellite velocity of about 7 km/s

points, which are regularly spaced at a distance of 7 km, consistently with the sampling rate of the GOCE observations. Concerning the data noise along the satellite orbit, the potential is degraded by a white noise with a standard deviation of 0.3 m2/s2, while the second radial derivatives have a coloured noise with a spectrum

derived from a realistic simulation of the GOCE mission (Catastini et al. 2007). The result of the inversion is shown in Figs. 25.1 and 25.2. It can be seen that low frequencies (such as biases and trends) are completely lost when using Tzz observations, but are recovered from T observations. On the other hand, the high resolution details

210

M. Reguzzoni and D. Sampietro

Fig. 25.4 Estimated Moho using satellite T and Tzz or combining satellite data with sparse ground Tz observations. Reference Moho in black solid line

(the spike in our simulation) are caught by Tzz observations. Obviously, combining the two solutions it is possible to estimate both the low and the high frequencies of the signal. This analysis is confirmed by the comparison between the signal and error spectra shown in Fig. 25.2. Here we consider as “well reconstructed” those frequencies for which the error spectrum is lower than 20% of the signal spectrum. According to this threshold, it can be stated that the maximum resolvable resolution with GOCE data only is about 250 km. This result can be improved by considering more than one satellite orbit flying over the Moho profile. Assuming independent observations for each orbit and considering for example ten orbits (corresponding to a reasonable repeat period of two months for the GOCE mission), the maximum resolution becomes 200 km, see Fig. 25.3. Ground gravity data add information at higher frequencies: a numerical simulation shows that if also ground gravity anomalies were regularly sampled as the GOCE observations, then the corresponding error spectrum could be computed (see Fig. 25.3), making possible to reconstruct Moho details up to about 70 km. A second simulation scenario is set up in order to investigate the improvement in the result due to the combination of gridded satellite data and sparse ground gravity data. The algorithm described in Sect. 25.2 is used. A high resolution detail, i.e. a

smaller Gaussian spike, is added to the previously simulated Moho model and we assume to have few (40) gravity anomaly observations at random points at ground level concentrated over this new feature (see Fig. 25.4). A white noise with a standard deviation of 1 mGal is used for these observations. As one can see, these additional data locally improve the Moho estimation coming from satellite-only data; in particular the estimation error decreases from 0.36 to 0.16 km. On the other hand it has been tested that, if the ground data are far from the higher resolution detail, they have no impact on the final solution, thus showing the numerical stability of the method. Finally a test on a realistic Moho profile over the Alpine region has been studied. In this case the reference Moho has been computed starting from the actual topography and applying the Airy isostasy theory (see Watts 2001). The result ranges from 30 to 46 km and clearly shows the presence of the root due to the Alps (see Fig. 25.5). Since gravity anomaly observations are usually placed in flat areas, we simulate a set of 40 ground observations in the Po valley. Again we can notice that local ground data contribute to improve high frequencies: in fact where the point-wise ground observations are present the error r.m.s. significantly decreases (from 0.72 to 0.36 km), while where they are not available the satellite-only solution and the combined solution have practically the same smooth behaviour.

25

Moho Estimation Using GOCE Data: A Numerical Simulation

211

Fig. 25.5 Estimated Moho profile over the Alps using satellite T and Tzz or combining satellite data with sparse ground Tz observations. Reference Moho in black solid line

25.4

A Two-Dimensional Numerical Example in the Alpine Area

In the previous section, some synthetic Moho profiles have been used to test the performances of the proposed inversion algorithm. However the final goal is to estimate the Moho discontinuity surface and not only a profile along the satellite orbit. For this reason the method has been generalized to the two-dimensional case (Sampietro 2009) and has been tested on a realistic numerical example in a region of 10 7 in the centre of Europe (latitude between 44 and 51 North and longitude between 5 and 15 East). This area presents a complex topography characterized by the presence of the Alps. Again, the Airy isostasy theory is used to simulate a reference Moho over the region of interest. Starting from this surface, the gravitational potential and its second radial derivatives have been numerically computed at satellite altitude on a grid of 0.2 0.2 by using (25.1)–(25.3), see Table 25.1. Note that these simulated observables are realistic as for the Moho contribution, but do not represent the full gravity signal. In this experiment, the noise is derived by applying the so called space-wise approach (Migliaccio et al. 2004b; Reguzzoni and Tselfes 2009) to realistic alongtrack simulated GOCE observations (Catastini et al.

Table 25.1 Statistics on simulated signal and corresponding errors on a grid at satellite altitude T signal [m2/s2] T error [m2/s2] Tzz signal [mE] Tzz error [mE]

Min 0.57 0.19 12.62 1.70

Max 1.24 0.29 17.68 1.86

Mean 0.17 0.02 0.14 0.10

Std 0.43 0.10 7.57 0.70

2007), obtaining grids of potential and second radial derivatives as intermediate results. Some statistics of the error of these grids are reported in Table 25.1. Note that the GOCE signal generated by the simulated Moho over the selected Alpine area is about one order of magnitude larger than the grid error, especially in the case of the second radial derivatives. In order to apply a Fourier analysis, it is required that the error covariance matrices of the gridded data have a Toeplitz–Toeplitz structure (Grenander and Szeg€o 1958). This is not the case because the gridding is performed by least squares collocation in spherical approximation (Tscherning 2004). For this reason the error covariance matrix has been approximated by averaging covariances along diagonals for each block of the Toeplitz–Toeplitz structure, namely: Cout i;iþk ¼

N k 1 X Cin N k j¼1 j;jþk

i ¼ 1; 2; :::; N k k ¼ 0; 1; :::; N 1 (25.10)

212

where Cin is the original error covariance matrix, Cout is the approximated Toeplitz–Toeplitz error covariance matrix and N is the block dimension depending on the grid definition. The differences between the original and the approximated matrix are about one order of magnitude smaller than the original covariances, numerically justifying this approximation.

M. Reguzzoni and D. Sampietro

The estimated Moho with and without the aid of additional ground gravity anomalies is shown in Fig. 25.6. These ground observations are located in the centre of the study area, i.e. close to the Alps and along the valleys in order to make the simulation more realistic. Note that the region covered by satellite data is generally larger due to the smoothing effect of

Fig. 25.6 Estimated Moho surface over the Alps using satellite data only (top) or combining satellite data and ground gravity data (bottom)

25

Moho Estimation Using GOCE Data: A Numerical Simulation

the gravity field with increasing altitude. As expected, the satellite data fix the long wavelengths of the Moho, while the ground data improve the details, In particular, the error r.m.s. is reduced from 0.95 to 0.86 km in the area where ground data are available.

25.5

Conclusions and Perspectives

The inverse gravimetric problem of reconstructing the shape of the Moho using GOCE satellite observations has been studied, as well as the integration of additional ground data in a numerically efficient way. In this work we assume a simple two layer model (to guarantee the uniqueness of the solution) with constant densities of mantle and crust in planar approximation; under these hypotheses, positive results have been obtained. In particular the long wavelengths are well recovered thanks to the use of the gravitational potential, while higher resolution details come from gradiometric observations. Sparse gravity anomaly observations at ground level further improve the resolution of the final model. All in all, the estimated Moho over the Alpine area presents errors of less than 1 km for a grid of 0.2 0.2 resolution. However, it has to be stressed that the GOCE-only solution, without integrating any ground observations, is just 10% worse than the combined solution. This shows that reasonable Moho models can be computed at lower resolution also in areas where ground observations are not available, just using GOCE data. Beyond the positive results there are still open issues that need to be solved. First of all, the hypothesis of planar approximation has to be replaced by the spherical one in order to apply the inversion algorithm over larger areas. This implies that the whole theory, including filter design, has to be revisited. Furthermore, it seems interesting to proceed with the research by including additional geophysical information, such as geological models and bounds. This should contribute to get a more reliable solution from the physical point of view. Last but not least, the application of the method to real GOCE data, being the mission in its operational phase. However this requires to face the main problem of every inversion algorithm, that is how to disentangle the different gravimetric signals mixed up into the observed data. The results presented in this paper

213

are based on the assumption that the only source of the gravimetric signal was the Moho discontinuity. As a matter of fact modelling crustal dishomogeneities, as well as unwrapping the contributions of large deep features from those closer to the surface, is a problem that can be solved only with additional geological information. Acknowledgements The present research has been partially funded by the Italian Space Agency (ASI) through the GOCEITALY project.

References Benedek J, Papp G (2009) Geophysical inversion of on board satellite gradiometer data – a feasibility study in the ALPACA region, Central Europe. Acta Geodaetica et Geophysica Hungarica 44(2):179–190 Catastini G, Cesare S, De Sanctis S, Dumontel M, Parisch M, Sechi G (2007) Predictions of the GOCE in-flight performances with the End-to-End System Simulator. In: Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Italy, pp 9–16 Cesare S (2002) Performance requirements and budgets for the gradiometric mission, Technical Note, GOC-TN-AI-0027, Alenia Spazio, Turin, Italy Gangui AH (1998) A combined structural interpretation based on seismic data and 3-D gravity modeling in the Northern Puna, Eastern Cordillera, Argentina. Dissertation FU Berlin, Berliner Geowissenschaftliche Abhandlungen, Reihe B, Band 27, Berlin, Germany Grad M, Tiira T (2009) The Moho depth map of the European Plate. Geophys J Int 176(1):279–292 Grenander U, Szeg€o G (1958) Toeplitz forms and their applications. University of California Press, Berkeley, Los Angeles Heiskanen WA, Moritz H (1967) Physical geodesy. Springer, Wien, Austria Jekeli C (1999) The determination of gravitational potential differences from satellite to satellite tracking. Celestial Mech Dyn Astron 75:85–101 Migliaccio F, Reguzzoni M, Sanso´ F, Zatelli P (2004a) GOCE: dealing with large attitude variations in the conceptual structure of the space-wise approach. In: Proc. of the 2nd International GOCE User Workshop 8–10 March 2004, Frascati, Italy Migliaccio F, Reguzzoni M, Sanso´ F (2004b) Space-wise approach to satellite gravity field determination in the presence of coloured noise. J Geod 78(4–5):304–313 Migliaccio F, Reguzzoni M, Sanso´ F, Tselfes N (2007). On the use of gridded data to estimate potential coefficients. Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Rome, Italy, pp 311–318 Pail R (2005) A parametric study on the impact of satellite attitude errors on GOCE gravity field recovery. J Geod 79 (4–5):231–241 Papoulis A (1977) Signal analysis. McGraw-Hill, New York

214 Reguzzoni M, Sampietro D (2008) An inverse gravimetric problem with GOCE data. In: IAG Symposia “Gravity, Geoid and Earth Observation”, Mertikas SP (ed), vol 135, Springer, Berlin, pp 451–456 Reguzzoni M, Tselfes N (2009) Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis. J Geod 83(1):13–29 Sampietro D (2009) An inverse gravimetric problem with GOCE data. PhD Thesis, Doctorate in Geodesy and Geomatics, Politecnico di Milano, Italy Sampietro D, Sanso´ F (2009) Uniqueness theorems for inverse gravimetric problems. Proc. of the VII Hotine-Marussi Symposium, 6–10 June 2009, Rome, Italy (in press) Sanso´ F (1980) Internal collocation. Memorie dell’Accademia dei Lincei. vol XVI, N. 1 Shin YH, Xu H, Braitenberg C, Fang J, Wang Y (2007) Moho undulations beneath Tibet from GRACE-integrated gravity data. Geophys J Int 170(3):971–985

M. Reguzzoni and D. Sampietro Sideris MG (1996) On the use of heterogeneous noisy data in spectral gravity field modeling methods. J Geod 70 (8):470–479 Tscherning CC (1974) A FORTRAN IV Program for the Determination of the Anomalous Potential Using Stepwise Least Squares Collocation. Reports of the Department of Geodetic Science, No. 212, The Ohio State University, Columbus, Ohio Tscherning CC (2004) Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares Collocation using simulated data. In: IAG Symposia “A Window on the Future of Geodesy”, Sanso´ F (ed), vol 128, Springer, Berlin, pp 277–282 Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geod 77(3–4):207–216 Watts AB (2001) Isostasy and flexure of the lithosphere. Cambridge University Press, Cambridge

CHAMP, GRACE, GOCE Instruments and Beyond

26

P. Touboul, B. Foulon, B. Christophe, and J.P. Marque

Abstract

The electrostatic accelerometers of the CHAMP satellite as well as of the GRACE two ones have provided the necessary information to distinguish the satellite actual trajectories from the pure gravitational orbits. By providing the measurements of the satellite non-gravitational forces, one can distinguish the position or velocity fluctuations of the satellite due to the Earth gravity anomalies from those due to the drag fluctuations. In-orbit calibration and validation of onboard instruments, bandwidth, bias stability and resolution proof the success of the mission scientific geodesic return. The basic principle of these sensors stays on the servo-control of one solid mass, maintained motionless from the instrument highly stable structure. Care is paid for the mass motion detection, down to tenth of Angstrom, and to the fine measurement of the servo-controlled forces applied on the mass through electrostatic pressures. With the same concept and technologies, the GOCE inertial sensors have been designed, produced and tested to reach even better performances in order to deal with the milli-E€otv€os gradiometer objectives. The performance of the instrument and the interest of the obtained measurements do not only depend on the sensor accuracy itself but also on the onboard environment (magnetic, thermal, vibrational. . .), on the satellite attitude motions and on the in-orbit configuration and aliasing aspects. Future missions will have also to consider these aspects, especially when envisaging cryogenic electrostatic sensors which can exhibit better self accuracy or when considering satellite to satellite laser tracking.

26.1

Introduction

Space gradiometry missions have been already envisaged in the 1980s with different types of instruments demonstrating the interest of low altitude

P. Touboul (*) B. Foulon B. Christophe J.P. Marque ONERA - The French Aerospace Lab, F-91761 Palaiseau, France e-mail: [email protected]

satellites with drag compensation systems to eliminate the effects of the non-gravitational forces on the satellite orbital motion [1, 2]. To that aim the concept and the technology of a three axis linear electrostatic accelerometer have been developed in view of providing both the measurement of the satellite drag and the Earth gravity gradient effect [3]. From the experience acquired with the three flights of the ASTRE instrument, developed for the on-board shuttle microgravity environment survey [4], the

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_26, # Springer-Verlag Berlin Heidelberg 2012

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STAR ultra-sensitive space accelerometer has been designed, based on the accurate electrostatic levitation of a parallelepiped proof-mass [5]. This accelerometer has been integrated at the center of mass of the CHAMP satellite, first geodesic mission of the 2000 decade. For now quite 10 years, STAR performs the measurement of the non-gravitational accelerations of the satellite which perturb its low altitude trajectory at less than 500 km. The satellite orbit fluctuations are finely determined taking advantage of the on-board GPS receiver, aiming at the accurate recovery of the low spherical harmonics of the Earth’s gravity field. The concept of operation, the design and the technology of the accelerometer have been especially selected for space applications with the potential to optimize the definition for different mission performance and on-board accommodation requirements. So, the Super-STAR instrument has been designed in view of the two GRACE satellites with a reduced full scale range and a resolution ten times better leading to 1010 ms2 over 1 Hz bandwidth. The launch of the GRACE satellite in 2002 has allowed for the first time to compare the in orbit outputs of two accelerometers quite in same operational conditions: both spacecrafts are identical, flying on the same orbit with a 200 km distance from each other, corresponding to less than half a minute, the radiation pressure and the atmospheric drag being thus similar. The difference of velocities between the two satellites is provided with an accuracy of about 1 mm/s by a microwave Doppler device which links the two satellites and is finely analyzed in conjunction with the accelerometer outputs. This leads to the determination of the Earth’s gravity potential, every month and even more, allowing us to evaluate the secular and seasonal fluctuations of large mass distribution like the Antarctic and Greenland ices or the hydrologic basins of large rivers like Mississippi or Amazon [6]. Another way of improving the accuracy of the static Earth field with a much better geographical resolution is the accommodation of a space gradiometer on-board a drag-free satellite at an altitude lower than 300 km, like in the recently launched GOCE mission [7]. The gravity gradiometer is composed of several identical sensors mounted on a rigid and stable structure. In addition to the linear and angular acceleration of the satellite, it provides the fine measurements of the three diagonal components of the Earth’s gravity gradient tensor for the determination of the higher

P. Touboul et al.

harmonics of the potential. Taking advantage of the fine active thermal control of the instrument case and of the drag-free compensation system of the satellite, the full range of the sensors is limited to a few 106 ms2 to the benefit of the resolution of better than 2 1012 ms2Hz1/2 in the frequency measurement bandwidth, from 5 103 to 0.1 Hz. A resolution of a few milli-E€otv€os is thus expected at an altitude as low as 260 km. After the in-orbit switch on of the six accelerometers in April 2009, first gradiometric measurements have been recently obtained showing new space signatures of the Earth gravity field that will provide absolute reference for ocean circulation studies, geophysics, water or atmospheric changes [8]. With nine electrostatic space accelerometers, now in orbit on board four satellites at altitudes between 450 and 260 km, the performance of the concept, its robustness and flexibility to different functioning environment is clearly demonstrated to the benefit of the observation of the Earth gravity field, which is not only the static reference for many other disciplines but also the variation of mass distribution and transportation. Furthermore, improvements of the instrument can be expected as well as improvements of its accommodation on board future satellites and of involved space measurement techniques. So, beyond these three missions, other solid Earth missions are envisaged but also applications in fundamental physics or planetology.

26.2

The STAR and Super STAR Accelerometer for the CHAMP and GRACE Mission

The German CHAMP mission (CHAllenging Microsatellite Payload for geophysical research and application) was dedicated to Earth’s observation: global magnetic and gravity fields mapping. The satellite has been launched on July 15th, 2000 from the cosmodrome Plesetsk by a Russian COSMOS rocket at an altitude of 454 km in a circular orbit with a 87.3 inclination. CHAMP performed for the first time the combination of uninterrupted three dimensional high low tracking of its low orbit perturbations by the satellites of the GPS constellation and a high-precision three-axes measurement of the satellite surface forces: residual drag, solar and Earth radiation pressures and attitude manoeuvre thrusts are measured by the STAR

26

CHAMP, GRACE, GOCE Instruments and Beyond

217

(Space Three-axis Accelerometer for Research) accelerometer integrated at the centre of mass of the satellite. A by-product of the accelerometer measurements is the determination of the atmospheric density variations during the decade of the mission. STAR is a six-axis accelerometer providing the three linear accelerations along the instrument sensitive axes and the three angular accelerations about these axes (see Fig. 26.1). STAR presents a measurement range of 104 ms2 and exhibits a resolution of better than 3 109 ms2 for the y and z axes and 3 108 ms2 for the x axis within the measurement bandwidth from 104 to 101 Hz. The measurements are integrated over 1 s before delivery to the satellite data bus. The configuration of the instrument is compatible with ground tests which demand specific characteristics of the less accurate x-axis for the operation under 1 g gravity field. For example, measurements of residual low frequency vibrations on a specific testing pendulum platform have been performed along the y and z axes of the accelerometer controlled horizontal with a resolution of 108 ms2 Hz1/2 at frequencies lower than 0.1 Hz corresponding to the requirement of 3 109 ms2 rms in the measurement bandwidth. The accelerometer low level of bias (less than 105 ms2) of the same y and z sensitive axes has also been verified in free fall in the Bremen drop tower [9]. The following GRACE mission put in evidence the temporal variability of the Earth’s gravity field. In

addition to the High-Low GPS satellite tracking and to the accelerometer surface force measurement, the GRACE mission is based on low–low K-band tracking between two identical satellites separated by about 220 km on the same quasi circular orbit. The twin GRACE satellites were launched on March 17th 2002 from Plesetsk by Rockot launch vehicle at a initial altitude of 500 km on a quite polar orbit (89.0 inclination). The configuration of the two SuperSTAR accelerometers is quasi identical to STAR and takes advantage of the CHAMP mission experience. The concept of electrostatic servo-controlled accelerometer is well suited for space applications: the electrostatic forces give the possibility to generate very weak but accurate accelerations while the capacitive sensing offers a high position resolution with negligible backaction. The accelerometer proof-mass is fully suspended with six servo-control loops acting along its six degrees of freedom, suppressing any mechanical contact to the benefit of the resolution and yielding to a six-axis accelerometer. The internal cage of the accelerometer, constituted by three electrode-plates made of silica glass, surrounds the solid parallelepiped proof-mass, of 4 4 1 cm sides and 72 g weight, made in Chromium coated Titanium alloy (see Fig. 26.2). The accelerometer performance depends mainly on the geometrical accuracy and the stability of the core mechanical assembly. In operation, the mass is maintained motionless at the centre of the cage with

Fig. 26.1 STAR sensor unit before integration in the CHAMP satellite

Fig. 26.2 GRACE accelerometer mechanical core

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stability up to a few 1012 m/Hz1/2. The fine measurements of the applied voltages on the electrodes provide the accelerometer outputs. Independent analyses performed on the in orbit data confirm that the instrument specifications are achieved for both GRACE accelerometers in particular, along the sensitive axes Y and Z, the bias being less than 2 106 ms2 [10] and the noise PSD: S(f) ¼ 1020 (1 + 5 103/f) m2s4/Hz [11]. In fact, the instrument is so sensitive that the on board environment has to be precisely decorrelated from the self behavior of the instrument to exploit the sensor limit of performance.

26.3

From GRACE Accelerometer to GOCE Gradiometer

The aim of ESA’s Gravity Field and Steady-State Ocean Circulation Explorer mission is also to generate accurate global mapping of the Earth’s gravity field for oceanographic, geophysical, hydrological or climatologic applications. Compare with previously launched missions, GOCE seeks particularly to better understand the highest spherical harmonics of the gravity field model. Launched on March 17th 2009 from Plesetsk, the GOCE mission exploits for the first time a three-axis gradiometer consisting of six electrostatic accelerometers offering an outstanding resolution of 2 1012 ms2Hz1/2. The accelerometers were successfully activated on April 6th 2009. The satellite is on a sun-synchronous, quasi circular and quasi polar (96.5 ) orbit at an altitude near 260 km. Weighting 1 ton with a length of 5 m, the spacecraft has a very rigid structure without moving parts and exhibiting a limited cross section (1 m diameter) in order to reduce its drag. Its drag compensation system counteracts all non- gravitational forces acting on the spacecraft along its velocity vector through two ion thrusters. The electrostatic gravity gradiometer is composed of three pairs of accelerometer sensor head including the Platinum test masses, on one Carbon–Carbon ultra stable structure, three front end electronics units for the masses sensing and actuations, one gradiometer interface unit for the experiment control and the mass control laws, contained in three accurately thermal controlled stages. The outputs delivered by each pair of accelerometers are combined to provide the gravity

gradient along the gradiometer axis: this is the main science data, to be separated from residual angular accelerations of the satellite. This angular acceleration about the three axes, and by integration the angular rate, can be deduced from the transverse axes of the accelerometers, symmetrically distributed around the gradiometer centre which is also the satellite centre of inertia. In addition the common measured acceleration provides the external forces applied on the satellite to be cancelled along the velocity vector by the drag compensation system. The GOCE sensors are similar to the CHAMP and GRACE ones except the very dense proof-mass made in PtRh10 alloy, the eight pairs of electrodes instead of six for redundancy, plus the digital control loops instead of the previous analogue circuits. The main characteristics of the three accelerometers are compared in Table 26.1. On April 6th 2009, the six accelerometers were switched on and the proof-masses were immediately all levitated at the centre of their cage. The next day, the six accelerometers passed in Science Mode with a more limited full range and better resolution. On May 29th 2009, the GOCE satellite was successfully commanded in Drag-Free mode demonstrating the performing association of the accelerometer measurements and the electric ion propulsion system [12]. The residual along-track external acceleration (red spectrum on Fig. 26.3) is around 109 ms2Hz1/2 in the measurement bandwidth, more than one order of magnitude less than the specified value. Estimation of the accelerometer noise can be deduced from redundant sensor measurement (blue spectrum of Fig. 26.3) and is perfectly in line with the predicted noise computed from on ground tests and evaluations (black curve): the coherence of the two curves between 1 and 5 Hz, where the position sensing source is the main contributor confirms the right operation of the capacitive sensors; similarly, the coherence of the curves between 10 and 100 mHz where the electrostatic actuation is the main contributor, confirms the right operation of the electrode voltage driving amplifiers. This result, associated to the coherence of the data provided by the 36 channels of the six accelerometers is a very good clue for the success of the future data processing of the mission. The very low satellite altitude, facilitated by the Sun’s exceptionally weak activity and associated in consequence to a longer

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219

Table 26.1 Comparison between the mains characteristics of the CHAMP, GRACE and GOCE accelerometers Mission PM material PM mass Gap X Gap Y,Z Vd Vp Ge Acceleration sensitivity Y,Z Position sensitivity Y,Z Measurement range Y,Z Specified resolution Y,Z Specified freq. bandwidth

CHAMP TA6V 0.072 60 75 5 20 1.8 104 0.4 104 6 years) with high spatial resolution (comparable to that provided by GOCE) and high temporal resolution (weekly or better, so to reduce the level of aliasing of the high frequency phenomena found in the time series of the Earth’s gravity field variation provided by GRACE), and it must improve the separability of the observed geophysical signals (Koop and Rummel 2007). In one of the preparatory studies carried out for ESA by Thales Alenia Space I (TAS-I) (Laser Doppler 2005), a resolution of

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0.1 mm/year of the geoid height variation rate at spherical harmonic degree ¼ 200 (or ~100 km spatial resolution) was preliminarily identified satisfying the most stringent geophysical performance requirement.

27.2

FD2

D2 g2

Satellite 1

FD1

D1

Δd = ΔdG + ΔdD

g1

Earth

Measurement Techniques and Mission Scenario

Fig. 27.1 Principle of the LL-SST technique

The monitoring of the temporal variations of the Earth’s gravity field over a long period of time requires flying at an altitude higher than that of GOCE, in order to compensate the drag forces with an affordable amount of propellant. Since the gravitational potential U of degree rapidly decreases with the orbit radius r, i.e. U‘ / r ð‘þ1Þ ;

d

ZJ2000 S2

ZO 1

XO 1 YO 1

S1

(27.1)

it would be necessary to increase proportionally the gradiometer baseline (and/or the sensitivity of the accelerometers) in order to maintain the same signalto-noise ratio (see Cesare et al. 2009). Considering also that the signal of the temporal variations of the gravity field is at least one order of magnitude weaker than that of the static geopotential, for altitudes above ~300 km at the moment the most consolidated technique to measure the Earth’s gravity field is the “LowLow Satellite-to-Satellite Tracking” (LL-SST), which exploits the satellites themselves as the “proof masses” immersed in the Earth’s gravity field, as in the GRACE mission. Assuming a loose formation of two satellites, the distance variation between their centres of mass (Dd, produced by both gravitational and non-gravitational forces together) is gauged via length metrology. The drag accelerations (D1, D2) experienced by the two satellites are separately measured by means of accelerometers. From of Dd€D ¼ D1 D2 , the non-gravitational component RR the distance variation is obtained ðDdD ¼ Dd€D Þ and subtracted from Dd to isolate the component (DdG) produced by Earth’s gravity field only (see Fig. 27.1). Thanks to the separation between the satellites, this “measurement instrument”, which can be regarded as a kind of one-dimensional gradiometer with a very long baseline, has a higher sensitivity for the phenomena being targeted than a gradiometer embarked on a single satellite.

sun

i

γ = 90°

XJ2000

line of nodes

YJ2000

orbit

Fig. 27.2 Basic mission scenario

The simplest scenario implementing LL-SST consists of a loose formation of two co-orbiting satellites. In particular, the main parameters of the mission scenario defined in Laser Doppler study (2005) are (see Fig. 27.2): • Circular orbit with mean altitude of 325 km • Orbit inclination is 96.78 (sun-synchronous) • Local time of the ascending node: 6 am (dawn-dusk orbit) or 6 pm (dusk-dawn orbit) • Inter-satellite distance d ¼ 10 km These parameters and the requirements on the fundamental observables utilized for the determination of the Earth’s gravity field (Dd and Dd€D , as shown in Fig. 27.3), have been so chosen to fulfill the requirements on the geoid variation rate (0.1 mm/ year) at 100 km spatial resolution. They have been derived using analytical models and numerical simulations of the gravity field determination through the LL-SST technique, with the support of geodesy institutes (Laser Doppler 2005; Cesare et al. 2006).

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The Future of the Satellite Gravimetry After the GOCE Mission

Fig. 27.3 Requirements on Dd€D (above) and inter-satellite (or “relative”) Dd (below) measurement error spectral density. These requirements are strictly applicable to the frequency

Although this basic mission scenario still does not meet all the objectives of a NGGM, in particular with respect to the improvement of the temporal resolution and the separability of the geophysical signals, it can provide the “building block” for the realization of more complex scenarios that can potentially fulfill these objectives. For instance, a dense and uniform coverage of the Earth surface that allows generating gravity field solutions in less than 1 week can be obtained by flying two pairs of co-orbiting satellites

225

range [1100 mHz], the identified measurement bandwidth (MBW) of the NGGM

in two circular orbits with altitude at 312 km and with 90 and 62.7 inclination respectively, as in Bender et al. (2008).

27.2.1 Laser Metrology System The metrology system designed for the NGGM (Laser Doppler 2005; Laser Interferometry 2008) is based on a Michelson-type heterodyne laser interferometer (see

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Fig. 27.4 Functional scheme of the laser metrology

Fig. 27.4). A measurement laser beam produced by a frequency-stabilized Nd:YAG source (wavelength ¼ 1,064 nm, output optical power ¼ 750 mW) is emitted by satellite 1 (follower) towards satellite 2 (leader). Here it is retro-reflected by a corner-cube optical system towards satellite 1, where it is superimposed to a

reference laser beam generated by the same source but frequency shifted relatively to the measurement beam. The measurement beam is amplitude modulated in on-off mode with a period corresponding to the roundtrip time to avoid the occurrence of spurious signals and non-linearity caused by the unbalance between the

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The Future of the Satellite Gravimetry After the GOCE Mission

optical power of the strong reference beam and of the weak returning measurement beam. The distance variation is obtained from the phase variation of the interference signal formed by the measurement and the reference laser beams. A breadboard of such an interferometer has been realized (as in Fig. 27.5) and subject to functional and performance tests up to ~90 m distance. The test results shown in Fig. 27.6 prove that the requirement (as depicted in Fig. 27.3) can be met in the MBW. The laser metrology system is completed by: • An Angle Metrology for measuring the angles of the two satellites relative to the laser beam

Fig. 27.5 Breadboard of the laser interferometer Fig. 27.6 Results of the performance test (distance measurement error spectral density)

227

• A Beam Steering Mechanism (BSM), i.e. a device in charge of the acquisition and maintenance of the optical link between the satellites and of the laser beam fine pointing during the measurement phase • A Lateral Displacement Metrology for measuring the lateral displacement of the satellite 2 relative to the beam sent by the satellite 1 and driving in closed loop the BSM The Angle Metrology is merged with the Lateral Displacement Metrology and is realized by three small telescopes focusing the collected light on three Position Sensing Detectors (PSDs). Each PSD measures the position and the energy of the laser beam spot focused by the optics on the detector plane. The orientation and the lateral shift of satellite 2 relative to the laser beam are derived from the spot position and from the optical power measured by the three PSDs respectively. Breadboards of the Angle and Lateral Displacement Metrology and of the BSM have been realized (Fig. 27.7) and tested by TAS-I (Laser Doppler 2005; System Support 2008), including a test of the whole laser beam pointing system (BSM driven in closedloop by the Lateral Metrology). The elements of the laser metrology system described above are arranged on an optical bench, shared with the accelerometers utilized for the drag accelerations measurement (as shown in Fig. 27.8).

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satellite 2 can take its role, after the two satellites exchange positions.

27.2.2 Control Requirements

Fig. 27.7 Breadboard of the BSM

Angle Metrology

Beam Steering Mechanism Retro Reflectors

Interferometer core Accelerometers Angle/Lateral Metrology

Fig. 27.8 Configuration of the optical bench installed on each satellite

Two accelerometers, which can be similar to those of GOCE, are used on each satellite, arranged symmetrically about the centre of mass (COM). So doing, it is possible to measure the drag acceleration of the satellite COM, leaving the space in proximity of the COM free for the accommodation of the laser retro-reflector. The same optical bench is replicated on both satellites even if the laser is emitted only by satellite 1. Then in case of failure of the interferometer on satellite 1,

A complete assessment of the control requirements was done in Laser Doppler (2005), and revised and updated along the following studies: the control boundaries were conceived for the baseline configuration of two satellites chasing each other in a drag-free environment and flying in loose formation flying. The modeled metrology is the consolidated LL-SST technique, enhanced by the BSM. More in detail, in order to operate the accelerometers at the performance level required for the NGGM, each satellite must be endowed with a drag-free control system to reduce the level of the non-gravitational 2 accelerations below 106m/s and below a spectral 2 pﬃﬃﬃﬃﬃﬃ 8 density level of 10 m s Hz between 1 and 10 mHz, along each axis. The drag compensation is realized using ion thrusters in the same way as in GOCE. The formation control is in charge of keeping satellite 2 in the “control box” established by the working range of the optical metrology system: DdX ¼ 500 m (with respect to the nominal distance, d ¼ 10 km); DdY, DdZ ¼ 50 m along the Y, Z axes of the satellite 1 Local Orbital Reference Frame. The challenge of the formation control design for this mission consists in keeping the relative motion within these boundaries without interfering with the scientific measurements, which require that the satellites must be “free” to move under the effect of the gravity field over time scales of 1,000 s, and also without spoiling the drag-free environment (formation control accelerations must fulfil the drag-compensation requirements too), while minimizing thrusters use in terms of dynamic range, propellant consumption. Control architecture and recommended actuator technology is elaborated in Cesare et al. (2009).

27.3

Parallel Studies

27.3.1 NGGM All the illustrated studies have constituted a solid background for the preparation of a new pre-Phase A study, so called “Assessment of a Next Generation

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The Future of the Satellite Gravimetry After the GOCE Mission

Gravity Mission to Monitor the Variations of the Earth’s Gravity Field” (ESA ITT 1-5914/09/NL/CT). Thus, the ESA Future Missions Division has started two parallel studies led by two competitive consortia (Astrium D and ThalesAlenia Space I) in the second quarter of 2009, with the main objective of establishing mission architecture aimed at the optimal recovery of the Earth variable gravity field, through satellite-to-satellite tracking observation techniques.

27.3.2 Mass Transport Study The “Mass transport study” (Monitoring and Modelling Individual 2008) aimed at monitoring and modeling individual sources of mass distribution and transport in the Earth mass model (11 years, 0.5 0.5) was compiled including mass fields stemming from atmosphere, oceans, continental water, ice (ice sheets and glaciers) and the solid Earth. This model was used to optimize the ability to observe changes in mass over the continents and oceans. Detailed and realistic close-loop simulations were performed to study separability and aliasing characteristics inherent in different satellite mission concepts. One of the conclusions was that temporal aliasing is intrinsic to observing gravity field changes by satellites, but e.g. leads to relatively smaller distortions for hydrology than for oceanography. Furthermore, the combination of LL-SST and orbit observables does not allow the precise determination of the spherical harmonic degree 1 terms or geo-centre variations. Finally, flying more than one pair of LLSST satellites can significantly reduce the gravity field retrieval errors.

27.4

Parallel Technology Developments

27.4.1 Atomic Clocks (ESA studies) The European Space Agency is currently carrying on two parallel activities on atomic clocks. The ACES (Atomic Clock Ensemble in Space) mission on the International Space Station fosters research on coldatom sensor technology for space, in view of applications in different domains, e.g. fundamental physics tests (violation of special relativity, search for drifts of fundamental constants) and relativistic

229

geodesy (till 10 cm accuracy) with a time and frequency metrology at 1016 stability and accuracy level. ACES will be ready for launch in 2013. Secondarily, a Pre-phase A study of an atomic clock ensemble in space based on the optical transitions of strontium and ytterbium atoms has started in 2006. The so-called SOC (Space Optical Clocks, ESA-AO-2004-100) will take advantage of the ACES heritage and will push stability and accuracy of atomic frequency standards down to the 1018 regime.

27.4.2 Atom Interferometers The European Space Agency has been funding a number of technology development activities and studies in the area of atom interferometry in view of the everincreasing requirements for future inertial satellite platforms. Recently, a study has been initiated to explore the “Applications and Implementations of Atom-Based Inertial Quantum Sensors” in future space missions, in the fields of Space Science, Earth Observation, Fundamental Physics, Microgravity, and Navigation. As part of this ongoing study, the fundamental limits of atom interferometry were translated into the expected performance of a realistic gravity gradiometer payload for a future gravity mission. Gravity gradients can be measured by comparing the time evolution of two clouds of atoms (106 to 108 or more atoms), suitably cooled and prepared by means of lasers as well as other (time varying) electromagnetic fields, and separated by a given baseline. The time evolution of each cloud is measured by sequentially inducing suitable atomic transitions by means of lasers. This constitutes the atom interferometric measurement (see Peters et al. 1999). Comparison of the two atomic clouds is also achieved optically by ensuring the optical phase locking of the interferometric measurements effected on the two clouds. A major advantage compared to a macroscopic gravity gradiometer, such as the one onboard GOCE, is that the absolute local value of gravity is measured for each atomic cloud. Therefore, the atom interferometry gravity gradiometer does not suffer from drift but exhibits constant performance across the measurement frequency band. The acceleration sensitivity of an atom interferometer increases, amongst others, with the square of the interrogation time. This time is effectively limited on

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the ground to the achievable free fall time of an atomic cloud, which amounts to a few 100 ms in a typical vacuum chamber. By contrast, space operation, characterized by the intrinsic microgravity environment, would enable interrogation times of several tens of seconds leading to a sensitivity improvement of four orders of magnitude. Assuming a baseline of 1 m, this would amount to sensitivities of the order of 1 mE/Hz in space. This is one order of magnitude better than the GOCE specifications for measurements at frequencies above ~1 mHz, and, thanks to the absence of drift, well below, by several orders of magnitude, for frequencies below 1 mHz. The above estimate, however, does not yet represent a fundamental limit. Performance also increases with the magnitude of the probing laser wave vector and the signal-tonoise ratio, which is a function of the number of atoms in the cloud. Both these quantities can be effectively increased using quantum entanglement techniques. Although these still need to be demonstrated, the potential sensitivity for the same 1 m baseline atom interferometer gravity gradiometer would then be in the order of 1mE/Hz. In order to become suitable for future space mission, atom interferometry technology requires further development in the area of miniaturization and integration of both optical and laser components, as well as vacuum chambers. Furthermore, space worthiness also needs testing. These developments and investigations are currently underway in a number of European institutes (see Schmidt et al. 2009; Stern et al. 2009).

27.5

Conclusions

Since 2003 a number of studies have been initiated by ESA focusing on various technological, scientific and mission aspects of a potential future gravity field mission. It is important that the technology, not only related to the payload but also to satellite and/or constellation control, reaches the right readiness level. In addition, tools need to be developed to allow full mission performance analysis. Besides LL-SST other interesting technologies are under development but at different levels of maturity. The road towards availability of these techniques for future missions is still long and full of challenges.

Acknowledgements The authors gratefully acknowledge Prof. Achim Peters, Head, and Malte Schmidt, of the Humboldt University of Berlin, for their expert advice on atom interferometry.

References Bender PL, Wiese DN, and Nerem RS (2008) “A Possible DualGRACE Mission with 90 Degree and 63 Degree Inclination Orbits”, In: Proceedings of the 3 rd International Symposium on Formation Flying, Missions and Technologies, Noordwijk (NL) Cesare S, Sechi G, Bonino L, Sabadini R, Marotta AM, Migliaccio F, Reguzzoni M, Sanso` F, Milani A, Pisani M, Leone B, Silvestrin P (2006) “Satellite-to-satellite laser tracking mission for gravity field measurement”. In: Proceedings of the 1st International Symposium of IGFS, 28 Aug–1 Sept. 2006, Istanbul, Turkey. Harita Dergisi, Special Issue 18, pp 205–210 Cesare S, Allasio A, Aguirre M, Leone B, Massotti L, Muzi D, and Silvestrin P (2009) The Measurement of Earth’s Gravity Field after the GOCE Mission. In: Proceedings of 60th International Astronautical Congress, Daejeon, Korea, 2009, IAC-09.B1.2.7 Enabling Observation Techniques for Future Solid Earth Missions (2004) ESA Contract No: 16668/02/NL/MM, Final Report, Issue 2, 15 July 2004 Koop R, Rummel R (2007) The Future of Satellite Gravimetry, Final Report of the Future Gravity Mission Workshop, 12–13 April 2007 ESA/ESTEC, Noordwiik, Netherlands Laser Doppler Interferometry Mission for determination of the Earth’s Gravity Field (2005), ESA Contract 18456/04/NL/ CP, Final Report, Issue 1, 19 December 2005 Laser Interferometry High Precision Tracking for LEO (2008), ESA Contract No. 0512/06/NL/IA, Final Report, July 2008 Monitoring and Modelling Individual Sources of Mass Distribution and Transport in the Earth System by Means of Satellites (2008), ESA Contract No. 20406/06/NL/HE, Final report, November 2008 Peters A, Chung KY, Chu S (1999) Measurement of gravitational acceleration by dropping atoms. Nature 400:849–852 Schmidt M, Senger A, Gorkhover T, Grede S, Kovalchuk E, and Peters A (2009) “A Mobile Atom Interferometer for High Precision Measurements of Local Gravity”, In: Frequency Standards and Metrology – Proceedings of the 7th Symposium pp 511–516 Stern G, Battelier B, Geiger R, Varoquaux G, Villing A, Moron F, Carraz O, Zahzam N, Bidel Y, Chaibi W, Pereira Dos Santos F, Bresson A, Landragin A, Bouyer P (2009) Lightpulse atom interferometry in microgravity. Eur Phys J D 53:353–357 System Support to Laser Interferometry Tracking Technology Development for Gravity Field Monitoring (2008), ESA Contract No. 20846/07/NL/FF, Final report, September 2008

Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

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Abstract

Modern satellite gravity field recovery missions use accelerometric, intersatellite tracking or gradiometric observables for deducing gravity field related data. In this study an alternative observable type for gravity field recovery, the relativistic frequency shift, is investigated. As Einstein stated in his general theory of relativity, gravity can be considered as attribute of space-time. In this view mass alters the geometric shape of the metric tensor. Moreover mass, respectively gravity, has effects on electromagnetic wave propagation [Einstein (Annalen der Physik 35:898–908 1911)]. Although these relativistic effects are quite small and difficult to measure, with upcoming atomic clocks which have sufficient accuracy and short-term stability it will be possible to derive meaningful gravity related information. Since relativistic effects are used this method is called Post-Newtonian method. The main target of this paper is to demonstrate the validity of the derived relativistic equations. The scientific quality of the relativistic frequency shift observed by means of highly accurate atomic clocks is investigated. In our basic scenario a low earth orbit (LEO) sends an electromagnetic wave to a receiver. The reference station determines the frequency shift of the signal, which is connected to the time dilatation between the atomic clock of the satellite and an identical atomic clock nearby the receiver. A simplified, mathematical model for numerical simulations of this configuration is presented. The effect of different error sources are investigated by numerical closed-loop simulations. Thus, the performance requirements of atomic clocks, position and velocity determination and limiting factors for deducing earth’s gravity field can be derived.

28.1

R. Mayrhofer (*) R. Pail Institute of Navigation and Satellite Geodesy, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria e-mail: [email protected]

Introduction

In this study the principle of the Post-Newtonian method for earth gravity field determination is presented. The observable for gravity field reconstruction is the frequency shift of an electromagnetic signal transmitted from a satellite to a receiver station. This frequency shift is caused by relativistic time dilatation

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which is related to the gravity potential. By numerical simulations the spatial and spectral performance of a LEO satellite mission equipped with atomic clocks, which will be available within the next tow decades, is investigated. There have been done some studies which are related to future satellite gravity field missions and general relativistic effects. There is for example the Einstein Gravity Explorer mission proposal (Schiller et al. 2009) in which testing methods of relativistic effects and physical constants based on atomic clock measurements are investigated. M€ uller et al. (2007) explored the impact of relativity on various geodetic topics like the geoid, reference systems, geodynamics, Global Positioning System (GPS), Satellite Laser Ranging (SLR), and Very Long Base Interferometry (VLBI). Gulkett (2003) investigated relativistic effects on GPS and LEO and showed how to implement them correctly within a relativistic framework. The IAU already included relativistic effects for frame transformations (Soffel et al. 2003). For being able to realise a satellite mission as proposed in this paper, atomic clocks with sufficient quality will be needed. Actual atomic clocks achieve a short term stability of 1016 s (between two measurement epochs) on earth (Schiller 2007) and are expected to achieve 1018 s within the next 15 years. According to Cacciapuoti (2006), actual space-borne atomic clocks achieve an accuracy of 1015 s. Thus, a satellite mission with an atomic clock with 1018 s stability should be possible within the next 30 years. In this study, the equations for the Post-Newtonian method are derived in Chaps. 28.2 and 28.3. In Chap. 28.4 the simulation setup is described in more detail. Chapter 28.5 shows the simulation results, a performance analysis, and the analysis of the spectral and spatial error behaviour. Finally a conclusion and outlook is given in Chaps. 28.6 and 28.7.

28.2

Relativistic Time Dilatation

The metric tensor gmn describes the curvature of spacetime. Thus, it can be used for deducing a description of the relativistic frequency shift. The line element ds can be described by ds2 ¼ gmn ðxÞdxm dxn

(28.1)

Here Einstein’s tensor convention has been applied (Einstein 1916). Double upper and lower indices describe a summation. The gradient dxk ¼ ½c dt dx1 dx2 dx3 T of the scalar vector field x ¼ ðxk Þ contains position and time information. c is the speed of light in vacuum, dt the time element of a chosen time system and dxi describe the three dimensional coordinate elements. The metric tensor gmn ðxÞ is a function of xk ðm; n; k ¼ ½0; 1; 2; 3Þ, which means that the curvature of space-time depends on time and position. A solution of Einstein’s field equations delivers the elements of the metric tensor. A series expansion representation of the tensor elements is (Soffel et al. 2000) 2 FðxÞ 2 F2 ðxÞ þ Oðc5 Þ c2 c4 4 Fi ðxÞ g0i ¼ þ Oðc5 Þ c3 2 FðxÞ 2 F2 ðxÞ þ Oðc6 Þ gij ¼ dij 1 þ c2 c4 (28.2)

g00 ¼ 1 þ

where FðxÞ is the gravity potential and i; j ¼ ½1; 2; 3. The earth’s static gravity field potential is usually expressed by a spherical harmonic series expansion (Heiskanen and Moritz 1967): FE ðr; y; lÞ ¼

1 lþ1 GM X R R l¼0 r

l X

ðClm cos ml þ Slm sin mlÞ Plm ðcos yÞ

m¼0

(28.3) Here r, y and l are spherical coordinates, R is the earth’s reference radius, GM the gravitational constant times mass of the earth, l and m are the degree and order of the fully normalized spherical coefficients Clm ; Slm , and Plm ðcos yÞ represents the fully normalized Legendre function. Equation (28.3) describes the functional model for setting up the design matrix for least squares adjustment in our simulation environment. The spherical coefficients Clm ; Slm are the system parameters, while the gravity potential F, which is derived from relativistic frequency shifts is the observable of our system.

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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

The local line element dsclock of a clock moving within a gravity field affects the displayed time dtclock and frequency uclock of the clock. dtclock ¼

1 uclock

¼

dsclock 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ gmn ðxÞdxm dxn (28.4) c c

x ¼ ðxk Þ and dxk are related to the coordinates of the clock. The coordinate elements of a moving clock can be described by dxk ¼ ½ c dt

v1 dt

v2 dt

v3 dt

(28.5)

Here, vi ¼ vi ðxÞ denotes the velocity of the clock and is related to its coordinates, too. Merging (28.2), (28.4) and (28.5) and omitting elements smaller then c3 leads to a description of the inherent time of a moving body: 2 FðxÞ v2 ðxÞ dt2 dt2 ¼ 1 þ þ c2 c2

(28.6)

v ðxÞ is the local scalar velocity of the clock. The asterisk ‘*’ is used for underlining that v is scalar and preventing to mix it up with the velocities from (28.5). dt represents a virtual time of an non-moving, gravityfree (inertial) body located at infinite distance.

28.3

Functional Model

A LEO satellite transmits an electromagnetic signal via microwave link to a receiver station. This receiver station could be a geostationary satellite or a reference station located on earth’s surface. By comparing the local frequency of the transmitted signal with the local frequency of the received signal the time dilation between receiver and transmitter is defined. By using (28.6) and again omitting elements smaller then c3 the ratio of receiver and transmitter frequency is obtained by uR dtT ¼ uT dtR vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 2 u1 þ 2F2 T þ vT2 c c t ¼ þ Oðc4 Þ v2R 2FR 1 þ c2 þ c2

DuRT ¼

(28.7)

The lower indices R and T describe a receiver, respectively a transmitter related variable. The gravity

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potentials are related to the receiver or transmitter position FR ¼ FðxkR Þ; FT ¼ FðxkT Þ. This equation is used for synthesizing relativistic frequency shifts in our simulation environment. Moreover it is the fundamental equation for the Post-Newtonian approach. As the relativistic time dilation, which is not modeled by the Newtonian framework, is taken into account, the nomenclature ‘Post-Newtonian’ has bee chosen to describe this method. The gravity potential FT deduced from the frequency shift DuRT is the prime observable for the Post-Newtonian method. A general description of (28.7) is DuRT

uR dtT ¼ ¼ ¼ uT dtR

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ gmn ðxT ÞdxmT dxnT gmn ðxR ÞdxmR dxnR

(28.8)

By combining (28.8) with the description of the metric tensor elements in (28.2), (28.7) can be achieved. Based on this function the gravity potential FT at the satellite position can be calculated from a measured frequency shift DuRT . After some reformulations a function, which is used in our simulation environment as functional model for deducing the gravity potential at transmitter position from the frequency shift DuRT , is achieved: 2 v2T c 2 FT ¼ DuRT OR 2 þ 1 c 2

(28.9)

Here a support variable OR has been introduced. It contains all position, velocity and gravity potential information of the receiver station: OR ¼ 1 þ

2 FR v2R þ 2 : c2 c

Equations (28.9) and (28.10) describe two relativistic effects. The first one is time dilatation caused by relative movement of receiver and transmitter, the second one time dilatation caused by the gravity potential difference between transmitter and receiver location. The gravity potential FðxÞ is composed of all occurring gravity potentials. As a first approximation in our simulations, all non-earth gravity potentials and tide signals have been neglected. Beside the special relativistic and general relativistic effects the Doppler shift is the third large effect which influences the

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frequency observations of the receiver station. As a satellite in a LEO achieves large velocities, the radial velocity between receiver and transmitter cause a Doppler frequency shift which has to be modeled. The Doppler shift (Doppler 1842) is defined by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c vTR uR ¼ uT c þ vTR

(28.10)

where vTR is the scalar radial velocity between transmitter and receiver. The Doppler shift is applied to the measured frequency at the receiver station. It has to be mentioned that as we are working in a relativistic framework, the radial velocity has to be derived in a coordinate system located at the center of the receiver station.

28.4

Simulation Setup

Figure 28.1 shows the schematic set-up of our simulation software. In a configuration file the orbit properties, computation switches and observation noise types are defined. The software computes based on this configuration, the orbit positions and the frequency related effects by using (28.7) and (28.10). In our simulation environment (28.9) has been used to calculate the gravity potential at the satellite position from the synthesized frequency shifts. The simulation environment has been designed to determine the influence of data noise on the reproduced gravity field model. Therefore it is possible to manually add realistic noise on frequency and velocity measurements. Following effects on the frequency shift observable have been simulated:

• Special relativistic frequency shift caused by relative velocity of receiver and transmitter clock (28.7). • General relativistic frequency shift caused by relative potential difference at receiver and transmitter clock positions (28.7). • Doppler Effect caused by relative radial velocity of receiver and transmitter clock (28.10). For every effect a realistic stochastic noise signal was added on noise-free frequency and velocity measurements. Shin et al. (2008) suppose a coloured noise for H-maser clocks with increasing amplitudes at low and high frequencies and linear behaviour in-between. Figure 28.2 shows the power spectrum density function of the atomic clock noise with amplitude 1017 and 1018 s generated for our simulations based on this information. For the velocity error, white noise has been assumed. In the frame of gravity field adjustment, the stochastic models, which define the metric of the normal equation system, have been consistently incorporated by correspondingly designed digital filters applied to both, the observation time series and the columns of the design matrix (Schuh, 2001). A nearly polar (89.5 inclination), circular repeat orbit with 25 days and 403 cycles at 300 km mean height with 10 s sampling interval has been chosen for all simulations.

28.5

28.5.1 Performance Analysis The main observable of the Post-Newtonian method is the frequency shift. Equation (28.7) shows that beside the atomic clock noise the velocity determination noise

Orbit Definition

Simulation Definition

Fig. 28.1 Schematic presentation of the simulation environment used in this study

Simulation Results

Orbit Synthesis

Signal Synthesis

Noise Definition

Gravity Field Coefficients equation (28.3) least squares adjustment

Gravity Field Analysis equation (28.9)

equation (28.7), (28.10)

Potential Reconstruction

28

Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

235

Fig. 28.2 Smoothed noise amplitude spectrum of coloured clock noise with amplitude 1017 s and 1018 s

Fig. 28.3 Degree error median plot of simulations with frequency noise with 1017 s and 1018 s amplitude and velocity noise with 105 m/s and 106 m/s white noise

of the transmitter is a stochastic variable, too and is the second limiting factor for the Post-Newtonian method. Based on the frequency shift the transmitter potential can be determined by using (28.10). Thus, first a noise free data set has been defined, and the calculation method has been verified by using closed-loop computations. Next, realistic coloured clock noise (Fig. 28.2) with amplitudes of 1016 up to 1018 s has been applied to the synthetic observations. Finally, signals including white velocity noise at amplitudes of 104 to 106 m/s have been used instead. Figure 28.3 shows the degree error median of the by least squares adjustment reproduced gravity field

coefficients up to d/o (degree and order) 150. It can be seen that the velocity noise of 105 m/s has an effect on the gravity field reconstruction error which is comparable to 1017 s clock noise, while 106 m/s velocity noise has a similar influence as 1018 s clock noise. Moreover it can be seen that future atomic clocks (Cacciapuoti 2006; Schiller 2007) with an expected short term stability of 1018 s clock noise, and positioning precision of 106 m/s velocity noise it would be possible to deduce earth’s gravity field up do d/o 120. It has to be mentioned that following effects, which will in practice have additional contributions on the total error budget, have been neglected in our simulations:

236

• • • • • •

Tidal and non-tidal temporal variable effects. Non-conservative potentials Relativistic effects on frame transformations. Non-uniformly rotating earth. Non-inertial potentials (satellite rotation). Receiver position related errors (position, velocity and the gravity potential at receiver position are assumed to be error-free). • Dispersive atmosphere related effects. However, it can be expected that the error terms included in this study are the dominant ones.

Fig. 28.4 Geoid height error of simulation with 1018 s clock noise applied on frequency observations. A homogeneous and isotropic error structure is provided

Fig. 28.5 Degree error median plot of Doppler velocity error with 104 m/s noise simulation. Compared to the simulations shown in Fig. 28.3 the influence of Doppler noise is negligibly small

R. Mayrhofer and R. Pail

28.5.2 Spatial and Spectral Error Structure As the gravity potential at the transmitter position, which is deduced from frequency measurements, is the observable for the least squares adjustment, the error structure of the recovered earth gravity field corresponds to the error structure of direct potential observations. Thus, in the case of white noise, the error amplitude increases with higher degree and order of deduced spherical coefficients and the slope of the degree error median is related to the chosen orbit height. Figure 28.4

28

Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

shows the homogeneous and isotropic spatial error distribution of a simulation configuration with 1018 s clock noise. Simulations with white velocity noise superposed show similar spatial error structures.

28.5.3 Doppler Shift A simulation setup has been defined, where the Doppler shift has been applied on synthetic frequency observations based on noise-free data with which relativistic influences were modeled before. White noise with amplitude of 104 m/s has been superposed on the noise-free velocities (28.10). Figure 28.5 shows that the Doppler shift can be modeled very well even at high velocity noise amplitudes. Moreover, the influence on the recovered gravity field is negligibly small up to d/o 250. So the Doppler shift is no limiting factor. The reason for this is because compared to (28.9), the radial velocity in (28.10) is not scaled by c2 .

28.6

Conclusions

It has been shown that the Post-Newtonian method is a feasible method for reproducing earth’s gravity field for lower and medium frequencies, provided that the technological development of space-borne atomic clocks proceeds in the future. Additionally it has been shown that the derived equations are valid within the defined mission scenario. The two dominant error contributions of this method are the atomic clock noise and the satellite velocity error of the precise orbit determination. The Doppler-effect, which also influences the frequency measurements, can be modeled with sufficient precision, so its error does not leak into the recovered gravity coefficients. It has to be mentioned that the simulations done here should be seen as a concept study and some more realistic simulations will be done to provide more information about the behaviour of the method. A velocity determination precision up to 106 m/s and an atomic clock short term stability of 1018 s is required to resolve the gravity field up to d/o 120. The main advantage of this method is its homogeneous and isotropic spatial error structure.

28.7

237

Outlook

With upcoming atomic clocks below 1016 s short term stability and improving positioning and velocity determination methods, the presented method can be an additional piece for a global earth gravity field monitoring framework in a not too far future. Beside the single-satellite mission presented in this study, various satellite constellations and formations can be designed. In what extent satellite formations like Pendulum, Cartwheel or LISA-like lead to improved precision still has to be investigated by numerical simulations. There is no doubt that multi-satellite missions would underline the possible power of the Post-Newtonian method. One, two or three geostationary satellites could be used as reference stations. Additional rover satellites could be placed in different orbit types. A dense network of satellites could improve the time resolution, so the time variable gravity field could be optimally mapped. These satellites could be equipped with two or more RF-antennas, so one satellite could establish a connection to two or more other satellites, which would further increase the measurement density of the network. As the equations used in this study are strongly simplified, the influences of other effects have to be further investigated. First real orbits and non-conservative forces have to be modeled. Next, the influence of satellite rotation has to be investigated. Additional attitude information from star-tracker measurements and its noise behaviour have to be simulated. All observations and calculations have to be done in a relativistic framework. So the influences of frame transformations in this relativistic framework have to be investigated. All other effects listed in Chap. 28.5.1 will be further investigated in upcoming simulations. This will lead to a much more complex mathematical description which will be harder to linearize, but will also be closer to reality. The main target of upcoming simulations will be to set up a more realistic environment and to design multi-satellite missions that support the advantages of the Post-Newtonian method. Moreover there will be done simulations concerning the time variable gravity field. It will be investigated how different mission design affects the quality of the static and time variable gravity field and if there is

238

a possibility to deduce models with higher spatial and temporal resolution. Acknowledgements We would like to thank L. Vitushkin and an anonymous reviewer for their valuable comments which helped to improve the manuscript.

References Cacciapuoti L (2006) Atomic Clocks in Space. ESA-ESTEC (SCI-SP), Frascati Doppler C (1842) Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirnen des Himmels. Abhandlungen der k. b€ ohm. Gesellschaft der Wissenshaften Folge V Band 2, In Commision bei Borrosch & Andre´, Prag € Einstein A (1911) Uber den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes. Annalen der Physik 35, Verlag von Johann Ambrosius Barth, Leipzig pp 898–908 Einstein A (1916) Die Grundlage der allgemeinen Relativit€atstheorie. Annalen der Physik 49, Verlag von Johann Ambrosius Barth, Leipzig pp 769–821 Gulkett M (2003) Relativistic effects in GPS and LEO. Rapport for the Cand. Scient. degree, University of Copenhagen, Department of Geophysics, The Niels Bohr Institute for Physics, Astronomy and Geophysics, Denmark Heiskanen WA, and Moritz H (1967). Physical Geodesy. W. H. Freeman & Co Ltd, San Francisco, ISBN-13 978–0716702337

R. Mayrhofer and R. Pail M€uller J, Soffel M and Klioner SA (2007) Geodesy and relativity. Journal of Geodesy 82 Number 3, Springer, Berlin, ISSN 0949–7714, pp 133–145 Schiller S (2007) Gravimetry with optical clocks. Workshop on The Future of Satellite Gravimetry 12–13 April 2007, ESTEC Schiller S, Tino GM, Gill P, Salomon C, Sterr U, Peik E, Nevsky A, G€orlitz A, Svehla D, Ferrari G, Poli N, Lusanna L, Klein H, Margolis H, Lemonde P, Laurent P, Santarelli G, Clairon A, Ertmer W, Rasel E, M€uller J, Iorio L, L€ammerzahl C, Dittus H, Gill E, Rothacher M, Flechner F, Schreiber U, Flambaum V, Ni, Wei-Tou, Liu, Liang, Chen, Xuzong, Chen, Jingbiao, Gao, Kelin, Cacciapuoti L, Holzwarth R, Heß MP, Sch€afer W (2009) Einstein gravity explorer – a medium class fundamental physics mission. Exp Astron 23 (2):573–610 Shin MY, Park C and Lee SJ (2008) Atomic Clock Error Modeling for GNSS Software Platform. Position, Location and Navigation Symposium, IEEE/ION Plans 2008, pp 71–76, 1-4244-1537-03/08 Schuh W-D (2001) Improved modeling of SGG-data sets by advanced filter strategies. ESA-Project (Hg.): From E€otv€os to mGal+, WP2, Midterm-Report, pp 113–181 Soffel M, Klioner A, Petit G, Wolf P, Kopeikin SM, Bretagnon P, Brumberg VA, Capitaine N, Damour T, Fukushima T, Guinot B, Huang T-Y, Lindegren L, Ma C, Nordtvedt K, Ries JC, Seidelmann PK, Vokrouhlicky D, Will CM, Xu C (2003) The IAU 2000 resolution for astrometry, celestial mechanics, and metrology in the framework: explanatory supplement. Astron J 126:2687–2706, The American Astronomical Society. USA

Local and Regional Comparisons of Gravity and Magnetic Fields

29

C. Jekeli, O. Huang, and T.L. Abt

Abstract

A long recognized connection between the gravitational gradients of the Earth’s crust and its magnetic anomalies, known as Poisson’s relationship, is the object of investigation in this paper. We develop the mathematical and theoretical basis of this relationship in both the space and frequency domains. Anomalies of the magnetic field thus implied by the gravitational gradients (or other derivatives of the gravitational potential) are called pseudo-magnetic anomalies; and, they assume a linear relationship between the mass density of the source material and its magnetization induced by the Earth’s main magnetic field. Tests in several regions of the U.S. that compare gravitational gradients derived from the highresolution model, EGM08, and a continental magnetic anomaly data base reveal that the correlation implied by Poisson’s relationship is not consistent. Some areas exhibit high positive correlation at various frequencies, while others have even strong negative correlation. Therefore, useful applications of Poisson’s relationship depend on the validity of the underlying assumptions that, conversely, may also be investigated and studied using a combination of gradiometric and magnetic data.

29.1

Introduction

The gravitational and magnetic fields of the Earth are connected by the fact that both are generated in part by the same source material, gravitationally because this material has mass density, and magnetically because it is magnetized by the Earth’s main magnetic field due to its liquid outer core. At the microscopic level, the atomic structure of the material has both mass and

C. Jekeli (*) O. Huang T.L. Abt Division of Geodetic Science, School of Earth Sciences, The Ohio State University, 125 South Oval Mall, Columbus, OH 43210, USA e-mail: [email protected].edu

electric dipoles that respond to an applied magnetic field and, in turn, generate an induced field. Mathematically, both fields satisfy Laplace’s differential equation and the solutions are similar, one involving a Green’s function for a distribution of monopoles, the other a Green’s function for a distribution of dipoles. The putative relationship between gravity and magnetic fields was developed already by Poisson (1826), was popularized by Baranov (1957), and essentially posits that magnetic anomalies are gravitational gradients, that is, higher-frequency signals. Mathematical elaborations were carried out by Gunn (1975) and Klingele et al. (1991), and others. The idea that gravitational gradients are intimately connected to the magnetization of the Earth’s crust has motivated

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_29, # Springer-Verlag Berlin Heidelberg 2012

239

240

C. Jekeli et al.

geophysicists to use more easily obtained magnetic data to make gravimetric-type interpretations (Fedi 1989), and to combine (e.g., Dindi and Swain 1988; Ates and Keary 1995) or contrast (Briden et al. 1982) magnetic and gravimetric data for an improved characterization of subsurface geologic structures. While most previous investigations used gravimetric data to model the gravitational gradients, the correlation between gravitation and magnetism can now also take advantage of various gradiometric data sets (e.g., Jekeli et al. 2008). The purpose of this paper, however, is to investigate this correlation on the basis of the new very-high-degree gravitational model, EGM08 (Pavlis et al. 2008), from which gravitational gradients are readily computed.

29.2

Basic Theory

The gravitational potential, V, at some point, x, due to a mass point (monopole) with mass, m, located at x0 , is well known and given by V ðxÞ ¼ G

m ; jx x0 j

(29.1)

where G ¼ 6:674 1011 m3 /(kg s2 Þ is Newton’s gravitational constant. For the (anomalous) magnetic field, the most elemental source is a dipole, modeled as arising from a microscopic loop current. It can be shown (Telford et al. 1990) that the corresponding magnetic potential, A, is AðxÞ ¼

m0 p e ; 4p jx x0 j2

(29.2)

where m0 ¼ 4p 107 kg m/ ðamp sÞ2 is the magnetic permeability of free space, p is the magnetic dipole moment (vector), and e is the unit vector from x0 to x. For a continuous volume distribution of mass monopoles, we define the mass density as rðx0 Þ ¼ dm=dv; and, analogously, for a continuous distribution of magnetic dipole moments, we define the magnetic dipole moment density as z ðx0 Þ ¼ dp=dv. This density is also called the magnetic polarization or the magnetization intensity. The field is given in each case above, according to the law of superposition, by

ððð V ð xÞ ¼ G v

m A ð xÞ ¼ 0 4p

ððð v

rðx0 Þ dv; jx x0 j z ðx 0 Þ e jx x0 j 2

dv;

(29.3)

(29.4)

where dv ¼ dx1 0 dx2 0 dx3 0 . It is easily verified that the scalar magnetic potential is also expressed as A ð xÞ ¼

m0 4p

ðð ð

z ðx0 Þ:rx

v

1 dv jx x0 j

(29.5)

The magnetization, z, of the crust material, in part, is induced by the main field flux density, H0 , that is generated by the Earth’s outer liquid core, and includes a remnant (remanent) magnetization left from the time of rock formation: z ðx0 Þ ¼ wðx0 ÞH0 ðx0 Þ þ z R ðx0 Þ;

(29.6)

where w is the (presumably scalar) magnetic susceptibility. For present purposes, we assume that the Koenigsberger ratio, Q ¼ z R =wH 0 , is sufficiently small so as to make z R negligible. Furthermore, locally, H0 is almost constant in magnitude and direction. For a given volume of material (x0 2 v), we assume that the mass density and the magnetic susceptibility are linearly related: wðx0 Þ ¼

w0 rðx0 Þ; r0

(29.7)

where r0 and w0 are constants; then z ðx0 Þ ¼ k0

z0 rðx0 Þ; r0

(29.8)

where z 0 ¼ w0 H0 , H0 ¼ H0 k0 , and k0 ¼ ð cos cos x sin cos x

sin x ÞT ;

(29.9)

and where x is the dip angle (inclination) with respect to the local horizon, and is the strike angle (azimuth, or declination) with respect to local geodetic north. With (29.8), the magnetic potential becomes the pseudo-magnetic potential:

29

Local and Regional Comparisons of Gravity and Magnetic Fields

m z AðxÞ ¼ 0 0 k0 rx 4p r0

ððð v

rðx0 Þ dv; jx x0 j

(29.10)

where k0 rx is the directional derivative in the (constant) direction of the magnetization of the material. Combining (29.3) and (29.10), we thus have Poisson’s relation: A ð xÞ ¼

m0 z0 k0 rx V ðxÞ; 4pGr0

(29.11)

or, in terms of the force fields, g ¼ rV, F ¼ rA (the signs are conventional, and the subscript on the gradient operator is now redundant): F ð xÞ ¼

m0 z0 rgT ðxÞk0 : 4pGr0

(29.12)

241

Therefore, from (29.13), we have DB ¼

m0 z 0 T k GðxÞk0 ; 4pGr0 0

which directly relates magnetometry and gravity gradiometry. Technically, the right side of (29.16) is called the pseudo-magnetic anomaly. Under the assumption of a flat Earth, it is possible to relate the gravitational and magnetic fields locally also in the spatial frequency domain. Consider a potential that satisfies Laplace’s equation in the exterior space, z0 G ; > > < tGD1 p tG r D1 Ii3 þ D1 Ii3 ; > r > Ii3 ð1 ð1 pÞ tGD1 Þ; > D2 > : r pIi3 ;

0 t

International Association of Geodesy Symposia Michael G . Sideris, Series Editor

Symposium 101: Global and Regional Geodynamics Symposium 102: Global Positioning System: An Overview Symposium 103: Gravity, Gradiometry, and Gravimetry Symposium 104: Sea SurfaceTopography and the Geoid Symposium 105: Earth Rotation and Coordinate Reference Frames Symposium 106: Determination of the Geoid: Present and Future Symposium 107: Kinematic Systems in Geodesy, Surveying, and Remote Sensing Symposium 108: Application of Geodesy to Engineering Symposium 109: Permanent Satellite Tracking Networks for Geodesy and Geodynamics Symposium 110: From Mars to Greenland: Charting Gravity with Space and Airborne Instruments Symposium 111: Recent Geodetic and Gravimetric Research in Latin America Symposium 112: Geodesy and Physics of the Earth: Geodetic Contributions to Geodynamics Symposium 113: Gravity and Geoid Symposium 114: Geodetic Theory Today Symposium 115: GPS Trends in Precise Terrestrial, Airborne, and Spaceborne Applications Symposium 116: Global Gravity Field and Its Temporal Variations Symposium 117: Gravity, Geoid and Marine Geodesy Symposium 118: Advances in Positioning and Reference Frames Symposium 119: Geodesy on the Move Symposium 120: Towards an Integrated Global Geodetic Observation System (IGGOS) Symposium 121: Geodesy Beyond 2000: The Challenges of the First Decade Symposium 122: IV Hotine-Marussi Symposium on Mathematical Geodesy Symposium 123: Gravity, Geoid and Geodynamics 2000 Symposium 124: Vertical Reference Systems Symposium 125: Vistas for Geodesy in the New Millennium Symposium 126: Satellite Altimetry for Geodesy, Geophysics and Oceanography Symposium 127: V Hotine Marussi Symposium on Mathematical Geodesy Symposium 128: A Window on the Future of Geodesy Symposium 129: Gravity, Geoid and Space Missions Symposium 130: Dynamic Planet - Monitoring and Understanding … Symposium 131: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles Symposium 132: VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy Symposium 133: Observing our Changing Earth Symposium 134: Geodetic Reference Frames Symposium 135: Gravity, Geoid and Earth Observation

Geodesy for Planet Earth Proceedings of the 2009 IAG Symposium, Buenos Aires, Argentina, 31 August - 4 September 2009

Edited by Steve C. Kenyon Maria C. Pacino Urs J. Marti

Editors Steve Kenyon National Geospatial-Intelligence Agency SN L-41 Vogel Rd. 3838 63010-6238 Arnold Montana USA Urs Marti Federal Office of Topography swisstopo Geodesy Department Seftigenstrasse 264 3084 Wabern Switzerland

Maria Cristina Pacino Universidad Nacional de Rosario (UNR) Facultad de Ciencia Exactas Ingenierı´a y Agrimensura (FCEIA) Av. Pellegrini 250 2000 Rosario Argentina

ISBN 978-3-642-20337-4 e-ISBN 978-3-642-20338-1 DOI 10.1007/978-3-642-20338-1 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011941192 # Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The International Association of Geodesy IAG2009 “Geodesy for Planet Earth” Scientific Assembly was held 31 August to 4 September 2009 in Buenos Aires, Argentina. The theme “Geodesy for Planet Earth” was selected to follow the International Year of Planet Earth 2007–2009 goals of utilizing the knowledge of the world’s geoscientists to improve society for current and future generations. The International Year started in January 2007 and ran thru 2009 which coincided with the IAG2009 Scientific Assembly, one of the largest and most significant meetings of the Geodesy community held every 4 years. The IAG2009 Scientific Assembly was organized into eight Sessions with SubSessions in five of them. Four of the Sessions of IAG2009 were based on the IAG Structure (i.e. one per Commission) and covered Reference Frames, Gravity Field, Earth Rotation and Geodynamics, and Positioning and Applications. Since IAG2009 was taking place in the great Argentine city of Buenos Aires, a Session was devoted to the Geodesy of Latin America. A Session dedicated to the IAG’s Global Geodetic Observing System (GGOS), the primary observing system focused on the multidisciplinary research being done in Geodesy that contributes to important societal issues such as monitoring global climate change and the environment. A Session on the IAG Services was also part of the Assembly detailing the important role they play in providing geodetic data, products, and analysis to the scientific community. A final Session devoted to the organizations ION, FIG, and ISPRS and their significant work in navigation and earth observation that complements the IAG. This volume contains the proceedings of the eight Sessions which are listed below: Session 1: Reference frames implementation for geosciences’ applications: From local to global scales Convenors: Zuheir Altamimi, Claudio Brunini Session 2: Gravity of the Planet Earth Convenors: Yoichi Fukuda, Pieter Visser Session – 2.1: Physics and Geometry of Earth: Focus on satellite altimetry and InSAR Convenors: Cheinway Hwang, Jose´ Luis Vacaflor Session – 2.2: Gravity – An Earth Probing Tool: Focus on CHAMP/GRACE/ GOCE missions, relative/absolute/superconducting gravimetry, and their applications Convenors: Roland Pail, Leonid F. Vitushkin

v

vi

Session – 2.3: Modern Height Datum: Focus on definition and realization of GPS-levelling and gravity-base height datum Convenors: Sı´lvio R.C. de Freitas, Dru A. Smith Session – 2.4: Gravity and Geoid Modelling: Focus on global and regional gravity and geoid modelling Convenors: Urs Marti, Yan Ming Wang Session 3: Geodesy and Geodynamics: Global and Regional Scales Convenors: Mike Bevis, Sylvain Bonvalot Session – 3.1: Rotation of the Planet Earth Convenors: Richard Gross, Rodrigo, Abarca del Rio Session – 3.2: Sea level changes and post-glacial rebound Convenors: Juan Fierro, Michael Bevis Session – 3.3: Ocean loading and global water distribution / geophysical fluids Convenors: Tonie van Dam, Richard Gross Session – 3.4: Geodesy, crustal motions and geodynamic processes Convenors: Juan Carlos Baez, Sylvain Bonvalot, Arturo Echalar Session – 3.5: Geodesy and the near-field solid earth response to cryospheric mass changes Convenors: Jim Davis, Gino Casassa Session 4: Positioning and remote sensing of land, ocean and atmosphere Convenors: Sandra Verhagen, Pawel Wielgosz Session – 4.1: Technology and land applications Convenors: Dorota Grejner-Brzezinska, Xiaoli Ding Session – 4.2: Modelling and remote sensing of the atmosphere Convenors: Marcelo Santos, Cathryn Mitchell, Jens Wickert Session – 4.3: Multi-satellite ocean remote sensing Convenors: Shuanggen Jin, Ole Baltazar Andersen Session 5: Geodesy in Latin America Convenors: Denizar Blitzkow, Claudia Tocho Session 6: JOINT ION/FIG/ISPRS session on Navigation and Earth Observation Convenors: Dorota Grejner-Brzezinska, Charles Toth Session – 6.1: Navigation (FIG, ION) Convener: Dorota Grejner-Brzezinska Session – 6.2: Earth Observation (ISPRS)C onvener: Charles Toth Session 7: The Global Geodetic Observing System: Science and Applications Convenors: Richard Gross, Hans-Peter Plag, Luiz Paulo Fortes Session – 7.1: Past Progress and Future Plans Convenors: Hans-Peter Plag, Richard Gross, Luiz Paulo Fortes Session – 7.2: Science and Applications Convenors: Richard Gross, Hans-Peter Plag, Luiz Paulo Fortes Session 8: The IAG International Services and their role for Earth observation Convenors: Ruth Neilan, Rene Forsberg The number and quality of contributions for the eight Sessions clearly demonstrated the important and vital role that Geodesy plays in understanding the earth and its dynamical processes. Satellite, airborne, and terrestrial systems and networks are

Preface

Preface

vii

continually measuring and analyzing the earth for global change. For this reason the name of these proceedings is “Geodesy for Planet Earth” which reflects the role fourdimensional geodesy plays in understanding the changes to Planet Earth. The 2009 Assembly attracted nearly 500 oral and poster presentations from 370 geodesists from 45 countries and clearly shows the interest and importance of geodesy globally. The approximately 130 papers that are included in these proceedings (about 25% of the total) are intended to cover much of the latest research and projects on-going in the field. These proceedings would not be possible without the tremendous work by the Convenors of each of the Sessions and Sub-Sessions. They devoted a enormous amount of time and energy in organizing the reviews and final acceptance of the papers for their Sessions. We are very grateful to IAG Secretary General Hermann Drewes and IAG President Michael Sideris for all their guidance and help with these proceedings. The Local Organizing Committee in Buenos Aires was invaluable in helping arrange a very memorable Assembly and provided essential support in the development of these proceedings. And lastly, sincere thanks go out to all the participating scientists and graduate students who made the IAG 2009 “Geodesy for Planet Earth” Scientific Assembly and these proceedings a tremendous success. Steve Kenyon Maria Cristina Pacino Urs Marti

.

Contents

Session 1

Reference Frames Implementation for Geoscience’s Applications: From Local to Global Scales Convenors: Z. Altamimi, C. Brunini 1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 C. Brunini, L. Sanchez, H. Drewes, S. Costa, V. Mackern, W. Martı´nez, W. Seemuller, and A. da Silva

2

Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alvaro Santamarı´a-Go´mez, Marie-Noe¨lle Bouin, and Guy Wo¨ppelmann

3

4

5

6

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bruyninx, Z. Altamimi, M. Becker, M. Craymer, L. Combrinck, A. Combrink, J. Dawson, R. Dietrich, R. Fernandes, R. Govind, T. Herring, A. Kenyeres, R. King, C. Kreemer, D. Lavalle´e, J. Legrand, L. Sa´nchez, G. Sella, Z. Shen, A. Santamarı´a-Go´mez, and G. Wo¨ppelmann Enhancement of the EUREF Permanent Network Services and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bruyninx, H. Habrich, W. So¨hne, A. Kenyeres, G. Stangl, and C. Vo¨lksen

11

19

27

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ulla Kallio and Markku Poutanen

35

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008, the ignwd08 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Willis, M.L. Gobinddass, B. Garayt, and H. Fagard

43

7

Towards a Combination of Space-Geodetic Measurements . . . . . . . . . . . A. Pollet, D. Coulot, and N. Capitaine

8

Improving Length and Scale Traceability in Local Geodynamical Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Jokela, P. H€akli, M. Poutanen, U. Kallio, and J. Ahola

51

59

ix

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9

10

11

12

How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Drewes Transforming ITRF Coordinates to National ETRS89 Realization in the Presence of Postglacial Rebound: An Evaluation of the Nordic Geodynamical Model in Finland . . . . . . . P. H€akli and H. Koivula Global Terrestrial Reference Frame Realization Within the GGOS-D Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Angermann, H. Drewes, and M. Seitz Comparison of Regional and Global GNSS Positions, Velocities and Residual Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Legrand, N. Bergeot, C. Bruyninx, G. Wo¨ppelmann, A. Santamarı´a-Go´mez, M.-N. Bouin, and Z. Altamimi

67

77

87

95

13

GPS Metrology: Bringing Traceable Scale to a Local Crustal Deformation GPS Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 H. Koivula, P. H€akli, J. Jokela, A. Buga, and R. Putrimas

14

Impact of Albedo Radiation on GPS Satellites . . . . . . . . . . . . . . . . . . . . . . . . 113 C.J. Rodriguez-Solano, U. Hugentobler, and P. Steigenberger

Session 2 Gravity of the Planet Earth Convenors: Y. Fukuda, P. Visser 15

On the Determination of Sea Level Changes by Combining Altimetric, Tide Gauge, Satellite Gravity and Atmospheric Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 G.S. Vergos, I.N. Tziavos, and M.G. Sideris

16

Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite and Submarine Measurements . . . . . . . . . . . . 131 J. Calvao, J. Rodrigues, and P. Wadhams

17

The Impact of Attitude Control on GRACE Accelerometry and Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 U. Meyer, A. J€aggi, and G. Beutler

18

Using Atmospheric Uncertainties for GRACE De-aliasing: First Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 L. Zenner, T. Gruber, G. Beutler, A. J€aggi, F. Flechtner, T. Schmidt, J. Wickert, E. Fagiolini, G. Schwarz, and T. Trautmann

19

Challenges in Deriving Trends from GRACE . . . . . . . . . . . . . . . . . . . . . . . . . 153 A. Eicker, T. Mayer-Guerr, and E. Kurtenbach

20

AIUB-GRACE02S: Status of GRACE Gravity Field Recovery Using the Celestial Mechanics Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 A. J€aggi, G. Beutler, U. Meyer, L. Prange, R. Dach, and L. Mervart

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21

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.C. Gunter, T. Wittwer, W. Stolk, R. Klees, and P. Ditmar

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination of LEO Orbit Based on GNSS Observations . . . . . . . . . 179 A. Shabanloui and K.H. Ilk

23

Pure Geometrical Precise Orbit Determination of a LEO Based on GNSS Carrier Phase Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 187 A. Shabanloui and K.H. Ilk

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field: Boundary Problems and a Target Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 P. Holota and O. Nesvadba

25

Moho Estimation Using GOCE Data: A Numerical Simulation . . . . . 205 M. Reguzzoni and D. Sampietro

26

CHAMP, GRACE, GOCE Instruments and Beyond . . . . . . . . . . . . . . . . . 215 P. Touboul, B. Foulon, B. Christophe, and J.P. Marque

27

The Future of the Satellite Gravimetry After the GOCE Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 P. Silvestrin, M. Aguirre, L. Massotti, B. Leone, S. Cesare, M. Kern, and R. Haagmans

28

Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 R. Mayrhofer and R. Pail

29

Local and Regional Comparisons of Gravity and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 C. Jekeli, O. Huang, and T.L. Abt

30

Combination of Local Gravimetry and Magnetic Data to Locate Subsurface Anomalies Using a Matched Filter . . . . . . . . . . . . 247 T. Abt, O. Huang, and C. Jekeli

31

On the Use of UAVs for Strapdown Airborne Gravimetry . . . . . . . . . . 255 Richard Deurloo, Luisa Bastos, and Machiel Bos

32

Updating the Precise Gravity Network at the BIPM . . . . . . . . . . . . . . . . . 263 Z. Jiang, E.F. Arias, L. Tisserand, K.U. Kessler-Schulz, H.R. Schulz, V. Palinkas, C. Rothleitner, O. Francis, and M. Becker

33

Precise Gravimetric Surveys with the Field Absolute Gravimeter A-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 R. Falk, Ja. M€ uller, N. Lux, H. Wilmes, and H. Wziontek

34

Reconstruction of a Torsion Balance and the Results of the Test Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 L. Vo¨lgyesi and Z. Ultmann

xii

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35

The Superconducting Gravimeter as a Field Instrument Applied to Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 C.R. Wilson, H. Wu, L. Longuevergne, B. Scanlon, and J. Sharp

36

Local Hydrological Information in Gravity Time Series: Application and Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 M. Naujoks, S. Eisner, C. Kroner, A. Weise, P. Krause, and T. Jahr

37

Signals of Mass Redistribution at the South African Gravimeter Site SAGOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 C. Kroner, S. Werth, H. Pflug, A. G€ untner, B. Creutzfeldt, M. Thomas, H. Dobslaw, P. Fourie, and P.H. Charles

38

Gravity System and Network in Estonia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 To˜nis Oja

39

Evaluation of EGM2008 Within Geopotential Space from GPS, Tide Gauges and Altimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 N. Dayoub, P. Moore, N.T. Penna, and S.J. Edwards

40

Fixed Gravimetric BVP for the Vertical Datum Problem . . . . . . . . . . . . 333 ˇ underlı´k, Z. Fasˇkova´, and K. Mikula R. C

41

Realization of the World Height System in New Zealand: Preliminary Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 R. Tenzer, V. Vatrt, and M. Amos

42

Comparisons of Global Geopotential Models with Terrestrial Gravity Field Data Over Santiago del Estero Region, NW: Argentine . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 L. Galva´n, C. Infante, E. Laurı´a, and R. Ramos

43

Intermap’s Airborne Inertial Gravimetry System . . . . . . . . . . . . . . . . . . . . 357 Ming Wei

44

Galathea-3: A Global Marine Gravity Profile . . . . . . . . . . . . . . . . . . . . . . . . . 365 G. Strykowski, K.S. Cordua, R. Forsberg, A.V. Olesen, and O.B. Andersen

45

Dependency of Resolvable Gravitational Spatial Resolution on Space-Borne Observation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 P.N.A.M. Visser, E.J.O. Schrama, N. Sneeuw, and M. Weigelt

46

A Comparison of Different Integral-Equation-Based Approaches for Local Gravity Field Modelling: Case Study for the Canadian Rocky Mountains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 R. Tenzer, I. Prutkin, and R. Klees

47

Global Topographically Corrected and Topo-Density Contrast Stripped Gravity Field from EGM08 and CRUST 2.0 . . . . . . . . . . . . . . . 389 R. Tenzer, Hamayun, and P. Vajda

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48

Local Gravity Field Modelling in Rugged Terrain Using Spherical Radial Basis Functions: Case Study for the Canadian Rocky Mountains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 R. Tenzer, R. Klees, and T. Wittwer

49

A Sensitivity Analysis in Spectral Gravity Field Modeling Using Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 Vassilios D. Andritsanos and Ilias N. Tziavos

50

Investigation of Topographic Reductions for Marine Geoid Determination in the Presence of an Ultra-High Resolution Reference Geopotential Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 C. Tocho, G.S. Vergos, and M.G. Sideris

51

Effects of Hypothetical Complex Mass-Density Distributions on Geoidal Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 Robert Kingdon, Petr Vanı´cˇek, and Marcelo Santos

52

Evaluation of Gravity and Altimetry Data in Australian Coastal Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 S.J. Claessens

53

Development and User Testing of a Python Interface to the GRAVSOFT Gravity Field Programs . . . . . . . . . . . . . . . . . . . . . . . . . . 443 J. Nielsen, C.C. Tscherning, T.R.N. Jansson, and R. Forsberg

54

Progress and Prospects of the Antarctic Geoid Project (Commission Project 2.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Mirko Scheinert

55

Regional Geoid Improvement over the Antarctic Peninsula Utilizing Airborne Gravity Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 J. Schwabe, M. Scheinert, R. Dietrich, F. Ferraccioli, and T. Jordan

56

Auvergne Dataset: Testing Several Geoid Computation Methods . . . 465 P. Valty, H. Duquenne, and I. Panet

57

In Pursuit of a cm-Accurate Local Geoid Model for Ohio . . . . . . . . . . . 473 K.R. Edwards, Dorota Grejner-Brzezinska, and Dru Smith

58

Adjustment of Collocated GPS, Geoid and Orthometric Height Observations in Greece. Geoid or Orthometric Height Improvement? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 I.N. Tziavos, G.S. Vergos, V.N. Grigoriadis, and V.D. Andritsanos

Session 3 Geodesy and Geodynamics: Global and Regional Scales Convenors: M. Bevis, S. Bonvalot 59

Regional Geophysical Excitation Functions of Polar Motion over Land Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 J. Nastula and D.A. Salstein

60

Geophysical Excitation of the Chandler Wobble Revisited . . . . . . . . . . 499 Aleksander Brzezin´ski, Henryk Dobslaw, Robert Dill, and Maik Thomas

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On the Origin of the Bi-Decadal and the Semi-Secular Oscillations in the Length of the Day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 S. Duhau and C. de Jager

62

Future Improvements in EOP Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 W. Kosek

63

Determination of Nutation Coefficients from Lunar Laser Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 L. Biskupek, J. M€ uller, and F. Hofmann

64

A Set of Analytical Formulae to Model Deglaciation-Induced Polar Wander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 W. Keller, M. Kuhn, and W.E. Featherstone

65

Stabilization of Satellite Derived Gravity Field Coefficients by Earth Orientation Parameters and Excitation Functions . . . . . . . . . 537 Andrea Heiker, Hansjo¨rg Kutterer, and J€ urgen M€uller

66

The Statistical Characteristics of Altimetric Sea Level Anomaly Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 T. Niedzielski and W. Kosek

67

Testing Past Sea Level Reconstruction Methodology (1958–2006) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 J. Viarre and R. Abarca-del-Rı´o

68

Precise Determination of Relative Mean Sea Level Trends at Tide Gauges in the Adriatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 M. Repanic´ and T. Basˇic´

69

Quantile Analysis of Relative Sea-Level at the Hornbæk and Gedser Tide Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 S.M. Barbosa and K.S. Madsen

70

Assessment of the FES2004 Derived OTL Model in the West of France and Preliminary Results About Impacts of Tropospheric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 F. Fund, L. Morel, and A. Mocquet

71

Gravimetric Time Series Recording at the Argentine Antarctic Stations Belgrano II and San Martı´n for the Improvement of Ocean Tide Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 Mirko Scheinert, Andre´s F. Zakrajsek, Lutz Eberlein, Reinhard Dietrich, Sergio A. Marenssi, and Marta E. Ghidella

72

Mass-Change Acceleration in Antarctica from GRACE Monthly Gravity Field Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Lo´ra´nt Fo¨ldva´ry

73

Mass Variations in the Siberian Permafrost Region from GRACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 Holger Steffen, J€ urgen M€ uller, and Nadja Peterseim

Contents

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74

Seasonal Variability of Land Water Storage in South America Using GRACE Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Claudia Tocho, Luis Guarracino, Leonardo Monachesi, Andre´s Cesanelli, and Pablo Antico

75

Water Storage Changes from GRACE Data in the La Plata Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 A. Pereira, S. Miranda, M.C. Pacino, and R. Forsberg

76

Second and Third Order Ionospheric Effects on GNSS Positioning: A Case Study in Brazil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 H.A. Marques, J.F.G. Monico, G.P.S. Rosa, M.L. Chuerubim, and Ma´rcio Aquino

77

Advanced Techniques for Discontinuity Detection in GNSS Coordinate Time-Series. An Italian Case Study . . . . . . . . . . . . 627 A. Borghi, L. Cannizzaro, and A. Vitti

78

Traditional and Alternative Network Adjustment Approach for the TAMDEF GPS in Antarctica . . . . . . . . . . . . . . . . . . . . . . . 635 G. Esteban Va´zquez, Dorota A. Grejner-Brzezinska, and Burkhard Schaffrin

79

Impact of Loading Phenomena on Velocity Field Computation from GPS Campaigns: Application to ResPyr GPS Campaign in the Pyrenees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 J. Nicolas, F. Perosanz, A. Rigo, G. Bliguet, L. Morel, and F. Fund

80

Comparison of the Coordinates Solutions Between the Absolute and the Relative Phase Center Variation Models in the Dense Regional GPS Network in Japan . . . . . . . . . . . . . . . 651 S. Shimada

81

The 2009 Horizontal Velocity Field for South America and the Caribbean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 H. Drewes and O. Heidbach

82

New Estimates of Present-Day Crustal/Land Motions in the British Isles Based on the BIGF Network . . . . . . . . . . . . . . . . . . . . . . 665 D.N. Hansen, F.N. Teferle, R.M. Bingley, and S.D.P. Williams

83

GURN (GNSS Upper Rhine Graben Network): Research Goals and First Results of a Transnational Geo-scientific Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 M. Mayer, A. Kno¨pfler, B. Heck, F. Masson, P. Ulrich, and G. Ferhat

84

Determination of Horizontal and Vertical Movements of the Adriatic Microplate on the Basis of GPS Measurements . . . . . . 683 M. Marjanovic´, Zˇ. Bacˇic´, and T. Basˇic´

85

Determination of Tectonic Movements in the Swiss Alps Using GNSS and Levelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 E. Brockmann, D. Ineichen, U. Marti, S. Schaer, A. Schlatter, and A. Villiger

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A Compilation of a Preliminary Map of Vertical Deformations in New Zealand from Continuous GPS Data . . . . . . . . . . . . . . . . . . . . . . . . . . 697 R. Tenzer, M. Stevenson, and P. Denys

87

Detection of Vertical Temporal Behaviour of IGS Stations in Canada Using Least Squares Spectral Analysis . . . . . . . . . . . . . . . . . . . . 705 James Mtamakaya, Marcelo C. Santos, and Michael Craymer

Session 4 Positioning and Remote Sensing of Land, Ocean and Atmosphere Convenors: S. Verhagen, P. Wielgosz 88

Positioning and Applications for Planet Earth . . . . . . . . . . . . . . . . . . . . . . . . 713 S. Verhagen, G. Retscher, M.C. Santos, X.L. Ding, Y. Gao, and S.G. Jin

89

Report of Sub-commission 4.2 “Applications of Geodesy in Engineering” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 G. Retscher, A. Reiterer, and G. Mentes

90

A Fixed-s Digital Representation of a Random Scalar Field . . . . . . . . 725 K. Becek

91

The Impact of Adding SBAS Data on GPS Data Processing in Southeast of Brazil: Preliminary Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733 W.C. Machado, F. Albarici, E.S. Fonseca Junior, J.F.G. Monico, and W.G.C. Polezel

92

First Results of Relative Field Calibration of a GPS Antenna at BCAL/UFPR (Baseline Calibration Station for GNSS Antennas at UFPR/Brazil) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739 S.C.M. Huinca, C.P. Krueger, M. Mayer, A. Kno¨pfler, and B. Heck

93

Medium-Distance GPS Ambiguity Resolution with Controlled Failure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 D. Odijk, S. Verhagen, and P.J.G. Teunissen

94

Toward a SIRGAS Service for Mapping the Ionosphere’s Electron Density Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 ´ ngel, C. Brunini, F. Azpilicueta, M. Gende, A. Arago´n-A M. Herna´ndez-Pajares, J.M. Juan, and J. Sanz

95

Assisted Code Point Positioning at Sub-meter Accuracy Level with Ionospheric Corrections Estimated in a Local GNSS Permanent Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761 M. Crespi, A. Mazzoni, and C. Brunini

96

Semi-annual Anomaly and Annual Asymmetry on TOPEX TEC During a Full Solar Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 F. Azpilicueta, C. Brunini, and S.M. Radicella

97

Numerical Simulation and Prediction of Atmospheric Aerosol Extinction Using Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 775 J. Shin, S. Lim, C. Rizos, and K. Zhang

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98

Impact of Atmospheric Delay Reduction Using KARAT on GPS/PPP Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781 Ryuichi Ichikawa, Thomas Hobiger, Yasuhiro Koyama, and Tetsuro Kondo

99

Modelling Tropospheric Zenith Delays Using Regression Models Based on Surface Meteorology Data . . . . . . . . . . . . . . . . . . . . . . . . . . 789 Tama´s Tuchband and Szabolcs Ro´zsa

100

Calibration of Wet Tropospheric Delays in GPS Observation Using Raman Lidar Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795 P. Bosser, C. Thom, O. Bock, J. Pelon, and P. Willis

101

Generation of Slant Tropospheric Delay Time Series Based on Turbulence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801 M. Vennebusch and S. Scho¨n

102

Fitting of NWM Ray-Traced Slant Factors to Closed-Form Tropospheric Mapping Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809 Landon Urquhart, Marcelo Santos, and Felipe Nievinski

103

Estimation of Integrated Water Vapour from GPS Observations Using Local Models in Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 Sz. Ro´zsa

104

GNSS Remote Sensing in the Atmosphere, Oceans, Land and Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825 Shuanggen Jin

105

Mean Sea Surface Model of the Caspian Sea Based on TOPEX/Poseidon and Jason-1 Satellite Altimetry Data . . . . . . . . . 833 S.A. Lebedev

Session 5 Geodesy in Latin America Convenors: D. Blitzkow, C. Tocho 106

Combination of the Weekly Solutions Delivered by the SIRGAS Processing Centres for the SIRGAS-CON Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845 L. Sa´nchez, W. Seem€uller, and M. Seitz

107

Report on the SIRGAS-CON Combined Solution, by IBGE Analysis Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853 S.M.A. Costa, A.L. Silva, and J.A. Vaz

108

Processing Evaluation of SIRGAS-CON Network by IBGE Analysis Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 S.M.A. Costa, A.L. Silva, and J.A. Vaz

109

ProGriD: The Transformation Package for the Adoption of SIRGAS2000 in Brazil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869 Marcos F. Santos, Marcelo C. Santos, Leonardo C. Oliveira, Sonia A. Costa, Joa˜o B. Azevedo, and Maurı´cio Galo

xviii

Contents

110

The New Multi-year Position and Velocity Solution SIR09P01 of the IGS Regional Network Associate Analysis Centre (IGS RNAAC SIR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877 W. Seem€ uller, M. Seitz, L. Sa´nchez, and H. Drewes

111

Analysis of the Crust Displacement in Amazon Basin . . . . . . . . . . . . . . . 885 G.N. Guimara˜es, D. Blitzkow, A.C.O.C. de Matos, F.G.V. Almeida, and A.C.B. Barbosa

112

The Progress of the Geoid Model for South America Under GRACE and EGM2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 D. Blitzkow, A.C.O.C. de Matos, J.D. Fairhead, M.C. Pacino, M.C.B. Lobianco, and I.O. Campos

113

Combining High Resolution Global Geopotential and Terrain Models to Increase National and Regional Geoid Determinations, Maracaibo Lake and Venezuelan Andes Case Study . . . . . . . . . . . . . . . . 901 E. Wildermann, G. Royero, L. Bacaicoa, V. Cioce, G. Acun˜a, H. Codallo, J. Leo´n, M. Barrios, and M. Hoyer

114

Evaluation of a Few Interpolation Techniques of Gravity Values in the Border Region of Brazil and Argentina . . . . . . . . . . . . . . . 909 R.A.D. Pereira, S.R.C. De Freitas, V.G. Ferreira, P.L. Faggion, D.P. dos Santos, R.T. Luz, A.R. Tierra Criollo, and D. Del Cogliano

115

RBMC in Real Time via NTRIP and Its Benefits in RTK and DGPS Surveys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 S.M.A. Costa, M.A. de Almeida Lima, N.J. de Moura Jr, M.A. Abreu, A.L. de Silva, L.P. Souto Fortes, and A.M. Ramos

Session 6 Joint ION/FIG/ISPRS Session on Navigation and Earth Observation Convenors: D.A. Grejner-Brzezinska, C.K. Toth 116

Bootstrapping with Multi-frequency Mixed Code Carrier Linear Combinations and Partial Integer Decorrelation in the Presence of Biases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925 P. Henkel

117

Real Time Satellite Clocks in Precise Point Positioning . . . . . . . . . . . . . 935 R.J.P. van Bree, S. Verhagen, and A. Hauschild

118

Improving the GNSS Attitude Ambiguity Success Rate with the Multivariate Constrained LAMBDA Method . . . . . . . . . . . . . . 941 G. Giorgi, P.J.G. Teunissen, S. Verhagen, and P.J. Buist

119

An Intelligent Personal Navigator Integrating GNSS, RFID and INS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949 G. Retscher

120

Integration of Image-Based and Artificial Intelligence Algorithms: A Novel Approach to Personal Navigation . . . . . . . . . . . . . 957 Dorota A. Grejner-Brzezinska, Charles K. Toth, J. Nikki Markiel, Shahram Moafipoor, and Krystyna Czarnecka

Contents

xix

121

Modernization and New Services of the Brazilian Active Control Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 L.P.S. Fortes, S.M.A. Costa, M.A. Abreu, A.L. Silva, N.J.M Ju´nior, K. Barbosa, E. Gomes, J.G. Monico, M.C. Santos, and P. Te´treault

122

magicSBAS: A South-American SBAS Experiment with NTRIP Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973 I. Alcantarilla, J. Caro, A. Cezo´n, J. Ostolaza, and F. Azpilicueta

Session 7 The Global Geodetic Observing System: Science and Applications Convenors: R. Gross, H.-P. Plas, L.P. Forles 123

Scientific Rationale and Development of the Global Geodetic Observing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987 G. Beutler and R. Rummel

124

GGOS Bureau for Standards and Conventions: Integrated Standards and Conventions for Geodesy . . . . . . . . . . . . . . . . 995 U. Hugentobler, T. Gruber, P. Steigenberger, D. Angermann, J. Bouman, M. Gerstl, and B. Richter

125

VLBI2010: Next Generation VLBI System for Geodesy and Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999 W.T. Petrachenko, A.E. Niell, B.E. Corey, D. Behrend, H. Schuh, and J. Wresnik

126

The New Vienna VLBI Software VieVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 J. Bo¨hm, S. Bo¨hm, T. Nilsson, A. Pany, L. Plank, H. Spicakova, K. Teke, and H. Schuh

127

Estimating Horizontal Tropospheric Gradients in DORIS Data Processing: Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013 P. Willis, Y.E. Bar-Sever, and O. Bock

Session 8

The IAG International Services and their Role for Earth Observation Convenors: R. Neilan, R. Forsberg 128

The BIPM: International References for Earth Sciences . . . . . . . . . . 1023 E.F. Arias

129

Development of the GLONASS Ultra-Rapid Orbit Determination at Geodetic Observatory Pecny´ . . . . . . . . . . . . . . . . . . . . . 1029 J. Dousa

130

AGrav: An International Database for Absolute Gravity Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 H. Wziontek, H. Wilmes, and S. Bonvalot

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1043

.

Session 1 Reference Frames Implementation for Geoscience’s Applications: From Local to Global Scales Convenors: Z. Altamimi, C. Brunini

.

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame C. Brunini, L. Sanchez, H. Drewes, S. Costa, V. Mackern, W. Martı´nez, W. Seemuller, and A. da Silva

Abstract

The SIRGAS reference system is at present realized by the SIRGAS Continuously Operating Network (SIRGAS-CON) composed by about 200 stations distributed over Latin America and the Caribbean. SIRGAS member countries are improving their national reference frames by installing continuously operating GPS stations, which have to be consistently integrated into the continental network. As the number of these stations is rapidly increasing, the analysis strategy of the SIRGAS-CON network is based on two hierarchy levels: a) A core network with homogeneous continental coverage and stable site locations ensures the long-term stability of the reference frame. This network is processed by DGFI (Germany) as the IGS RNAAC SIR. b) Several densification sub-networks (corresponding to the national reference networks) improve the accessibility to the reference frame in the individual countries. Currently, the SIRGAS-CON stations are classified in three densification sub-networks (a southern, a middle, and a northern one), which are processed by the SIRGAS Local Processing Centres CIMA (Argentina), IBGE (Brazil), and IGAC (Colombia). These four Processing Centres deliver loosely constrained weekly solutions for the assigned sub-networks, which are integrated in a unified solution by the SIRGAS Combination Centres (DGFI and IBGE). The main SIRGAS products are: loosely constrained weekly solutions in SINEX format for further combinations of the

C. Brunini (*) Universidad Nacional de La Plata (UNLP), La Plata, Argentina e-mail: [email protected] L. Sanchez H. Drewes W. Seemuller Deutsches Geod€atisches Forschungsinstitut (DGFI), Munich, Germany S. Costa A. da Silva Instituto Brasileiro de Geografia e Esta´tistica (IBGE), Rio de Janeiro, Brazil V. Mackern Universidad Nacional de Cuyo, Mendoza (CIMA), Argentina W. Martı´nez Instituto Geogra´fico Agustı´n Codazzi (IGAC), Bogota, Colombia S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_1, # Springer-Verlag Berlin Heidelberg 2012

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C. Brunini et al.

network, weekly positions aligned to the ITRF as reference for GPS positioning in Latin America; and multi-year solutions (positions + velocities) for practical and scientific applications requiring time-dependent coordinates. This paper describes the analysis of the SIRGAS-CON network as the current realization of the SIRGAS reference system, its quality and consistency, as well as the planned activities to continue improving this reference frame.

1.1

Introduction

SIRGAS (Sistema de Referencia Geoce´ntrico para las Ame´ricas) as a reference system is defined identical with the ITRS (International Terrestrial Reference System). It is realized by means of a regional densification of the global ITRF (International Terrestrial Reference Frame) in Latin America and the Caribbean. The SIRGAS reference frame is in the same way extended to the countries through national densifications, which provide accessibility to the reference frame at national and local levels (Sanchez and Brunini 2009). SIRGAS has three realizations: two by means of episodic GPS campaigns and one by means of a network of continuously operating GPS stations. The first realization of SIRGAS (SIRGAS95) refers to the ITRF94, epoch 1995.4. It is given by a high-precision GPS network of 58 points distributed over South America (SIRGAS, 1997). In 2000, this network was re-measured and extended to the Caribbean, Central and North American countries. This second realization (SIRGAS2000) includes 184 GPS stations and refers to the ITRF2000, epoch 2000.4 (Drewes et al. 2005). The third realization of SIRGAS is the SIRGAS Continuously Operating Network (SIRGAS-CON), which is at present composed by more than 200 permanently operating GPS sites. The SIRGAS-CON network is weekly computed by the SIRGAS Analysis Centres; main products of this computation are: loosely constrained weekly solutions for station positions to be included in the IGS (International GNSS Service) global polyhedron and in multi-year solutions of the network; weekly station positions aligned to the ITRF for further applications in Latin America; and multi-year solutions providing station positions and velocities for high-precise practical and scientific applications. The SIRGASCON weekly positions refer to the observation epoch and to the current frame in which the GPS satellite orbits (i.e. IGS final orbits, Dow et al. 2009) are

given, at present the IGS05, the IGS realization of the ITRF2005 (see IGSMAIL 5447, http://igscb.jpl.nasa. gov/). The coordinates of the multi-year solutions refer to the latest available ITRF and to a specified epoch, e.g. the most recent SIRGAS-CON multi-year solution SIR09P01 refers to IGS05, epoch 2005.0 (Seemuller et al. 2009). The relationship between the different SIRGAS realizations is given by the transformation parameters between the corresponding ITRF solutions they refer and by taking into account the station position variations with time through a velocity (deformation) model (Drewes and Heidbach 2005). In this way, realizations or densifications of SIRGAS associated to different ITRFs and reference epochs materialize the same reference system and, after reducing them to the same frame and epoch, their positions are compatible at the mm-level. The present paper summarizes the improvement of the SIRGAS realization by means of the SIRGASCON network, the applied procedures for generating weekly solutions of this reference frame, quality and consistency of the obtained coordinates, as well as ongoing activities to avoid some disadvantages of the actual analysis strategy. The multi-year solutions for the SIRGAS-CON reference frame are presented by e.g. Seemuller et al. 2008, 2009.

1.2

The SIRGAS-CON Reference Frame

The initial realizations of SIRGAS based on pillars have been replaced by an increasing number of continuously operating GPS stations (Fig. 1.1), which all together constitute the SIRGAS-CON network (Fig. 1.2). 48 of these stations belong to the IGS global network, while the others (about 160) correspond to the national reference frames.

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame

Fig. 1.1 SIRGAS continuously operating stations since 2000

5 IGS

No. of stations

Regional

Total

200 150 100 50 0 2000

2001

2002

2003

2004

2005

2006

2007

2008 06/2009

Year

To guarantee the consistency of the national reference frames with the global reference frame in which the GPS orbits are computed, the national reference stations are integrated into the SIRGAS-CON network and all together are processed in a common way. This provides homogeneous consistency and accuracy of their coordinates on a continental level. Until GPS week 1495 (August 2008), the Deutsches Geod€atisches Forschungsinstitut (DGFI, Germany), as the IGS RNAAC SIR (IGS Regional Network Associate Analysis Centre for SIRGAS), processed the entire SIRGAS-CON network in one block only (Seemuller and Drewes 2008). However, given the large number of SIRGAS-CON stations, this usual one-block processing became unfeasible and it was necessary to redefine the analysis strategy of the network. The new analysis strategy is based on (1) defining a core continental network (SIRGAS-CON-C) as the primary densification of the ITRF in Latin America, and (2) improving the geographical density of this core network by means of densification sub-networks (SIRGASCON-D). The core network ensures the long-term stability of the continental reference frame, while the densification sub-networks make it available at national and local levels. Although, they appear as two different categories, core and densification stations match requirements, characteristics, performance, and quality of the ITRF stations. The SIRGAS-CON-D sub-networks shall correspond to the national reference frames, i.e., as an optimum there shall be as many sub-networks as countries in the region. Since at present not all of the countries are operating a Processing Centre, the existing stations are classified in three densification sub-networks (Fig. 1.2): a northern one covering Mexico, Central America, the Caribbean, Colombia, and Venezuela; a

middle one comprising stations installed on Brazil, Ecuador, Bolivia, Suriname, French Guyana, Guyana, Peru, and Bolivia; and a southern one including the stations located in Uruguay, Paraguay, Argentina, Chile, and Antarctica. Each densification sub-network includes a minimum number of IGS and SIRGASCON core stations as overlapping points for the combination.

1.3

Analysis of the SIRGAS-CON Network

The SIRGAS-CON-C network is computed by DGFI. The densification sub-networks are processed by the active SIRGAS Local Processing Centres until new ones become operational. At present, they are: Centro de Procesamiento Ingenierı´a-Mendoza-Argentina at the Universidad Nacional del Cuyo (CIMA, Argentina), Instituto Brasileiro de Geografia e Estatistica (IBGE, Brazil), and Instituto Geogra´fico Agustı´n Codazzi (IGAC, Colombia). These four Processing Centres apply a common procedure established by SIRGAS (in agreement with the standards of the IGS and the IERS – International Earth Rotation and Reference Systems Service) to generate loosely constrained weekly solutions for station positions (see e.g. Natali et al. 2009; Seemuller and Sanchez 2009). In these solutions satellite orbits, satellite clock offsets, and Earth orientation parameters are fixed to the final weekly IGS solutions (Dow et al. 2009) and all station positions are constrained to 1 m. The individual contributions are integrated in a unified solution by the SIRGAS Combination Centres DGFI and IBGE (Fig. 1.3). The DGFI combinations are provided to the users as the SIRGAS official

6

C. Brunini et al.

Fig. 1.2 SIRGAS-CON network (status August 2009)

products (Sanchez et al. 2011), while the IBGE combinations assure redundancy and control for those products (Costa et al. 2009). At present, all SIRGAS Analysis Centres use the Bernese GPS Software (Dach et al. 2007) for processing the individual sub-networks and for their combination. Before combining the individual solutions, the constraints included in the delivered normal equations

are removed and the solutions are separately aligned to the IGS05 reference frame. The obtained standard deviations are analysed to establish the quality of the individual solutions and to determine variance factors, when it is necessary to compensate differences in the stochastic models of the Processing Centres. The station positions computed from each solution are compared by means of a similarity transformation to the

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame

7

Fig. 1.3 Data flow in the weekly processing of the SIRGASCON reference frame

IGS weekly values and to each other to identify possible outliers. Once inconsistencies and outliers are reduced from the individual free normal equations, a combination for a loosely constrained weekly solution for station positions (all of them constrained to 1 m) is computed. This solution is submitted in SINEX format to IGS for the global polyhedron and it is stored to be included in the next multi-year solution of the SIRGAS-CON network. A solution aligned to the IGS05 reference frame is also computed to provide weekly positions of all SIRGAS-CON stations for further applications (Fig. 1.4). Different criteria are applied to establish the quality of the contributing solutions delivered by the SIRGAS Processing Centres. The first one relates to the determination of mean standard deviation of station positions by solving the individual normal equations with respect to the IGS05 frame. These standard deviations represent the formal errors of the individual solutions. Secondly, the analysis of station position time series allows ascertaining the consistency of the individual contributions from week to week (repeatability). Then, the comparison by means of a similarity transformation of the individual solutions referring to the IGS05 with the IGS weekly positions provides information about their compatibility with the IGS global network. Figure 1.5 presents the mean values

Fig. 1.4 Combination procedure applied to generate the weekly solution of the SIRGAS-CON reference frame

for each criterion and for each sub-network for the period between the GPS weeks 1495 and 1538. These results indicate that the individual solutions are at the same level of precision: the formal error of the station positions is about 1.6 mm and the repeatability of the weekly coordinates is estimated to be ~ 2.0 mm for the horizontal component and 4.0 mm in the height.

1.4

Weekly Processing of the SIRGAS-CON Reference Frame

Regional and national reference frames supporting GNSS positioning must be consistent with the reference frame in which the GPS orbits are determined. For that reason, the IGS RNAAC SIR yearly generates a new multi-year solution referred to the current ITRF realization and including the SIRGAS-CON stations operating more than 2 years (e.g. Seemuller et al 2008, 2009). The latest solution SIR09P01 contains 128 stations with positions and velocities referring to IGS05, epoch 2005.0 (Seemuller et al. 2009). However, as mentioned before, the SIRGAS-CON network

8

Fig. 1.5 Evaluation of the solutions computed for the SIRGASCON individual sub-networks (mean values for the period GPS weeks 1495–1538)

is composed by more than 200 stations and those stations (about 80) that are not included in SIR09P01 can be used as reference points only, if their weekly positions linked to the ITRF (i.e. IGS05) are available. In this way, weekly solutions of the SIRGAS-CON network aligned to the IGS05 frame are necessary. Usually, epoch solutions (daily, weekly, multi-year) of regional reference networks are aligned to the ITRF using a set of fiducial stations with known positions and constant velocities; i.e. they consider linear coordinate changes only. However, GPS stations show significant seasonal position variations (mainly in the up component) resulting from a combination of geophysical loading and systematic errors. Ignoring these seasonal variations at reference stations can introduce systematic errors in the datum realization and the reference networks can be significantly deformed. These effects are larger in regional networks than in global ones, especially in zones with strong seasonal variations as the SIRGAS region. In this way, with the objective of minimizing the influence of seasonal

C. Brunini et al.

variations in the weekly realization of the SIRGASCON frame, the SIRGAS Working Group I (Reference System) analyzed different strategies for the datum definition taken into account the minimal network deformation, the weekly repeatability of station positions, and the consistency with the IGS weekly solutions for the global network. This analysis basically consisted of solving the same free normal equations applying two different sets of reference coordinates for the datum definition: the first one corresponds to the IGS05 positions at epoch 2000.0 extrapolated to the observation epoch using the ITRF2005 constant velocities. The second set corresponds to the weekly positions determined for the IGS05 reference stations within the IGS weekly combination (igsyyPwwww. snx). After comparing the loosely constrained solutions (in which the network is not deformed) with the constrained ones, the main conclusion shows that applying constant velocities to the reference coordinates introduces the largest distortions (more than 5 mm) into the station positions, mainly at the fiducial points (Fig. 1.6). This is a consequence of constraining a seasonal signal to be a linear trend. In this way, the SIRGAS-CON weekly solutions are aligned to the IGS05 by constraining the positions of the reference IGS05 stations to the values resulting of the IGS weekly combinations (Sanchez et al. 2011). The quality control of the SIRGAS-CON weekly solutions takes into account (1) the mean standard deviation for station positions to estimate the formal error of the final values; (2) the coordinate repeatability after combining the individual solutions to evaluate the internal consistency of the combined network; (3) station position time series analysis to determine the consistency of the combined solutions from week to week; and the (4) comparison with the IGS weekly coordinates and the IBGE weekly combinations to ascertain the reliability of the weekly solutions as well as to guarantee the required redundancy for the generation of the final SIRGAS-CON weekly positions. Table 1.1 summarizes the mean values resulting from the evaluation criteria for the period covering the GPS weeks 1495–1538. The mean standard deviation of the combined solutions is very similar to those obtained for the individual contributions (Fig. 1.5), i.e. their quality is maintained and their combination does not generate distortions in the SIRGAS-CON weekly realization. The weekly repeatability of the resulting

1

Improved Analysis Strategy and Accessibility of the SIRGAS Reference Frame

Fig. 1.6 3D residuals derived after comparing the primary loosely constrained solutions for the SIRGAS-CON network with the same solutions aligned to the IGS05 frame applying two different sets of reference coordinates for the datum definition. In (a) the IGS05 station positions at 2000.0 are

9

extrapolated to the observation epoch by means of the ITRF2005 constant velocities. In (b) the station positions computed within the IGS weekly combinations for the IGS05 stations are directly introduced as reference coordinates. Values presented on the maps are mean values for 117 weeks

Table 1.1 Evaluation of the SIRGAS-CON weekly realizations positions provides an estimate of the internal consis(mean values for the period between GPS weeks 1495–1538) tency of approximately 0.8 mm in the horizontal Criteria Component Value in components and 2.5 mm in the vertical one. The [mm] RMS values derived from the time series for station Mean standard deviation 1.64 coordinates and with respect to the IGS weekly 0.61 Mean RMS of residuals for coordinate N positions indicate that the external accuracy of the repeatability in the weekly combination E 0.87 network is about 1.5 mm in the horizontal position Up 2.51 and 3.8 mm in the height. Total 2.73 Mean RMS of residuals derived from N 1.50 time series of station positions E 1.36 Up 3.80 1.5 Closing Remarks and Outlook Total 4.33 RMS of residuals wrt IGS weekly N 1.39 The processing strategy described in this paper for the solutions E 1.75 SIRGAS-CON network is applied since GPS week Up 3.69 1495. As already mentioned, before (since June 1996 Total 4.35 to August 2008), the entire SIRGAS-CON network RMS of station coordinate differences N 1.10 was computed by DGFI in one adjustment. In order between DGFI and IBGE combinations E 1.10 to establish the consistency of the current combined Up 1.40 solutions with the previous computations, residual Total 2.20

position time series were generated from the weekly

10

solutions available between January 2000 and January 2009. Discontinuities or jumps at the epoch in which the analysis strategy was changed (last week of August 2008) are not identifiable. Results show that the current weekly combined solutions are at the same accuracy level and totally consistent with the previous computations (when the network was calculated in one block). Nevertheless, the present sub-network distribution has two main disadvantages: (1) Not all SIRGASCON stations are included in the same number of individual solutions, i.e., they are unequally weighted in the weekly combinations, and (2) since there are not enough Local Processing Centres, the required redundancy (each station processed by at least three processing centres) is not fulfilled. Therefore, SIRGAS promotes the installation of more Local Processing Centres hosted by Latin American countries. In this frame, institutions interested to install a SIRGAS Processing Centre shall pass a test period of one year. In this period, they have to align their processing strategies with the SIRGAS guidelines and meet the delivering deadlines. At present, there are five Experimental Processing Centres: Instituto Geogra´fico Militar of Ecuador (IGM, Ecuador), Laboratorio de Geodesia Fı´sica y Satelital at the Universidad del Zulia (LGFS-LUZ, Venezuela), Servicio Geogra´fico Militar of Uruguay (SGM, Uruguay), Instituto Nacional de Estadı´stica y Geografı´a (INEGI, Mexico), and Instituto Geogra´fico Nacional de Argentina (IGN, Argentina). They will become Official Processing Centres in the near future and a redistribution of the SIRGAS-CON stations between the operative SIRGAS Analysis Centres will allow including each regional station in the same number of individual solutions. This will significantly improve the reliability and quality control of the weekly solutions for the SIRGAS-CON reference frame.

References Costa SMA, da Silva AL, Vaz JA (2009) Report of IBGE Combination Centre. Period of SIRGAS-CON solutions:

C. Brunini et al. from week 1495 to 1531. Presented at the SIRGAS 2009 General Meeting. Buenos Aires, Argentina. September. Available at www.sirgas.org. Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS Software Version 5.0 – Documentation. Astronomical Institute, University of Berne, p 640 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a hanging landscape of Global Navigation Satellite Systems. J Geodesy 83:191–198. doi:10.1007/s00190-0080300-3 Drewes H, Kaniuth K, Voelksen C, Alves Costa SM, Souto Fortes LP (2005) Results of the SIRGAS campaign 2000 and coordinates variations with respect to the 1995 South American geocentric reference frame. In: Sanso F (ed) A window on the future of geodesy, vol 128, IAG symposia. Springer, Heidelberg, pp 32–37 Drewes H, Heidbach O (2005) Deformation of the South American crust estimated from finite element and collocation methods. In: Sanso F (ed) A window on the future of geodesy, vol 128, IAG Symposia. Springer, Heidelberg, pp 544–549 Natali MP, M€uller M, Ferna´ndez L, Brunini C (2009) CPLat: the pilot processing center for SIRGAS in Argentina. In: Drewes H (ed) Geodetic reference frames, vol 134, IAG Symposia. Springer, Heidelberg, pp 179–184 Sanchez L, Brunini C (2009) Achievements and challenges of SIRGAS. In: Drewes H (ed) Geodetic reference frames, vol 134, IAG symposia. Springer, Heidelberg, pp 161–166 Sanchez LW, Seem€uller MS, Seitz M (2011) Combination of the weekly solutions delivered by the SIRGAS Processing Centres for the SIRGAS-CON reference frame. In: Kenyon S et al (eds) Geodesy for planet Earth, Buenos Aires Argentina, vol 136, IAG symposia. Springer, Heidelberg Seemuller W, Drewes H (2008) Annual Report 2003–2004 of IGS RNAAC SIR. In: IGS 2001–02 Technical Reports, IGS Central Bureau, (eds) Jet Propulsion Laboratory, Pasadena, CA. Available at http://igscb.jpl.nasa.gov/igscb/resource/ pubs/2003-2004_IGS_Annual_Report.pdf Seemuller W, Krugel M, Sanchez L, Drewes H (2008) The position and velocity solution DGF08P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 79. DGFI, Munich. Available at www.sirgas.org Seemuller W, Seitz M, Sanchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at www.sirgas.org Seemuller W, Sanchez L (2009) SIRGAS Processing Centre at DGFI: report for the SIRGAS 2009 General Meeting. Presented at the SIRGAS 2009 General Meeting. Buenos Aires, Argentina, September. Available at www.sirgas.org SIRGAS (1997). SIRGAS Final Report; Working Groups I and II IBGE, Rio de Janeiro, p 96. Available at www.sirgas.org

2

Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring Alvaro Santamarı´a-Go´mez, Marie-Noe¨lle Bouin, €ppelmann and Guy Wo

Abstract

The University of La Rochelle (ULR) TIGA Analysis Center (TAC) completed a new global reprocessed solution spanning 13 years with more than 300 GPS permanent stations, 216 of them being co-located with tide gauges. A state-of-theart GPS processing strategy was applied, in particular, the station sub-networks used in the daily processing were optimally built. Station vertical velocities were estimated in the ITRF2005 reference frame by stacking the weekly position estimates. Outliers, offsets and discontinuities in time series were carefully examined. Vertical velocities uncertainties were assessed in a realistic way by analysing the type and amplitude of the noise content in the residual position time series. The comparison shows that the velocity uncertainties have been reduced by a factor of 2 with respect to previous ULR solutions. The analysis of this solution and its by-products shows the high geodetic quality achieved in terms of homogeneity, precision and consistency with respect to other top-level geodetic solutions.

2.1

A. Santamarı´a-Go´mez (*) Instituto Geogra´fico Nacional, c/ General Iban˜ez Ibero 3, 28071, Madrid, Spain Institut Ge´ographique National, LAREG/GRGS, 6–8 av. Blaise Pascal, 77455, Champs-sur-Marne, France e-mail: [email protected] M.-N. Bouin Institut Ge´ographique National, LAREG/GRGS, 6–8 av. Blaise Pascal, 77455, Champs-sur-Marne, France CNRM/CMM, Me´te´o France, 13 rue du Chatellier, CS 12804, 29228, Brest, France G. W€oppelmann Universite´ de La Rochelle-CNRS, UMR 6250 LIENSS, 2 rue Olympe de Gouges, 17000, La Rochelle, France

Introduction

In order to estimate long-term geocentric sea level rise, tide gauges trends must be corrected for the long-term vertical displacements of the land upon which they are settled. In addition, for proper satellite altimeter calibration purposes, tide gauges trends must be referred to a common, global and stable reference frame, such as the latest realization of the International Terrestrial Reference Frame (ITRF) (Altamimi et al. 2007). These long-term vertical displacements can be corrected by modelling geological processes as the Global Isostatic Adjustment (GIA) (e.g. Douglas 2001) or directly from continuous geodetic observations at or near tide gauges. This second method should be preferred as it takes into account local displacements (geological, anthropogenic or whatever), not accounted

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_2, # Springer-Verlag Berlin Heidelberg 2012

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A. Santamarı´a-Go´mez et al.

12

Fig. 2.1 Number of daily available stations (grey) and percentage of daily resolved ambiguities (black)

for in the GIA models. Within the different geodetic techniques used for this purpose (GPS, DORIS and absolute gravity), GPS is the most widespread. Recent studies (W€ oppelmann et al. 2009, Bouin and W€oppelmann 2010) have shown that correcting the tide gauge trends using continuous GPS stations ([email protected]) improves the consistency of the sea level rates. To this aim, the International GNSS Service (IGS) Tide Gauge Benchmark Monitoring Pilot Project (TIGA) was established in 2001 (Sch€ one et al. 2009). Since 2002, the ULR consortium contributes to the TIGA project as an Analysis and Data Center (W€oppelmann et al. 2004). Several global vertical velocity field solutions (ULR solutions hereafter) were released with different station networks, time spans and processing strategies (W€oppelmann et al. 2007, 2009). In this paper, we present the fourth ULR solution based on an homogeneous reprocessing of a larger global network of 316 stations, spanning an increased period of 13 years (January 1996 to December 2008). This solution comes out with a new data analysis strategy, including a new sub-network design and combination. The troposphere and ocean tide modelisation were also improved. Both GPS processing and vertical velocity estimation strategies are described; realistic uncertainties are estimated by analysing the noise content of time series. Finally, the quality of the solution is assessed and discussed.

2.2

Data Analysis Strategy

2.2.1

Data

The global tracking network consists of 316 GPS stations. 216 of them are [email protected], including 81 stations committed to TIGA. Also 124 of them are

IGS reference frame (RF) stations used for realizing the reference frame (Kouba et al. 1998) and for improving the network geometry. This network was processed over the period 1st January 1994 to 31st December 2008. Small RINEX files (less than 5 h of observation) were rejected. This quality check procedure yielded a number of daily available stations between a minimum of 25 in 1994 (53 in 1996) and a maximum of 239 in 2006 (grey line in Fig. 2.1). 1994 and 1995 were finally not retained in the solution due to a lack of fixed ambiguities and therefore quality (black line in Fig. 2.1) and they will not be further considered.

2.2.2

Improved Network Geometry

GPS processing time increases exponentially with the number of stations. To overcome this limitation, it is usual to split the whole network in several subnetworks, to process each sub-network independently and then to combine the sub-network solutions into a unique daily solution. Historic ULR solutions (ULR1 to ULR3 solutions) used five global, manually-selected, permanent subnetworks over the entire data span (“static subnetworks” hereafter). Using this approach, the a priori stations included in each sub-network were always the same, whether or not their data were available for a specific day, making the geometry worse when their data were missing, and therefore, possibly yielding an unnecessary large number of sub-networks in the processing (always five). This static configuration was changed in the ULR4 solution into a new station distribution approach resulting in global, automatic, daily-variable sub-networks (“dynamic sub-networks” hereafter), with up to 50 stations per sub-network.

2

Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring

Shorter baselines improves ambiguity resolution (Steigenberger et al. 2006). With the dynamic approach, all daily available stations were distributed into the strictly necessary number of sub-networks, ensuring optimal dense sub-networks. Thus, the number of dynamic sub-networks used grows from 1 in 1996 to 6 in 2003. Moreover, to obtain global geometrically well-distributed sub-networks for optimal orbit estimation, each station is assigned to the sub-network where it is more isolated, i.e. reducing the baselines. In this way, “deserted” areas of each sub-network are iteratively being “populated”. In addition, six daily-variable common IGS RF stations, with more than 12 h of observation, are included in each dynamic sub-network to combine the solutions. Northernmost and southernmost stations are always selected and then four other globally welldistributed stations are added. Static versus dynamic approaches were compared by processing two solutions using the same stations and processing strategy except for the stations distribution. Figure 2.2 shows that using dynamic sub-networks clearly increases the percentage of resolved ambiguities as the number of available stations decreases, up to 20% in 1997 (Fig. 2.2). The 10% offset in the percentage of resolved ambiguities observed at the end of 1999 for both approaches is related to the use of code bias corrections (see Sect. 2.2.3), only available for post2000 year period when the test was performed.

2.2.3

Models and Parameterization

Double-differenced ionosphere-free carrier phase data is analysed using GAMIT software version

Fig. 2.2 Resolved ambiguities for static (grey), dynamic subnetworks (top black) and the difference (bottom black)

13

10.34 (Herring et al. 2006a). The elevation cut-off angle is set to 10 , avoiding mismodelling of lowelevation troposphere and phase center variations (PCV) of relative-to-absolute antenna calibration. Sampling rate is set to 3 minutes. Carrier phase observations are weighted in two iterations: by elevation angle first and then by elevation angle and by station, accounting for the station phase residuals from the first iteration. Code bias corrections are applied for the whole period using monthly tables from the Astronomical Institute of the University of Bern (AIUB) [IGSMAIL-2827 (2000) at http:// igscb.jpl.nasa.gov/mail/). Real-valued double differenced phase cycle ambiguities are adjusted except when they can be resolved confidently. In this case, they are fixed using the MelbourneW€ubbena wide-lane to resolve L1–L2 cycles and then estimation to resolve L1 and L2 cycles. For satellite antennas, satellite-specific z-offsets (Ge et al. 2005) and block-specific nadir angledependent absolute PCV (Schmid et al. 2007) are applied. For receiver antennas, L1/L2 offsets and azimuth-dependent, when available, and elevationdependent absolute PCV are applied. A priori zenith hydrostatic (dry) delay values are extracted by station from the ECMWF meteorological model through the VMF1 grids (Boehm et al. 2006). Residual delays are adjusted for each station assuming mostly dominated by the wet component and parameterized by a piecewise linear, continuous model with 2 h intervals. Both dry and wet VMF1 mapping functions are used. One gradient is estimated for each day and each station. Solid Earth tides are corrected following IERS Conventions (2003) (McCarthy and Petit, 2004). Ocean tide loading is corrected using FES2004 model (Lyard et al. 2006). No atmospheric tide nor non-tidal corrections were applied. Earth orientation parameters (EOP) are daily estimated as a piecewise, linear model with a priori values from IERS Bulletin B. UT1–UTC offsets are highly constrained to their a priori values. Satellite positions and velocities are adjusted in 24 h arcs taking IGS final orbits (Dow et al. 2005) as a priori. Solar radiation pressure parameters are estimated using the Berne model (Beutler et al. 1994).

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2.2.4

Data Processing Scheme and Reference Frame

Each dynamic sub-network is processed independently using GAMIT software. The daily sub-network solutions are combined into a daily solution (by estimating only translations and rotations) using GLOBK (Herring et al. 2006b) by means of the estimated orbital parameters, the estimated positions of the six common stations and their estimated zenith tropospheric path delays. Daily loose solutions are constrained by no-net-rotation (NNR) constraints with respect to ITRF2005 and combined into a weekly solution using CATREF software (Altamimi et al. 2007). These weekly solutions are aligned to ITRF2005 using NNR constraints with all IGS RF stations available, whereas inner constraints (Altamimi et al. 2007) are used for scale and translation, in order to preserve the weekly apparent geocenter motion information. All the weekly solutions for the whole period (GPS weeks 0834–1512), are then combined into a longterm solution using CATREF. This long-term solution (ULR4) is aligned to ITRF2005 using minimal constraints over all the transformation parameters with a selected set of IGS RF stations called datum. The 68 stations retained in the datum were selected based on their data availability (at least present in 80% of the whole processed period) and their quality as follows. Firstly, stations with known or suspected velocity discontinuities were rejected, and secondly, in an iterated process, stations showing large position and velocity residuals with respect to ITRF2005 values were also rejected. Thresholds for positions were set to 0.5 cm in horizontal and 1.5 cm in vertical. The larger value in the vertical component is due to the fact that ITRF2005 GPS coordinates were estimated with a relative PCV model. Station differences using the absolute PCV model are estimated to be within this range. Thresholds for velocity residuals were set to 1.5 mm/year and 2 mm/year respectively. The residual position time series of each station were visually examined. To avoid biased velocities, all discontinuities (significant offsets and velocity changes) were detected, identified if possible, and removed using ITRF2005 discontinuities as a priori. Then, all outliers were removed in an iterative process, from bigger to smaller magnitude (depending on the time series noise), down to a minimum of 2 cm for residuals and 4 for normalized residuals.

2.3

Results

2.3.1

Vertical Rates

The vertical velocity fields of ULR4 and ULR3 (W€oppelmann et al. 2009)) solutions were compared using a common set of 170 stations with more than 4.5 years of data. Figure 2.3 shows that most of the velocity differences are below 1 mm/year (RMS of 0.8 mm/year), except some stations for which larger differences are due to different discontinuities on their time series. The mean difference between both velocity fields is 0.16 0.06 mm/year which is related to the different datum used to aling the solutions. This misalignment is under the internal precision of the ITRF2005. From the complete ULR4 solution, 224 stations with more than 4.5 years of data were retained. For these stations, their estimated velocities are confidently not influenced by seasonal signals (Blewitt and Lavalle´e 2002). Nevertheless, the rate uncertainties estimated with a standard least squares algorithm (based on a Gaussian white noise process) are clearly optimistic by a factor of 3–11 (Zhang et al. 1997; Mao et al. 1999). More realistic uncertainties of the estimated velocities must account for correlated noise present in the time series. A noise analysis was performed using the Maximum Likelihood Estimation (MLE) technique (CATS software, (Williams, 2008)). Vertical velocity uncertainties were estimated using a white noise plus power law noise model. To avoid biased adjustments, time series were previously examined for periodic signals. Besides the annual and semi-annual terms, we also found and removed up to six harmonics of

Fig. 2.3 Vertical velocity difference between ULR3 and ULR4. Dashed lines represent 1 mm/year

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Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring

15

Fig. 2.4 Histogram of estimated uncertainties for ULR4 (grey) and ULR3 (black) solutions and their median values

the GPS “draconitic” period described by Ray et al. (2007). Figure 2.4 shows the histogram of the realistic vertical velocity uncertainties of the ULR4 solution with respect to the realistic uncertainties estimated for the ULR3 solution also using CATS. The improvement is close to a factor of 2. Also the factor of optimism of the formal uncertainties with respect to the realistic ones is 2–3, quite smaller than the abovementioned values. This is due to the improvement and consistency of the processing strategy presented here, which results in a noticeable reduction of the correlated noise content for the ULR4 solution compared to previous solutions.

2.3.2

Weekly Repeatability

The internal quality of the ULR4 solution was assessed by analysing the repeatability of the weekly position solutions. Figure 2.5 shows the repeatability of the time series (mean values of the weighted RMS of the weekly positions with respect to the long-term combined positions) for ULR4 and ULR3 solutions. Horizontal and vertical repeatabilities are improved in the ULR4 solution. Moreover, for the whole reprocessed period vertical repeatabilities are more stable, showing the improved ULR4 time consistency. ULR4 repeatability values are between 1 and 3 mm for the horizontal and between 4 and 6 mm for the vertical component (3D weighted RMS between 2 and 4 mm). These values are fully consistent with those of the IGS

Fig. 2.5 Horizontal (bottom) and vertical (top) weighted RMS of the weekly solutions with respect to the long-term solution for both ULR4 (black) and ULR3 (grey) solutions

combined solution (Altamimi and Collilieux, 2008), showing that ULR4 solution is comparable in quality with the ITRF2005.

2.3.3

Origin and Scale

As a satellite technique, GPS estimated origin should be coincident with the Earth’s center of mass. However this affirmation is not completely fulfilled due to remaining GPS-specific systematic errors, as the modelling of the solar radiation pressure coefficients or the unaccounted effect of higher ionospheric orders (Herna´ndez-Pajares et al. 2007). We have estimated here apparent geocenter motion using the network shift or geometric approach

A. Santamarı´a-Go´mez et al.

16

(Lavalle´e et al. 2006). Figure 2.6 shows the translation and scale parameters of the weekly solutions with respect to the long-term combined solution aligned to the ITRF2005. Translation trends are not significant, showing the consistency of the secular origin definition with respect to the ITRF2005. The scale shows no trend either, as this parameter is completely dependent on the ITRF2005 scale definition through the satellites antenna z-offset corrections. For intercomparison purposes, an annual signal was estimated for each transformation parameter (Table 2.1).

Compared to SLR results (Collilieux et al. 2009), the annual amplitudes of the equatorial components (X and Y) and the scale are fully consistent. However, the amplitude of the Z component is twice larger. Regarding the annual phase, the scale parameter is fully consistent, but all translational parameters show a shift of about 137º (4.5 months). Compared to other GPS results (Lavalle´e et al. 2006), the amplitude of the Z component and both equatorial phases are consistent. The phase of Z component exhibits larger solution-dependent variations. Both issues point probably at the above-mentioned GPS systematic errors and also at the poor performance of the network shift method used with a not-well distributed global network (Lavalle´e et al. 2006).

2.3.4

Orbits

The estimated ULR4 orbits were compared with the current official non-reprocessed IGS final orbits (Dow et al. 2005). A classic 7-parameter Helmert transformation was applied between both 24 h-arc sets. 1D RMS differences (the average of the three RMS components) were estimated for each common observed satellite and then the median daily RMS value was extracted and traced (black line, Fig. 2.7). We show that ULR and IGS orbits are in good agreement with each other, from 8.5 cm in 1996 to 1.5 cm in 2009. The same range of differences was obtained between IGS orbits and reprocessed orbits from SIO/SOPAC IGS Analysis Center (light grey line). Some smaller differences were obtained with

Fig. 2.6 Weekly translation and scale parameters with respect to the ITRF2005. Also their trends and annual signal are traced

Table 2.1 Annual signal of apparent geocenter and scale TX TY TZ Scale

Amplitude (mm) 2.3 0.2 4.2 0.3 9.9 0.8 1.8 0.1

Phase (deg) 164.6 5.4 122.2 3.5 171.3 3.5 243.2 1.6

Fig. 2.7 7-day smoothed daily RMS between final IGS orbits and ULR (black), SIO/SOPAC (light grey) and CODE/AIUB (dark grey) reprocessed orbits

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Improved GPS Data Analysis Strategy for Tide Gauge Benchmark Monitoring

reprocessed CODE/AIUB IGS Analysis Center (dark grey line) for the post-2000 period. This demonstrates that the ULR4 orbits are of the same quality as the reprocessed orbits of some of the IGS Analysis Centers.

2.4

Concluding Remarks

The new ULR4 solution is based on an homogeneous reprocessing of a global GPS network of 316 stations spanning up to 13 years of data. The processing strategy was improved with respect to past ULR solutions. Special attention was paid to the sub-network geometry distribution, which clearly improves the quality of the reprocessing by increasing the number of resolved ambiguities. The analysis of the results and by-products of this solution (vertical velocities, repeatability, transformation parameters and orbits) shows the high geodetic quality achieved. The stateof-the-art GPS processing strategy implemented fulfils the IGS requirements and recommendations. Thereby, in addition to the IGS TIGA project, the ULR consortium is participating with its latest solution to the first IGS reanalysis campaign, enabling an invaluable extension of IGS and ITRF reference frames towards tide gauges. Also, the ULR consortium is contributing to the Working Group on Regional Dense Velocity Fields of the International Association of Geodesy Subcommision 1.3. (Bruyninx 2011). Further studies will be carried out in order to assess the geophysical usefulness of this solution. For example, this global and accurate vertical velocity field may be used to separate vertical land motion trends from relative sea level trends as recorded by tide gauges. Acknowledgements The authors acknowledge two unknown reviewers who contributed to an improved paper. We also thank the invaluable technical support given by Mikael Guichard, Marc-Henri Boisis-Delavaud and Frederic Bret from the IT centre of the University of La Rochelle (ULR). The ULR computing infrastructure used for the reprocessing of the GPS data was partly funded by the European Union (Contract 31031-2008, European regional development fund). This work was also feasible thanks to all institutions and individuals worldwide that contribute to make GPS data and products freely available.

17

References Altamimi Z, Collilieux X (2008) IGS contribution to ITRF. J Geod 83(3–4):375–383 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401 Beutler G, Brockmann E, Gurtner W, Hugentobler U, Mervart L, Rothacher M (1994) Extended orbit modeling techniques at the CODE Processing Center of the International GPS Service for Geodynamics (IGS): theory and initial results. Manuscripta Geodaetica 19:367–386 Blewitt G, Lavalle´e D (2002) Effect of annual signals on geodetic velocity. J Geophys Res 107:B02145 Boehm J, Werl B, Schuh H (2006) Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for Medium-Range Weather Forecasts operational analysis data. J Geophys Res 111:B02406 Bouin M-N, W€oppelmann G (2010) Land motion estimates from GPS at tide gauges: a geophysical evaluation. Geophys J Int 180:193–209 Bruyninx C (2011) A dense global velocity field based on GNSS observations: preliminary results. In: Kenyon S et al (eds) Geodesy for planet Earth. Springer, Heidelberg Collilieux X, Altamimi Z, Ray J, van Dam T, Wu X (2009) Effect of the satellite laser ranging network distribution on geocenter motion estimation. J Geophys Res 114:B02145 Douglas B (2001) Sea level change in the era of the recording tide gauge, vol 75, International geophysics series. Academic, San Diego, CA Dow JM, Neilan RE, Gendt G (2005) The International GPS Service: celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36:320–326 Ge M, Gendt G, Dick G, Zhang F, Reigber C (2005) Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys Res Lett 32:L06310 Herna´ndez-Pajares M, Juan JM, Sanz J, Oru´s R (2007) Secondorder ionospheric term in GPS: Implementation and impact on geodetic estimates. J Geophys Res 112:B08417 Herring TA, King RW, McClusky SC (2006a) GAMIT: Reference Manual Version 10.34. Internal Memorandum, Massachusetts Institute of Technology, Cambridge Herring TA, King RW, McClusky SC (2006b) GLOBK: Global Kalman filter VLBI and GPS analysis program Version 10.3. Internal Memorandum, Massachusetts Institute of Technology, Cambridge Kouba J, Ray J, Watkins M (1998). IGS Reference Frame realization. IGS 1998 Analysis Center Workshop – Proceedings Darmstadt. p 139 Lavalle´e D, van Dam T, Blewitt G, Clarke P (2006) Geocenter motions from GPS: A unified observation model. J Geophys Res 111:B05405 Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: modern insights from FES2004. Ocean Dyn 56:394–415 Mao A, Harrison CGA, Dixon TH (1999) Noise in GPS coordinate time series. J Geophys Res 104:2797–2816

18 McCarthy D, Petit G (2004) IERS Technical Note 32 – IERS Conventions (2003) Technical report, Verlag des Bundesamts fur Kartographie und Geodasie, Frankfurt am Main, Germany Ray R, Altamimi Z, Collilieux X, Van Dam T (2007) Anomalous harmonics in the spectra of GPS position estimates. GPS Solut 12(1):55–64 Schmid R, Steigenberger P, Gendt G, Ge M, Rothacher M (2007) Generation of a consistent absolute phase-center correction model for GPS receiver and satellite antennas. J Geod 81:781–798 Sch€one T, Sch€on N, Thaller D (2009) IGS Tide Gauge Benchmark Monitoring Pilot Project (TIGA): scientific benefits. J Geod 83:249–261 Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111:B05402 Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solut 12(2):147–153

A. Santamarı´a-Go´mez et al. W€oppelmann G, McLellan S, Bouin M-N, Altamimi Z, Daniel L (2004) Current GPS data analysis at CLDG for the IGS TIGA Pilot Project. Cahiers du Centre Europe´en Ge´odynamique & de Sismologie 23:149–154 W€oppelmann G, Martı´n Mı´guez B, Bouin M-N, Altamimi Z (2007) Geocentric sea-level trend estimates from GPS analyses at relevant tide gauges world-wide. Glob Planet Change 57(3–4):396–406 W€oppelmann G, Letretel C, Santamarı´a A, Bouin M-N, Collilieux X, Altamimi Z, Williams S, Martı´n Mı´guez B (2009) Rates of sea-level change over the past century in a geocentric reference frame. Geophys Res Lett 36:L12607 Zhang J, Bock Y, Johnson H, Fang P, Williams S, Genrich J, Wdowinski S, Behr J (1997) Southern California Permanent GPS Geodetic Array: error analysis of daily position estimates and site velocities. J Geophys Res 102:18035–18056

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results C. Bruyninx, Z. Altamimi, M. Becker, M. Craymer, L. Combrinck, A. Combrink, J. Dawson, R. Dietrich, R. Fernandes, R. Govind, T. Herring, A. Kenyeres, R. King, C. Kreemer, D. Lavalle´e, J. Legrand, L. Sa´nchez, G. Sella, Z. Shen, A. Santamarı´a-Go´mez, and €ppelmann G. Wo

Abstract

In a collaborative effort with the regional sub-commissions within IAG subcommission 1.3 “Regional Reference Frames”, the IAG Working Group (WG) on “Regional Dense Velocity Fields” (see http://epncb.oma.be/IAG) has made a first attempt to create a dense global velocity field. GNSS-based velocity solutions

C. Bruyninx (*) Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium e-mail: [email protected] Z. Altamimi Institut Ge´ographique National, Service de la recherche/ LAREG, c/o ENSG, 6 - 8 Ave. Blaise Pascal, 77455 Champssur-Marne, France M. Becker Institute of Physical Geodesy, Technische Universit€at Darmstadt, Petersenstr, 13, 64287 Darmstadt, Germany M. Craymer Geodetic Survey Division, Natural Resources Canada, 615 Booth Str., Ottawa Ontario K1A 0E9, Canada L. Combrinck HartRAO, PO Box 443, 1740 Krugersdorp South Africa, Canada A. Combrink HartRAO, PO Box 443, 1740 Krugersdorp South Africa, Canada J. Dawson Geoscience Australia, Cnr Jerrabomberra Ave. and Hindmarsh Drive, Symonston, ACT 2609, Australia R. Dietrich TU Dresden, Institut f€ ur Planetare Geod€asie, Mommsenstr, 13, 01062 Dresden, Germany R. Fernandes University of Beira Interior, IDL, R. Marqueˆs d’Aacute;vila e Bolama, 6201-001 Covilha˜, Portugal and Delft University of Technology, DEOS – PSG, PO Box 5058, 2600 GB Delft, The Netherlands

R. Govind Geoscience Australia, Cnr Jerrabomberra Ave. and Hindmarsh Drive, Symonston, ACT 2609, Australia T. Herring Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA A. Kenyeres FOMI Satellite Geodetic Observatory, POBox 585 1592 Budapest, Hungary R. King Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA C. Kreemer Nevada Bureau of Mines and Geology, and Seismological Laboratory, University of Nevada, 1664 N. Virginia Str., MS178, Reno, NV 89557, USA D. Lavalle´e Nevada Bureau of Mines and Geology, and Seismological Laboratory, University of Nevada, 1664 N. Virginia Str., MS178, Reno, NV 89557, USA J. Legrand Delft University of Technology, Faculty of Aerospace Engineering, DEOS – PSG, PO Box 5058 2600 GB Delft, Netherlands L. Sa´nchez Royal Observatory of Belgium, Ave. Circulaire 3, 1180, Brussels, Belgium G. Sella Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_3, # Springer-Verlag Berlin Heidelberg 2012

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for more than 6,000 continuous and episodic GNSS tracking stations, were proposed to the WG in reply to the first call for participation issued in November 2008. The combination of a part of these solutions was done in a two-step approach: first at the regional level, and secondly at the global level. Comparisons between different velocity solutions show an RMS agreement between 0.3 and 0.5 mm/year resp. for the horizontal and vertical velocities. In some cases, significant disagreements between the velocities of some of the networks are seen, but these are primarily caused by the inconsistent handling of discontinuity epochs and solution numbers. In the future, the WG will re-visit the procedures in order to develop a combination process that is efficient, automated, transparent, and not more complex than it needs to be.

3.1

Introduction

The Working Group “Regional Dense Velocity Fields” has been created in 2007 at the IUGG (International Union of Geodesy and Geophysics) General Assembly in Perugia, Italy. It is embedded within IAG (International Association of Geodesy) sub-commission 1.3 on “Regional Reference Frames” where it co-exists with the regional sub-commissions for Europe, South and Central America, North America, Africa, South-East Asia and Pacific, and Antarctica (Drewes et al. 2008). The long-term goal of the Working Group is to provide a globally referenced dense velocity field based on GNSS observations and linked to the multitechnique global conventional reference frame, the ITRF (International Terrestrial Reference Frame, Altamimi et al. 2007a).

Z. Shen NOAA-National Geodetic Survey (NGS), 1315 East-West Hwy, Silver Spring, MD 20910, USA A. Santamarı´a-Go´mez University of California, Department of Earth and Space Sciences, 595 Charles E Young Drive, Los Angeles, CA 900951567, USA G. W€oppelmann Instituto Geogra´fico Nacional, c/ General Iban˜ez Ibero 3, 28071 Madrid, Spain and Institut Ge´ographique National, Service de la recherche/LAREG, c/o ENSG, 6–8 Ave. Blaise Pascal, 77455 Champs-sur-Marne, France

3.2

Working Group Objectives and Work Plan

3.2.1

Objectives

The Working Group on “Regional Dense Velocity Fields” joins the efforts of groups processing local/ regional/global CORS or repeated GNSS campaigns and set up the following action items: – Define specifications and quality standards for the regional SINEX solutions and relevant meta-data (e.g. description of GNSS equipment and position/ velocity discontinuities) – Collect SINEX solutions and their meta-data – Study in-depth the individual strengths and shortcomings of local/regional and continuous/ epoch GNSS solutions to determine site velocities – Define optimal strategies for the combination of regional and global SINEX solutions – Provide dense regional velocity fields – Provide the densification of the ITRF2005 (or its successors) – Encourage participation in related symposia – Implement a web site in order to provide information on the activities and access to the products of the WG – And prepare recommendations and a comprehensive final report on the WG activities at the next IUGG General Assembly in 2011

3.2.2

Work Plan

The work plan of the WG has been divided into two major parts. During the first part, covering 2007–2009, the WG set up the initial strategy and submission

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results

guidelines, collected a first set of test solutions, and performed a first preliminary velocity combination. The working group closely links its activities with the regional sub-commissions within IAG sub-commission 1.3. Their expertise, coordination role for their region, and their capability to generate a unique cumulative solution for their region including velocity solutions from third parties (even campaigns) is essential for the WG. The initial WG strategy consisted therefore in a two-step approach. First, region coordinators (one for each region corresponding to the regions of the different regional sub-commissions) gathered sub-regional velocity solutions for their region (in accordance with the WG requirements) and combined these with, where available, the velocity solution from the regional subcommissions (e.g. EUREF, SIRGAS. . .) in order to produce one regional combined velocity solution in the SINEX format. Secondly, two combination coordinators -T. Herring (MIT, US) and D. Lavalle´e (TU Delft, Netherlands)- combined these regional SINEX solutions with the long-term solutions from global networks to generate a preliminary velocity solution tied to the ITRS. The main goal of this preliminary solution was to identify the problems that would arise and help to set strategic choices and guidelines for the future. These guidelines will be used to issue a new solution in the second part of the WG term, 2010–2011. The results of the 2007–2009 period will be presented in this paper. As mentioned in the introduction, the WG accepts velocity solutions based on CORS and repeated GNSS campaigns (under specific conditions, see Sect. 3.1). One of the strategic choices the WG group had to make from the start was to decide whether to (1) stack weekly combined regional and global (position) SINEX solutions to compute the velocities or to (2) combine cumulative regional and global (position + velocity) SINEX solutions. Considering that the WG does not have access to the weekly SINEX of many cumulative velocity solutions, it was decided to go for approach (2). This will allow us combining, if necessary, only velocities (without the positions). In addition, it will allow us a stepwise combination of regional and global solutions; it will also facilitate meta-data management and outlier detection, as these will be done at regional level. And finally, it perfectly fits in the initial frame (using region and combination coordinators) that was set up. The disadvantages of combining cumulative (position+) velocity solutions are

21

however that no coordinate time series will be available to the WG and that it will be necessary to consistently handle discontinuities, especially on frame-attachment sites.

3.3

Call for Participation

3.3.1

Initial Submission Guidelines

In order to allow inclusion of a maximum number of velocity solutions, the WG set up the following guidelines for the contributing solutions: – Minimum 2 years of continuous data or 2 campaign epochs over a 4 year period – Minimum 2.5 years of continuous data if significant seasonal signals are present – Significant number of “frame-attachment” sites, preferably observed over a period exceeding 5 years – Position/velocity discontinuities should be identical to the ones used by the (IGS) International GNSS Service (Dow et al. 2009) – Velocity constraints should be minimal or removable – SINEX format should contain full covariance information (an exception is allowed for PPP solutions only providing correlations between individual station coordinates) The detailed submission guidelines are available from the Working Group web site: http://www. epncb.oma.be/IAG/.

3.3.2

Call for Participation

A first Call for Participation (CfP) was issued at the end of 2008. Analysts, producing regional and global velocity solutions, were invited to submit their SINEX files to the Working Group. Figure 3.1 shows the map with the sites for which solutions that have been proposed following this CfP (black dots); in total more than 6,000 sites were proposed.

3.4

Input for Preliminary Combination

A preliminary velocity combination has been computed in the summer of 2009. This solution contained contributions from the region coordinators as well as one global solution.

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Fig. 3.1 Map with the sites for which velocity solutions have been proposed to the Working Group up to July 2009 (black dots). In total more than 6,000 sites have been proposed. In red: sites used in the first preliminary combination (see Sect. 3.4)

3.4.1

Regional Contributions

Several of the region coordinators prepared a velocity solution for their region to be included in the preliminary combination. The African and South & Central American contributions are based on the contribution of a single analysis center, while the solutions from Europe and South-East Asia & Pacific are combined solutions based on input from several analysis groups. More details are given below: – Africa (see Fig. 3.2): The solution includes 93 CORS and covers the period from Jan. 1996 till June 2009 (Fernandes et al. 2007). The GNSS data analysis, has been done using GIPSY/OASIS II (Zumberge et al. 1997) by applying the PPP strategy with ambiguity resolution (Blewitt 2008). GIPSY tools were also used to derive the velocity solutions. – Europe (see Fig. 3.3): The solution includes the velocities estimated from a reprocessing of the EUREF Permanent Network (EPN), maintained by the regional sub-commission for Europe (Bruyninx et al. 2009), complemented with several sub-regional velocity solutions. In total, velocity estimates for 525 sites were obtained, which is more than twice the number of the sites presently included in the EPN. All of the contributing sub-regional solutions were available in the SINEX format and the combination was done with the CATREF software (Altamimi et al. 2007b). The main problem encountered during the combination was the fact that some of the submitted sub-regional solutions did not use any discontinuities at all (more about this in Sect. 3.6).

Fig. 3.2 Sites contributing to the African velocity solution

Fig. 3.3 Sites and solutions included in the European solution: reprocessed EPN (‘96-’09) in black, AGNES (‘98-09’) in pink, AMON (‘01-’09) in green, ASI (’97-’09) in red, IGN (‘98-’09) in brown, and CEGRN (‘94-’07) blue triangles

– South and Central America (see Fig. 3.4): The solution includes about 128 CORS from the

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results

23

In addition, as indicated in Fig. 3.5, not all solutions were available in full SINEX format.

3.4.2

Fig. 3.4 Sites contributing to the South and Central America velocity solution

One global solution was included in this first test combination. This solution, from the ULR consortium (Universite´ de La Rochelle and IGN/LAREG) is based on a reprocessing of 299 CORS from Jan. 1996 till Jan. 2009 using the GAMIT software (Herring et al. 2007), see Fig. 3.6. Its main objective is the correction of vertical land movements that affect the tide gauge records (W€oppelmann et al. 2009). A key issue to achieve the accuracy requirement of the sea-level application (submm/year) is the realization of a stable and accurate reference frame. The ULR solution therefore includes a global set of reliable reference frame stations from the IGS. It includes three additional years of data and an improved data analysis strategy with respect to the previous solution (W€oppelmann et al. 2009). See details in Santamaria-Gomez et al. (this issue). The stacking of the solutions was done with CATREF. The ULR network has several sites common to the regional solutions.

3.5 Fig. 3.5 Sites and solutions contributing to the East-Asia and Pacific velocity solution. Solutions with full SINEX information are indicated with circles: PCGIAP (‘97-’06) in dark blue, SW Austr. seism. zone (‘02-’06) in light blue, and GeoScience Australia TIGA (‘97-’09) in black. The triangles indicate the stations belonging to networks providing velocity-only solutions: Tibet (‘98-’04) in green, Asia (‘94-04) in pink, Global (‘95-’07) in orange, and Indonesia (‘91-’01) in red

SIRGAS network (Seem€ uller et al. 2009) and covers the period from 2000 till 2009. The GNSS data processing, as well as the cumulative SINEX solution, have been computed using the Bernese V5.0 software (Beutler et al. 2007). – South-East Asia and Pacific (see Fig. 3.5): The solution comprises 1,156 sites resulting of a combination of several sub-regional networks. The combination was done using the CATREF software. In this solution, ensuring the consistent use of station names, particularly four-character identifiers, was a major struggle for many stations.

Global Contribution

Test Combination

The submitted regional networks and the global network were combined using two different approaches. First approach was a step-wise one: first a combination of sites that are present in at least three solutions was done. Based on the common sites from this combination, re-weighting factors for each SINEX file were estimated. The solution is then iterated with the final weights coming from the w2 of the individual solution velocity estimates with respect to the combination, using sites common to at least two solutions. The variance weights vary from about 200 to 1.6. The differences are most probably caused by the usage of different software packages and will be investigated in more detail in the future. The second approach consisted in attaching each regional network to the global network (ULR) using frame-attachment sites. This means that the global sites are not changed by the attachment of the regional sites but the regional networks are adjusted. More details on this approach are given in Davis and Blewitt (2000).

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Fig. 3.6 Sites contributing to the global ULR velocity solution

Figure 3.7 and Table 3.1 present the first comparisons between each of the regional velocity solutions and the global ULR solution. The comparison was done after performing a Helmert transformation on both positions and velocities (estimating translations and scale, together with their rate). The results clearly show that the European and South-Asia & Pacific solutions agree better with the ULR solution than the SIRGAS and African solutions. This does not necessarily mean that the quality of the latter solutions is worse than the first ones, but reflects more the fact that inconsistent discontinuity epochs and solution numbers are used between the last two solutions and the ULR. This is confirmed by the fact that the SIRGAS and African solutions also have a significantly larger position RMS w.r.t. to the ULR solution compared to the European and South-Asia & Pacific solution. Detailed maps of the comparison between the different solutions are available from the WG web site at http://epncb.oma.be/IAG/

3.6

Difficulties

Not all sites included in the contributing solutions have official DOMES numbers assigned by the IERS (International Earth Rotation and Reference Systems Service) and this can make SINEX combination software fail. As a large number of these sites are third party sites without detailed monumentation information, it is impossible to request official IERS DOMES numbers for them. Therefore, the WG implemented a coordinated approach for attributing virtual DOMES

Fig. 3.7 Differences between global ULR velocity solution and each of the regional velocity solutions after a Helmert transformation. In purple: vertical velocity differences and in blue: horizontal velocity differences. Top: Europe, middle-left: Africa, middleright: South and Central America, bottom: East-Asia and Pacific

numbers. Moreover, in the case of duplicate station names, a new station identification and virtual DOMES number was assigned in a coordinated way,

3

A Dense Global Velocity Field Based on GNSS Observations: Preliminary Results

avoiding overlaps and inconsistencies between the different regions. Typically, when a position change occurs at a CORS, two different positions are estimated: one before the discontinuity epoch and one after it. Independently of the fact that the position change is associated with a velocity change or not, most stacking is done in a way to estimate, in addition to the two position solutions, also two velocity solutions. Separate velocities may also be estimated but can be linked through constraints if they are statistically compatible, as usually the case for instrument-related discontinuities. This principle is illustrated in Fig. 3.8. When discontinuities occur at reference frame sites or at sites common to different solutions, it is imperative that the same discontinuity epochs and solution numbers are applied during the analysis Table 3.1 Agreement between global ULR velocity solution and each of the regional velocity solutions after a Helmert transformation Solution

RMS Pos. (mm) Hor. Up. Europe 1.68 2.58 Africa 4.54 4.14 South & Central America 3.85 4.41 South-Asia & Pacific 2.12 3.83 1997

1998

1999

2000

North-component

15 7.5 0 -7.5 -15

# Common (excluded sites) 43(10) 12(2) 25(3) 26(13)

2001

2002

before combining the solutions. As in this first test, cumulative position + velocity solutions have been combined, the WG asked the analysts producing these solutions to apply the station discontinuities identified by the IGS/ITRF (ftp://macs.geod.nrcan.gc.ca/ pub/requests/sinex/discontinuities/ALL.SNX). However, the treatment of discontinuities and velocity changes are subject to interpretation and analysis groups may not necessarily agree on discontinuity epochs. In practice, the different contributors did not strictly follow the IGS discontinuities, obviously influencing the estimated positions as well as velocities and most probably causing some of the outliers seen in Fig. 3.8. This exercise demonstrated the need to come up with a consensus on the discontinuities. To test a collaborative approach, a first attempt was made to merge the discontinuities reported by the different groups in one file which can be edited and maintained by the different groups. In addition, to the “bookkeeping” problems described above, some sub-regional solutions consisted of precise velocity estimates with only approximate coordinates. The implication is that inter-site correlations (not always negligible) are neglected which caused failure of some combination software. Other numerical instabilities were seen due to the equating (or heavily constraining) of velocities before and after a position jump. 2003

2004

2005

2006

2007

2008

2009

15 7.5 0 -7.5 -15

Discontinuity Epoch

[mm]

[mm]

1996

15 7.5 0 -7.5 -15

RMS Vel. (mm/year) Hor. Up 0.28 0.44 0.92 1.24 0.74 1.26 0.22 0.47

25

East-component

15 7.5 -7.5 -15

25

25

0

0

[mm]

-25

-25

-50

Position Solution 1

-50

Position Solution 2

-75

-75

-100

Up-component

-125 850

900

950

1000

-100

Heavily constrained velocities

-125 1050

1100

1150

1200

1250

1300

GPS WEEK

Fig. 3.8 Principle of the introduction of discontinuity epochs and solution numbers

1350

1400

1450

1500

1550

26

3.7

C. Bruyninx et al.

Summary and Outlook

The IAG Working Group on “Regional Dense Velocity Fields” preformed a first test combination of a set of cumulative velocity solutions from regional and global networks in order to identify the main problems when producing a dense velocity field based on multiple cumulative position and velocity solutions. The test identified the urgent need for a consensus on the attribution of discontinuity epochs for stations common to several solutions. Due to the use of different analysis strategies and software packages by the individual contributors, finding such a consensus is a challenge as most probably not the same discontinuities are seen by different people. In addition, the treatment of the post-seismic signals is also subject to interpretation. A possible way to go ahead for the Working Group could be to combine solutions at the weekly level and only deal with the attribution of the discontinuity epochs at the combination level. This would mean that in a first step the weekly global solutions would be combined with to generate a global core network with reliable velocity solutions. In a second step, weekly combined regional solutions (including the sub-regional solutions providing weekly contributions) could be added to this global core network on a weekly basis resulting in a densified core network which could then be used for velocity estimation using one single set of agreed-upon discontinuities. Finally, the remaining cumulative velocity solutions (for which the WG does not have access to weekly position solutions) could be attached to the cumulative densified core network. This approach is one of the alternative procedures which are presently under discussion within the WG. Acknowledgements The authors of this paper would like to thank the groups who submitted velocity solutions. The full list of contributors is available from http://epncb.oma.be/IAG/

References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007a) ITRF2005: a new release of the international terrestrial reference frame based on time series of station positions and earth orientation parameters. J Geophys Res 112: B09401. doi:10.1029/2007JB004949

Altamimi Z, Sillard P, and Boucher C (2007b). CATREF software: Combination and analysis of terrestrial reference frames. LAREG, Technical, Institut Ge´ographique National, Paris, France Beutler G, Bock H, Brockmann E, Dach R, Fridez P, Gurtner W, Habrich U, Hugentobler D, Ineichen A, Jaeggi M, Meindl L, Mervart M, Rothacher S, Schaer R, Schmid T, Springer P, Steigenberger D, Svehla D, Thaller C, Urschl R, Weber (2007) Bernese GPS software version 5.0. ed. Urs Hugentobler, R. Dach, P. Fridez, M. Meindl, Univ. Bern Blewitt G (2008) Fixed point theorems of GPS carrier phase ambiguity resolution and their application to massive network processing: Ambizap. J Geophys Res 113:B12410. doi:10.1029/2008JB005736 Bruyninx C, Altamimi Z, Boucher C, Brockmann E, Caporali A, Gurtner W, Habrich H, Hornik H, Ihde J, Kenyeres A, M€akinen J, Stangl G, van der Marel H, Simek J, S€ohne W, Torres JA, Weber G (2009). The European Reference Frame: Maintenance and Products, IAG Symposia Series, “Geodetic Reference Frames”, Springer, vol 134, pp 131–136, DOI: 10.1007/978-3-642-00860-3_20 Davis Ph, Blewitt G (2000) Methodology for global geodetic time series estimation: A new tool for geodynamics. J Geophys Red 105(B5):11083–11100 Dow JM, Neilan RE, Rizos C (2009) The International GNSS Service in a changing landscape of Global Navigation Satellite Systems. J Geod 83:191–198. doi:10.1007/s00190008-0300-3 ´ da´m J, Ro´zsa S (eds) (2008). The Drewes H, Hornik H, A geodesist handbook 2008. Journal of Geodesy, Springer, 82 (11):661–846 Fernandes RMS, Miranda JM, Meijninger BML, Bos MS, Noomen R, Bastos L, Ambrosius BAC, Riva REM (2007) Surface velocity field of the Ibero-Maghrebian Segment of the Eurasia-Nubia plate boundary. Geophys J Int 169:315–324. doi:10.1111/j.1365-246X.2006.03252.x Herring TA, King RW, McClusky SC (2007) Introduction to GAMIT/GLOBK, Release 10.3, Mass. Instit. of Tech., Cambridge Santamaria-Gomez A, Bouin M-N, and W€oppelmann G. An improved GPS data analysis strategy for tide gauge benchmark monitoring. (Submitted) Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85. DGFI, Munich. Available at http://www.sirgas.org/index.php?id¼97 W€oppelmann G, Letetrel C, Santamaria A, Bouin M-N, Collilieux X, Altamimi Z, Williams SDP, Martin Miguez B (2009) Rates of sea-level change over the past century in a geocentric reference frame. Geophys Res Lett 36:L12607. doi:10.1029/2009GL038720 Zumberge J, Heflin M, Jefferson D, Watkins M, Webb F (1997) Precise point positioning for the efficient and robust analysis of GPS data from large networks. J Geophys Res 102:5005–501

4

Enhancement of the EUREF Permanent Network Services and Products €hne, A. Kenyeres, G. Stangl, C. Bruyninx, H. Habrich, W. So €lksen and C. Vo

Abstract

This paper describes the EUREF Permanent Network (EPN) and the efforts made to monitor and improve the quality of the EPN products and services. It is shown that the EPN is becoming a multi-GNSS tracking network and that the EPN Central Bureau and the Analysis Centers are preparating to include the new satellite signals in their routine operations. Thanks to the EPN Special Project on “Reprocessing”, set up early 2009, EPN products with much better quality and homogeneity will be generated. The Special Project on “Real-time analysis” will improve the reliability of the EPN real-time data streams and develop new EPN real-time products.

4.1

C. Bruyninx (*) Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium e-mail: [email protected] H. Habrich W. S€ ohne Bundesamt f€ur Kartographie und Geod€asie, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany A. Kenyeres FOMI Satellite Geodetic Observatory, P.O. Box 585, 1592 Budapest, Hungary G. Stangl Department of Satellite Geodesy, Institute of Space Research, Schmiedlstrasse 6, 8042 Graz, Austria C. V€olksen Bayerische Kommission f€ ur die Internationale Erdmessung, Bayerische Akademie der Wissenschaften, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany

Introduction

The IAG (International Association of Geodesy) Regional Reference Frame sub-commission for Europe, EUREF, is responsible for defining, providing access and maintaining the European Terrestrial Reference System (ETRS89), which is recommended by the European Commission for use in all EU member states (Bruyninx et al. 2009a). In 1996, EUREF created the EUREF Permanent Network (EPN) based on a partnership with site operators of permanent GNSS sites who are willing to share their data with the public. The EPN cooperates closely with the International GNSS Service (IGS, Dow et al. 2005); EUREF members are participating to the IGS Real-Time Pilot Project, the IGS GNSS Working Group, the IGS Antenna Calibration Working Group and the IGS Infrastructure Committee. The EPN network contains more than 220 stations (as of October 2009) and its GNSS observations are used extensively by the public, national mapping agencies, and researchers (Bruyninx 2004). EPN

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_4, # Springer-Verlag Berlin Heidelberg 2012

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stations provide daily (mandatory) and hourly (93% of the stations) data in the RINEX format as well realtime data (48% of the stations) in RTCM, RTIGS or raw formats where 15-min high-rate RINEX files are derived from. Two Regional Data Centres (at BKG and OLG) provide access to all hourly and daily data, one data centre (at EPN Central Bureau) to provides access to historical daily EPN observation data, and one regional broadcaster (www.euref-ip.net at BKG) broadcasts all real-time EPN data streams. 16 Local Analysis Centers (LAC) deliver weekly (some also daily and hourly) site position estimates in SINEX format and hourly zenith path delays. These results are the basis to generate the following EPN products: – Weekly combined site positions (in SINEX format) – Daily combined site zenith path delays – Cumulative site positions and velocities, in both the ITRS (International Terrestrial Reference System) and ETRS89, and a discontinuity table – Fully analysed residual position time series The day-to-day management of EPN is performed by the EPN Central Bureau (http://epncb.oma.be). In this paper we describe the efforts recently undertaken within the EPN to keep the pace with new user demands and GNSS modernization, and to continuously improve the EPN products and services.

4.2

Modernization of the EPN

4.2.1

Tracking Network

Since 1996, the EPN has grown from a network of continuously observing dual frequency GPS stations to a network where 49% of the stations (Oct. 2009) are observing GPS as well as GLONASS satellites. In addition, 48 EPN stations have the GPS L2C tracking capability and 31 are equipped with an L5-capable receiver. The new observation types imply switching to RINEX 3 (e.g. ftp://ftp.unibe.ch/aiub/rinex/rinex300. pdf) and an update of the quality checking routines. Since Dec. 2006, each antenna/radome introduced in the EPN (new stations or replacements at existing stations) must have true absolute calibrations. Thanks to this requirement, the number of antenna/radome combinations without true absolute calibrations decreased from 31 to 18%. In addition, more than 90% of the new antennas installed within the EPN are multi-GNSS antenna. From these, 75% are designed for observing GPS and GLONASS signals

and 25% are in addition Galileo-ready. Taking into account that due to local site conditions and near-field multipath antenna/radome replacements continue to introduce jumps in the station positions (even with absolutely calibrated antenna/radome combinations) and that the large majority of new EPN antennae are multi-GNSS capable, the EUREF Technical Working Group (TWG) -which is the EUREF Steering Committee- strongly recommends all station managers, in the case an antenna has to be replaced, to replace it directly with a multi-GNSS antenna.

4.2.2

Data Analysis

The large majority of the EPN Analysis Centers uses data analysis software from third parties and therefore depends on the multi-GNSS policy of these software providers. In practice, 15 of the 16 LAC use GNSSanalysis software that is GLONASS-ready. However, even while EPN stations have been observing GLONASS signals since several years, only 4 of these 15 EPN Analysis Centers presently include GLONASS observations in their data analysis. But, the majority of the LACs plan to start with GPS +GLONASS data analysis within the next year. To guarantee consistency with the IGS, EUREF asks the EPN LACs to use IGS products (e.g., satellite orbits & clocks, and antenna calibrations) within their data analysis. However, to accommodate the missing combined GPS/ GLONASS IGS orbits, most of the four LACs processing GLONASS use the fully consistent combined GPS/GLONASS orbit files from individual IGS ACs, such as CODE (Dach et al. 2009). In addition, contrary to the IGS, the EPN has allowed the introduction of individual receiver antenna calibrations. They allow to introduce new antenna-specific calibrations while, to maintain the consistency of the reference frame, the type calibrations cannot always be updated.

4.3

Quality of the EPN Products

4.3.1

IGS Global Combination of Regional Networks

The Massachusetts Institute of Technology (MIT), Cambridge, USA is an IGS Analysis and Associate Analysis Centre. It generates the global MIT T2 solution as a combination of eight IGS ACs solutions

4

Enhancement of the EUREF Permanent Network Services and Products

(Herring 2008). In addition, MIT also generates the MIT T2 RNAAC solution by adding three regional solutions (EPN, SIRGAS -Sistema de Referencia Geoce´ntrico para Las Ame´ricas- and GSI –Geographical Survey Institute-) and one global solution (CNES Centre National d’Etudes Spatiales- Toulouse) to the MIT T2 solution. The delay of the publication of the mentioned two products is 8 weeks after the end of observations and it requires publishing the weekly EPN combination well in time. The global solution from CNES largely overlaps with the MIT T2 solution. The EPN, SIRGAS and GSI solutions densify the IGS network in Europe, South America and Japan. The coordinate comparison between the EPN and the MIT T2 solutions may be considered as a kind of accuracy estimate and interpreted as a measure of the global alignment of the EPN network. The computation of such r.m.s. numbers for 10 weeks (week 1,511–1,520) confirm that the global alignment is in the range from 2 to 4 mm.

4.3.2

Residual Position Time Series

The main target of the EPN coordinate time series analysis and monitoring is to strengthen the EPN as a geodetic reference network and to offer various products for geodesists and geophysicists. Using the CATREF software (Altamimi et al. 2007b), each 15 weeks, updated EPN cumulative solutions are released (while each 5 weeks an internal solution is created) based on the weekly combined EPN SINEX solutions. During the time series analysis all station specific events (coordinate outliers and discontinuities), confirmed by the log files are identified and considered. A cumulative solution is associated with the following products: – The EPN cumulative solution in SINEX format tied to the actual ITRS reference frame realization (currently ITRF2005) using minimum constraints – EPN station positions and velocities, as the most accurate and up-to-date solutions for the EPN sites. They are used for the maintenance of the regional densification of the ITRFyy between two releases and also for the maintenance of the ETRS89 (see Sect. 4.2) – An up-to-date list of station discontinuities fully harmonized with the IGS/ITRF discontinuity table – Residual coordinate time series as the Helmert difference between the positions in the cumulative

29

solution and the ones in the weekly input SINEX solutions – Harmonic analysis of the time series to detect seasonal coordinate variations – Sophisticated statistical analysis of the position residuals using the CATS software (Williams 2008) to estimate reliable velocity uncertainties and station-specific noise characteristics More information on these products is available from the EPN CB web page at http://www.epncb.oma.be/ _dataproducts/products/timeseriesanalysis/ (Fig. 4.1).

4.3.3

Tropospheric Zenith Path Delays

Since June 2001 (GPS week 1108) the EPN LACs are delivering the estimated Zenith Path Delay (ZPD) parameters to the BKG Data Centre. Daily files (in the so-called SINEX TRO format) are produced on a weekly basis, together with the weekly coordinate solution. These ZPD-estimates were originally combined using a procedure based on the strategy developed for the IGS (Gendt 1997), but since then several changes have been introduced regarding, e.g., reference frame realization, software versions, processing options, absolute antenna calibration, etc. The results have been improved step by step, leading to an internal precision of 2–3 mm ZPD (Fig. 4.2). The most significant progress can be seen from November 2006 on (GPS week 1400) when the absolute antenna models were introduced, especially for stations in the south of Europe. Thanks to the Memorandum of Understanding between EUREF and EUMETNET (Pottiaux et al. 2009) EUREF has now access to radiosonde observations and synoptic data. Applying these data, GNSS processing and analysis may be improved in the future. Moreover, a validation of the GNSS-derived ZPD parameters by those derived from radiosonde observations and vice versa can be performed. Figure 4.3 shows the time series of the differences between ZPD parameters derived from radiosonde data and the EPN combined solution for the station YEBE. One can see that the jump in November 2006 reduced the bias between both observables and the scattering increased during summer time. In 2008 the EPN Special Project “Troposphere Parameter Estimation” was closed and the ZTD processing moved towards routine operation.

30

C. Bruyninx et al.

Fig. 4.1 Residual position time series of the EPN station CHIZ. Top: raw time series; bottom: cleaned time series after removal of seasonal signals and introduction of multiple position and velocity estimates

4.4

Maintenance of and Access to ETRS89

4.4.1

Historical EUREF Campaigns

Immediately after defining the ETRS89 and realizing the ETRF89 (with ETRF, the EUREF Terrestrial Reference Frame), European countries started the

densification of the ETRF using GPS campaigns. These campaigns had typically a minimal duration of three full days so that the comparison of the daily position estimates allowed to get an impression of the precision of the positions. Based on this, EUREF campaigns are divided in three classes: Class A is (only for permanent stations) requiring a 1 cm r.m.s. for the position over a time period exceeding the

4

Enhancement of the EUREF Permanent Network Services and Products

Fig. 4.2 Daily ZPD bias time series from EPN combination for YEBE

Fig. 4.3 Time series of differences between radiosonde and GNSS ZPD parameters for YEBE

campaign period. A class B campaign requires an r.m.s. of 1 cm for the estimated positions at the epoch of observations, whereas Class C means the precision is worse than 1 cm. Not all of these campaigns have been accepted by EUREF, but most of them have been adopted by the National Mapping Agencies of the different countries (Fig. 4.4). Some countries have in the mean time replaced the old campaigns of the 1990s by newer ones, using a set of permanent stations. These pseudo-campaigns are usually based on 1 week of permanent observations. Campaign coordinates are valid for the mean epoch of the measurements within the then used reference frame, e.g., ETRF2000. The advantage of campaigns is that a number of national markers get coordinates validated by an international organization.

4.4.2

Densification of the ETRF

Only a selected number of EPN sites (mostly the ones belonging to the IGS) have coordinates included in recent ITRF realizations released by the IERS

31

(International Earth Rotation and Reference Systems Service). In addition, the latest realization of the ITRS, ITRF2005 (Altamimi et al. 2007a) is based on observations from space geodetic techniques (GNSS, DORIS, VLBI, SLR) up to December 2005 and does not take into account any of the IGS/EPN data gathered after Jan 1st, 2006. The problem of relative antenna models, the limited number of stations, and the lack of frequent updates consequently restricts the use of the ITRF for EUREF densifications The antenna modeling problem will be resolved with the release of the ITRF2008 (expected for 2010) which will be compatible with absolute GNSS antenna models. To take full advantage of the EPN and its most recent GNSS observation data, the EUREF TWG decided, to release regular official updates of the ITRS/ETRS89 positions/velocities of the EPN stations. A first step in this process consisted in a densification of the ITRF2005 using all EPN data up to Dec. 2005 (the same observation period as covered by the ITRF2005). Since early 2009, this realization is updated each 15 weeks. The advantage of the regular update is that the most recent EPN results are taken as much as possible into account. In order to provide the most reliable products, the EPN stations are categorized taking the station quality and the length of available observation span into account (Kenyeres 2009) (Fig. 4.5): – Class A stations with positions at the 1 cm precision and velocities at the 1 mm/yr precision at all epochs. – Class B stations with positions at the 1 cm precision at the epoch of minimal position variance of each station; where no velocities are provided. Exclusively the Class A stations can be used as reference stations for the computation of EUREF campaigns and densification of the ETRF (see Bruyninx et al. 2009b).

4.5

Recent Initiatives

4.5.1

Real-Time Analysis Project

At the end of 2007, the EUREF-IP pilot project for real-time data streaming was successfully transferred to routine operation. Within this pilot project the EPN

32

C. Bruyninx et al.

Fig. 4.4 Geographical distribution of accepted EUREF campaigns Fig. 4.5 In green: EPN class A stations; in red: EPN class B stations (status Sept. 2009)

guidelines for real-time data have been developed (Dettmering et al. 2006) and disseminating streams via NTRIP (Weber et al. 2005) became a widely used and implemented feature. From Jan. 2006 to

May 2009 the number of EPN real-time stations grew from 16 to 103 and the number of registered users at EUREF-IP broadcaster at BKG grew from 500 to 1,250 . To pursue this success, the EUREF

4

Enhancement of the EUREF Permanent Network Services and Products

TWG launched a new Special Project “EPN RealTime Analysis” with the primary goals to develop and set-up a re-dissemination of GNSS real-time data/products in Europe via NTRIP broadcasters, validate satellite orbit and clock correctors to broadcast ephemeris and establish backups for all critical realtime service components. In that frame, a new realtime dissemination concept was developed (S€ohne et al. 2009). While today EPN stations stream realtime data to EUREF-IP broadcaster maintained by BKG, the goal is to maintain a series of top-level casters distributing the real-time data and sharing workload. This activity will be supported by a number of relay casters distributing real-time data to, e.g., special communities. In the final stage it is envisaged that almost every reference station is streaming realtime data to at least two different broadcasters to overcome the “single point of failure” issue.

4.5.2

Reprocessing Special Project

Reprocessing of regional networks (V€ olksen 2008) or the complete EPN became an important issue during recent years due to the availability of reprocessed orbit and clock products (Steigenberger et al. 2006). A reprocessing of the complete EPN has been done in 2008 by the MUT (Warsaw) and ROB (Brussels) LAC, clearly demonstrating an improvement of the EPN time series which were previously inconsistent due to analysis and modeling changes (Kenyeres et al. 2009). In order to coordinate the reprocessing of the full EPN between all EPN LAC a new EPN Special Project was created at the EUREF TWG meeting held in Budapest on Feb. 26–27, 2009 (V€ olksen 2009). During its pilot phase, from 2009 till 2010, a test reanalysis of the year 2006 will be performed in order to test and develop new strategies and standards for the data analysis. It is expected that each LAC setup the facilities for the reprocessing of the 2006 data before the end of 2009. At the end of the pilot phase, it is expected to have identified suitable sets of reprocessed products, which will be generated during re-analysis as well as the day-to-day analysis. The next step will then consist of a re-analysis of the complete EPN data set (1996-200x) applying the new strategies and standards. The combination of the different solutions by the analysis coordinator will provide

33

reports and feedback to the working group and the LACs.

4.6

Summary

In response to evolving user needs and new satellite signals, the EPN is continuously improving its tracking network, products and services. Examples are the new EPN real-time analysis and reprocessing special projects, and the fact that new cumulative EPN coordinates are now updated each 15 weeks. While the EPN tracking network is preparing to track new and modernized GNSS signals, monitoring and processing the new and modernized GNSS signals imposes the usage of updated exchange formats, antenna calibrations, consistent multi-GNSS satellite orbits and clocks, and requires additional software developments. For several reasons, the EPN Central Bureau (resp. Analysis Centers) are not yet ready for a real multi-GNSS data monitoring resp. analysis, but this situation is expected to improve dramatically in the next year.

References Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007a) ITRF2005: a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:B09401. doi:10.1029/2007JB004949 Altamimi Z, Sillard P, Boucher C (2007b) CATREF software: combination and analysis of terrestrial reference frames. LAREG, Technical, Institut Ge´ographique National, Paris, France Bruyninx C (2004) The EUREF Permanent Network: a multidisciplinary network serving surveyors as well as scientists. GeoInformatics 7:32–35 Bruyninx C, Altamimi Z, Boucher C, Brockmann E, Caporali A, Gurtner W, Habrich H, Hornik H, Ihde J, Kenyeres A, M€akinen J, Stangl G, van der Marel H, Simek J, S€ohne W, Torres JA, Weber G (2009a) The European reference frame: maintenance and products, IAG Symposia Series, “Geodetic Reference Frames”, vol 134. Springer, Heidelberg, pp 131–136. doi: 10.1007/978-3-642-00860-3_20 Bruyninx C, Altamimi Z, Caporali A, Kenyeres A, Lidberg M, Stangl G, Torres JA (2009b) Guidelines for EUREF Densifications, ftp://epncb.oma.be/pub/general/Guidelines_ for_EUREF_Densifications.pdf Dach R, Brockmann E, Schaer S, Beutler G, Meindl M, Prange L, Bock H, J€aggi A, Ostini L (2009) GNSS processing at CODE: status report. J Geodesy 83(3–4):353–366. doi:10.1007/s00190-008-0281-2

34 Dettmering D, Weber G, Bruyninx C, v.d.Marel H, Gurtner W, Torres J, Caporali A (2006) Real-time GNSS in routine EPN operations: concept. http://epncb.oma.be/_organisation/ guidelines/EPNRT_WhitePaper.pdf Dow JM, Neilan RE, Gendt G (2005) The International GPS Service (IGS) celebrating the 10th anniversary and looking to the next decade. Adv Space Res 36(3):320–326. doi:10.1016/j.asr.2005.05.125 Gendt G (1997) IGS Combination of Tropospheric Estimates (Status Report). In: Proceedings of the IGS Analysis Centre Workshop, Pasadena, CA Herring T (2008) RNAAC combinations in the IGS, presented at IGS Analysis Center Workshop 2008, Miami Beach, FL, 2–6 June 2008 Kenyeres A (2009) Maintenance of the EPN ETRS89 coordinates. In: Minutes of EUREF TWG meeting, Budapest, Hungary, 26–27 Feb 2009. http://www.euref.eu/ Kenyeres A., Figurski M, Legrand J, Kaminski P, Habrich H (2009) Homogeneous reprocessing of the EPN: first experiences and comparisons. Bull Geod Geomatics 3:207–218 Pottiaux E, Brockmann E, .S€ ohne W, Bruyninx C (2009) The EUREF – EUMETNET collaboration: first experience and potential benefits. Bull Geod Geomatics 3:269–288

C. Bruyninx et al. S€ohne W, St€urze A, Weber G (2009) Increasing the GNSS stream dissemination capacity for IGS and EUREF. http://epncb.oma. be/_dataproducts/data_access/real_time/BroadcastConcept.pdf Steigenberger P, Rothacher M, Dietrich R, Fritsche M, R€ulke A, Vey S (2006) Reprocessing of a global GPS network. J Geophys Res 111(B5):B05402. doi:10.1029/ 2005JB003747 V€olksen C (2008) Reprocessing of a regional GPS network in Europe. In: Sideris MG (eds) Observing our changing Earth: proceedings of the 2007 IAG General Assembly, Berlin, pp 57–64 V€olksen C (2009) Draft Charter for the EUREF Working Group on reprocessing of the EPN. http://epn-repro.bek.badw.de/ Documents/charter_repro.pdf Weber G, Dettmering D, Gebhard H (2005) Networked Transport of RTCM via Internet Protocol (NTRIP). In: Sanso F (ed) Proceedings of the IAG symposia ‘A window on the Future of Geodesy’, vol 128. Springer, Heidelberg, pp 60–64 Williams SDP (2008) CATS: GPS coordinate time series analysis software. GPS Solut 12(2):147–153. doi:10.1007/s1029-0070086-4

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna Ulla Kallio and Markku Poutanen

Abstract

For the next-generation geodetic VLBI network a 1 mm positioning accuracy is anticipated. The accuracy should be site-independent consistent, reliably controlled, and traceable over long time periods. There are a number of remaining limitations. These include random and systematic components of the delay observable itself, various antenna-related errors, and especially a proper handling of local ties at multi-technique sites. At the Mets€ahovi Fundamental Station operated by the Finnish Geodetic Institute there are a CGPS and Glonass receivers, both in IGS network, a SLR (currently under renovation), a DORIS beacon, a superconducting gravimeter, and a 14.5 m radio telescope owned by Mets€ahovi Radio Observatory of the Helsinki University of Technology. Between five and eight geodetic VLBI sessions are conducted annually. We tested a method to simultaneously measure the tie of the VLBI antenna to the GPS network by tracking the movement of two GPS antennas attached to the radio telescope during geodetic VLBI sessions. We used kinematic trajectory solutions of the two GPS antennas to calculate the orientation and the reference point of the VLBI antenna. In this paper we describe the data acquisition, calculation model, some error sources and test results of four campaigns. The position of the reference point is time, temperature, antenna elevation and azimuth dependent. We propose that in the future, the position should be tracked permanently during geo-VLBI campaigns with attached GPS antennas.

5.1

U. Kallio M. Poutanen (*) Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02431 Masala, Finland e-mail: [email protected]

Introduction

The IERS Working Group on co-location and local ties gives recommendations and guidelines for tie vector measurements, computation and transformation for multi-technique sites in the global reference frame (IERS 2005, 2009). The idea of local tie measurements is to solve for the reference points and orientations of the all instruments in a fundamental station with

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_5, # Springer-Verlag Berlin Heidelberg 2012

35

36

respect to one common marked point or solve for the reference point and orientation of one instrument relative to another instrument in a local reference frame. The tie vector must then be transformed to the global reference frame. Local ties should be determined at regular intervals. There are different practices and recommendations on the repeat rate, ranging from twice per year up to several years. How often ties can be completed in practice depends on the workload and difficulties of the measurement. It is a totally different question, how often such ties should be made to reliably detect possible movement or deformation of an instrument. The measurement methods in different ties depends on the case and the instruments used in tracking have varied from tachymeters and levelling instruments or laser trackers to GPS (Dawson et al. 2007; IERS 2005; ITRF 2009; Sarti et al. 2004; L€ osler and Haas 2009). The methods have been space intersection with two or more theodolites or kinematic or static GPS measurements, and the use of a laser tracker. As pointed out e.g., in (Sarti et al. 2008), there are basically three surveying approaches: a direct method, a hybrid method and an indirect method. In the direct method the reference point of an instrument is an actual geodetic marker which can be surveyed directly. In the hybrid method (mainly applicable to some VLBI and SLR telescopes) the reference point itself is not identified by a geodetic marker but defined by the axes of rotation of the telescope. Some surveying targets are fixed on the telescope to make the axis visible for pointing of the surveying instrument. The indirect method is entirely based on observed positions of targets fixed on solid structure of the telescope. The positions of the targets are measured with surveying techniques when the telescope is turned in different positions. In our approach, the indirect method was applied. Actually there is the same local tie problem with reference points and orientations of the measuring instruments as there is with the VLBI antenna itself. One needs a local reference network around the fundamental station. Another question is how to connect the local network to the global reference frame reliably with controlled error estimates, and how to combine different measurements together. Additionally, a local geoid model is needed to tie the local measurements in a global frame.

U. Kallio and M. Poutanen

In our measurements the main goal was to determine the vector between the VLBI antenna reference point and the IGS GPS point. Beside that we tied these two points to the benchmarks of the local network. We also determined the orientation and the offset of axes of the radio telescope.

5.2

Modelling the Antenna Rotation

The IVP (Invariant Point) of a VLBI antenna is defined as a fixed point (in a reference frame) to which VLBI observations are referred. The IVP is the intersection of the primary axis with the shortest vector between the primary and secondary axis (Dawson et al. 2007). The radio telescope is an instrument with two rotation axes, and like the other instruments with axes (e.g., tachymeters, laser trackers, cameras, SLR) has errors such as eccentricity or offset between axes and axes misalignment. In general the directions of axes are not perpendicular. The secondary axis (elevation) rotates about the primary axis (azimuth) and they don’t intersect. In this case, the reference point is the projection of the secondary axis on the primary axis. IVP is not a marked point and no line of sight from outside the telescope to the reference point is possible. The determination of the IVP coordinates must be done indirectly by tracking the positions of some points on the antenna structure in different antenna positions. Let’s assume that the antenna is rotated around its axes. If the azimuth is kept fixed then a point in the antenna structure rotates about the elevation axis and the track forms a circle in space. Similarly, if the elevation is locked then the point rotates about the azimuth axis. In an ideal case, all the tracks together form a sphere. In recent determinations of the IVP, the model of three dimensional circles has been the most popular. (Dawson et al. 2007; ITRF 2009) The order of observations has been planned so that the circle fitting is possible. Observing all circles which are needed for reliable fitting means a large number of VLBI antenna positions. The whole process is typically very time consuming. A different approach for calculating the reference point was presented by L€osler (L€osler and Hennes 2008; L€osler and Haas 2009). In the L€osler’s model

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna

37

one computes the transformation of the tracked coordinates from the instrument coordinate system to the VLBI-antenna coordinate system. It uses the telescope angle readings as observations but the order of observations is arbitary. Therefore the model can be applied also to the sparse and scattered data and it includes antenna axes errors.

5.3

€hovi Model The Metsa

We developed a model to be used in connection of kinematic GPS measurements which were made during the normal use of the VLBI telescope. GPS data were collected during an ordinary 24-h geodetic VLBI session. Therefore, paths of the two GPS-antennas attached on the radio telescope dish do not form such kind of circles that could be used in three dimensional circle fitting model to solve for the azimuth and the elevation axes and the IVP coordinates. The idea of using telescope angles as observations is the same as in L€ osler’s model, but we reformulated the movement of the antenna points and the rotation matrices. Our model is suitable also for different types of telescope mounting other than the elevationazimuth mount described here. The basic assumptions of our model are that points in antenna structure rotate about the elevation axis and the elevation axis rotates about the azimuth axis. The axes need not intersect or be perpendicular. The position vector of the GPS-antenna X (in an arbitrary reference frame) is the sum of three vectors: the position vector of the reference point X0, the axis offset vector (E-X0) rotated by angle a about the azimuth axis a and a vector from the eccentric point E to the antenna point p rotated about the elevation axis e by angle b and about the azimuth axis by angle a (Fig. 5.1). Unknown parameters are X0, E, a, e, and p. Observations are coordinates X for each antenna point and epoch, and VLBI antenna angle readings a and b for every epoch. The estimated values of E, e and p are those of an antenna initial position which may be zero for both angles. The basic equation of our model is X0 þ Ra;a ðE X0 Þ þ Ra;a Rb;e p X ¼ 0

(5.1)

Fig. 5.1 The model parameters in a local reference frame. IVP denotes the Invariant Point, to which VLBI observations are referred. Vectors X, p1 and p2 denote the position vectors of the two GPS antennas attached on side of the VLBI antenna

Rotation matrices Ri,j are formed by applying the Rodrigues’ rotation formula which uses rotation axis and angle. The rotated X is Xr ¼ cosðaÞX þ ð1 cosðaÞÞaaT X þ sinðaÞa X (5.2) The rotation matrix is then 0

1 Rða; aÞ ¼ cosðaÞ@ 0 0 0 xx þ ð1 cosðaÞÞ@ xy xz 0 0 z þ sinðaÞ@ z 0 y x

1 0 0 1 0A 0 1 1 xy xz yy yz A yz zz 1 y x A 0

(5.3)

where a is the rotation angle, and the components of the unit vector of a rotation axis a are x, y and z. In our model the angles in the rotation matrices are the VLBI antenna azimuth and elevation angle readings. Azimuth and elevation axes are described with unknown unit direction vectors. There are four parameters in rotation matrix: three for axis and one

38

U. Kallio and M. Poutanen

for the angle, although only three are independent parameters because the axis is a unit vector. Therefore, we need two condition equations between parameters in our adjustment, one for each axis. aT a 1 ¼ 0

(5.4)

e e1¼0

(5.5)

T

The other two conditions between parameters are needed for finding the shortest vector between the azimuth and elevation axes. The offset vector between the axes is perpendicular to both axes. ðE X0 ÞT a ¼ 0

(5.6)

ðE X0 ÞT e ¼ 0

(5.7)

We do not need any condition equation between the GPS-antenna points: the model itself includes the condition of the constant distance between the points. The number of unknown parameters in the model is u ¼ 12 þ m 3

(5.8)

and the number of observations n¼tm3þt2

1 0 0 0 0 0 FðX0 ; E ; a ; e ; p ; ai ; bi ; X1 Þ C B .. yi ¼ @ A . 0

0

(5.10)

The solution of unknown parameters is reached by iteration in a linearized least squares mixed model with conditions between the parameters: 0 t h 11 i P T 1 T 1 T x D A ðB P B Þ A i i i i i A ¼@ i k h D 0 0 t h i1 P T 1 T Ai ðBi P1 i Bi Þ yi A @ i GðX0 0 ; E0 ; a0 ; e0 ; p0 Þ (5.11)

0

0

0

(5.12)

0

FðX0 ; E ; a ; e ; p ; ai ; bi ; Xm Þ

The matrix G includes condition equations with approximate values of parameters. 0

1 g1 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ B g2 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ C C G¼B @ g3 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ A g4 ðX0 0 ; E0 ; a0 ; e0 ; p0 Þ

(5.13)

Jacobian matrices A and B for epoch i are 0 B Ai ¼ B @

(5.9)

where t is the number of epochs and m is the number of points to be tracked. The degree of freedom in our mixed model least squares adjustment with four conditions between parameters is r ¼ 3m t þ 4 u

where xh is the correction to the approximate values of parameters after h iteration and yi includes the basic equation for all points in epoch i with approximate values of parameters. The apostrophes here denote the approximate value.

0 B Bi ¼ B @

@F @X0

@F @X0

@F @azi

@F @azi

@F @E

@F @a

.. .

@F @e

@F @p1

@F @E

@F @a

@F @e

@F @eli

@F @X1

.. .

.. .

@F @eli

0

.. .

0

1 0 .. C . C A (5.14) @F @pm

1 0 .. C . C A

(5.15)

@F @Xm

and D¼

@G @G @G @G @G @G ... @X0 @E @a @e @p1 @pm

(5.16)

The weight matrix of the epoch i is 0

s2a @ Pi ¼ 0 0

0 s2b 0

11 0 0A Ci

(5.17)

s2a and s2b are variances of azimuth and elevation observations, respectively, and Ci is the covariance matrix of coordinate observations in epoch i.

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna

5.4

Tie Measurements Using Kinematic GPS

5.4.1

Installation of the GPS Antennas

Due to restrictions on telescope time and the anticipated time-consuming procedure with traditional surveying methods or static GPS, we ended up testing a kinematic approach. We had two Ashtech Dorne Margolin-type GPS antennas attached to the edge of theVLBI antenna and we gathered data with two Ashtech Z-12 receivers during the following geo-VLBI campaigns: EUROPE-96 (27–28. 11.2008), IVS-T2059 (16–17.12.2008), EUROPE-97 (19–20.1.2009), EUROPE-98 (25–26.3.2009), and IVST2061 (7–8.4.2009). The GPS-antennas were attached to rotating holders on both sides of the radio telescope dish, and they had the counterweights that forced them to point to the zenith regardless of the position of the radio telescope. The GPS antenna reference point (ARP) was in the rotation axis of the antenna holder (Fig. 5.2). The antennas were assembled so that the rotation axis of the holder of a GPS-antenna pointed to the second antenna and this direction was approximately the same as the elevation axis. We measured the angle between the antenna north arrow and the direction of the elevation axis. It is possible to calculate the GPS antenna orientation and phase centre corrections for every epoch. For this we can use the results of an individual antenna calibration.

5.4.2

39

same time we gathered GPS data on six pillar points to form a local network. In this paper we concentrate on the tie to the IGS point. Kinematic co-ordinate solutions were computed with the Trimble TTC software on the fly strategy. We accepted only the fixed solutions with the pdop values less than 8 and the standard deviations of coordinates less than 0.1 m. If the left hand side GPS antenna was rejected we also rejected the right hand side of the same epoch for symmetry. Because the mutual distance of the antennas did not change, we rejected both observations where the distance between the antennas differed more than 0.05 m from the median value. The height difference between the antennas was allowed to differ from the median not more than 0.1 m. Azimuth and elevation angles of the VLBI antenna were dumped into binary disk files every 15 s and v tagged with UTC time, one file for each epoch. Subsequently, those files were afterwards combined into one ASCII file. Due to a system error during the first campaign, the actual VLBI antenna information was missing. The telescope angle readings are used as observations. The trajectory co-ordinate data of the GPSantennas and the VLBI-antenna angle readings must be combined. The synchronization error of one second in time may cause up to 0.01 (depending on the angular velocity) orientation error to the estimated elevation axis. We obtained good data only when the VLBI-antenna was tracking a radio source. When it changed the source the movement was too quick for fixed solutions.

Preparation of the Kinematic Data

The GPS observation interval was 30 s. This is also the interval for the Mets€ahovi IGS GPS receiver which was our main target for the tie measurement. At the

Fig. 5.2 GPS antenna holder on the VLBI antenna

5.4.3

GPS Antenna Phase Center Variation

The orientation of the GPS antennas on the VLBI antenna was changed when the VLBI antenna rotated. We used individually calibrated antennas to eliminate the effect of antenna phase center variation in the computation. Only in one azimuth position the orientation of the GPS antennas was the same as used in the calibration. If we were using the existing antenna calibration tables in the kinematic solution in the ordinary way it would cause systematic errors in the coordinates of GPS antennas. This would propagate

40

U. Kallio and M. Poutanen

to the solution of the reference point coordinates, and especially to the axis offset. One possible solution for the problem is to use only elevation dependent tables for phase variations and let the north and east components of the offset vectors be zeros. If the distribution of VLBI antenna positions over azimuth angles is homogeneous and number and directions of satellites is equal to every direction, then the systematic error will be canceled in the calculation of the reference point. However, it affects still in the solution of axis offset. Unfortunately the distribution of VLBI antenna positions was not equal because of the distribution of the tracked radio sources during the geodetic VLBI session. The VLBI antenna dish also blocked observations so that it was not possible to get fixed solution for trajectory points in every azimuth and elevation positions. We tested the possibility to apply antenna correction tables from the absolute calibration directly to the RINEX data and then use the antenna calibration tables with zero offsets and zero phase variations during the GPS data processing. The GPS antenna orientation was calculated for every epoch from known angle between elevation axis and the north arrow of the GPS antenna and the azimuth angle reading of the telescope. The antenna offset in the direction of a satellite was then subtracted in the RINEX code and phase data.

about 2/3, but we had still data enough for the final adjustment (Table 5.1).

5.4.4

Table 5.2 The local tie vectors between METS GPS and VLBI reference point and the axis offsets in the four campaigns in meters

Computing and Outlier Detection

We programmed the linearized least squares mixed model with conditions between parameters using Octave language. Over two iterations the residuals were checked and outliers removed. Because there still existed some bad data after the preprocessing and filtering we rejected those observations which had associated residuals larger than 1 m. Also angle readings with residuals larger than 0.1 were rejected. After two more iterations we repeated the rejection procedure with limit values 0.2 m and 0.001 . After this we repeated again the adjustment and rejected those observations which had a standardized residual larger than 3. For symmetry we rejected observations from both sides of the VLBI antenna. The loss of the data during the process was

5.5

Results

The maximum difference between the solutions of the reference point in the four campaigns was 2 mm in each vector component. The dependence of the axis offset values on the GPS antenna orientation changes is clear. The VLBI axis offset vector E-X0 rotates about the azimuth axis at the same time and the same amount as the GPS antenna phase center offset vector. In our basic equation it is impossible to distinguish these offsets from each other. In the Trimble TTC processing the control point was METS with coordinates in ITRF 2005 at epoch 1.1.2009. In the results presented in the Table 5.2. the Table 5.1 The loss of the data during the processing Campaign IVS-T2059 EUROPE-97 EUROPE-98 IVS-T2061

GPS antenna Left Right Left Right Left Right Left Right

IVS- T2059

Trajectory points 2,495 2,674 2,642 2,810 2,676 2,823 2,807 2,787

Points in the final adjustment 809 809 642 642 698 698 499 499

EUROPE-97 EUROPE-98 IVS- T2061

N 37.6086 2 37.6080 3 37.6071 3 37.6094 3 E 122.4009 2 122.4006 2 122.4019 2 122.3995 3 U 14.6781 4 14.6780 5 14.6776 6 14.6786 8 Axis 0.0045 6 0.0050 7 0.0052 9 0.0020 11 offset Differences to ITRF2005 (2009.0) N 0.0021 0.0027 0.0035 0.0012 E 0.0005 0.0008 0.0005 0.0019 U 0.0239 0.0239 0.0234 0.0245 Differences to the determination by Jokela N 0.0061 0.0067 0.0076 0.0053 E 0.0032 0.0028 0.0041 0.0018 U 0.7262 0.7262 0.7267 0.7256

Single numbers denote the standard error, in 0.0001 m. We also show a comparison to the vectors from ITRF2005 (2009.0), and to the earlier determination by Jokela

5

Can We Really Promise a mm-Accuracy for the Local Ties on a Geo-VLBI Antenna

GPS antenna phase center variations have been corrected in the RINEX data epoch by epoch before the kinematic trajectory computing. For comparison, the vector from the METS GPS ARP to the Mets€ahovi VLBI reference point in ITRF 2005 coordinates at epoch 1.1.2009 was calculated from the ITRF data set. Differences to our results are presented in Table 5.2. The agreement in horizontal components is good but the vertical component differs more than 2 cm. The comparison to the earlier direct determination by Jorma Jokela in 2004 (Jokela et al. 2009) is shown in Table 5.2. The vertical component differs more than 70 cm because the directly achievable point in Jokela’s determination was not the intersection point of the axes but the pinhole on the floor of the upper platform of the VLBI telescope. Beside the local tie vector we computed the antenna axis offset and orientation. The orientation was modeled as unit direction vectors of the axes. The orientation angles of the telescope, non-perpendicularity of axes and the tilt of the azimuth axis was calculated. Because our model doesn’t take into account epoch-wise variation of the azimuth axis tilt, the direction vector of azimuth axis varies between campaigns. The antenna axes offset differs from the offset determined by Leonid Petrov in 2007: 0.0051 0.0037 m (Petrov 2007). The absolute value from the first three campaigns is about the same, but the sign is opposite to that of Petrov.

41

expect more reliable results for the axes orientation and alignment. We have also used traditional terrestrial surveying methods combined with space intersection technique by measuring reflecting targets on antenna construction. This method was very time consuming and we had telescope for that purpose only during bad weather conditions (e.g., heavy rain, snowing) when it was not in normal use. The results will be published in a future paper. The kinematic method has proved to be suitable for monitoring the IVP of a radio telescope simultaneously during a VLBI session. Especially for sites without a radome, there will be less noisy data. Based on our experience, we may propose that GPS tracking should be done permanently during VLBI sessions. This will give an instantaneous tie to the co-located CGPS antenna. This technique should be considered especially for the VLBI2010 plan. Acknowledgements We would like to thank T. Lindfors, A. Mujunen, M. Tornikoski, J. Kallunki, E. Oinaskallio and H. R€onnberg of the Mets€ahovi Radio Observatory of the Aalto University for their invaluable help. Elevations and azimuths of GPS satellites were computed using “wheresat” of GPStk. (http://www.gpstk.org/bin/view/ Documentation/WebHome). The Mets€ahovi model was programmed with Octave language (http://www.octave.org). This work was partly supported by the Academy of Finland project 134952.

References 5.6

Conclusions and Future Actions

The test shows that a millimeter accuracy is possible to achieve in local tie vector determination with kinematic GPS. Our calculation model is suitable for sparsely scattered data as well as for the data with preplanned circles. The angle readings of the telescope must be synchronized carefully with the GPS observations. Systematic errors like GPS antenna phase center variations must be taken into account although they are smaller than the accuracy of the trajectory point in kinematic solution. Future studies will continue with the analysis of the static GPS observation with two hour sessions of each 90 pre-planned VLBI antenna position. Observations that have been made in March and April 2009. We

Dawson J, Sarti P, Johnston GM, Vittuari L (2007) Indirect approach to invariant point determination for SLR and VLBI systems: an assessment. J Geodesy 81(6–8):433–441 IERS (2005) In: Richter B, Schwegmann W, Dick WR (eds) Proceedings of the IERS workshop on site co-location, Matera, Italy, 23–24 Oct 2003. IERS technical note No. 33. Verlag des Bundesamts f€ur Kartographie und Geod€asie, Frankfurt am Main, p 148 IERS (2009) http://www.iers.org/MainDisp.csl?pid¼68-38. (8.10.2009) ITRF (2009) Co-location survey: online reports, in ITRF web site: http://itrf.ensg.ign.fr/local_surveys.php. (8.10.2009) Jokela J, H€akli P, Uusitalo J, Piironen J, Poutanen M (2009) Control measurements between the Geodetic Observation Sites at Mets€ahovi. In: Drewes H (ed) Geodetic reference frames. IAG Symposium Munich, Germany. Springer, Heidelberg, pp 101–106, 9–14 Oct 2006 L€osler M, Haas R (2009) The 2008 local-tie determination at the Onsala Space Observatory. In: Proceedings of the EVGA, 2009, Bordeaux, France, 24–25 Mar 2009

42 L€osler M, Hennes M (2008) An innovative mathematical solution for a time-efficient ivs reference point determination. In: Measuring the changes, 2008. 4th IAG Symposium on Geodesy for Geotechnical and Structural Engineering, Lisbon, Portugal, 12–15 May 2008 Petrov L (2007) VLBI antenna axis offsets. http://geminig. sfc. nasa.gov/500/oper/solve_save_files/2007c.axo (27.10.2009)

U. Kallio and M. Poutanen Sarti P, Sillard P, Vittuari L (2004) Surveying co-located space geodetic instruments for ITRF computation. J Geod 78(3):210–222 Sarti P, Abbondanza C, Vittuari L (2008) Terrestrial Surveying Applied to Large VLBI Telescopes and Eccentricity Vectors Monitoring. 13th FIG symposium on deformation measurement and analysis. LNEC, Lisbon

6

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008, the ignwd08 Solution P. Willis, M.L. Gobinddass, B. Garayt, and H. Fagard

Abstract

In preparation for the computation of ITRF2008, the DORIS IGN analysis center has undertaken the task of a complete reprocessing of all DORIS data from 1993.0 to 2009.0, using all available DORIS data as well as the most recent models and estimation strategies. We provide here a detailed description of the major improvements recently made in the DORIS data processing, mainly in terms of solar radiation pressure, atmospheric drag, gravity field, and tropospheric correction. We address here the impact of the new IGN time series (ignwd08) on geodetic products using comparison to the previous IGN solutions (ignwd04). In particular, previous artifacts, such as 118-day or 1-year periodic errors in the TZ-geocenter solution or in the vertical component of high latitude DORIS tracking stations, have now disappeared, leading to more precise and reliable time series of DORIS station coordinates. Finally, possible future improvements are discussed proposing new investigations for the future.

6.1

P. Willis (*) Institut Ge´ographique National, Direction Technique, 2 avenue Pasteur, 94165 Saint-Mande´, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris Cite´, UFR STEP, 35 rue He´le`ne Brion, 75013 Paris, France e-mail: [email protected] M.L. Gobinddass Institut Ge´ographique National, LAREG, 6-8 avenue Blaise Pascal, 77455 Marne-la-Valle´e, France Institut de Physique du Globe de Paris, PRES Sorbonne Paris Cite´, UFR STEP, 35 rue He´le`ne Brion, 75013 Paris, France B. Garayt H. Fagard Institut Ge´ographique National, Service de Ge´ode´sie et de Nivellement, 2 avenue Pasteur, 94165 Saint-Mande´, France

Introduction

DORIS (Doppler Orbitography and Radiopositioning Integrated on Satellite) is one of the four geodetic techniques participating in the realization of the International Terrestrial Reference System (ITRS) (Willis et al. 2006). Figure 6.1 presents the current DORIS permanent network, demonstrating a dense and homogenous geographical distribution of 57 tracking stations (Fagard 2006). Since 2003, an International DORIS Service (IDS) was created in order to foster international cooperation (Tavernier et al. 2002; Willis et al. 2010a). The Institut Ge´ographique National (IGN) is one of the seven IDS Analysis Service, providing products on a weekly basis (Willis et al. 2010b). In preparation of ITRF2008 (Altamimi and Collilieux 2010), a new DORIS time

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_6, # Springer-Verlag Berlin Heidelberg 2012

43

44

P. Willis et al. Table 6.1 Main differences in models and analysis strategies used for the two latest DORIS/IGN weekly solutions Model/strategy Gravity field Tropospheric mapping function Elevation cut-off Solar radiation pressure coefficient Atmospheric drag parameter

ignwd04 GGM01C Lanyi 15 estimated Reset/6h

ignwd08 GGM03S GMF 10 fixed Reset/1 h

Fig. 6.1 DORIS permanent tracking network (as of September 2009)

series (ignwd08) was reprocessed using data from 1993.0 to 2009.0. Since then, new IGN weekly solutions using the same processing strategy are regularly delivered at the IDS data center, on average once a week. The goal of this article is to present the major differences in terms of data analysis between this new reprocessed solution (ignwd08) and the previous DORIS solution (ignwd04), and to provide an overview of current available geodetic products from the IGN Analysis Center (AC): weekly time series of station coordinates, velocity field, terrestrial reference frame and polar motion.

6.2

DORIS Data Analysis

Table 6.1 summarizes the main differences between the ignwd04 and the ignwd08 analysis strategies. A more recent GGM gravity was used (Tapley et al. 2005). C21 and S21 rates were corrected to follow the IERS 2003 conventions (McCarthy and Petit 2004). No annual correction was taken into account in the gravity field coefficients. Using the more recent GMF mapping function (Boehm et al. 2006) allowed us to use DORIS data at a lower elevation without any drawback. Following recent investigations (Gobinddass et al. 2009a, 2009b), solar radiation pressure models were rescaled using a constant empirical parameter per satellite. Atmospheric drag parameters were reset every 1-hour for the SPOT and Envisat satellites (Gobinddass et al. 2010). This new strategy can be used for all days, even during high geomagnetic activity (Willis et al. 2005a). For the previous ignwd04 solution, a specific strategy was earlier required

(resetting the drag parameter for estimation every minute instead of every 6 h). This involved some non-automated processing for these few days, while the new procedure is fully automated.

6.3

Time Series of ignwd08 Station Coordinates

Unlike other IDS Analysis Centers, DORIS data are processed automatically as soon as they appear at the IDS data centers. Weekly station coordinates in SINEX format are available at these data centers within a few hours, and are provided in free-network (loosely constrained) for the IDS combination but also after projection and transformation in the latest ITRF solution (currently ITRF2005 but soon in ITRF2008 as this transformation is straightforward and does not require any DORIS data processing). These results are freely available to the scientific community at the following URL address for further geophysical investigations: http://ids.cls.fr/html/doris/ids-stationseries.php3. This technique provides SINEX results for geodesists (including full covariance information in a loosely constrained terrestrial reference frame). It also provides tabulated results in STCD format (Noll and Soudarin 2006) for geophysicists directly expressed in ITRF2005 (Altamimi et al. 2007). Figure 6.2 provides an example of such results for the Rio Grande station in Argentina. Different colors indicate the different occupations of this station, as related to equipment upgrades. The positions of successive occupation at the same site were tied together using geodetic local tie information. Smaller scatter after 2002.4 indicates an improvement in repeatability when 4 or 5 DORIS satellite are available. No large discontinuity can be noticed, showing a good

6

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008

45

Fig. 6.2 Weekly time series of ignwd08 Rio Grande station coordinates expressed in ITRF2005 (from STCD files). In North (a) and Up (b) (in mm)

agreement between the DORIS results and the geodetic local tie vectors. However a closer inspection may indicate possible problem after first occupation or after data gap (vertical). The East component (not displayed here) is noisier due to the North-South tracks of the sun-synchronous satellites (SPOTs and Envisat), especially for mid-latitude stations. Other authors already showed that previous artifacts in the DORIS time series at 118 days (TOPEX draconitic period) are no longer visible in the vertical component for the high-latitude station when solar radiation pressure models are empirically rescaled (Amalvict et al. 2009; Kierulf et al. 2009). The previous periodic effects visible in the ignwd04 time series were due to an improper handling of the solar radiation pressure (Gobinddass et al. 2009a, 2009b) and were observed in previous time series (le Bail 2006; Williams and Willis 2006) but were never fully appreciated. These problems were rather serious for altimetry as they affected mostly the Zcomponent of the stations, which is the major factor for mean sea level determination (Morel and Willis 2002, 2005; Beckley et al. 2007). Table 6.2 provides some information on annual signals present in DORIS/ IGN time series in the polar region. Table 6.2 demonstrates that the previous annual signals around 10 mm in the ignwd04 time series of vertical coordinates of high-latitude station are not present anymore in the new ignwd08 time series. Currently observed annual signals around 3 mm could easily be explained by real geophysical reasons, as geocenter motion is estimated to be at this level.

Table 6.2 Amplitude of annual signals in DORIS vertical time series (in mm) Station SPJB THUB ADEA ROTA SYOB BEMB

Lat (deg) 78 550 76 320 66 400 67 340 69 000 77 520

ignwd04 13.7 11.6 3.4 6.6 6.5 11.2

ignwd08 2.7 2.7 1.9 3.0 3.9 5.0

A similar problem was also previously detected at 1 year (draconitic periods of SPOT and Envisat, being sun-synchronous satellites) and also disappeared in this new ignwd08 solution (Willis et al. 2010b).

6.4

ign09d02 Derived Velocity Field

At regular intervals (every 6 months to 1 year), we also stack all the available DORIS weekly solutions and provide a cumulative position and velocity solution, making full use of geodetic local ties between successive DORIS occupations, using proper a priori variance, as provided by IGN. The latest IGN velocity field (ign09d02) available at both IDS data centers (CDDIS in USA and IGN in France). In most cases, these results can be explained by plate tectonics (Soudarin and Cre´taux 2006; Argus et al. 2010). However, in other cases, such as the Socorro Island (Mexico), the station displacement is not linear and can be attributed to local volcanic deformations (Briole et al. 2009). While GPS is the

46

P. Willis et al.

key player for geodynamics due to its easy densification, DORIS may still have a role to play in a few cases such as Africa where the geodetic infrastructure is still sparse (Nocquet et al. 2006; Argus et al. 2010). We do not estimate the DORIS-derived velocity more often than every 6 months, as a large number of DORIS observations is already available (16 years) For most stations, formal errors of 0.15–0.30 mm/yr are typical.

6.5

Terrestrial Reference Frame

As our DORIS weekly solutions are provided in free-network form, we can compute for each week the 7-parameters for the TRF (origin, orientation and scale), looking at the best transformation fit into the ign09d02 position/velocity solution already aligned on ITRF2005, but including all DORIS stations. Figure 6.3 displays results for the ignwd08 weekly scales with respect to ITRF2005. No antenna map correction was used for DORIS (Willis et al. 2005b) to map these results toward any ITRF. While the weekly scatter is small, a significant drift can be seen toward ITRF2005 realization. This is currently visible in all results from all DORIS Analysis Centers, sometimes with lower values (Valette et al. 2010) and no convincing explanation has yet been proposed.

Fig. 6.3 Weekly determination of the TRF scale between the ignwd08 solution and ITRF2005 (through ign09d02)

No significant discontinuity can be seen in these results. In our opinion, the discontinuities detected at the end of 2004 by Altamimi and Collilieux (2010) and Valette et al. (2010) may be related to results from other DORIS ACs that could map into the IDS-3 combination, which is the DORIS combination submitted for ITRF2008. Such problems could be related to the end of the TOPEX/DORIS data or to a software modification in the Envisat satellite (Willis et al. 2005b, 2007). The IGN weekly solutions also contain more DORIS stations than the IDS-3 solution, as a preselection was done for IDS-3, based on recommendation from DPOD2005 (Willis et al. 2009). Finally, when transforming into ITRF2005, we do not use the original ITRF2005 coordinates and velocities but our internal ign09d02 long-term solution. We also disregard some DORIS stations (six in total) on a week-byweek basis, taking into account possible temporary problems with these data, as done recently in a more sophisticated way using genetically modified networks (Coulot et al. 2009). It is certainly too early to have a definite conclusion on this difficult problem and more tests are required to understand the exact nature of such possible discontinuities. In Fig. 6.3, a different behavior may be observed for the very early data. This is still under investigation but it could be linked to data availability (only two DORIS satellites : TOPEX/Poseidon and SPOT-2) or

.

6

Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008

to a preprocessing problem as detected earlier in the case of SPOT4 for most of the 1998 data and some early 1999 data (Willis et al. 2006).

6.6

Polar Motion

We also update every week polar motion estimation derived using the DORIS data. Tables 6.3 and 6.4 display direct comparisons of these daily DORIS results with the JPL/GPS time series. When considering the full data set (from 1993.0) in Table 6.3, a clear improvement can be seen for the most recent ignwd08 solution. When considering only the best DORIS results, when four or five satellites are available after 2002.4, the improvement is even more pronounced and RMS of 0.5 mas are now achievable (Table 6.4), even if a small bias of 0.3 mas in X still needs to be explained. This is a significant improvement when compared to earlier determination (Gambis 2006): RMS of 1.74 mas for XPole and 0.99 mas for YPole, with a 0.24 offset for recent data (2000.0–2004.0). Part of these improvements is due to the fact that no daily polar rates are estimated in the recent ignwd08 solution. Another improvement is related to systematic errors at the 5.2 day period (SPOT sub-cycle), linked again to the solar radiation pressure estimation as demonstrated in Willis et al. (2010).

Table 6.3 DORIS polar motion external precision compared to IGN/JPL time series (mean removed) XPole RMS (mas) ignwd04 1.864 ignwd08 1.228

YPole RMS (mas) 1.440 1.290

XPole mean (mas) 0.287 0.126

YPole mean (mas) 0.164 0.373

6.7

47

Discussion on Future Improvements

While the ignwd08 solution is still very new and regularly updated, future possible improvements are already considered: • Early analysis of Jason-2 data showed that it could improve the realization of the terrestrial reference frame (Zelensky et al. 2010) and that no effect related to the South Atlantic Anomaly (Willis et al. 2004) is observed in these data. • New DORIS satellites will be launched soon (Cryosat-2 from ESA and Altika from India and France). Direct use of the new DORIS phase and pseudorange (Mercier et al. 2010) should be investigated. • While only minor improvement is expected at the altitude of the DORIS satellite from a GOCEderived gravity field (Visser et al. 2009), timevarying effects , new tide models and AOD (Atmospheric and Ocean De-aliasing) corrections could improve the DORIS orbits. • For the tropospheric correction, VMF (Boehm 2004) could be used instead of GMF. Early tests showed that horizontal tropospheric gradients could be considered for DORIS data processing as well (Flouzat et al. 2009). • Previous studies using Laser data also demonstrated possible time tagging problems in the DORIS data files (Zelensky et al. 2010). • Finally, the recent inter-comparisons between the 7 IDS Analysis Centers (Valette et al. 2010) should certainly lead to new investigations, for example for problems related to the South Anomaly (Bock et al. 2010, Stepanek et al. 2010).

6.8

Conclusions

Period 1993.0–2002.4

Table 6.4 DORIS polar motion external precision compared to IGN/JPL time series (mean removed) XPole RMS (mas) ignwd04 1.387 ignwd08 0.584 2002.4–2008.7

YPole RMS (mas) 0.740 0.525

XPole mean (mas) 0.003 0.289

YPole mean (mas) 0.287 0.012

In conclusion, the new ignwd08 solution is a clear improvement over the previous ignwd04 solution. In particular, due to a better analysis strategy concerning the solar radiation pressure, previous artifacts at 118 days and 1 year have now disappeared in the Zgeocenter as well in the vertical component of highlatitude station time series. Significant improvements were also obtained for polar motion for which 0.5 mas

48

comparison with GPS results can be observed, when 4 or more DORIS satellites are available. Finally, with the use of more recent satellites (Jason-2, Cryosat-2, and Altika) equipped with new digital DGXX equipments, more improvements are already foreseen and currently under investigation. Acknowledgement This work was supported by the Centre National d’Etudes Spatiales (CNES). It is based on observations with DORIS embarked on SPOTs, TOPEX/Poseidon, ENVISAT and Jason satellites. This paper is IPGP contribution number 2593.

References Altamimi Z, Collilieux X (2010) DORIS contribution to ITRF2008. Adv Space Res 45(12):1500–1509 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005, a new release of the International Terrestrial Reference Frame based on time series of station positions and Earth orientation parameters. J Geophys Res 112(B9), art. B09401 Amalvict M, Willis P, W€ oppelmann G, Ivins ER, Bouin MN, Testut L (2009) Isostatic stability of the East Antarctica station Dumont d’Urville from long-term geodetic observations and geophysical models. Polar Res 28 (2):193–202 Argus DF, Gordon R, Heflin M, Ma C, Eanes R, Willis P, Peltier WR, Owen S (2010) The angular velocities of the plates and the velocity of the Earth’s center from space geodesy. Geophys J Int 180(3):916–960. doi:10.1111/j.1365246X.2009.04463 Beckley BD, Lemoine FG, Luthcke SB, Ray RD, Zelensly NP (2007) A reassessment of global and regional mean sea level rends from TOPEX and Jason-1 altimetry ased on revised reference frame and orbits. Geophys Res Lett 34(14):L14608 Bock O, Willis P, Lacarra M, Bosser P (2010) An intercomparison of tropospheric delays estimated from DORIS and GPS data, Adv Space Res 46(12):1648–1660 Boehm J (2004) Vienna mapping functions in VLBI analysis. Geophys Res Lett 31:L01603 Boehm J, Niell A, Tregoning P, Schuh H (2006) Global Mapping Function (GMF), a new empirica mapping function based on numerical weather model data. Geophys Res Lett 33(7), art. L07304 Briole P, Willis P, Dubois J, Charade O (2009) Potential applications of the DORIS system. A geodetic study of the Socorro Island (Mexico) coordinate time series. Geophys J Int 178(1):581–590 Coulot D, Collilieux X, Pollet A, Berio P, Gobinddass ML, Soudarin L, Willis P (2009) Genetically modified networks. A genetic algorithm contribution to space geodesy, application to the transformation of SLR and DORIS EOP time series. In: European Geocience Union meeting, Vienna, Austria, EGU2009-7988 Fagard H (2006) Twenty years of evolution for the DORIS permanent network, from its initial deployment to its renovation. J Geod 80(8–11):429–456

P. Willis et al. Flouzat M, Bettinelli P, Willis P, Avouac JP, Heriter T, Gautam U (2009) Investigating tropospheric effects and seasonal position variations in GPS and DORIS time series from the Nepal Himalaya. Geophys J Int 178(3):1246–1259 Gambis D (2006) DORIS and the determination of the Earth’s polar motion. J Geod 80(8–11):649–656 Gobinddass ML, Willis P, de Viron O, Sibthorpe AJ, Zelensky NP, Ries JC, Ferland R, Bar-Sever YE, Diament M (2009a) Systematic biases in DORIS-derived geocenter time series related to solar radiation pressure mis-modelling. J Geod 83 (9):849–858 Gobinddass ML, Willis P, de Viron O, Sibthorpe A, Zelensky NP, Ries JC, Ferland R, Bar-sever YE, Diament M, Lemoine FG (2009b) Improving DORIS geocenter time series using an empirical rescaling of solar radiation pressure. Adv Space Res 44(11):1279–1287 Gobinddass ML, Willis P, Diament M, Menvielle M (2010). Refining DORIS atmospheric drag estimation in preparation of ITRF2008. Adv Space Res 46(12):1566–1577 Kierulf HP, Pettersen BR, MacMillan DS, Willis P (2009) The kinematics of Ny-Alesund from space geodetic data. J Geodyn 48(1):37–46 le Bail K (2006) Estimating the noise in space-geodetic positioning, the case of DORIS. J Geod 80(8–11):541–565 McCarthy D, Petit G (eds) (2004) IERS 2003 Conventions. In: IERS Techn Note 32, Frankfurt-am-Main, Germany Mercier F, Cerri L, Berthias JP (2010) Jason-2 DORIS phase measurement processing. Adv Space Res 45 (12):1441–1454. doi:10.1016/j.asr.2009.12.002 Morel L, Willis P (2002) Parameter sensitivity of TOPEX orbit and derived mean sea level to DORIS stations coordinates. Adv Space Res 30(2):255–263 Morel L, Willis P (2005) Terrestrial reference frame effects on global sea level rise determination from TOPEX/Poseidon altimetric data. Adv Space Res 36(3):358–368 Nocquet JM, Willis P, Garcia S (2006) Plate kinematics of Nubia-Somalia using a combined DORIS and GPS solution. J Geod 80(8–11):591–607 Noll C, Soudarin L (2006) On-line resources supporting the data, products, and information infrastructure for the International DORIS Service. J Geod 80(8–11):419–427 Soudarin L, Cre´taux JF (2006) A model of present-day tectonic plate motions from 12 years of DORIS measurements. J Geod 80(8–11):609–624 Stepanek P, Dousa J, Filler V (2010) DORIS data analysis at Geodetic Observatory Pecny using single-satellites and multisatellites geodetic solutions. Adv Space Res 46 (12):1578–1592 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F (2005) GGM02, an improved Earth gravity field model from GRACE. J Geod 79(8):467–478 Tavernier G, Soudarin L, Larson K, Noll C, Ries J, Willis P (2002) Current status of the DORIS Pilot experiment. Adv Space Res 30(2):151–156 Valette JJ, Lemoine FG, Ferrage P, Yaya P, Altamimi Z, Willis P, Soudarin L, (2010) IDS contribution to ITRF2008 Adv Space Res 46(12):1614–1632 Visser PNAM, van den IJssel J, van Helleputte T et al (2009) Orit determination for the GOCE satellite. Adv Space Res 43 (5):760–768

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Recent Improvements in DORIS Data Processing at IGN in View of ITRF2008

Williams SDP, Willis P (2006) Error analysis of weekly station coordinates in the DORIS network. J Geod 80 (8–11):429–456 Willis P, Haines B, Berthias JP, Sengenes P, Le Mouel JL (2004) Behavior of the DORIS/Jason oscillator over the South Atlantic Anomaly. CR Geosci 336(9):839–846 Willis P, Deleflie F, Barlier F, Bar-Sever YE, Romans L (2005a) Effects of thermosphere total density perturbations on LEO orbits during severe geomagnetic conditions (Oct – Nov 2003). Adv Space Res 36(3):522–533 Willis P, Desai SD, Bertiger WI, Haines BJ, Auriol A (2005b) DORIS satellite antenna maps derived from long-term residuals time series. Adv Space Res 36(3):486–497 Willis P, Jayles C, Bar-Sever YE (2006) DORIS, from altimeric missions orbit determination to geodesy. CR Geosci 338 (14–15):968–979 Willis P, Haines BJ, Kuang D (2007) DORIS satellite phase center determination and consequences on the derived scale of the Terrestrial Reference Frame. Adv Space Res 39 (10):1589–1596 Willis P, Ries JC, Zelensky NP, Soudarin L, Fagard H, Pavlis EC, Lemoine FG (2009) DPOD2005: realization of a DORIS

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terrestrial reference frame for precise orbit determination. Adv Space Res 44(5):535–544 Willis P, Fagard H, Ferrage P, Lemoine FG, Noll CE, Noomen R, Otten M, Ries JC, Soudarin L, Tavernier G, Valette JJ (2010a) The International DORIS Service, toward maturity. Adv Space Res 45(12):1408–1420. doi:10.1016/j.asr.2009. 11.018 Willis P, Boucher C, Fagard H, Garayt B, Gobinddass ML (2010b) Contributions of the French Institut Ge´ographique National (IGN) to the International DORIS Service. Adv Space Res 45(12):1470–1480. doi:10.1016/j.asr.2009.09.019 Zelensky NP, Berthias JP, Lemoine FG (2006) DORIS time bias estimated using Jason-1, TOPEX/Poseidon and Envisat orbits. J Geod 83(9):497–506 Zelensky NP, Berthias JP, Lemoine FG (2006) DORIS time bias estimated using Jason-1, TOPEX/Poseidon and ENVISAT orbits. J Geod 80(8–11):497–506 Zelensky NP, Lemoine FG, Chinn DS, Rowlands DD, Luthcke SB, Beckley D, Pavlis D, Ziebart A, Sibthorpe A, Willis P, Luceri V (2010) DORIS/SLR POD modeling improvements for Jason-1 and Jason-2. Adv Space Res 46(12): 1541–1558

7

Towards a Combination of Space-Geodetic Measurements A. Pollet, D. Coulot, and N. Capitaine

Abstract

The International Terrestrial Reference Frame (ITRF), the Earth Orientation Parameter (EOP) time series, and the International Celestial Reference Frame are obtained separately and may thus present inconsistencies. To solve this problem, a first step has been made with the latest ITRF realization (ITRF2005), which has been computed, for the first time, with consistent EOP time series. Another approach to better understand this issue is to directly estimate, in the same process, station positions and EOP time series, from all the space-geodetic measurements. In the framework of the French Groupe de Recherche de Ge´ode´sie Spatiale (GRGS) activities, this latter approach has been studied for several years. For this purpose, the observations of VLBI, SLR, GPS, and DORIS techniques are combined using the same models and software for all the individual data processing. In this paper, we study methodological issues regarding the definition and the consistency of the weekly combined terrestrial frames.

7.1

Introduction

The International Earth rotation and Reference systems Service (IERS) provides different geodetic products as the International Terrestrial Reference Frame (ITRF), the Earth Orientation Parameters (EOP), and the International Celestial Reference Frame (ICRF), the EOP providing the link between

A. Pollet (*) D. Coulot Institut Ge´ographique National, LAREG & GRGS, 6-8 Avenue Blaise Pascal, 77455 Champs-sur-Marne, Marne-la-Valle´e, France e-mail: [email protected] N. Capitaine SYRTE, Observatoire de Paris, CNRS, UPMC & GRGS, 61 Avenue de l’Observatoire, 75014 Paris, France

ITRF and ICRF. These products are computed separately; this may introduce inconsistencies. The latest ITRF realization, ITRF2005 (Altamimi et al. 2007), has been a major step toward the consistency between IERS products. Indeed, for the first time, the ITRS Product Center (PC) has provided the ITRF2005 together with consistent EOP time series. Another approach is under investigation, namely the combination of space-geodetic techniques (DORIS/GPS/SLR/VLBI) at the measurement level. This method permits us to use the same models and software for all techniques in order to obtain consistent results. Furthermore, such a combination enables the introduction of new common parameters and technique links, in addition to the local ties. It should thus allow us to evidence the possible systematic errors of each technique in order to understand and

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_7, # Springer-Verlag Berlin Heidelberg 2012

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Table 7.1 Configurations of GPS, VLBI, DORIS, and SLR processing Data

GPS Double differences

Satellites

GPS constellation

Sampling rate Elevation cutoff angle Network

10 min 10 10 150 stations (IGS core sites + colocated stations) Absolute ITRF2005 IERS 05 C04 (Bizouard and Gambis 2009) IERS conventions (McCarthy and Petit 2004) FES2004 (Lyard et al. 2006) FES2004 (Lyard et al. 2006) Not modeled ECMWFa (Guo and Langley 2004) 48 h with a 24-h overlapping

Antenna model Stations EOP Solid Earth tides Ocean tides Ocean loading Atm. loading Tropospheric model Mapping function Orbital arc length a

VLBI IVS-R1, IVS-R4, intensive sessions

DORIS

SLR

SPOT 2,4,5 ENVISAT Jason 1

LAGEOS 1 LAGEOS 2

12 8 All available stations

ITRF2005 rescaled

30 h with a 6-h overlapping

(Mendes and Pavlis 2004) (Mendes et al. 2002) 9 days with a 2-day overlapping

http://www.ecmwf.int

to reduce them. Several studies have been carried out on this subject. Andersen (2000) has applied this kind of combination with a stochastic approach and a square root formation filtering and smoothing to VLBI sessions. Thaller et al. (2007) have combined GPS and VLBI data and got promising results regarding EOP and Zenithal Tropospheric Delays (ZTD). A proof of the great interest aroused by such combination is the creation of a new IERS working group (COL, for Combination at the Observation Level) in 2009. The present study is the continuation of Coulot et al. (2007), who have combined DORIS, GPS, SLR, and VLBI normal systems to evidence the interest of this approach for EOP (and, in particular, Universal Time – UT). We focus here on the definition and the homogeneity of the combined terrestrial frame. In the first section, we present the data used and the different parameters estimated during the combination. Then, we test different possible approaches to combine the observation systems with a particular emphasis on the consistency of the combined frame. Finally, we underline the major role of the local ties in the computation and we provide some conclusions and prospects.

7.2

Data, Software, and Parameters

We carry out tests over nearly 3 months of data obtained by the four space-geodetic techniques, available at the IVS, ILRS, IGS, and IDS data centers (between January 9 and March 19, 2005). The CNES1/GRGS software GINS provides the observation systems for each technique. Indeed, this software, which is also designed for gravity field determination (Bruinsma et al. 2009), has the capability to process DORIS, GPS, SLR, and VLBI data (cf. Bourda et al. 2007). It is used by the IDS LCA-CNES/CLS, the IGS CNES/CLS, and the ILRS GRGS Analysis centers (AC), and by the Bordeaux Observatory for VLBI data processing. Table 7.1 details the processing configurations for each technique. These configurations are close to those applied by the AC for the satellite techniques and by the Bordeaux Observatory team for VLBI. The LAREG/GRGS LOCOMOTIV software combines weekly individual observation systems, uses the degree of freedom method (Sillard 1999) in

1

Centre National d’Etudes Spatiales, French institute.

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Towards a Combination of Space-Geodetic Measurements

order to provide optimal weights for each satellite technique (one weight per satellite observation set) and each VLBI session observation set in the combination, and estimates the following parameters: – Weekly station positions, at the middle of the GPS week. – Daily EOP (polar motion and UT), at noon. – Orbital parameters in agreement with the three GRGS AC configurations. – ZTD, every 2 h, except for SLR. – Technique specific biases in agreement with the three GRGS AC configurations and with the Bordeaux Observatory VLBI configuration. In addition, depending on the model applied for the combination, we estimate or not estimate Helmert parameters (translations and scale factor for each satellite technique and only scale factor for VLBI, cf. Sect. 7.2).

7.3

Combination, Referencing, and Consistency

Coulot et al. (2007) directly combined the normal equation systems derived from the technique observations. In this work, only the EOP were used as common parameters. The minimum constraints were applied over four sub-networks (one per technique). These minimum constraints were related to the parameters evidenced, per technique, in loose constrained combined solutions, by the reference system effect criterion (Sillard and Boucher 2001). This combination provided heterogeneous combined frames. Indeed, each technique realized its own reference frame. This may be caused by the lack of links between the technique networks; common EOP indeed only link techniques regarding orientation. In the next sections, we thus introduce the local ties and we carry out different combinations (from C1 to C4) in order to obtain a solution for which the systematic errors of the techniques are well handled.

7.3.1

Model for an Ideal Case

We introduce local ties in the combination through observation equations:

53

X01 þ dXc1 ðX02 þ dXc2 Þ ð12Þ

¼ LocTieX

ðSLocTie12 Þ ;

(7.1)

with X01 (resp. X02 ) being the station 1 (resp. 2) a priori positions, dXc1 (resp. dXc2 ) the estimated position ð12Þ the local tie offsets of station 1 (resp. 2), LocTieX vector between the stations 1 and 2 and SLocTie1–2 its variance-covariance matrix. Theoretically speaking, each technique is sensitive to the scale of the terrestrial frame. Each satellite technique is sensitive to the terrestrial frame origin (the geocenter), via the dynamical orbits of its dedicated satellites. Through local ties, we can transmit this definition of the origin to the VLBI station network. Finally, we must conventionally define the orientation of the frame; no technique is sensitive to this orientation (EOP are estimated). We should thus theoretically obtain a homogeneous combined solution by gathering the normal equation systems per technique, with local ties and three minimum constraints (one per rotation) applied over a GPS sub-network to define the orientation of the combined frame w.r.t. ITRF2005. However, as shown in Table 7.2 (C1 test), we still have inconsistent results. Indeed, we estimate the transformation parameters between different networks in the Fc combined frames (all the stations and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005 in order to evaluate the homogeneity of the combined solutions. Each technique network realizes its own reference frame. For example, we notice a 3D discrepancy of about 9.6 mm between the DORIS and GPS translations and a scale difference of about 0.7 ppb (4 mm) between SLR and VLBI. Two reasons may explain these heterogeneities. Systematic errors exist between the techniques, due to problems in the models used, in particular the models specific to a particular technique [antenna model for GPS (Ge et al. 2005), solar radiation pressure for DORIS (Gobinddass et al. 2009), etc.] and/or introducing local ties on a weekly basis may be problematic. Indeed, on a weekly basis, we can get poor network distributions, especially regarding the co-located station networks for VLBI or SLR. To investigate these inconsistencies, we directly introduce Helmert parameters to take into account possible mismodellings.

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Table 7.2 Transformation parameters – mean value (formal error of the mean value) standard deviation value in mm for translations and scale factor Tx, Ty, Tz, and D and m as for rotations Rx, Ry, and Rz – estimated between different networks in the Fc combined frames (all the stations and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005, for C1, C2, C3, and C4 combinations over 3 months of data (see text) Combination C1 C2 Helmert parameters No Yes Local tie handling Equation (7.1) Equation (7.1) Local tie weights Provided by ITRF PC Provided by ITRF PC Transformation parameters between all stations and ITRF2005 Tx 5.2 (0.7) 3.5 0.1 (0.2) 0.1 Ty 1.5 (0.5) 1.6 0.0 (0.2) 0.1 Tz 6.9 (0.9) 5.3 0.2 (0.3) 0.3 D 2.1 (0.4) 1.1 0.0 (0.3) 0.3 Rx 40 (5) 28 2 (3) 2 Ry 36 (17) 34 2 (5) 2 Rz 32 (15) 26 2 (6) 3 Transformation parameters between DORIS network and ITRF2005 Tx 2.6 (0.7) 3.5 5.9 (0.6) 2.1 Ty 0.1 (0.7) 2.5 1.7 (0.6) 1.0 Tz 0.8 (0.8) 3.7 2.9 (0.7) 3.0 D 5.9 (0.5) 3.7 0.5 (0.7) 3.3 Rx 63 (22) 60 29 (27) 73 Ry 105 (25) 72 298 (37) 139 Rz 175 (28) 93 146 (24) 58 Transformation parameters between GPS network and ITRF2005 Tx 5.1 (0.7) 3.2 0.1 (0.3) 0.1 Ty 2.3 (0.7) 2.2 0.0 (0.2) 0.0 Tz 9.8 (1.0) 5.3 0.1 (0.3) 0.1 D 1.3 (0.4) 0.9 0.0 (0.2) 0.1 Rx 20 (16) 31 2 (2) 0 Ry 29 (16) 30 1 (2) 0 Rz 25 (14) 24 3 (3) 0 Transformation parameters between SLR network and ITRF2005 Tx 3.9 (0.7) 3.9 0.5 (0.7) 1.6 Ty 2.4 (0.5) 1.9 3.4 (0.6) 1.6 Tz 0.8 (0.9) 4.1 3.9 (0.7) 0.6 D 16.2 (0.6) 2.4 2.4 (0.7) 1.2 Rx 34 (23) 85 36 (24) 86 Ry 16 (26) 92 29 (27) 91 Rz 47 (24) 66 51 (19) 57 Transformation parameters between VLBI network and ITRF2005 Tx 3.8 (0.9) 3.5 1.5 (0.9) 3.2 Ty 5.5 (0.9) 4.8 6.9 (0.9) 4.6 Tz 7.2 (0.9) 3.5 10.8 (1.0) 4.1 D 12.1 (0.7) 3.8 6.2 (0.9) 4.4 Rx 126 (25) 75 126 ( 25) 62 Ry 76 (28) 94 36 (31) 95 Rz 87 (38) 172 64 (43) 181

C3 No Equation (7.7) Equation (7.8)

C4 Yes Equation (7.7) Equation (7.8)

5.3 (0.7) 3.6 1.9 (0.5) 1.5 5.9 (0.8) 5.3 3.3 (0.4) 1.5 36 (5) 27 41 (18) 39 36 (16) 30

0.1 (0.2) 0.2 0.0 (0.2) 0.2 0.4 (0.3) 0.4 0.1 (0.2) 0.3 5 (3) 6 2 (5) 3 1 (7) 5

4.9 (0.6) 3.0 1.8 (0.6) 2.0 5.3 (0.8) 4.0 3.1 (0.4) 1.3 26 (19) 42 42 (17) 36 31 (16) 30

0.3 (0.4) 0.6 0.7 (0.4) 0.5 0.3 (0.4) 0.7 0.5 (0.4) 1.5 4 (13) 22 8 (8) 8 1 (12) 16

5.1 (0.7) 3.3 0.1 (0.6) 1.4 7.3 (0.9) 5.0 1.8 (0.4) 1.0 30 (15) 28 33 (17) 34 29 (15) 26

0.0 (0.2) 0.1 0.0 (0.2) 0.0 0.0 (0.2) 0.0 0.0 (0.2) 0.1 0 (2) 0 0 (2) 0 0 (2) 0

5.2 (0.6) 3.2 1.2 (0.5) 1.7 0.3 (0.8) 3.6 12.0 (0.4) 2.4 13 (17) 40 36 (17) 39 40 (16) 39

1.4 (0.4) 0.3 0.2 (0.4) 1.2 0.4 (0.4) 1.0 0.6 (0.5) 1.2 9 (14) 33 13 (14) 29 10 (11) 25

5.0 (0.7) 2.8 1.5 (0.7) 2.5 2.0 (0.8) 3.3 6.3 (0.5) 1.8 6 (22) 58 64 (19) 44 51 (19) 45

0.9 (0.5) 1.3 1.5 (0.5) 1.5 1.5 (0.5) 1.2 2.2 (0.5) 1.8 8 (19) 43 32 (19) 45 24 (18) 39

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Towards a Combination of Space-Geodetic Measurements

7.3.2

55

3 dXc 6 dEOPc 7 7 6 6 a 7 7 6 7 Y ¼ B6 6 yDORIS 7; 6 yGPS 7 7 6 4 ySLR 5 yVLBI 2

Improved Model

A possible way to ensure the homogeneity of the combined solution is to introduce the Helmert (transformation) parameters in the observation systems of each technique. We consider the following observation system: 2

3 dX Y ¼ A4 dEOP 5; a

(7.2)

with Y being the pseudo-observations, A the design matrix, dX the station position offsets w.r.t. a frame Fk, dEOP the EOP offsets consistent with the orientation of Fk and a the other parameter offsets such as orbital parameters, ZTD, etc. We then introduce the transformation parameters, using the following equations: dX ¼ dXc þ T þ DX0 þ RX0 ; dEOP ¼ dEOPc þ R0 ;

(7.3)

with dXc being the station position offsets w.r.t. Fc, the combined frame, T, D, R, and R’ the scalars, vectors, and matrices related to the transformation parameters between the frames Fc and Fk, X0 the a priori station positions, and dEOPc the EOP offsets consistent with the orientation of Fc. In practice, no technique is sensitive to the orientation of Fc (we estimate EOP). The estimated EOP offsets thus align w.r.t. the orientation we define (ibid regarding the frame origin for the VLBI technique). We thus estimate translations and scale factor for satellite techniques: dX ¼ dXc þ T þ DX0 ; dEOP ¼ dEOPc ;

where ytech (tech corresponding respectively to DORIS, GPS, SLR or VLBI) is the vector of the transformation parameters per technique (a scale factor for each technique and three translation parameters for each satellite technique) and B is the new design matrix, deduced from the matrix A. These transformation parameters are estimated in addition to all the other parameters. Due to the lack of information regarding the Fc combined frame definition, the normal system deduced from this observation system presents rank deficiency. As local ties link the technique networks, this rank deficiency correspond to the seven parameters needed to define the combined frame. This definition is obtained by using constraints: either constraints on the estimated transformation parameters and/or minimum constraints. Indeed, we can apply seven minimum constraints w.r.t. ITRF2005 to define Fc. We can also take advantage of the Helmert parameters. For example, if we want to define the origin of Fc as the SLR origin, we do not estimate the SLR translations. In this case, we define the origin of the combined frame independently of ITRF.

7.3.3

Numerical Tests

(7.4)

and only scale factor for VLBI technique: dX ¼ dXc þ DX0 ; dEOP ¼ dEOPc :

(7.6)

(7.5)

We thus obtain the following global observation system:

We compare here two combinations: the “ideal case” combination, called C1 (cf. Sect. 7.3.1), and a similar one with Helmert parameters, called C2 (cf. Sect. 7.3.2). For local ties, we use the values and the associated standard deviations provided by the ITRF PC.2 More than 50 local ties are used per week. For the

2

See tab.

http://itrf.ensg.ign.fr/ties/ITRF2005/ITRF2005-localties.

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combination C1, we apply three minimum constraints w.r.t. ITRF2005 over a GPS sub-network to define the orientation of Fc. Consequently, Fc has the orientation of ITRF2005 but not necessarily the same origin and scale. For the combination C2, we use seven minimum constraints w.r.t. ITRF2005 over the same GPS subnetwork to define Fc, which is thus theoretically fully expressed in ITRF2005. To check the homogeneity of the combined solutions, we compute the transformation parameters between different networks in the Fc combined frame (all the stations, and DORIS, GPS, SLR, and VLBI sub-networks) and ITRF2005. Table 7.2 shows that the estimation of the Helmert parameters in the combination increases the homogeneity of the solution. For example, the 3D discrepancy between the DORIS and GPS translations decreases from 9.6 to 6.7 mm. However, there are still some heterogeneities. The VLBI and SLR networks present the largest ones, probably due to poor co-located networks with a weekly sampling.

7.4

A Possible Approach of Referencing

In order to obtain an homogeneous solution, we present another way of introducing local ties in the combination by the use of equality constraints between estimated co-located station position offsets: dXc1 dXc2 ¼ 0 ðSTie12 Þ :

(7.7)

with dXc1 (resp. dXc2 ) the estimated position offsets of station 1 (resp. 2). Doing so, we consider that the value of the local tie is the difference between the ITRF2005 a priori station ð12Þ positions [X01 X02 ¼ LocTieX – cf. (7.1)]. We thus switch from rescaled ITRF2005 to ITRF2005 for SLR a priori station positions, in order to get consistent values. The standard deviations sTie12 (on which STie12 is based) are computed from the variancecovariance matrix of the ITRF2005 a priori station positions X01 andX02 , at the considered epoch t:

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Var X01 ðtÞ X02 ðtÞ ¼ Var X01 ðtÞÞ þ VarðX02 ðtÞ 2 cov X01 ðtÞ; X02 ðtÞ

sTie12 ¼

i X0i ðtÞ ¼ X0i ðt0 Þ þ dt X_ 0 ðt0 Þ with dt ¼ t t0 hence : Var X0i ðtÞ ¼ Var X0i ðt0 Þ þ dt2 Var X0i ðt0 Þ i þ 2 dt cov X0i ðt0 Þ; X_ 0 ðt0 Þ and : cov X01 ðtÞ; X02 ðtÞ ¼ cov X01 ðt0 Þ; X02 ðt0 Þ 1 2 2 þ dt2 cov X_ 0 ðt0 Þ; X_ 0 ðt0 Þ þ dt cov X01 ðt0 Þ; X_ 0 ðt0 Þ 1 þ dt cov X_ 0 ðt0 Þ; X02 ðt0 Þ

(7.8) In practice, sTie12 sLocTie12 . Furthermore, this approach is ITRF-dependent: if an event has changed a co-located station position after 2005, this kind of local ties cannot be used for the considered co-location site. A secular solution, estimated from our data and used as the a priori solution, could avoid this dependence. Due to our short data span, here we have used the ITRF2005 to evaluate these constraints. We compute two combinations based on this approach, C3 and C4. Regarding the estimated parameters and the minimum constraints used, C3 (resp. C4) is similar to C1 (resp. C2). In addition to C1 and C2, Table 7.2 also provides the statistics for the transformation parameters for the C3 and C4 combinations. The homogeneity of the solutions is improved for C3 and C4 but only the C4 combination gives a real consistent combined solution. Indeed, the heterogeneities for the scale and Tz still exist in the C3 combination. Regarding C4, these heterogeneities are embedded in the estimated Helmert parameters. For example, the difference between the VLBI and SLR estimated scale factors is about 0.7 ppb over the 3 months of studied data (Mean of estimated VLBI/SLR scale factor: 2.1/2.8 ppb). In the same way, we find a 3D discrepancy of about 10.2 mm between the DORIS and GPS translations in the estimated Helmert parameters. They consequently do not disturb the referencing of the solution anymore. On the one hand, the local ties are essential to link the technique networks in the combination but, on the another hand, they must be used carefully as they have a great impact.

7

Towards a Combination of Space-Geodetic Measurements

7.5

Conclusions and Prospects

Through the direct estimation of transformation parameters in the combination process, we get an homogeneous combined solution. With our new approach, the problem of datum inconsistency evidenced by (Coulot et al. 2007) is solved. But the introduction of the colocation information on a weekly basis appears to be problematic and it can twist the combined frame, due to poor weekly co-located networks. From the tests carried out in the present study, we recommend to use equality constraints between estimated co-located station position offsets together with a rigorous weighting [cf. (7.7) and (7.8)] to obtain an homogeneous result. Even with this method, the definition of the combined frame will still inevitably depend on this link between the technique networks. Consequently, the introduction of supplementary common parameters and links could be helpful to decrease this dependence. At a terrestrial level, we could use common geodynamic signals (such as loading effects, for instance), ZTD, etc., and, at a space level, the use of multi-technique satellites should be of great interest. With our combination model, the inconsistencies between the techniques are embedded in the estimated Helmert parameters. The understanding of these inconsistencies should help to improve some models. Gradually, such improvements should lead to more consistency and, consequently, to definitions of combined frame more independent of any external terrestrial reference frame. However, as long as technique discrepencies exist, the estimation of the Helmert parameters is essential to insure the homogeneity of the combined solutions.

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8

Improving Length and Scale Traceability in Local Geodynamical Measurements J. Jokela, P. H€akli, M. Poutanen, U. Kallio, and J. Ahola

Abstract

Traceability is a feature that is required more frequently in local geodetic highprecision measurements. This basic term of metrology, a measurement science, describes the property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty (BIPM International vocabulary of metrology – basic and general concepts and associated terms (VIM). JCGM 200:2008. Joint Committee for Guides in Metrology, 2008b). GPS measurements are widely used in local geodynamical research. From the viewpoint of metrology, their traceability is uncontrollable because the scale cannot be unambiguously conducted based on the definition of the metre. In particular, atmospheric effects on a GPS signal cannot be modelled or calibrated along the path of the signal. We are testing a method to bring the traceable scale to small GPS networks using high-precision electronic distance measurement (EDM) instruments, the scales of which have been corrected and validated in calibrations at the Nummela Standard Baseline. The traceable scale of EDM is expected to explain the annual scale variations that have been found in GPS time series and to improve results of episodic GPS campaigns. The scale of a standard baseline is validated and maintained through regular interference measurements with the V€ais€al€a interference comparator, in which a quartz gauge conveys the traceable scale. The results from the interference measurements in 2005 and 2007 in Nummela are presented here together with a brief description of the present state of the renowned measurement standard. A standard uncertainty of 0.08 ppm was obtained again for the baseline length of 864 m, and the results confirm the good long-term stability of the baseline. The scale is transferred further to geodetic and geophysical applications by using calibrated high-precision EDM instruments as transfer standards.

J. Jokela (*) P. H€akli M. Poutanen U. Kallio J. Ahola Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland e-mail: [email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_8, # Springer-Verlag Berlin Heidelberg 2012

59

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Local geodynamical measurements will profit from the reduced and accurately estimated uncertainty of the measurement, and therefore we seek further innovations to improve their traceability. We present here a topical example of calibrations and scale transfer for a baseline and monitoring network around a nuclear power plant. We compare simultaneously measured EDM and GPS results and show a scale bias of approximately 1 ppm between them. By using a traceable length in the network, the bias could be reduced, e.g. by improving the processing strategy of GPS observations. This paper focuses on the metrological part of EDM. Some related results and analysis of GPS measurements are discussed in another paper in this volume (Koivula et al. GPS metrology – bringing traceable scale to local crustal deformation GPS network. IAG Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, 2010).

8.1

The Nummela Standard Baseline

The length and scale traceability is based on the Nummela Standard Baseline of the Finnish Geodetic Institute (FGI). The baseline is widely known as the longest, most accurate and stable measurement standard in the world for traceable geodetic length measurements. It serves in national and international scale transfer measurements for other national or accredited calibration services for EDM instruments, as well as for scientific purposes. The accuracy of the baseline, in terms of standard uncertainty, is better than 0.1 mm/km according to the interference measurements with the V€ais€al€a comparator. Since 1997 the traceable scale has been transferred from Nummela to about 20 baselines or test fields in more than ten countries. The FGI is one of the National Standards Laboratories in Finland.

8.1.1

The 80-Years History

The 864-m geodetic baseline in Nummela, northwest of Helsinki, Finland, was established in 1933, replacing an older measurement standard in Santahamina, southeast of Helsinki. Since 1947, when the first successful interference measurement with the wellknown V€ais€al€a (white light) interference method was performed, it has been called the Nummela Standard Baseline. Originally it was used for the calibration of invar wires for triangulation, and later on for EDM instruments. The scale of the Finnish first-order triangulation network was conducted from Nummela.

The importance of the V€ais€al€a interference method was already recognized in an IAG motion in 1951 and in an IUGG resolution of 1954, which recommended the use of similar methods for assuring a uniform scale in geodetic networks. Since then, the FGI has measured 12 similar baselines on almost all continents. Interference measurements of a standard baseline are still performed using the classical V€ais€al€a comparator. Even today, the method is superior to modern techniques in terms of accuracy for outdoor baselines up to a few kilometres. The scale is traceable to the definition of the metre through a quartz gauge system. The lengths of one-metre-long quartz gauges are known from comparisons and absolute calibrations with better than 40 nm standard uncertainty and multiplied with the V€ais€al€a comparator. Also, the history of the quartz metres and their comparisons and calibrations dates back more than 80 years.

8.1.2

The Present State

Between 1947 and 2007 the Nummela Standard Baseline has been measured 15 times using the V€ais€al€a method, complemented by regular comparisons and absolute calibrations of quartz gauges. The latest absolute calibrations have been performed by the PTB, Braunschweig, in Germany, and MIKES, Helsinki, in Finland (Lassila et al. 2003), showing equal results with smaller than 40 nm standard uncertainties for a set of quartz gauges. Standard uncertainties of comparisons, performed at the Tuorla Observatory of the University of Turku and, since 2005, by the FGI, are a few nm (Fig. 8.1).

8

Improving Length and Scale Traceability in Local Geodynamical Measurements

61

151,5 mm + 151,0 1m 150,5 1950

1960

1970

1980

1990

2000

2010

Year

Fig. 8.1 Length of quartz metre no. VIII, which conducts the scale in interference measurements at the Nummela Standard Baseline, from comparisons at Tuorla. The black spot at the year 2000 is the latest absolute calibration of this particular gauge Table 8.1 Baseline lengths at the Nummela Standard Baseline from the 15 interference measurements from 1947 to 2007 Epoch 1947.7 1952.8 1955.4 1958.8 1961.8 1966.8 1968.8 1975.9 1977.8 1983.8 1984.8 1991.8 1996.9 2005.8 2007.8

024 mm + 24 m – – – – – – – – 33.28 0.02 33.50 0.02 33.29 0.03 33.36 0.04 33.41 0.03 33.23 0.04 33.22 0.03

072 mm + 72 m – – – – – – – – 15.78 0.02 15.16 0.02 15.01 0.03 14.88 0.04 14.87 0.04 14.98 0.04 14.95 0.02

0216 mm + 216 m – – – – – – – – 54.31 0.02 53.66 0.04 53.58 0.05 53.24 0.06 53.21 0.04 53.20 0.04 53.13 0.03

0432 mm + 432 m 95.46 0.04 95.39 0.05 95.31 0.05 95.19 0.04 95.21 0.04 95.16 0.04 95.18 0.04 94.94 0.04 95.10 0.05 95.03 0.06 94.93 0.06 95.02 0.05 95.23 0.04 95.36 0.05 95.28 0.04

0864 mm + 864 m 122.78 0.07 122.47 0.08 122.41 0.09 122.25 0.08 122.33 0.08 122.31 0.06 122.37 0.07 122.33 0.07 122.70 0.08 – 122.40 0.09 122.32 0.08 122.75 0.07 – 122.86 0.07

The number following the symbol is the numerical value (in mm) of the combined standard uncertainty

The results of our latest interference measurements in 2005 and 2007 are presented in Table 8.1, together with the previous results. In calibrations they are used as true values with a known uncertainty in the traceability chain. They are lengths between underground benchmarks, to which (and from which, for calibrations) the lengths between reference points on less permanent observation pillars are projected using precision tacheometry and mechanical plumbing and probing. To maintain sub-mm uncertainties, optimal measurement geometry is essential in these studies. The standard (1-s) uncertainties of the lengths from 24 to 864 m ranged from 0.02 to 0.07 mm in the 2007 measurements, which means they were about the same as previous measurements. The difference between the first interference measurement in 1947 and the latest one in 2007 is 0.08 mm for the full length of 864 m, and the variation during the

60-year time span is 0.6 mm. This proves in excellent fashion the stable location of the baseline on a forested non-frozen sandy ridge. The small, but sometimes significant variations in the results are impossible to detect with any other metrological or geodetic method in field conditions, and they are small enough not to disturb calibrations of the most precise EDM instruments. The variations are caused by the settling of the markers after they have been cast in the ground, by later construction projects and other disturbing activities in the neighbourhood, and by slightly improved methods in processing the measurements. Results for the distances up to 216 m before 1977 have not been published. In 1983 and 2005 measurements for the length 864 m were not possible because of unfavourable weather conditions. The latest publications giving more detailed information on the baseline are by Jokela and Poutanen (1998), Jokela et al. (2009) and Jokela and H€akli (2010).

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Working conditions and security at the unique measurement standard were greatly improved by construction projects in 2004. The old office and store buildings were replaced with a new house. A part of the 0.12 km2 baseline property was fenced, including all observation pillars and underground benchmarks. Roofs were built over all observation pillars. Most of the observation pillars were reconditioned before the interference measurements in 2007, and a new subsurface drainage system keeps the soil dry around the comparator house, preventing possible “floods” caused by melting snow. The old baseline and the interference method were acknowledged again in 2008 when the FGI, together with eight other European institutes, entered a joint research project called “Absolute long distance measurement in air” (Wallerand et al. 2008). This project, which is a part of the European Metrology Research Programme (EMRP), is financially supported by the European Union. The baseline will be utilized in testing and validating new absolute distance measurement (ADM) techniques and instruments, including improved determination of refraction. New instruments are expected to improve distance measurement traceability up to several kilometres, which will allow for more reliable measurements in local deformation networks or other geodetic applications requiring a traceable scale.

measured - true, mm

measured - true, mm

0.0

–0.5

216

432

648

2008-10-31... 2008-11-06

1.0

0.5

0

Calibration of Transfer Standards

High-precision EDM instruments (typically a Kern Mekometer ME5000, since other suitable instruments are even fewer) are used as transfer standards from the Nummela Standard Baseline to other applications. The principles for how to perform EDM observations and make the necessary reductions and corrections to them are well known, and only a few details noteworthy in our scale transfer measurements are discussed here. All measurements are performed in field conditions. The calibration of EDM equipment for a scale transfer takes a few days, including several measurements of all available baseline distances in both directions. This allows for possible daily changes in the equipment and makes the estimation of uncertainty more reliable. Projection measurements at the Nummela Standard Baseline are performed before and after every important calibration, usually a few times a year. More frequent projections are needed during interference measurements, which typically last about three months. Calibrations are performed immediately before and after the transfer, which reveals possible changes in the equipment during the procedure (Fig. 8.2). In addition to the calibration measurements, similar possible sources of uncertainty have to be taken into account at the transfer site. One of them is the centring

2008-08-28... 2008-09-03

1.0

–1.0

8.2

864

m

Fig. 8.2 Example of calibration of transfer standard (Kern ME5000) before and after a scale transfer. Scale correction (+0.151 mm/km 0.049 mm/km, 1-s) is determined from the

0.5

0.0

–0.5

–1.0

0

216

432

648

864

m

differences between measured and true distances. The additive constant (+0.079 mm 0.014 mm, 1-s) has been corrected. Variation and corrections for this equipment are especially small

8

Improving Length and Scale Traceability in Local Geodynamical Measurements

of instruments. Common commercial fixing methods are widely used with many kinds of surveying instruments and also at geodetic baselines, but in integrated geodynamical measurements self-made solutions are often needed. The GPS antenna is often centred and adjusted with a standard forced-centring plate or directly attached to a pillar. But does the uncertainty really remain smaller than 1 mm? Local monitoring networks and tie measurements that include reference points of, for example, SLR or VLBI are even more challenging. Weather correction is another major source of uncertainty. At local networks the distances to be measured may be short, but at least ambient temperature, air pressure and relative humidity must be observed with sufficient accuracy and observed distances corrected with proper formulas, e.g. as recommended by the IAG (1999). The influences of all sources of uncertainty in the traceability chain must be carefully estimated and summarized in the estimate of combined uncertainty as described in metrological regulations (BIPM 2008a). This also includes of course uncertainties in the quartz gauge system and in interference measurements. In favourable conditions 0.5 to 1.0 mm/km values of expanded (k ¼ 2) uncertainty can be expected at the transfer site, depending on the application. By using traceable baseline lengths to adjust the geodetic networks, we can extend these values to a traceable scale of the network, as was done in triangulation. The method as such is probably not reasonable for GPS networks, though it is an interesting addition for estimating the uncertainty of measurements.

8.3

Traceable Scale in Local Geodynamical Measurements

GPS is an excellent tool for crustal deformation research. There is no need for inter-station visibility, distances are not limited to local measurements and accuracies are generally superior to the traditional methods, especially over longer distances. Metrological traceability is a property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty. The basic requirement of calibrations

63

under specified conditions (BIPM 2008b) makes the traceability and scale of a GPS measurement uncontrollable from the viewpoint of metrology. Heights and height changes are more difficult to monitor than horizontal motions. This is mainly due to the influence of the atmosphere on the GPS signal and the effect of the observing geometry. Some sources of uncertainty are random variables, and can be estimated more reliably by increasing the number of observations or the length of time series. Others are systematic effects, which cannot be quantified, thus causing bias. Although the scale is based on time and the frequency carried by the GPS signal, one is unable to unambiguously conduct the measured distances from the definition of the metre. This is mainly due to the effects of the atmosphere and local conditions, such as multipath. There is no unique standardized set of parameters to process GPS observations (and to obtain the specified conditions), which leads to slightly different results. A more severe limitation is that there is no uniquely traceable documented chain from the computed distances to the definition of the metre. In geodynamical research, one is in many cases mainly interested in changes. In such cases the absolute scale is not critical, but it is sufficient to assume that the scale remains constant between measurements. However, this cannot be guaranteed in GPS observations, and under severe ionosphere conditions scale variations up to 1 ppm have been observed in the Olkiluoto network (Ollikainen et al. 2004). Such results, especially in episodic campaigns, beg for a proper scale in GPS measurements in order to obtain as reliable results as possible. Epoch to epoch variation in scale may lead to misinterpretation in deformation research and one cannot reliably determine the size of this deformation. One solution is to bring the scale into the network using high-precision EDM instruments. Local ties at multi-technique space geodetic sites are another interesting application of dimensional metrology. Quite often these ties are (at least partly) done using GPS, and current demands for sub-mm accuracies in the ties require calibrated EDM baselines for proper scale. Adjacent research projects on GPS/ EDM metrology in the FGI (local ties and another baseline) are reported in other papers in this volume (Kallio and Poutanen 2011; Koivula et al. 2011).

64

Olkiluoto Deformation Network

The FGI and Posiva, an expert organisation responsible for the final disposal of spent nuclear fuel, started GPS measurements in the 10-station control network at the Olkiluoto nuclear power plants in 1995 (Fig. 8.3). The purpose is to monitor possible local crustal deformations with regularly repeated GPS observations. The original network of 10 observation pillars has been slightly changed and expanded through adding new pillars, due to construction projects in the area (Chen and Kakkuri 1995, 1996; Ollikainen et al. 2004; Kallio et al. 2009). The extent of the network, 2 km 4 km, is quite optimal for combining EDM and GPS measurements. The interpretation of results from repeated GPS measurements has been complicated by obvious variations in scale, which are often explained by deficiencies in ionospheric modelling. Since autumn 2002 most GPS campaigns give 1 ppm longer distances than the spring campaign of the same year (Fig. 8.4). Annual variation is also visible in the 196 km long vector between the permanent GPS stations at Olkiluoto and Mets€ahovi (Fig. 8.5). To better monitor the scale and variations to it, the 511-m line between the observation pillars GPS7 and GPS8 was cleared and equipped with a geodetic baseline for high-precision EDM. The visibility for terrestrial observations is not obtainable between the other pillars. Between 1995 and 2009 the network has been measured with episodic GPS campaigns 28 times. Dual-frequency Ashtech Z-12 and Ashtech mZ receivers and Dorne Margolin choke ring antennas have been used to collect data and, to reduce uncertainties caused by antennas, the same antennas have been used at the same stations in each campaign. All observation campaigns have been processed using Bernese software (the latest with version 5.0) and with equal processing principles, e.g. by using network solution, independent L1 and L2 observables to obtain lower measurement noise, and a local ionosphere model computed from the observations (e.g. Kallio et al. 2009). Observation sessions have taken a minimum of 24 h, with an observing interval of 30 s. Since 2002, GPS and EDM observations have been simultaneously conducted twice a year, when possible, for about one week in April and October. The expected movements are extremely small. The largest detected, but

Fig. 8.3 The network for deformation research around the nuclear power plants at Olkiluoto

511258,0 Olkiluoto baseline

GPS

511257,5

EDM

511257,0

Length [mm]

8.4

J. Jokela et al.

511256,5 511256,0 511255,5 511255,0 2002

2003

2004

2005 2006 Year

2007

2008

2009

Fig. 8.4 The GPS and EDM results for the baseline between pillars GPS7 and GPS8 at Olkiluoto

statistically significant, movements are 0.18 mm/a, with a standard deviation of 0.06 mm/a (Kallio et al. 2009). The scale for EDM in the 15 measurements between 2002 and 2009 has been determined in 13 calibrations at the Nummela Standard Baseline. In the 13 measurements between 2002 and 2008 the result for EDM measurements has 12 times been shorter than the result for GPS measurements (Fig. 8.4), with the maximum differences being about 1 mm and the average about 1 ppm. The GPS result is longer in October than in April in six cases out of seven. The scale difference between the GPS and high-precision EDM is obvious. The probable reasons causing such behaviour in the GPS results are uncertainties in GPS antenna calibration values, atmospheric modelling and site-specific effects like multipath conditions. Thus far, the EDM scale has been used for comparison only. By applying the EDM scale directly to the

8

Improving Length and Scale Traceability in Local Geodynamical Measurements

Fig. 8.5 A typical GPS time series between two permanent GPS stations (Mets€ahovi and Olkiluoto). From top to bottom: change in height, East and North component. One can see the annual variation in the vector components. Vertical scale is in millimetres (Kallio et al. 2009)

GPS analysis, the traceable scale can be brought into the GPS network too. Some further research is currently being conducted to better understand the reason for the GPS/EDM difference, and to eliminate any site or antenna dependency (Koivula et al. 2011).

8.5

Conclusions and Further Activities

Scale transfer from a standard baseline (a national geodetic measurement standard) is a method for improving length and scale traceability, which can also be used for local geodynamical measurements. The traceability chain includes the maintenance of the quartz metre system, baseline measurements with the V€ais€al€a interference comparator, baseline maintenance with projection measurements, the calibration of transfer standards (high-precision EDM) and

65

measurements at the transfer site. Most of these stages are included in the customary maintenance of the measurement standard and calibration service, and the effort needed for applications is simpler. The components of measurement uncertainty at every stage are estimated and, by combining them, an important part of a final measurement result, the uncertainty of measurement in the traceability chain, relative to the definition of the metre, is obtained. Up to 0.5 mm/km expanded uncertainty is obtained for distances of about 1 km. New results for the world-class measurement standard, the Nummela Standard Baseline of the FGI, are now available. They are frequently needed for scientific research and for scale transfers to other geodetic baselines and test fields and also for local geodynamical monitoring networks. The GPS/EDM network at the Olkiluoto nuclear power plants is presented as an example. Monitoring one GPS baseline with traceable EDM measurements cannot explain the scale variation in GPS results but can probably improve the processing strategy of episodic GPS campaigns in Olkiluoto. The scale difference between GPS and EDM has also provided motivation for further research. Measurements at Olkiluoto will be continued and there is a plan to expand the network. The amount of continuous GPS measurements may be increased; at the moment there is only one permanent station. A newly planned EastWest EDM/GPS baseline, in addition to the current North-South baseline, could be used to obtain better control over possible azimuth dependency. Acknowledgement For the scale transfer, we have used the Kern Mekometer ME5000 of Helsinki University of Technology’s (TKK) Laboratory of Geoinformation and Positioning Technology as a transfer standard. We thank Professor Martin Vermeer and the rest of the staff there for their willing cooperation.

References BIPM (2008a) Evaluation of measurement data – guide to the expression of uncertainty in measurement (GUM). JCGM 100:2008. Joint Committee for Guides in Metrology. http:// www.bipm.org/ BIPM (2008b) International vocabulary of metrology – basic and general concepts and associated terms (VIM). JCGM 200:2008. Joint Committee for Guides in Metrology. http:// www.bipm.org/

66 Chen R, Kakkuri J (1995) GPS Work at Olkiluoto for the Year of 1994. Work Report PATU-95-30e, Teollisuuden Voima Oy. Helsinki, p 11 Chen R, Kakkuri J (1996) GPS Operations at Olkiluoto, Kivetty, and Romuvaara in 1995. Work report PATU-96-07e, Posiva Oy. Helsinki, p 68 IAG (1999) IAG Resolutions adopted at the XXIIth General Assembly in Birmingham, 1999. http://www.gfy.ku.dk/~ iag/HB2000/part2/iag_res.htm Jokela J, H€akli P (2010) Interference measurements of the Nummela Standard Baseline in 2005 and 2007. Publications of the Finnish Geodetic Institute, no 144, p 85 Jokela J, Poutanen M (1998). V€ais€al€a baselines in Finland. Publications of the Finnish Geodetic Institute, no 127, p 61 Jokela J, H€akli P, Ahola J, Bu¯ga A, Putrimas R (2009) On traceability of long distances. XIX IMEKO World Congress Fundamental and Applied Metrology, Lisbon, Portugal, pp 1882–1887, 6–11 Sept 2009. http://www.imeko2009.it.pt/ Papers/FP_100.pdf Kallio U, Poutanen M (2011) Can we really promise a mmaccuracy for the local ties on a geo-VLBI antenna? (Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, Aug 31–Sept 4) Kallio U, Ahola J, Koivula H, Jokela J., Poutanen M (2009) GPS operations at Olkiluoto, Kivetty and Romuvaara in 2008.

J. Jokela et al. Working Report 2009–75. Posiva, Olkiluoto, p 216. http:// www.posiva.fi/en/databank/working_reports Koivula H, H€akli P, Jokela J, Buga A., Putrimas R (2011) GPS metrology – bringing traceable scale to local crustal deformation GPS network. (IAG Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, Aug 31–Sept 4) Lassila A,. Jokela J, Poutanen M, Xu J (2003) Absolute calibration of quartz bars of V€ais€al€a interferometer by white light gauge block interferometer. XVII IMEKO World Congress, Dubrovnik, Croatia, pp 1886–1889, 22–27 June 2003. http://www.imeko.org/publications/wc-2003/PWC-2003TC14-026.pdf Ollikainen M, Ahola J, Koivula H (2004). GPS operations at Olkiluoto, Kivetty and Romuvaara in 2002–2003. Working Report 2004–12. Posiva, Olkiluoto, p 268. http://www. posiva.fi/en/databank/working_reports Wallerand J-P, Abou-Zeid A, Badr T, Balling P, Jokela J, Kugler R, Matus M, Merimaa M, Poutanen M, Prieto E, van den Berg S, Zucco M (2008) Towards new absolute long distance measurement systems in air. 2008 NCSL International Workshop and Symposium, Orlando (USA), Aug 2008. http://www.longdistance project.eu/files/towards_ new_absolute.pdf

9

How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications? H. Drewes

Abstract

Geodetic parameters always correspond to a reference system defined by conventions and realized by a reference frame through materialized points with given coordinates. For the coordinate estimation one has to fix the geodetic datum, i.e. the origin and directions of the coordinate axes, and the scale unit. In geosciences applications, e.g. for geodynamics and global change research, the datum has to be fixed over a very long time period in order to refer time-dependent parameters to one and the same reference frame. The paper focuses on the methodology how to fix the datum by parameters independent of the measurements and deformations of the reference frame, and to hold it over a long time span. It is shown that transformations between reference frames at different epochs are not suited to realize the datum parameters because systematic network deformations may affect it. Independent parameters are in particular the first degree and order harmonic coefficients of the gravity field for fixing the origin, and external calibration for fixing the scale. The long-term stability is achieved by the permanent fixing of the datum parameters. Regional reference frames must refer to the global datum by using epoch station coordinates as fiducial values.

9.1

Introduction

The geodetic datum provides the origin, orientation and scale unit of a coordinate system with respect to the body of the Earth. All geodetic parameters refer to a given datum. Coordinates cannot be estimated from geodetic measurements without fixing the datum; there is a rank defect in the observation equation

H. Drewes (*) Deutsches Geod€atisches Forschungsinstitut, Alfons-Goppel-Str. 11, 80539 M€unchen, Germany e-mail: [email protected]

systems equal to the number of necessary datum parameters (datum defect). In a three-dimensional Cartesian coordinate system, the defect is seven: three to fix the position of the origin, three to fix the orientation of the coordinate axes, and one to fix the scale unit. The datum parameters cannot be measured; they must be given with the definition of the reference system (Latin “datum” ¼ given). If the geodetic datum is not unique or it changes in parameter estimations, the results cannot be compared among each other; e.g. the position coordinates referring to the origin in one location cannot be used together with coordinates referring to an origin in another location.

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_9, # Springer-Verlag Berlin Heidelberg 2012

67

68

H. Drewes

The most challenging task of modern geodesy is the accurate measurement and the reliable representation of parameters of global phenomena and processes within the Earth System for applications and interpretations in geosciences, e.g. geodynamics and global change research. As these processes are very slow and long-lasting, the parameters must refer to one and the same datum over very long time intervals for the unique and unequivocal representation (stability over decades).

9.2

Terminology

In order to avoid misunderstandings we shall use the following nomenclature in this paper: – Datum parameters fix the origin, orientation and the scale unit of a coordinate system. – Network (or reference frame) motions are named shifts, rotations and dilatation given with respect to one and the same geodetic datum. – Network (similarity) transformation parameters provide the translations, orientation changes and scale factor of one network with respect to another one with a different datum. We further shall consider the following facts: – Transformation of one network to another changes the geodetic datum of the transformed network with respect to the original one. – Transformation between two different epochs of a geodetic network with identical datum provides the average motion of the whole network (shifts, rotations and dilatation) and not a datum change. – A deformed network is in general mathematically not similar to the original network, i.e. the deformed network cannot generally be expressed as a similarity transformation of the original one. An example is the change of the flattening of the Earth which requires affine transformations.

9.3

Current Status of Datum Realization

The International Terrestrial Reference System (ITRS) is defined as follows (McCarthy and Petit 2004):

– The origin of the coordinate system is in the Earth’s centre of mass (geocentre). – The orientation of coordinate axes follows the Earth Orientation Parameters of the BIH 1984.0. – The scale unit is the metre consistent with TCG. The International Reference Frame (ITRF) is a realization of the ITRS. In global and regional analyses, solutions are often aligned to the ITRF by transformations of observed station networks to a preexisting reference frame or to a previously observed network, e.g. by the conditions of “no net translation”, “no net rotation”, and/or “no net scale” (NNT, NNR, NNS) (e.g. Thaller 2008; Sa´nchez et al. 2011). Doing so between different epochs of an identical network, the transformation parameters provide the average motion of the network and not different realizations of the datum. The transformation parameters may not be taken as a datum correction. Coordinates transformed in this way cannot be used for applications in geosciences because common motions have gone to the transformation parameters and can no longer be identified in the coordinate differences. Let us demonstrate this effect by two examples: The Earth’s surface undergoes long-term global deformations. The largest vertical deformations are caused by post-glacial rebound (glacial isostatic adjustment), and horizontal ones caused by tectonic processes (plate tectonics, intra- and inter-plate deformations). The station network for observing these deformations is very inhomogeneous (e.g. Collilieux et al. 2009); we have an agglomeration of stations in North America and Europe, and sparse station distribution in other regions such as Africa and Asia (Fig. 9.1). To detect the deformations, we need long-lasting measurements and a reference frame with a stable geodetic datum over decades. Under these prerequisites we discuss the effect of a similarity transformation between the global station network observed at time t and the same network at time t + Dt, and its relevance for geosciences applications.

9.3.1

Vertical Deformation

We start with the example of an uplift caused by postglacial rebound, which produces changes in vertical station coordinates (heights h). If the vertical motions

9

How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

69

Fig. 9.1 ITRF2005 station distribution (http://itrf.ensg.ign.fr)

Fig. 9.2 Vertical motions appear as transformation parameters

are the same in all observation points, they appear as a scale factor (a) in transformation parameters, not as displacements of individual stations (Fig. 9.2, above). If they are different in the northern and southern

hemispheres, they appear as a translation and a scale factor. Due to the inhomogeneous station distribution there may also appear changes in orientation (Fig. 9.2, below), although no network rotation occurred, because the transformation parameters become correlated if the origins of the coordinate systems to be transformed (geocentre) do not coincide with the centres of the networks. What is said here for displacements holds also for velocities in case of their transformation. The described effect can be seen in the results of ITRF2005 computations by different approaches. The computation at Institut Ge´ographique National (IGN), Paris (Altamimi et al. 2007) is based on similarity transformations of the epoch coordinate solutions (weekly from satellite techniques or daily from VLBI) to the combined solution, where the difference in epoch is taken into account by the estimated velocities. The Deutsches Geod€atisches Forschungsinstitut (DGFI), M€unchen, (Angermann et al. 2007, 2009) accumulates datum-free normal equations of the measurements. The comparison of both solutions shows systematic differences. These are predominantly negative in the northern and positive in the southern hemisphere (Fig. 9.3). The differences can be approximated by a sine function of latitude ’ (Fig. 9.4), which is equivalent to the Z-component (DZ ¼ Dh · sin ’). The associated differences in the Z-component are shown in Fig. 9.5. They are – as expected – predominately

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Fig. 9.3 Differences of vertical velocities between ITRF2005 computations at DGFI and IGN

Fig. 9.4 Latitude dependence of vertical velocity differences and fit by a sine function, which indicates a translation of the network in Z-direction

negative with an average of 1.4 mm/a, and show a longitude dependence. There are, however, accumulated effects affecting the Z-component, which will be discussed below. Figure 9.6 shows the longitude dependence of the Z-component of velocity differences. We again see a sine function with even larger amplitude. The interpretation is more difficult, because there are additional effects from horizontal deformations.

9.3.2

Horizontal Deformation

Horizontal deformations in a global scope are mainly caused by plate tectonics. The kinematic datum of the ITRF is defined by the condition of no net rotation with regard to horizontal tectonic motions over the whole

Earth (McCarthy and Petit 2004). It is realized by transforming the geodetic velocities to those derived from plate tectonic models. The ITRF2005 computation at IGN uses the geophysical model NNR NUVEL-1A (DeMets et al. 1990, 1994). This model does not fulfil the NNR condition at present, because it is based on observations over geological time scales, includes only 16 rigid plates, and does not consider the extended deformation zones along the plate boundaries. Therefore, DGFI uses the Actual Plate Kinematic and Crustal Deformation Model (APKIM2005) based on the structure of the geophysical model PB2002 (Bird 2003) with 52 (micro-) plates, the ten major ones being identical with NUVEL-1A, and 13 deformation zones. The plate velocities are computed from present-day geodetic observations (Drewes 2009b) and show significant differences with respect to the geophysical model

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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

71

Fig. 9.5 Differences of the Z-component of station velocities from ITRF2005 computations at DGFI and IGN

Fig. 9.6 Longitude dependence of the Z-component of station velocity differences, indicating a shift (1.4 mm/a) and a rotation of the network around an axis in the equatorial plane

(Fig. 9.7). The differences of the horizontal velocities from both ITRF computations based on these models are shown in Fig. 9.8 (mind the different velocity scales of Fig. 9.7 and Fig. 9.8). We clearly see the rotation around an axis from the Northern Atlantic to the Southern Indian Ocean which is nearly identical with the differences between NNR NUVEL-1A and APKIM2005. The global motions of all tectonic plates, which are modelled as rotations on a sphere, have a dominant trend towards northeast (Fig. 9.7). This causes a shift of the whole network with an average dX/dt ¼ 10 mm/a, dY/dt ¼ +13 mm/a, dZ/dt ¼ +15 mm/a (Drewes 2009b). This shift enters completely into the station velocities and does not affect the datum, if the origin of the reference frame is fixed to the geocentre and the NNR condition of the plate model is fulfilled. If we perform a transformation of the discrete geodetic

network with inhomogeneous point distribution to a slightly rotating geophysical plate model (like NNR NUVEL-1A with oX ¼ 0.04 mas/a, oY ¼ +0.03 mas/a, oZ ¼ 0.02 mas/a.), then the residual rotations appear also in the translation transformation parameters. The effect can be seen in the systematic change of all the ITRF translation parameters and its rates (Altamimi et al. 2002, 2007), in particular in the Y and Z components.

9.3.3

Regional Deformation

The examples presented so far refer to effects of global deformations on the global datum realization. Regional reference frames are normally defined as densifications of the ITRF, i.e. they are subject to the

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H. Drewes

Fig. 9.7 Comparison of the geophysical plate kinematic model PB2002 and the geodetic model APKIM2005

Fig. 9.8 Differences of horizontal velocities between ITRF2005 computations at DGFI and IGN

same effects. In addition, there are regional deformations due to tectonic, isostatic, sedimentary, atmospheric, hydrospheric, and other processes. Large effects are caused by seasonal loading. Figure 9.9 shows height variations of stations distributed over all the South American continent (from ’ ¼ 5 to ’ ¼ 35 ) computed from observations in the South

American reference frame (SIRGAS, Sa´nchez et al. 2011; Seem€uller et al. 2009). They clearly demonstrate a systematic periodic (seasonal) variation up to 2 cm. The datum of a regional reference frame is often realized by NNR, NNT, and NNS (i.e. similarity) transformations of the regional network to ITRF

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How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

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KOUR ( =5°, =-53°)

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Fig. 9.9 Station height variations in South America [cm]

(or IGS) coordinates extrapolated with linear velocities from the ITRF (or IGS) reference epoch to the regional reference epoch. If the reference stations do not move linearly (Fig. 9.9), the extrapolated coordinates do not refer to actual geocentric station positions at every time. We get then the same type of effects as described in chapter 3, and we cannot use coordinate variations

estimated in this way for geosciences applications. To avoid this effect, the reference frames in some regional projects are realized by transforming the epoch (weekly) coordinates to the weekly IGS solutions (Craymer et al. 2007; Sa´nchez et al. 2011). This procedure accounts for the regional seasonal effects, but still includes the global motions of the network (secularly

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due to post glacial rebound, seasonally due to different atmosphere and hydrosphere loads in the northern and southern hemispheres).

9.4

How to Fix the Geodetic Datum?

9.4.1

Fixing the Origin

The realization of the geodetic datum by similarity transformation to other networks or reference frames is not a suitable approach, because network deformations in the reference stations selected for the transformation change the datum definition. Fixing the origin to the geocentre must be done by a gravimetric approach as a network-independent method (Drewes 2009a). The Earth’s centre of mass is defined by (M ¼ total mass of the Earth) X0 X dm=M Y0 Y dm=M Z0 Z dm=M:

9.4.2 (9.1)

The first degree and order spherical harmonic coefficients of the Earth’s gravity field express the position of the geocentre with the semi-major axis a as a scaling factor: C11 X dm=a M S11 Y dm=a M C10 Z dm=a M:

do not change the geodetic datum. They must be interpreted as orbit errors and be reduced by improved orbit modelling. The geocentric coordinates of the ground stations are derived from the geocentric coordinates of the satellites using, e.g. SLR distance measurements. In differential approaches, like double differencing GNSS or Doppler methods, most of the relation to the geocentre is lost in the differencing. It can therefore not strictly be realized. As a consequence, SLR measurements should be included for any datum realization, also in regional reference frames. The fixing of the geocentre through the gravity field holds for any time. Thus it is clear that there is no time evolution of the origin. It remains always in the geocentre. If it is realized in this way, it guarantees the long-term stability of the reference frame for geosciences applications.

(9.2)

Using a gravity field model with C11¼S11¼C10¼0 in the coordinate estimation with dynamic satellite methods, as customary in satellite geodesy, fixes the origin of the coordinate system automatically to the geocentre. Satellite orbits are always geocentric unless additional constraints are introduced. Such constraints may enter by fixing the coordinates of tracking stations or an average of them (e.g. NNT constraints). Therefore, global orbit computations for reference frame determinations must not fix any terrestrial coordinates; the necessary minimum datum constraint is given by the gravity field. Wrong modelling of other gravitational or non-gravitational forces (e.g. solar radiation pressure) introduces additive constraints, too. They cause systematic errors in the orbits, but

Fixing the Orientation

The orientation of the coordinate system could also be realized by gravimetric methods through the principal axes of inertia, which are expressed by the second degree spherical harmonics of the gravity field (C21, S21, S22). These coefficients, however, can at present not be determined with the required accuracy. This may change by including data from future gravity field and global navigation satellite missions. Until these are available, the orientation must further be fixed conventionally (e.g. BIH84). The time evolution of the orientation of the coordinate system is characterized by the high correlation between station velocities and Earth Orientation Parameters (EOP). Systematic velocity changes may interchange with EOP. It is therefore indispensable to estimate velocities and EOP simultaneously in a global adjustment, and to use a present-day no-netrotation constraint by models derived from geodetic measurements instead of geologic-geophysical models. The orientation resulting from network transformations refers then to the same coordinate axes in velocities and EOP.

9.4.3

Fixing the Scale

The scale unit has to be fixed to the definition of the unit of length (metre) by calibrating the measuring

9

How to Fix the Geodetic Datum for Reference Frames in Geosciences Applications?

instruments and reduction of atmospheric effects. There is no time evolution of the scale; it is always the same unit of length. We must never perform a similarity transformation between networks at different epochs with a scale factor as a parameter. This would only absorb the network dilatation and exclude geosciences interpretations (e.g. to prove or disprove the expansion theory of the Earth, or to detect global postglacial rebound). Conclusions

If we realize the geodetic datum by network transformations, we change the datum between the original and the transformed reference frame. A geocentric frame is then loosing the geocentricity. This behaviour is often discussed as geocentre motion. Doing so, one has to keep in mind that it is a motion w. r. t. the always deforming crust-based reference frame. There is no stable reference. A consequence is that the non-geocentric reference frame is no longer consistent with the satellite orbit computations, which normally use a geocentric gravity field (C11 ¼ S11 ¼ C10 ¼ 0). One would then have to re-compute all SLR, GPS and DORIS orbits with the changed lower spherical harmonic coefficients. For users of such a moving reference frame, one would have to provide the corresponding time-dependent gravity fields together with the station coordinates. It is by far more practicable to keep the origin fixed in the geocentre and to include all the global motions in the station velocities. Experiences with inconsistent global reference frames were reported, e.g. from satellite altimetry studies on global sea level changes (e.g. Beckley et al. 2007; Morel and Willis 2005). There are considerable differences in the interpretation of results when using different ITRF realizations as a reference frame. The datum of regional reference frames has strictly to be realized as a densification of the global reference frame in order to be consistent with the satellite orbits. True coordinates (ITRF or IGS) have to be used as fiducial values, and not the coordinates extrapolated from a reference epoch with linear velocities. In epoch solutions (e.g. weekly), the coordinates of the same epoch have to be taken from the superior frame. As the global

75

network may undergo a common motion w. r. t. the geocentre, one has to fix the geocentric datum by external gravimetric parameters (C11 ¼ S11 ¼ C10 ¼ 0). This should be done by including always station coordinates derived from SLR in a combined network adjustment.

References Altamimi Z, Sillard P, Boucher C (2002) ITRF2000: a new release of the International Terrestrial Reference Frame for Earth science applications. J Geophys Res 107(B10) 2214:19. doi:10.1029/2001JB000561 Altamimi Z, Collilieux X, Legrand J, Garayt B, Boucher C (2007) ITRF2005: a new release of the International Terrestrial reference frame based on time series of station positions and Earth Orientation Parameters. J Geophys Res 112:19. doi:10.1029/2007JB004949, B09401 Angermann D, Drewes H, Kr€ugel M, Meisel B (2007) Advances in terrestrial reference frame computations, vol 130, IAG Symposia. Springer, Heidelberg, pp 595–602 Angermann D, Drewes H, Gerstl M, Kr€ugel M, Meisel B (2009) DGFI combination methodology for ITRF2005, vol 134, IAG Symposia. Springer, Heidelberg, pp 11–16 Beckley BD, Lemoine FG, Luthcke SB, Ray RD, Zelensky NP (2007) A reassessment of global and regional mean sea level trends from TOPEX and Jason-1 altimetry based on revised reference frame and orbits. Geophys Res Lett 34:14608. doi:10.1029/2007GL030002, 5pp Bird P (2003) An updated digital model for plate boundaries. G3 – Geochem Geophys Geosyst 4(3):52. doi:1010.1029/ 2001GC000252 Collilieux X, Altamimi Z, Ray J, Van Dam T, Wu X (2009) Effect of the satellite laser ranging network distribution on geocenter motion estimation. J Geophys Res 114:B04402. doi:10.1029/2008JB005727, 17pp Craymer MR, Piraszewski M, Henton JA (2007) The North American Reference Frame (NAREF) project to densify the ITRF in North America. Proceedings ION GNSS 20th International Technical Meeting of the Satellite Division, pp 2145–2154 DeMets C, Gordon RG, Argus DF, Stein S (1990) Current plate motions. Geophys J Int 101:425–478 DeMets C, Gordon R, Argus DF, Stein S (1994) Effect of recent revisions to the geomagnetic reversal time scale on estimates of current plate motions. Geophys Res Lett 21:2191–2194 Drewes H (2009a) Reference systems, reference frames, and the geodetic datum – basic considerations, vol 133, IAG Symposia. Springer, Heidelberg, pp 3–9 Drewes H (2009b) The Actual Plate Kinematic and crustal deformation Model (APKIM2005) as basis for a non-rotating ITRF, vol 134, IAG Symposia. Springer, Heidelberg, pp 95–99 McCarthy DD, Petit G (2004) IERS Conventions 2003. IERS Technical Note No. 32 Morel L, Willis P (2005) Terrestrial reference frame effects on sea level rise determined by TOPEX/Poseidon. Adv Space Res 36:358–368. doi:10.1016/j.asr.2005.05.113

76 Sa´nchez L, Seem€uller W, Seitz M (2011) Combination of the weekly solutions delivered by the SIRGAS Processing Centres for the SIRGAS-CON Reference Frame. In: Kenyon S et al (eds) Geodesy for Planet Earth, IAG Symposia. Springer, Heidelberg Seem€uller W, Seitz M, Sa´nchez L, Drewes H (2009) The position and velocity solution SIR09P01 of the IGS Regional

H. Drewes Network Associate Analysis Centre for SIRGAS (IGS RNAAC SIR). DGFI Report No. 85 Thaller D (2008) Inter-technique combination based on homogeneous normal equation systems including station coordinates, Earth orientation and troposphere parameters. GFZ Potsdam Sci Rep STR08/15

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence of Postglacial Rebound: An Evaluation of the Nordic Geodynamical Model in Finland P. H€akli and H. Koivula

Abstract

The IAG Reference Frame Sub-Commission for Europe (EUREF) created the European Terrestrial Reference System 89 (ETRS89) and fixed it to the Eurasian plate in order to avoid time evolution of the coordinates due to plate motions. However, the Fennoscandian area in Northern Europe is affected by postglacial rebound (PGR), causing intraplate deformations with respect to the stable part of the Eurasian tectonic plate. The Nordic countries created their national ETRS89 realizations in the 1990s and have adopted them as the basis for geospatial data. As the most accurate GNSS processing is done in ITRS realizations, an accurate connection to national ETRS89 realizations is required. If the official EUREF transformation is used, residuals are up to 10 cm in the Nordic countries. Therefore, the Nordic Geodetic Commission (NKG) has created a 3-D intraplate velocity model NKG_RF03vel over Fennoscandia and a new transformation procedure to correct for the deformations caused by PGR. This paper evaluates the NKG approach and compares it to the current recommendation given by EUREF with a 100-point ETRS89 realization in Finland. The results show that, by using a high-quality intraplate velocity model, the transformation residuals are reduced to the cm-level.

10.1

Introduction

National reference frames are usually fixed to a global or regional terrestrial reference frame (TRF) that in most cases is one of the ITRS realizations, ITRFyy. ITRF solutions are the most accurate global realizations, but the coordinates are also time-

P. H€akli (*) H. Koivula Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland e-mail: [email protected]

dependent due to site velocities. Site velocities are needed if the Earth’s dynamics (e.g., plate tectonic motion) are to be taken into account at the cm-level. Time variable coordinates, however, are not useful for geodetic practice. To avoid time evolution of the coordinates, the IAG Reference Frame Sub-Commission for Europe, EUREF, has created the European Terrestrial Reference System 89 (ETRS89), which is fixed to the stable part of the Eurasian tectonic plate and coincides with the ITRS at the epoch 1989.0 (Boucher and Altamimi 1992). However, the Fennoscandian area in Northern Europe is affected by postglacial rebound (PGR), causing intraplate

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_10, # Springer-Verlag Berlin Heidelberg 2012

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deformations. The uplift part of the PGR in Fennoscandia has been observed for centuries, first with tide gauges, later with repeated levellings and gravity observations and, recently, with permanent GPS stations as well. Intraplate deformations in Europe and Fennoscandia have been studied, for example, by Nocquet et al. (2001, 2005), Johansson et al. (2002) and Lidberg et al. (2007). The land uplift has a maximum of approx. 10 mm/year (from the reference ellipsoid). The horizontal part of the PGR, from the time series of permanent GPS stations, is up to 2.5 mm/year. These deformations can be seen, for example, in EPN (EUREF Permanent Network) velocities expressed in ETRS89, see Figs. 10.1 and 10.2. The figures were plotted from the EPN cumulative solution up to GPS week 1,540 of class A stations (EPN 2010). Class A stations are categorized as highquality fiducial stations meaning that the velocity estimates are known better than 0.5 mm/yr (Kenyeres 2010). The Nordic countries created their national ETRS89 realizations in the 1990s using the ITRF of that time (Denmark ITRF92, Norway ITRF93, Finland ITRF96 and Sweden ITRF97). The ETRS89 realizations have been adopted as the basis for geospatial data. The Finnish ETRS89 realization, called EUREF-FIN, was measured in 1996–1997 and realized through ITRF96(1997.0) with official

formulas provided by EUREF (Boucher and Altamimi 2008), yielding ETRF96 coordinates in 1997.0. The formulas correct the rigid plate motion to epoch 1989.0, but for intraplate deformations the epoch remains in 1997.0, which can be considered the reference epoch for EUREF-FIN. Since the most accurate GNSS processing has to be done in adequate ITRFyy, accurate transformations between ITRF and national ETRS89 realizations are needed. The connection between different threedimensional reference frames is generally presented with a linear 7-parameter similarity transformation. EUREF provides a 14-parameter similarity transformation approach (including time evolution of the parameters) from ITRS to ETRS89 that takes into account the rigid plate motion of the Eurasian tectonic plate (Boucher and Altamimi 2008). In the context of a GNSS campaign, EUREF does not recommend using velocity models. However, any similarity transformation alone cannot depict the influence of PGR since it deforms the crust unequally. Therefore, the EUREF transformation does not describe discontinuities and intraplate deformations; they remain as transformation residuals. In the Fennoscandian area, the EUREF approach is therefore not cm-level accurate when transforming from present ITRF to national ETRS89 realizations. The residuals are already up to 10 cm within a 10-year time span.

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10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence

ITRF2000 to ETRF96 is done in two steps. The first step transforms ITRF2000(tc) coordinates to ITRF96(tc), keeping the epoch of coordinates untouched. The transformation parameters are given at reference epoch t0 (for ITRF2000 t0 ¼ 1997.0) and the parameters have to be converted to the epoch of observations (tc) with the change rates of the parameters. In the second step, ITRF96(tc) coordinates are transformed to ETRF96(tc). Since no intraplate or epoch correction is recommended, the epoch of coordinates remains at tc.

To overcome this problem, the Nordic Geodetic Commission (NKG) has created a 3-D velocity model, NKG_RF03vel, and a procedure that also takes into account intraplate deformations. This study evaluates the NKG approach and compares it to the transformation recommended by EUREF. Transformations were evaluated using a 100-point network that is defining the Finnish ETRS89 realization, EUREF-FIN. EUREF-FIN was originally measured in 1996–1997 and the same network was re-measured in 2006. The re-measurement campaign was processed with Bernese 5.0 and the coordinates for fiducial stations at the central epoch of observations, tc ¼ 2006.5, were obtained from the official IERS ITRF2000 solution (ITRF 2010a). The resulting campaign coordinates were transformed using EUREF and NKG approaches and compared to the official EUREF-FIN coordinates.

10.2

10.2.2 Official NKG Transformation The NKG transformation was done according to the recommendations given by the NKG working group for positioning and reference frames (NKG WG). The NKG transformation takes intraplate deformations into account using velocity model NKG_RF03vel. The model corrects both horizontal and vertical intraplate deformations. The horizontal part originates from the GIA model by Milne et al. (2001) that was rotated to the GPS-derived velocities by Lidberg et al. (2007) (Fig. 10.3). The vertical part, NKG2005LU (ABS) model by NKG working group for height determination, is constructed from tide gauge, levelling and

Transformation Approaches

10.2.1 Official EUREF Transformation The EUREF transformation was made according to the recommendation and parameters given by Boucher and Altamimi (2008). The transformation from 340

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permanent GPS data (Fig. 10.4) (Nørbech et al. 2006; Lidberg 2008). The Nordic ETRS89 realizations are based on different ITRF solutions and reference epochs leading up to discrepancies of a few cm. In order to take into account these discrepancies, a common reference frame was needed. The common Nordic frame, NKG_RF03, was realized with a NKG2003 GPS campaign and computed in ITRF2000 at epoch 2003.75. The complete description of NKG_RF03 is given by Jivall et al. (2005). The transformation is determined through the NKG_RF03 frame in order to determine the transformation for each Nordic country. For Finland, the transformation consists of three steps: 1. ITRF2000(tc) ! ITRF2000(2003.75) (NKG_ RF03) 2. ITRF2000(2003.75) ! ITRF2000(2003.75)1997.0 (¼intraplate corrected to 1997.0) 3. ITRF2000(2003.75)1997.0 ! EUREF-FIN The first step is to transform ITRF2000 coordinates at the epoch of observations to ITRF2000 at epoch 2003.75 in which the NKG_RF03 is expressed. The

rigid plate motion during the period is reduced with the ITRF2000 rotation pole for the Eurasian plate (given by Boucher and Altamimi 2008). In addition to the rigid plate motion, also intraplate deformations during the period have to be taken into account in order to simulate the ITRF2000 coordinates at epoch 2003.75 accurately. For this purpose the NKG_RF03vel intraplate velocity model is used. In the second step, the resulting coordinates in ITRF2000(2003.75) are further corrected for intraplate deformations to the reference epoch (tr) of the national ETRS89 realization, in Finland tr ¼ 1997.0. This step corrects the internal geometry of the network at the epoch 2003.75 to the one at the reference epoch of EUREF-FIN. In the third step these intraplate corrected ITRF2000(2003.75) coordinates (we are using notation ITRF2000(2003.75)1997.0) are transformed to EUREF-FIN with a 7-parameter similarity transformation. The parameters were determined between the intraplate corrected NKG_RF03(tr) coordinates and the Nordic ETRS89 realizations by the NKG WG.

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Transforming ITRF Coordinates to National ETRS89 Realization in the Presence 0˚ 8˚ 12 6˚ 4 ˚ 16˚ 20˚ 24˚ 28˚ 32˚ 3

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The standard error of unit weight for the parameter estimation with 12 fitting points in Finland was 1.6, 1.5 and 3.2 mm for North, East and the up component, respectively. The formulas, parameters and more detailed explanation are given by Nørbech et al. (2006).

10.3

Evaluation of the Transformations

10.3.1 EUREF Transformation Residuals of the official EUREF transformation for the vertical coordinates are up to 7 cm (Fig. 10.6) and show the neglected effect of the PGR between 1997.0 and 2006.5 (compare to the contours of the land uplift model NKG_RF03vel, Fig. 10.4). The horizontal residuals of the transformation (Fig. 10.5) have approximately the same direction, but there is a large difference in magnitude between Southern and

Northern Finland. The residuals vary from a couple of mms to 3 cm (see also Table 10.1). In order to understand this feature, and despite the fact that the EUREF does not recommend using intraplate velocities in the context of a GPS campaign, we subtracted the effect of PGR between 1997.0 and 2006.5 from the transformed coordinates. After removing the effect of PGR, horizontal residuals are more equal in size and direction throughout Finland. All residuals are approximately 15–20 mm and the direction is N-NW (and slightly rotating from the North to the West when moving from Southern to Northern Finland). This bias likely originates from different plate models used in ITRF2000 and ITRF96. Altamimi et al. (2002) report a significant disagreement between ITRF2000 velocities and those predicted by NNR-NUVEL-1A that were used in ITRF96. The difference between ITRF2000PMM (plate motion model) and NNR-NUVEL-1A, illustrated by Altamimi and Boucher (2002) and

P. H€akli and H. Koivula

82 Fig. 10.5 Horizontal residuals of official EUREF transformation

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NUHP

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TSUA

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KEST

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PIHJ

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10

TAIN

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KASK

TAHK

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OLKI

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GETA

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VAAT

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PORL

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DRAG

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HARK

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HANK

© FGI/PH 2009

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Table 10.1 Statistics of transformation residuals from ITRF2000 to EUREF-FIN with different approaches (mm) n ¼ 95 Mean Std Rms Min Max

EUREF N 10.8 8.0 13.4 3.3 26.0

E 1.6 5.5 5.7 13.9 8.6

U 37.7 18.8 42.1 1.1 75.4

NKG-1 N 3.7 4.1 5.5 5.5 14.6

Altamimi et al. (2003), shows a strong agreement with our horizontal residuals after intraplate correction for the Fennoscandian area. Vertical residuals are reduced to a couple of cm when intraplate deformations were removed.

E 3.0 3.5 4.6 4.9 10.8

U 16.9 9.5 19.4 39.8 0.0

NKG-2 N 2.2 4.0 4.5 8.3 12.1

E 0.7 3.3 3.3 7.0 9.0

U 3.5 8.1 8.8 22.2 15.0

10.3.2 NKG Transformation The ITRF2000(2006.5) coordinates were transformed according to the official NKG transformation (solution labelled NKG-1). The solution includes intraplate

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence

Fig. 10.6 Vertical residuals of official EUREF transformation

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corrections of approx. 3–9 mm/year vertical and up to 2 mm/year horizontal velocities in Finland. The residuals that are at the level of a few cm are clearly smaller than those of the EUREF transformation without intraplate corrections. The horizontal residuals have a random nature, but the vertical residuals are systematically biased by an average of 16.9 mm (see Table 10.1). This is caused mainly by biased NKG_RF03 coordinates. The NKG2003 campaign was computed in ITRF2000(2003.75), but in Finland the vertical coordinates of NKG_RF03 differ approx. 13 cm from the ITRF2000 coordinates computed with positions and velocities published by the IERS (these were used in evaluation). The difference is of

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the same magnitude as the residuals (however, comparison was possible only at 4 Finnish EPN stations: METS, JOEN, VAAS and SODA). The NKG2003 campaign and resulting NKG_RF03 frame was computed using IGS00 cumulative solution. The NKG_RF03 was processed using GIPSY-OASISII, GAMIT/GLOBK and Bernese software and the final solution is a combination of these solutions. Constraining to ITRF2000 was done with GIPSY and GAMIT/GLOBK solutions, meaning that the NKG2003 campaign has a global alignment, through transformation parameters, to ITRF2000 (Jivall et al. 2005). The bias is partly caused by the cumulative IGS00 coordinates, but in Finland these are equal within a

P. H€akli and H. Koivula

84 Fig. 10.7 Horizontal residuals of NKG2 transformation

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few mm to the IERS-published ITRF2000 coordinates at epoch 2003.75. Instead, aligning the regional solution with transformation parameters to global ITRF2000 may lead to a biased solution since this method is sensitive to station selection (network effect), see e.g., Altamimi (2003) and Legrand and Bruyninx (2008). The ITRF positions and velocities published by the IERS are widely used, even if the positions usually need to be extrapolated outside of the temporal extent of the data used in the ITRF solution. Therefore, misaligned NKG_RF03 coordinates cannot be used for transformation, even if the solution itself is accurate. As a consequence, either new parameters from

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the IERS-published coordinates to national realization or an additional step from NKG_RF03 to ITRF2000 (2003.75) is needed. We solved for new national transformation parameters using the IERS-published positions and velocities. Figures 10.7 and 10.8 show the residuals for this solution (labelled NKG-2). This corrects most of the vertical bias in the NKG-1 solution. The accuracy (rms) of this transformation is better than 1 cm. The residuals of different transformations are summarized in Table 10.1. The NKG-2 transformation was also evaluated with ITRF2005, using only permanent FinnRef stations. The transformation has an additional step from ITRF2005(tc) to ITRF2000(tc) that was made

10

Transforming ITRF Coordinates to National ETRS89 Realization in the Presence

Fig. 10.8 Vertical residuals of NKG-2 transformation

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Table 10.2 Comparison of ITRF2005/ITRF2000 transformations to EUREF-FIN (mm) n ¼ 12 Mean Stdev Rms Min Max

ITRF2000(2006.50) N E U 1.1 1.9 1.8 2.7 2.3 5.2 2.8 2.9 5.3 3.0 2.0 12.0 4.7 5.2 3.8

ITRF2005(2008.56) N E U 1.3 4.3 3.1 4.2 1.3 7.6 4.2 4.4 7.9 11.7 7.1 18.6 3.1 2.4 7.1

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plate motion with the ITRF2000 rotation pole and intraplate deformations with the NKG_RF03vel model. Table 10.2 summarizes the residuals for 12 permanent FinnRef stations. The accuracy of the ITRF2005 transformation was slightly worse, as expected. However, an accuracy of 1 cm (rms) can also be achieved from ITRF2005 to EUREF-FIN.

10.4 according to EUREF’s MEMO. The IERS-published ITRF2005 coordinates were used for the fiducial station METS (ITRF 2010b). The epoch of ITRF2005 coordinates was 2008.56, so the transformation includes an additional 2.06 year correction for rigid

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Conclusions and Discussion

We have studied several ways of transforming ITRF coordinates back to national ETRS89 realization in Finland, where the reference frame is under the influence of postglacial rebound (PGR). The official

86

EUREF transformation led to transformation residuals up to 7 cm in vertical and up to 3 cm in horizontal components. The pattern of vertical residuals is consistent with the effect of PGR and the magnitude is too large for accurate geodetic purposes. The NKG has created a common reference frame NKG_RF03, expressed in ITRF2000 at epoch 2003.75, and a transformation through it back to the Nordic ETRS89 realizations. The official NKG transformation includes a high-quality intraplate velocity model for Fennoscandia and corrects most of the PGR effect. However, a bias of 16.9 mm was found in vertical residuals and was identified as being the consequence of a network effect. NKG_RF03 was created using minimum constraints approach that is known to be sensitive for site selection. By using IERS-published official ITRF2000 coordinates instead of NKG_RF03, this bias was greatly reduced. This NKG-2 solution gives transformation residuals below 5 mm for horizontal and 8.8 mm (rms) for vertical coordinates, which is acceptable for most geodetic purposes. The main outcome of the study is that there are several influencing factors (e.g., plate rigid rotation, intraplate deformations and constraining approach to the ITRF) that need to be taken into account appropriately in order to get cm-level transformation accuracies back to national ETRS89 realizations. Our results verify that we need a velocity model to map coordinates with cm-level accuracy from the current ITRF to the Finnish realization of ETRS89. However, guidelines for using such models are missing or their use is not recommended. There are several different ways to implement such models. Therefore, guidelines for incorporating deformation models in a unified and standardized way or even a pan-European velocity model would be desirable.

References Altamimi Z (2003) Discussion on how to express a regional GPS solution in the ITRF, EUREF Publication No. 12. Verlag des Bundesamtes f€ ur Kartographie und Geod€asie, Frankfurt am Main, pp 162–167 Altamimi Z, Boucher C (2002) The ITRS and ETRS89 Relationship: New Results from ITRF2000. EUREF Publication No. 10. Verlag des Bundesamtes f€ ur Kartographie und Geod€asie, Frankfurt am Main, pp 49–52 Altamimi Z, Sillard P, Boucher C (2002) ITRF2000: a new release of the International Terrestrial Reference Frame for

P. H€akli and H. Koivula earth science applications. J Geophys Res 107(B10):2214. doi:10.1029/2001JB000561 Altamimi Z, Sillard P, Boucher C (2003) The impact of a nonet-rotation condition on ITRF2000. Geophys Res Lett 30 (2):1064. doi:10.1029/2002GL016279, 2003 Boucher C, Altamimi Z (1992) The EUREF Terrestrial Reference System and its First Realization. Report on the Symposium of the IAG Subcommission for the European Reference Frame (EUREF) held in Florence 28–31 May 1990. Ver€offentlichungen der Bayerischen Kommission Heft 52, M€unchen Boucher C, Altamimi Z (2008) Memo: specifications for reference frame fixing in the analysis of a EUREF GPS campaign. Version 7: 24-10-2008 EPN (2010) EPN cumulative solution GPS weeks 860–1540. ftp://epncb.oma.be/pub/station/coord/EPN/ EPN_A_ETRF2000_C1540.SSC. Accessed 16 Mar 2010 ITRF (2010a) Primary ITRF2000 solution. http://itrf.ensg. ign. fr/ITRF_solutions/2000/sol.php. Accessed 16 Mar 2010 ITRF (2010b) ITRF2005. http://itrf.ensg.ign.fr/ITRF_solutions/ 2005/ITRF2005.php. Accessed 16 Mar 2010 Jivall L, Lidberg M, Nørbech T, Weber M (2005) Processing of the NKG 2003 GPS Campaign. LMV-rapport 2005:7, Reports in Geodesy and Geographical Information Systems, G€avle 2005. Available at http://www.lantmateriet.se/templates/ LMV_Page.aspx?id¼2688. Accessed 16 Mar 2010 Johansson JM, Davis JL, Scherneck H-G, Milne GA, Vermeer M, Mitrovica JX, Bennett RA, Jonsson B, Elgered G, Elo´segui P, Koivula H, Poutanen M, R€onn€ang BO, Shapiro II (2002) Continuous GPS measurements of postglacial adjustment in Fennoscandia 1. Geodetic results. J Geophys Res 107(B8):2157. doi:10.1029/2001JB000400 Kenyeres A (2010) Categorization of permanent GNSS reference stations. Bolletino di Geodesia e Scienze Affini Legrand J, Bruyninx C (2008) EPN Reference Frame Alignment: Consistency of the Station Positions. EUREF 2008 Symposium, Brussels, Belgium, 18–21 June 2008 Lidberg M (2008) Geodetic Reference Frames in Presence of Crustal Deformations. Integrating Generations, FIG Working Week 2008, Stockholm, Sweden, 14–19 June 2008 Lidberg M, Johansson JM, Scherneck H-G, Davis JL (2007) An improved and extended GPS-derived 3D velocity field of the glacial isostatic adjustment (GIA) in Fennoscandia. J Geodesy 2007(81):213–230. doi:10.1007/s00190-006-0102-4 Milne GA, Davis JL, Mitrovica JX, Scherneck H-G, Johansson JM, Vermeer M, Koivula H (2001) Space-geodetic constraints on glacial isostatic adjustments in Fennoscandia. Science 291:2381–2385 Nocquet J-M, Calais E, Altamimi Z, Sillard P, Boucher C (2001) Intraplate deformation in western Europe deduced from an analysis of the ITRF97 velocity field. J Geophys Res 106 (B6):11239 Nocquet J-M, Calais E, Parsons B (2005) Geodetic constraints on glacial isostatic adjustment in Europe. Geophys Res Lett 32:L06308. doi:10.1029/2004GL022174, 2005 Nørbech T, Engsager K, Jivall L, Knudsen P, Koivula H, Lidberg M, Madsen B, Ollikainen M, Weber M (2006) Transformation from a Common Nordic Reference Frame to ETRS89 in Denmark, Finland, Norway, and Sweden – status report. Also Paper C in Lidberg M (2007) Geodetic Reference Frames in Presence of Crustal Deformations. Ph.D. Thesis, Chalmers University of Technology, G€oteborg, Sweden

Global Terrestrial Reference Frame Realization Within the GGOS-D Project

11

D. Angermann, H. Drewes, and M. Seitz

Abstract

The GGOS-D terrestrial reference frame has been computed in a common adjustment of station positions and velocities together with the Earth orientation parameters and the quasar coordinates (celestial reference frame). The data were processed as datum-free normal equations from homogeneously generated VLBI, SLR and GPS observation time series using identical standards for the modelling and parameterization. A major focus was on the analysis of the station position time series, investigations regarding seasonal variations in station motions and on the combination methodology for the terrestrial reference frame computation.

11.1

Introduction

The project GGOS-D was funded by the German Federal Ministry of Education and Research (BMBF) in the GEOTECHNOLIEN-Programme under the topic “Observation of System Earth from Space”. GGOS-D involved four German institutions: Deutsches GeoForschungsZentrum Potsdam (GFZ), Bundesamt f€ur Kartographie und Geod€asie (BKG), Institut f€ur Geod€asie und Geoinformation, Universit€at Bonn (IGG-B) and DGFI. An overview of the project is given by Rothacher et al. (2010). The major goals and challenges of this project were (1) the definition and implementation of common GGOS-D standards and a unique modelling and parameterization in the different software packages,

D. Angermann (*) H. Drewes M. Seitz Deutsches Geod€atisches Forschungsinstitut (DGFI), AlfonsGoppel-Strasse 11, 80539 M€ unchen, Germany e-mail: [email protected]

(2) the generation of homogeneously re-processed observation time series from the different space geodetic observation techniques, (3) the computation of the GGOS-D terrestrial and celestial reference frames, and (4) the generation of consistent, high-quality time series of geodetic-geophysical parameters describing the Earth System. In this paper, we focus on the computation of the terrestrial reference frame (TRF). The input data comprise observation time series from the space techniques Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR) and the Global Positioning System (GPS) based on identical standards for the modelling and parameterization. The data were processed as datum-free normal equations in two major steps (1) Analysis and accumulation of time series normal equations per technique, (2) Intertechnique combination and computation of the final TRF solution. A major focus was thereby on the investigation of seasonal variations in station positions and on the development of advanced combination methods.

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_11, # Springer-Verlag Berlin Heidelberg 2012

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11.2

D. Angermann et al.

Data Used for TRF Computation

The computation of the GGOS-D terrestrial reference frame is based on homogeneously processed VLBI, SLR and GPS observation time series.Table 11.1 gives an overview of the input data. The time series of the different techniques were provided as unconstrained datum-free normal equations. The GGOS–D input data also comprise the local tie information, which is available at the ITRS Centre at http://itrf.ensg.ign.fr/ local_survey.php. The common models used in the various software packages were not only implemented but also validated in detail by intensive comparisons. Based on the experiences gathered a refined set of standards was agreed upon the participating institutions and was subsequently implemented for the second project phase (Rothacher et al. 2007; Steigenberger et al. 2010). Most recent state-of-the models were taken into account, e.g., the use of 6-hourly ECMWF grids for computing the hydrostatic delay and the use of the hydrostatic mapping function VMF1 (B€ ohm et al. 2006) for VLBI and GPS, the correction for higherorder ionospheric terms for GPS, and the modelling of thermal deformation for the VLBI telescopes (Nothnagel 2008). In order to allow for a comparison and validation of the results, for each of the techniques two solutions were computed with different software packages. Based on the individual observation time series, intratechnique combinations were performed for VLBI and SLR. For GPS such a combination was not performed since the EPOS solution was not fully consistent

Table 11.1 GGOS-D data used for the global TRF computation. For each technique two series were computed using two different software packages. In case of GPS the second series computed with EPOS was used for comparisons only, since the time resolution of this solution is 1 week and there are also small differences w.r.t. the GGOS-D standards Institution Software Institution Software Time interval resolution No. of stations

GPS GFZ Bernese GFZ EPOS 1994–2007 1/7 days 240

SLR DGFI DOGS GFZ EPOS 1993–2007 7 days 71

VLBI DGFI OCCAM IGGB Calc/Solve 1984–2007 1 h sessions 49

with the Bernese solution. For the same reason, DORIS (Doppler Orbitography and Radiopositioning Integrated by Satellite) data were not included in the TRF computation, since the standards and models applied for the DORIS processing were not fully consistent with the GGOS-D standards.

11.3

Combination Methodology for the TRF Computation

The methodology applied in the GGOS-D project was based on combining datum-free normal equations of the VLBI, SLR and GPS observation time series. The tropospheric parameters and low-degree spherical harmonic coefficients, which were also included in the time series normal equations, were reduced and presently not considered for the TRF computations. The station positions and velocities of the observing stations (TRF) were combined in a common adjustment with the quaser coordinates (CRF), and the Earth Orientation Parameters (EOP). The TRF computation has been performed with the DGFI Software package DOGS. The general procedure of the combination methodology is given in Fig. 11.1.

Input: Datum-free normal equations (NEQ) Epoch 1

VLBI NEQ

SLR NEQ

GPS NEQ

Epoch 2

VLBI NEQ

SLR NEQ

GPS NEQ

Epoch n

VLBI NEQ

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GPS NEQ

Accumulation of time series Multi-year NEQ‘s

VLBI NEQ

SLR NEQ

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Inter-technique combination TRF (station positions and velocities), EOP, and quasar coordinates)

Fig. 11.1 Combination methodology for the terrestrial reference frame computation

11

Global Terrestrial Reference Frame Realization Within the GGOS-D Project

11.3.1 Accumulation of Time Series per Technique In the first part of the terrestrial reference frame computation, the time series of the daily/weekly normal equations were accumulated to a multi-year normal equation for each technique. The procedure comprises two major steps (1) The generation and analysis of station position time series to detect non-linear behaviours, and (2) the computation of the cumulative multi-year normal equations per technique. The time series analysis and accumulation was performed in an iterative procedure. At first, constant velocity parameters were set up in the epoch normal equations to represent linear station motions and the epoch station positions were transformed to positions at the reference epoch 2000.0. From the analysis of position time series discontinuities were identified for many stations, which are often caused by equipment changes or geophysical events (e.g., earthquakes). The discontinuities were parameterized by setting up new position and velocity parameters after the jump. The total number of discontinuities could significantly be reduced compared to ITRF2005, in particular for GPS stations. The ITRF2005 solution computed at DGFI (ITRF2005-D) comprises 221 discontinuities for 332 stations (67%), whereas 124 discontinuities for 240 stations (52%) were identified within GGOS-D. This improvement was mainly achieved by the homogeneuosly processed GGOS-D data sets and, in the case of GPS, by the implementation of absolute antenna phase center corrections. Figure 11.2 shows the position time series for the GPS station Yuzhno-Sakhalin (YSSK), Russia, located in a geodynamically very active region, the Sakhalin seismic belt. Two large earthquakes both with a magnitude of 8.3, at Hokkaido (25 September 2003) and at Kuril island (15 November 2006) caused discontinuities of about 1–2 cm in the position time series, in particular in the north and east component. The station velocity in the north component was changed by about 5 mm/year after the first earthquake, changing again to its nominal value after about 2 years. The precision of repeated station positions obtained from the accumulation of the time series solutions for each space technique was used to estimate scaling factors for the technique-specific normal equations. For stations with discontinuities separate positions

89

were estimated for each segment. Statistical tests were applied to decide whether the estimated velocities can be equated or not.

11.3.2 Computation of the TRF Solution Input for the combination of different techniques were the accumulated intra-technique normal equations for GPS, SLR and VLBI. The parameters include station positions, velocities, daily EOPs, and (for VLBI) also quasar coordinates (right ascension and declination). The connection of the different techniques’ observations is given by geodetic local tie measurements between the instruments’ reference points at colocation sites. The selection of suitable local ties is a critical issue because the number and the spatial distribution of “high quality” co-location sites is not optimal. Furthermore, there are discrepancies between the difference vectors derived from the space geodetic techniques and the local ties, as shown for example in Fig. 11.3 for VLBI and GPS co-locations. The results are given for the GGOS-D TRF computation in comparison with ITRF2005-D (Angermann et al. 2007, 2009), which is in excellent agreement with ITRF2005 (Altamimi et al. 2007). The agreement of the space geodetic solutions with the local ties is better for most stations of the GGOS-D computation, in particular for those located on the southern hemisphere. In the ideal case (without systematic errors) the EOP estimates must be (statistically) identical for all space techniques. Thus, their estimates are used as a criterion to validate the selected local ties and to stabilize the inter-technique combination as additional “global ties”. The selection and implementation of local ties for the inter-technique combination as well as the equating of station velocities at co-location sites was done in an iterative procedure (see Kr€ugel and Angermann 2007). The selection of local ties is performed on the basis of two criteria (1) the pole coordinates as common parameters provide an ideal basis to measure the consistency: A first combination of station coordinates is computed using a set of local ties, while the EOP are not combined. The mean offsets between the estimated coordinates of the pole are then used to quantify the consistency of the TRF. (2) The r.m.s. values of the 7-parameter similarity transformations between the combined TRF solution and the single-technique

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JD2000 [Tage] Fig. 11.2 Position time series for the GPS station Yuzhno-Sakhalin (YSSK), Russia 25 ITRF 2005 GGOS D

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ONSA

5

Fig. 11.3 Comparison of the GGOS-D results with ITRF2005D. The 3D difference vectors (mm) between the VLBI and GPS solutions and the terrestrial difference vectors are given for 21

co-location sites. The stations located in the southern hemisphere are highlighted by yellow (grey) color

solutions are a measure of the deformation of the station networks (caused by the local tie implementation). The two above mentioned criteria were applied in various test computations by satisfying also that an optimal number of co-location sites with a good spatial distribution is selected.

To identify the best (most suitable) set of co-location sites, different solutions were computed, varying the co-locations and the assumed accuracy of the introduced local ties. As an example, the results for VLBI and SLR co-locations are given in Table 11.2. Shown are the mean pole offsets between the VLBI

11

Global Terrestrial Reference Frame Realization Within the GGOS-D Project

Table 11.2 Mean pole offsets and network deformation for VLBI and GPS co-locations obtained from the GGOS-D intertechnique combination in comparison with the DGFI solution for ITRF2005 (ITRF2005-D) Selected VLBI-GPS co-locations Mean pole difference (mas) Network deformation Epoch 2000.0 (mm)

GGOS-D 19

ITRF2005-D 13

0.035 0.3

0.041 1.0

and GPS estimates as a measure for the consistency. Furthermore, the r.m.s. position differences of a 7-parameter similarity transformation between the VLBI-only and the combined TRF solution (expressed at the reference epoch 2000.0) as a measure for the network deformation are shown. The GGOS-D results are compared with those obtained from ITRF2005-D. It shall be emphasized that the number of co-locations significantly increased for the GGOS-D computations, accompanied with a better agreement of the VLBI and GPS pole coordinates and a smaller network deformation. Major differences of the GGOS-D solution w.r.t. ITRF2005 are (1) an improved modelling of the individual space techniques, (2) a homogeneous re-processing by applying unified standards and models for all techniques, and (3) the inclusion of two more years of recent data until the end of 2007. Other tasks of the inter-technique combination include the weighting of the different techniques and the equating of station velocities of co-located instruments. The weighting was done by estimating variance factors for the normal equations based on the precision of repeated (daily/weekly) station positions. The station velocities of co-located instruments were estimated as separate parameters. The velocities were equated, if the differences are statistically not significant. To generate the TRF solution, minimum datum conditions were added to the combined normal equations, and the complete normal equation system was inverted. The origin was realized by SLR observations. As shown by R€ ulke et al. (2008), the network translation parameters derived from GPS are affected by orbital perturbances and cannot be used for the datum definition. The scale of the TRF solution was defined by SLR, VLBI and GPS observations. This was justified, since a comparison of the techniquespecific station networks w.r.t. the scale showed no significant differences. The orientation of the TRF was

91

defined by a No-Net-Rotation (NNR) condition w.r.t. ITRF2005. The kinematic datum of the TRF solution was given by an actual plate kinematic and crustal deformation model (APKIM) derived from observed station velocities (Drewes 2009). The GGOS-D combination results comprise the terrestrial reference frame (station positions and velocities), daily EOP and quasar coordinates, which were estimated in a common adjustment. The terrestrial reference frame results provide the basis for the generation of consistent, high-quality time series of geodetic-geophysical parameters describing the Earth System (Nothnagel et al. 2010).

11.4

Analysis of Seasonal Station Motions

In specific GGOS-D studies, the time series of co-located VLBI and GPS stations were analysed and compared (Tesmer et al. 2009). As an example, Fig. 11.4 shows the VLBI and GPS height time series for two co-location sites: Wettzell (Germany) and Ny-Alesund (Spitsbergen, Norway). The VLBI antenna of Wettzell has the most dense height series (two observations per week), which agrees rather well with GPS. In case of Ny-Alesund there are larger discrepancies between the VLBI and GPS height time series, which may be caused by solutionand/or technique-specific effects. The results given by Tesmer et al. (2009) show a rather good agreement of the height time series for most of the VLBI and GPS colocation sites. The strongest signals in the time series have mostly annually repeating patterns. The station height time series were compared with geophysical model results (Seitz and Kr€ugel 2009). It was found, that a large part of the observed annual signals can be explained by loading, which was computed from atmospheric, hydrospheric and non-tidal oceanic loading variations. Unfortunately, the geophysical models are not as accurate as necessary to reduce loading from the variations of the station positions. Thus, the reduction of loading effects from the original observations is problematic. However, these seasonal station motions will affect the terrestrial reference frame computations, if the temporal variations of station positions are described only by constant velocities, as it is done currently. The consequences are (1) Deviations of the station motions from a linear model (e.g., seasonal variations) will

92

D. Angermann et al. WTZR vs. WETTZELL: wmeans (each 7 days for −+35 days)

dR [cm]

1

0 VLBI GPS

−1 1997

1998

1999

2000

2001

2002

2003

2004

2005

time [year] NYAL vs. NYALES20: wmeans (each 7 days for −+35 days)

dR [cm]

1

0 VLBI GPS

−1 1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

time [year]

Fig. 11.4 Homogeneously re-processed VLBI (stars) and GPS (circles) height time series at two co-located sites: Wettzell (Germany) and Ny-Alesund (Spitsbergen, Norway). The figures illustrate 90-days moving weighted means computed each 7 days

produce errors in the combination results; (2) seasonal variations will affect the velocity estimations, in particular for stations with relatively short observation time spans (i.e., < P ðsn ð d zÞÞ2 Pn cos c þ kdz Dt for c þ kdz Dt p > = ; n¼0

> > : for c þ kdz Dt > p ; 0

(15.20)

where kdz is a conversion factor representing in the case of the time-varying SST the correlation time of the signal. This should be studied and determined in each region under study, since the characteristics of dz vary significantly for each area and in open or closed sea regions (Knudsen and Tscherning 2006). The degree variances sn ðd zÞ are determined as in (15.17) in order to fit the local characteristics of the time-varying SST. Since the time-varying SST is triggered by salinity and temperature variations, in-situ oceanographic data and climatology models can be used for the fit of the analytical model to empirical values. Given the analytical expressions for the covariance functions of all observations, it is possible to proceed to the simultaneous estimation of the deterministic parameters and of the signals z c, dz, T, hSLA nonsteric , etc., along with their prediction errors as:

1 ^x ¼ AT C1 A AT C1 yy yy y;

(15.21)

^zc ðPÞ ¼ Czs ðP; ÞC1 y Ax ^ ; yy

(15.22)

n o s^2c ðPÞ ¼ Czc zc ðP; PÞ Czc s C1 C1 AN1 AT C1 Cszc : yy yy yy z

(15.23)

15

On the Determination of Sea Level Changes by Combining Altimetric

^hSLA ðPÞ ¼ C SLA ðP; ÞC1 ðy A^ xÞ; hSTERIC s STERIC yy

(15.24)

SLA ðP; PÞ ChSLA s2^hSLA ðPÞ ¼ ChSLA STERIC hSTERIC STERIC s STERIC n o 1 1 T 1 C1 : yy Cyy AN A Cyy CshSLA STERIC

(15.25) The estimation of all other signals and prediction errors can be done according to (15.22)–(15.25). A short note on the estimation of variance components within this optimal combination scheme will be given. Given the general observation equation for LSC estimation with parameters and knowing that v ~ (0, Vy ¼ Cvv) we can write Cvv in a form as to depend on a unknown set ofPso-called variance components yi. Therefore, Vy ¼ yi Vi and the matrices Vi come from the original datai error CV matrices. Within the present combination scheme Vy would take the form " Vy ¼ yhalt Qhalt þ yhTG 1m mm

2

6 4

# yHISL TG q3

QhTG

0 QHISL TG

0

yhSLA 3 7 5

QhSLA

3q3q

þ yhSLA QhSLA þ yT QT þ yTrr QTrr GRACE GRACE 1k

kk

1p pp

1p pp

(15.26) The variance components can then be estimated by an iterative procedure such as the iterative almost unbiased estimation-IAUE (Rao and Kleffe 1988). According to IAUE, we first have to compute matrix W as T 1 1 T 1 1 W ¼ C1 A Cvv ; vv Cvv A A Cvv A

(15.27)

and then estimate the variance components ^y y ¼ Jþ k depending on according to ^y ¼ J1 k or ^ whether matrix J can be inverted. Matrices J and k can be analytical determined from the data error CV matrices Jij ¼ tr WCvi vi WCvj vj ;

(15.28)

1 ki ¼ ^ vT C1 v: vv Cvi vi Cvv ^

(15.29)

129

After the first variance components have been estimated, the iterations can be carried out according to ^y

a i

¼

^ya1 ^vT C1 Cv v C1 ^v i i i vv vv : trfWCvi vi g

(15.30)

where ^y a1 is the previous estimate and y^ ai is the i new one. When their difference is smaller than a a1 ^ ^ a certain threshold e, so that y y < e, then convergence is achieved, the variance components are estimated and the iterations can stop. Conclusions

A detailed combination scheme of satellite altimetry, tide-gauge, GRACE and GOCE data for the determination of various signals both static and variable has been presented. The latter, can be the disturbing potential, its second order derivatives, geoid heights, steric and non-steric sea level variations and the stationary and time-variable sea surface topography. It is clear that the versatility of LSC with parameters allows including other observations too, like gravity anomalies, salinity and temperature from oceanographic measurements, hydrological models, etc. Therefore, the proposed combination scheme can be extended easily as new data sources become available, so that better estimates will be derived. Another advantage of the combination strategy outlined is that rigorous signal estimation errors can be derived along with the possibility of variance component estimation so that the covariance matrices and errors models can be calibrated.

References Barzaghi R, Tselfes N, Tziavos IN, Vergos GS (2009) Geoid and high resolution sea surface topography modelling in the mediterranean from gravimetry, altimetry and GOCE data: evaluation by simulation. J Geod 83(8):751–772 Chambers DP, Wahr J, Nerem RS (2004) Preliminary observations of global ocean mass variations from GRACE. Geophys Res Lett 31:L13310. doi:10.1029/ 2004GL0204 61 Chelton DB, Ries JC, Haines BJ, Fu LL, Callahan PS (2001) Satellite altimetry. In: Fu LL, Cazenave A (eds) Satellite altimetry and earth sciences: a handbook of techniques and applications, vol 69, International Geophysics Series. Academic Press, San Diego, CA, pp 4–131 Chen JL, Wilson CR, Tapley BD, Famiglietti JS, Rodell M (2005) Seasonal global mean sea level change from satellite

130 altimeter, GRACE and geophysical models. J Geod 79:532–539 Garcia D, Chao BF, Rio JD, Vigo I (2006) On the steric and mass-induced contributions to the annual sea level variations in the Mediterranean Sea. J Geophys Res 111:C09030. doi:10.1029/2005JC002956 Garcia D, Ramillien G, Lombard A, Cazenave A (2007) Steric sea-level variations inferred from combined Topex/Poseidon altimetry and GRACE gradiometry. Pure Appl Geophys 164:721–731 Heiskanen WA, Moritz H (1967) Physical geodesy. W.H. Freeman, San Fransisco, CA Knudsen P (1987) Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data. Bull Geod 61:45–160 Knudsen P (1991) Simultaneous estimation of the gravity field and sea surface topography from satellite altimeter data by least-squares collocation. Geophys J Inter 104(2):307–317 Knudsen P (1992) Estimation of sea surface topography in the Norwegian sea using gravimetry and Geosat altimetry. Bull Ge´od 66:27–40 Knudsen P (1993) Integration of gravity and altimeter data by optimal estimation techniques. In: Rummel R, Sanso` F (eds) Satellite altimetry for geodesy and oceanography, vol 50, Lecture Notes in Earth Sciences. Springer, Berlin, pp 453–466 Knudsen P, Tscherning CC (2006) Error Characteristics of dynamic topography models derived from altimetry and GOCE Gravimetry. In: Tregoning P, Rizos C (eds) Dynamic Planet 2005 – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools, IAG Symposia, vol 130. Springer, Berlin, pp 11–16

G.S. Vergos et al. Lombard A, Garcia D, Ramillien G, Cazenave A, Biancale R, Lemoine JM, Fletcher F, Schmidt R, Ishii M (2007) Estimation of steric sea level variations from combined GRACE and Jason-1 data. Earth Planet Sci Lett 254:194–202 Moritz H (1980) Advanced physical geodesy. Wichmann, Karlsruhe Rao CR, Kleffe J (1988) Estimation of variance components and applications. North-Holland Series in Statistics and Probability, vol 3 Rio MH, Hernandez F (2004) A mean dynamic topography computed over the world ocean from altimetry, in-situ measurements and a geoid model. J Geoph Res 109(12): C12032 Sanso` F, Sideris MG (1997) On the similarities and differences between systems theory and least-squares collocation in physical geodesy. Boll di Geod e Scie Aff 2:174–206 Swenson S, Wahr J (2002) Methods for inferring regional surface-mass anomalies from GRACE measurements of time-variable gravity. J Geophys Res 107(B9):2193 Tscherning CC and Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations, and deflections of the vertical implied by anomaly degreevariance models. Reports of the Department of Geodetic Science, 208, The Ohio State University, Columbus, OH Tziavos IN, Sideris MG, Forsberg R (1998) Combined satellite altimetry and shipborne gravimetry data processing. Mar Geod 21:299–317 Vergos GS, Tziavos IN and Andritsanos VD (2005) On the determination of marine geoid models by least-squares collocation and spectral methods using heterogeneous data. In: Sanso´ F (ed) IAG Symposia, A Window on the Future of Geodesy, vol 128. Springer, Berlin, pp 332–337

Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite and Submarine Measurements

16

J. Calvao, J. Rodrigues, and P. Wadhams

Abstract

The L3H phase of operation of ICESat’s laser in the winter of 2007 coincided for about two weeks with the cruise of the British submarine “Tireless” which was equipped with upward-looking sonars. This provided a rare opportunity for a simultaneous determination of the sea ice freeboard and draft in the Arctic Ocean.

16.1

Introduction

The determination of the thickness of the sea ice layer in the Arctic Ocean and surrounding seas is an essential component of the study of the climate of the Arctic. On the one side, the Arctic environment as a whole and, in particular, its sea ice cover, respond quickly to global climatic changes due to some amplification mechanisms that tend to be present in the polar regions, like the well-known albedo effect. As such, a decline in the volume of sea ice can be considered as one of the best indicators of the warming of our planet. On the other hand, because the sea ice acts as a regulator of heat and moisture transfer between the ocean and the atmosphere, its thickness is likely to affect significantly the climate of the Arctic and nearby regions. Basin-wide measurements of the thickness of the sea ice in the Arctic Ocean began in 1958 when the

J. Calvao (*) Lattex, IDL, Faculdade de Cieˆncias da Universidade de Lisboa, Lisboa, Portugal e-mail: [email protected] J. Rodrigues P. Wadhams Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK

U.S. submarine “Nautilus” reached the North Pole for the first time. Since then, regular Arctic cruises by U.S. submarines have collected a vast amount of sea ice draft data, mostly in the so-called SCICEX box, which roughly coincides with the portion of the Arctic Ocean outside international waters (but including the region north of Alaska). Most of these data sets are available through the National Snow and Ice Data Center (Boulder, CO) archive and have been extensively analysed by Rothrock et al. (1999, 2008). In the early 1970s British submarines started cruising in the Arctic Ocean. For more than three decades they have been taking ice thickness data in regions rarely visited by U.S. boats, such as Fram Strait and the waters north of Greenland. The latter are of special importance in the understanding of the large scale sea ice dynamics of the Arctic Ocean. Through Fram Strait passes most of the ice, fresh water and heat exchanged between the Arctic Ocean and the rest of the world oceans. The complex and variable sea ice conditions in Fram Strait partly reflect the diverse origins of that ice; some of it was transported all the way from the coasts of East Siberia by the Transpolar Drift, other parts may have been advected from the west or the east, and some is formed locally. On the other hand, the region north of Greenland and Ellesmere Island is known to have the thickest ice in

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_16, # Springer-Verlag Berlin Heidelberg 2012

131

132

the Arctic. The last data collected by British submarines suggest that, unlike the rest of the Arctic, there is no decline in ice thickness here in recent years. Data from earlier cruises have been processed at the University of Cambridge and the results published in several papers. Wadhams (1990) provides the first direct evidence of the thinning of the sea ice north of Greenland. Later, he and, independently, Rothrock, observed a significant overall thinning of the Arctic sea ice by comparing results from cruises in the mid 1970 s and in the mid 1990 s (Rothrock et al. 1999, Wadhams and Davis 2000). The Polar Oceans Physics group of the University of Cambridge is currently in the last stages of the statistical analysis of the data gathered during the last two Royal Navy submarine cruises, in the winters of 2004 and 2007. Data from the March 2007 cruise will be particularly relevant because they were taken in several regions of the Arctic with very different ice regimes, some of which would later become ice-free during the exceptional summer of 2007. In this paper we show some results from these two cruises. The quality and scientific importance of the submarine data are unquestionable. However, the use of different sonar equipment in different cruises, the frequent impossibility of an independent evaluation of the accuracy of the measurements, the scarsity of the voyages, the non-coincidence of the tracks, the varied times of the year of the cruises, and the difficulty in merging British and U.S. data in a single global sonic data bank on Arctic sea ice thickness, suggest that some caution is needed when they are used to derive long-term trends. The first use of satellite altimetry to retrieve sea ice freeboard (elevation of the ice floes above the sea level) and thickness is due to Laxon et al. (2003). They used data from the European Space Agency (ESA) satellites ERS-1 and ERS-2 in order to explore the correlation between ice thickness and the length of the ice season. More recently, the same group analysed other radar altimetry data, this time from ESA’s Envisat, obtained between the winters of 2002/2003 and 2007/2008 (Giles et al. 2008). They found a significant reduction in sea ice thickness in the region south of 81.5 N after the record minimum ice extent of September 2007 but no particular trend (in the winter season) between 2003 and 2007. The launch of NASA’s ICESat in January 2003, equipped with its high accuracy Geoscience Laser

J. Calvao et al.

Altimeter System (GLAS), allowed the first determinations of the central Arctic sea ice thickness from freeboard retrievals. Methods to apply laser altimetry to measurements of ice freeboard have been developed by Forsberg and Skourup (2005), Kwok et al. (2007) and Zwally et al. (2008), among others. In a thorough analysis of ten ICESat campaigns between 2003 and 2008, Kwok et al. (2009) derived the evolution of the Arctic sea ice thickness during this five year period, and were able to separate the contribution of first and multi-year ice. They concluded that the average winter ice thickness in the Arctic decreased from 3.3 m in February-March 2005 to less than 2.5 m in February-March 2008 and showed that this decline is essentially due to the disappearance of multi-year ice which, in the winter of 2008, was responsible for only one third of the total volume of ice in the Arctic Ocean. During a period of two weeks in March 2007 the L3H phase of operation of ICESat’s laser coincided with the cruise of the British submarine HMS “Tireless”. Large scale simultaneous observations of the Arctic Ocean’s sea ice thickness are very rare events. To our knowledge, this is only the second occurrence of this type. The first one was reported by Kwok et al. (2009), who used U.S. submarine data collected during a cruise in October/November 2005. The simultaneous measurement of the sea ice freeboard and draft offers the possibility of a cross-check (hopefully crossvalidation) between the results for Arctic sea ice thickness obtained independently by the complex satellite and submarine measurements. In this paper we present results for the sea ice thickness in the Arctic Ocean in the winter of 2007 (which preceded the extraordinary minimum ice extent of September 2007) and in the winter of 2004 obtained from ICESat and submarine data. In Sect. 16.2 we briefly describe the methods used in the retrieval of ice freeboard from ICESat and present overall results for the ice freeboard during the L2B and L3H phases. In Sect. 16.3 we give an overview of the techniques used to calculate ice draft from submarine observations and give average draft values for each section of the two cruises. In Sect. 16.4 we compare the ice thickness obtained by the two platforms and identify the main changes between the ice distributions in 2004 and 2007. We end up with a discussion on the possible causes of the observed discrepancies between ICESat and submarine results.

16

Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite

16.2

ICES at Measurements

The laser altimeter system GLAS carried on board the ICESat satellite, especially dedicated to the observation of Polar Regions, generates profiles of the surface topography along the tracks with an accuracy of 15 cm. The beam width is 110 mrad and the pulse rate emission is 40 Hz, sampling the Earth surface with a footprint of 70 m diameter, spaced at 170 m intervals. Small-scale features inside the footprint modulate the amplitude and character of the waveforms, which can be used to determine sea-ice freeboard heights using a “lowest level” filtering scheme: specular laser returns from open water in leads between ice floes are used to define an ocean reference surface. The procedure applied to obtain the ice freeboard F ¼ h-N-MDT (where h is ICESat’s ellipsoidal height estimate of the surface, N is the geoid undulation and MDT is the ocean mean dynamic topography) for the whole Arctic basin (with the exception of points beyond 86 N) consisted of a high-pass filtering of the satellite data to remove low frequency effects due to the geoid and ocean dynamics (the geoid model ArcGP with sufficient accuracy to allow the computation of the freeboard was very recently made available (Forsberg et al. 2007). The original tide model was replaced by the tide model AOTIM5 and the tide loading model TPXO6.2. The inverse barometer correction was applied. As there are no MDT models with enough accuracy, the definition of the ocean surface was done through the identification of areas of open water (or thin ice, typically less than 30 cm thick) in each 20 km segment of the tracks to allow the interpolation of the ocean surface. Several solutions were tested to define the ocean surface and the computed freeboard values were interpolated on a 5 5 minute grid, where the submarine track was interpolated. Some waveform derived parameters in the ICESat data products were used to filter unreliable elevation estimates: surface reflectivity (i_reflctUncorr), detector gain (i_gain_rcv) and surface roughness (i_SeaIceVar). Figure 16.1a, b show the spatial distribution of the sea ice freeboard during the L2B (2004) and L3H (2007) phases of ICESat. Some comparisons between the ice freeboards in the two campaigns are now in order. The mean ice

133

freeboard decreased between 2004 and 2007 in some regions of the central Arctic Ocean such as north of the Barents and Kara Seas in the Russian Sector, and between 120 W and 180 in the Canadian/Pacific sectors. Assuming that the depth of the snow on sea ice is similar in the two periods, this indicates a decrease in the sea ice thickness. The volume of ice in Fram Strait is much larger in 2007 than in 2004. There appears to be more thick ice in the vicinity of the north coasts of Greenland and Ellesmere Island in 2007 than in 2004. These observations are consistent with the strengthening of the Transpolar Drift in recent years reported by several authors, with large quantities of ice being transported across the central Arctic Ocean from the Chukchi and Beaufort Seas. Part of this ice is advected to the Atlantic through Fram Strait while the remaining piles up along the north coasts of Greenland and Ellesmere Island.

16.3

Submarine Measurements

In March 2007 the British submarine HMS “Tireless” carried out the route shown in Fig. 16.1b. After entering the Arctic Ocean through Fram Strait, it rounded northeast Greenland, thence cruised north of Greenland and Ellesmere Island roughly along the 85 N parallel until approximately 92 W, when it started heading SW towards the S Beaufort Sea. In the vicinity of (85 200 N, 64 080 W) the boat ran a set of parallel lines at low speed and depth with total length of 200 km under an area that was simultaneously used as a base for experiments of the EU DAMOCLES project. The outgoing part of the cruise ended at the location of the SEDNA ice camp (73 070 N, 143 440 W), where several other ice thickness measurements were performed (Hutchings et al. 2008). The submarine returned to the UK following almost exactly the same route as in the outbound journey and also collected draft data. However, because it was travelling at higher speed and depth, the quality of the data was somewhat inferior and will not be considered here. The April 2004 track, also shown in Fig. 16.1a, included a transect to and from the Pole and a diversion to (85 N, 62 W) in order to survey under a region which a month later was used for an ice camp experiment as part of the EU GreenICE project. Draft data were collected between the marginal ice zone in Fram Strait and the westernmost point of the

134

J. Calvao et al.

Fig. 16.1 (a) L2B freeboard grid (m): max ¼ 0.97, mean ¼ 0.13, std ¼ 0.13. The track of the “Tireless” is shown in red. (b) L3H freeboard grid (m): max ¼ 0.84, mean ¼ 0.14, std ¼ 0.13. The track of the “Tireless” is shown in red

transect, including about 250 km under the GreenICE camp. Shortly after this survey the sonar stopped recording, except for a short period in the vicinity of the North Pole. In both cruises sea ice draft was measured with two upward-looking single-beam echosounders: the analogue system known as Admiralty Type 780 and the digital system known as Admiralty Type 2077. Because the latter malfunctioned during most of the 2007 voyage, in this paper we limit ourselves to the data obtained with AT780. Though the entire return from each ping is recorded in a paper chart, with the darkness of the trace a function of the intensity of the echo, we shall only take into account the first return from the insonified area (which is typically a circle with a diameter of a few metres for the 3 nominal beam width of the AT780 and ordinary cruise depths). This is a standard procedure in the analysis of submarine data: it is more reliable than to average over the entire pulse and it ensures compatibility with digital systems (where only the first return is recorded). Details of the techniques used to process this type of data can be found in Wadhams (1981).

Once the navigation data are included, the final product is the ice draft as a function of the position of the submarine or the along-track distance for the entire cruise. It is traditional to divide the full track into 50 km sections, as it has been shown that with such a length there is reliable statistics and a low probability of finding different ice regimes within one single section. Figure 16.2a, b show the mean ice draft obtained from the AT780 mounted in the “Tireless” for each of the sections of the 2004 and 2007 cruises. Notice that the 2004 cruise (which took place in April) did not coincide in time with ICESat’s winter campaign. However, because both measurements were made at the end of the winter, when the ice thickness is at or near its annual maximum, a comparison between them is, in our view, legitimate. This is also the reason why we used the freeboard values obtained throughout the whole ICESat winter campaign in 2007. A different approach, which the authors may follow in future work, would be to consider only the portion of the ICESat tracks that coincide in time and position with the track of the submarine.

16

Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite

135

Fig. 16.2 (a) Mean ice drafts in the 2004 submarine cruise. (b) Mean ice drafts in the 2007 submarine cruise

16.4

Results

The sea ice thickness hi can be calculated from either the freeboard f (which includes the snow layer of depth hs because the laser pulse is reflected at the snow-air

interface if there is snow on top of the sea ice) or from the ice draft d through the relations hi ¼

rw r rs f w hs rw ri rw ri

(16.1)

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hi ¼

rw r d s hs ri ri

which follow from the assumption of isostatic equilibrium. Here rw ¼ 1024 kg=m3 , ri ¼ 920 kg=m3 and rs ¼ 360 kg=m3 are our assumed densities of the sea water, sea ice and snow, respectively. Values for the density of sea ice ranging between 850 and 925 kg/m3 have been reported (the later is used by Kwok et al. (2009)) while the density of the snow at the end of the winter may range from 300 to 360 kg/m3. Our choices have been widely used in the literature. While the snow term in the second equation can be safely neglected, this is not the case in the retrieval of the ice thickness from the ice (plus snow) freeboard measured with altimetry techniques, where the snow depth plays a crucial role. Unfortunately, the algorithms that attempt to extract snow depth from passive microwave satellite imagery (such as the NASA Team algorithm used to process raw AMSRE data) suffer from severe limitations. As such, in this paper we choose to work with values observed during recent winter field campaigns in the Beaufort and Lincoln Seas, which, typically, vary from zero to 30 cm. Here we use the upper value of 30 cm. In this case, it should be understood that the minima of the freeboard used to define a reference level as explained in Sect. 16.2 correspond not to zero elevation but to 30 cm elevation. Consequently, the value of the freeboard used to retrieve the sea ice thickness from (16.1) is the value that appears in the maps shown in Fig. 16.1a, b with 30 cm added. In fact, it is reasonable to assume that at the end of the winter the minima do not coincide with open water but with refrozen leads, which would have some snow on top. Tables 16.1a and 16.1b show the mean ice thicknesses obtained from satellite and submarine measurements for different regions of the Arctic visited by the submarine. Central (North) Fram Strait is the portion of the strait between 80 N and 82 N (82 N and 84 N); North Greenland 1 is the path of the submarine in 2004 along the 85 N parallel roughly between 20 W and 70 W, which coincides with the path of 2007; North Greenland 2 is the portion of the trajectory of the boat between 85 N and 86 N when it was heading northeastwards. Note that within each section, or region, the amount of invalid data differs for the satellite and the

Table 16.1a Mean sea ice thickness in the winter of 2004 Region

Sections

Central Fram Strait W Central Fram Strait E North Fram Strait Northeast Greenland North Greenland 1 North Greenland 2

01–18 30–33 34–37 38–43 44–57 59–62

Thickness (ICESat) 2.28 3.73 3.67 4.03 4.41 4.14

Thickness (Sub) 3.37 2.13 2.95 3.70 5.90 5.42

Table 16.1b Mean sea ice thickness in the winter of 2007 Region

Sections

Central Fram Strait North Fram Strait Northeast Greenland North Greenland N Ellesmere Island Canadian Basin Beaufort Sea

06–10 11–14 15–18 19–32 35–36 38–52 53–64

Thickness (ICESat) 2.85 2.88 2.78 4.65 4.65 2.90 2.27

Thickness (Sub) 4.43 3.93 3.77 5.95 7.00 5.42 3.73

submarine, which may explain some of the differences in the retrieved ice thickness. Figure 16.3a, b show the mean ice thickness for each of the 50nmjh sections into which the transects of the submarine were divided. ICESat values for each section are taken from a “square” box whose boundaries are the extreme values of the latitude and longitude of the submarine within that section.

16.5

Discussion

As it can be seen from Fig. 16.3a, b, although the results obtained from satellite and submarine data processing have good correlation concerning the shape of the curves representing sea ice thickness, there are visible discrepancies. The mean ice thickness computed from submarine data is consistently higher, with differences that can reach 1.5 m. Such differences occur mostly in the heavily ridged zone north of Greenland and Ellesmere Island (sections 44–57 in 2004 and 19–36 in 2007). There is reasonably good agreement in the Southern Beaufort Sea (last sections in 2007), where the submarine measured the thinnest ice of the whole transect.

16

Arctic Sea Ice Thickness in the Winters of 2004 and 2007 from Coincident Satellite

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Fig. 16.3 (a) Mean ice thickness in the winter of 2004 for each section of the submarine cruise. (b) Mean ice thickness in the winter of 2007 for each section of the submarine cruise

Several reasons contribute to these results: (a) The comparison relates instantaneous data obtained from submarine draft data with gridded freeboard data computed from almost one month of satellite data, both converted into thickness, which means that this latter data set is a mean value for that period (moreover, the processing scheme produces some kind of attenuation of the freeboard grid). (b) The size of the footprints of the sonar and laser are quite different, which means that the integrated value and the resolution of the draft and freeboard are correspondingly different. (c) Beamwidth sonar correction were not performed, so the draft tends to be overestimated. Because of the comparatively narrow beam of the AT780 echosounder, such corrections have been considered

negligible in previous publications. However, a recent study by one of the authors suggests that this may not always be the case, namely in regions with a high density of pressure ridges. This might explain why the largest differences were found north of Greenland and Ellesmere Island. (d) A more sophisticated treatment of the snow may be required. We look forward to the release of version V12 of the NASA Team algorithm where, hopefully, some of the problems in the retrieval of the snow depth have been sorted. These should be considered as preliminary results. Clearly, more research is needed to fully understand the causes of the differences in the values of the sea ice thickness extracted from submarine sonars and satellite altimetry.

138

References Forsberg R, Skourup H (2005) Arctic Ocean gravity, geoid and sea ice freeboard heights from ICESat and Grace. Geophys Res Lett 32:L215502. doi:10.1029/2005GL023711 Forsberg R, Skourup H, Andersen OB, Knudsen P, Laxon SW, Ridout A, Johannsen J, Siegismund F, Drange H, Tscherning CC, Arabelos D, Braun A, Renganathan V (2007) Combination of spaceborne, airborne and in-situ gravity measurements in support of Arctic Sea ice thickness mapping, Danish National Space Center, Technical report Nº 7, 2007 Giles KA, Laxon SW, Ridout AL (2008) Circumpolar thinning of Arctic sea ice following the 2007 record ice extent minimum. Geophys Res Lett 35:L22502. doi:10.1029/ 2008GL035710 Hutchings J et al (2008) Role of ice dynamics in the sea ice mass balance. Eos Trans AGU. 89(50). doi:10.1029/ 2008EO500003 Kwok R, Cunningham GF, Zwally HJ, Yi D (2007) Ice, Cloud, and land Elevation Satellite (ICESat) over Arctic sea ice: retrieval of freeboard. J Geophys Res 112:C12013. doi:10.1029/2006JC003978 Kwok R, Cunningham GF, Wensnahan M, Rigor I, Zwally HJ, Yi D (2009) Thinning and volume loss of the Arctic Ocean

J. Calvao et al. sea ice cover: 2003–2008. J Geophys Res 114:C07005. doi:10.1029/2009JC005312 Laxon S et al (2003) High interannual variability of sea ice thickness in the Arctic region. Nature 425:947–950 Rothrock DA, Yu Y, Maykut GA (1999) Thinning of the Arctic sea-ice cover. Geophys Res Lett 26(23):3469–3472. doi:10.1029/1999GL010863 Rothrock DA, Percival DB, Wensnahan M (2008) The decline in Arctic sea-ice thickness: Separating the spatial, annual, and interannual variability in a quarter century of submarine data. J Geophys Res 113:C05003. doi:10.1029/ 2007JC004252 Wadhams P (1981) Sea ice topography of the Arctic Ocean in the region 70 W to 25 E. Philos Trans Roy Soc Lond A302: 45–85 Wadhams P (1990) Evidence for thinning of the Arctic ice cover north of Greenland. Nature 345:795–797 Wadhams P, Davis N (2000) Further evidence of ice thinning in the Arctic Ocean. Geophys Res Lett 27:3973–3976 Zwally HJ, Yi D, Kwok R, Zhao Y (2008) ICESat measurements of sea ice freeboard and estimates of sea ice thickness in the Weddell Sea. J Geophys Res 113, C020S15. doi:10.1029/2007JC004284.

The Impact of Attitude Control on GRACE Accelerometry and Orbits

17

U. Meyer, A. J€aggi, and G. Beutler

Abstract

Since March 2002 the two GRACE satellites orbit the Earth at relatively low altitude (500 km in 2002, still close to 460 km mid of 2009). GPS-receivers for orbit determination, star cameras and thrusters for attitude control, accelerometers to observe the surface forces, and a very precise microwave link (K-band) to measure the inter-satellite distance with micrometer accuracy are the principal instruments onboard the satellites. Determination of the gravity field of the Earth including its temporal variations from the satellites’ orbits and the inter-satellite measurements is the main goal of the mission. The accelerometers are needed to separate the gravitational acceleration from the surface forces acting on the satellites. They collect a wealth of information about the atmospheric density at satellite height as a by-product. These accelerations have not yet been analyzed thoroughly, because their interpretation is complicated due to numerous thruster spikes. We outline a method to model the thruster spikes and to clean the time series of the accelerations. The isolated effect of the modeled thruster pulses on the satellite orbits is studied and a first interpretation of the cleaned accelerations is given. A correlation between K-band residuals and regions of high atmospheric fluctuations was not observed, which is probably due to time variable signals of hydrological origin that dominate the residuals.

17.1

Introduction

The Gravity Recovery And Climate Experiment (GRACE, Tapley et al. 2004) satellites orbit the Earth at a height, where the atmospheric drag, in addition to solar radiation pressure and albedo effects,

U. Meyer (*) A. J€aggi G. Beutler Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland e-mail: [email protected]

still plays a major role in orbit dynamics. When estimating the Earth’s gravity field, these surface forces are considered as disturbing forces that either have to be modeled or preferably directly observed with an accelerometer onboard the satellite. In order to separate the surface forces from gravity, the accelerometer has to be located at the center of mass of the satellite. The position of the accelerometer relative to the center of mass is checked regularily by dedicated shaking maneuvers. If necessary, the center of mass is adjusted with movable masses inside the satellite.

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Direct solar pressure mainly depends on the satellite’s cross-section exposed to the sun and therefore varies twice per revolution. The influence of albedo is quite small in comparison to the other forces. As both satellites are identical and follow each other at a distance of 220 km or a timelag of 28 s on average in the same orbit, the influence of solar radiation pressure on the satellites will be greatly reduced in the difference of the accelerations between the two satellites. The highly variable part of atmospheric drag is the main signal remaining in the acceleration differences. Simultaneous acceleration differences are a measure for the spatial variability of the atmosphere’s density, at a distance of 220 km. If the timelag of the trailing satellite is taken into account when forming the difference, one obtains a measure for the temporal variability of the atmosphere’s density at one and the same location within 28 s. Both kinds of acceleration differences are studied subsequently. The accelerations originally measured in the satellite-fixed frames have to be transformed into a common reference frame before forming the differences. The transformation into a co-rotating frame with its axes pointing radially outward (R), normally to the orbital plane of the satellites (W), and orthogonally to these axes completing a right-handed frame (S, approximately in the direction of flight of the satellites) is performed using the attitude observations of the star cameras and the reduced dynamic satellite

orbits (J€aggi 2007). The interpretation of the resulting acceleration differences is complicated by numerous spikes (see Fig. 17.1), which usually may be attributed to one of the two satellites and which are highly correlated with thruster firings for attitude control, as recorded in the THR1B-files (Case et al. 2002). The thruster fire should, in principle, not be recorded by the accelerometers at all, because for each event two thrusters fire simultaneously and the corresponding thrusters are positioned in such a way, that their combined thrusts should only cause angular accelerations on the satellite. The observed linear accelerations may have two explanations: – The thrusters of a pair do not fire perfectly symmetrically and cause real linear accelerations. – The accelerometer is not positioned exactly in the center of mass of the satellite and therefore measures some artefacts due to thruster firings. As the thrusters fire frequently (500–800 times per day), it is necessary to develop a model for the thruster pulses to clean the observed accelerations.

17.2

Modeling Thruster Pulses

Six thruster pairs for attitude control and one thruster pair for orbit maneuvers are mounted on each satellite (the latter is not studied here). Time and duration of the thruster pulses are recorded in the THR1B-files.

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detail in Wu et al. (2006). Flury et al. (2008) discuss artefacts in the ACC1A data and their removal by the low-pass filter. The accelerometers do not sense the entire pulse. Therefore, we estimate scaling factors between the simulated and the actually measured accelerations for each thruster pair and each axis (R, S and W) of the reference frame. From the huge number of daily pulses we select only those which do not heavily overlap (which are separated by half the fit interval length of the low pass filter, i.e., by 70 s) and which have a certain minimal thrust duration (here 50 ms). The simulated accelerations by each of these pulses are fitted to the acceleration differences between the two satellites (the leading satellite shifted by 28 s) by estimating a bias, drift, and scale factor. The estimation of bias and drift parameters is necessary, because the signal of the thruster fire interferes with accelerations caused by atmospheric density variations. Because some of the pulses cannot be fitted very well, a screening step is needed. For all pulses of one and the same thruster pair a mean fit (RMS) is computed and badly fitted pulses are removed by a one sigma criterion. Mean scale factors are determined from the remaining pulses. Table 17.1 shows the results using data of the year 2003 and Table 17.2 for data of the year 2007, respectively. The thruster pairs 2 and 4 fire less frequently than the other thrusters and the estimation of scaling factors therefore is difficult (not shown). For all other thruster

The nominal thrust is 10 mN and the duration per thrust varies between 20 and several hundred milliseconds. To model that part of the thruster pulses, that is mapped to the measured accelerations, it is assumed that a constant scaling factor exists for each thruster pair and each axis of the reference frame. The processing procedure from the raw accelerations, measured onboard the satellites, to the ACC1B accelerations has to be understood first. An analog Butterworth filter of order 3 with a cut-off frequency of 3 Hz is applied onboard the satellites to the raw data to prevent aliasing. A shift by 0.14 s, called Butterworth-delay, is an unwanted artefact of this filter. Afterwards the data is sampled at a rate of 10 Hz, resulting in ACC1A-data. In the next steps the time scale is converted from On Board Data Handling (OBDH) to GPS time, the Butterworth-delay is removed and the data resampled at 1 Hz using a Charge Routing Network (CRN) class digital low-pass filter with a fit interval of 140.7 s and a target low-pass bandwidth of 0.035 Hz (Thomas 1999). This filter dampens the sharp spikes due to thruster firings and massively widens their shape. The effect of this transformation on orbit integration is not important, because the energy of the pulses is preserved. Together with the masses of the satellites, available in the MAS1B-files, we are now able to model the acceleration caused by a sharp 10 mN pulse on the satellites as in the ACC1A- or ACC1B-data (see Fig. 17.2). The filtering procedure is described in

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Table 17.1 Scaling factors of four thruster pairs, their error and the number of pulses considered, estimated from ACC1Baccelerations of 2003 2003 R Thruster 1 Thruster 3 Thruster 5 Thruster 6 S Thruster 1 Thruster 3 Thruster 5 Thruster 6

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pairs results of similar quality are obtained for 2003 and 2007. The scaling factors therefore can indeed be considered as constant over time (the change in sign in the S-direction from 2003 to 2007 is caused by the satellites’ switch of position in December 2005). All the pulses from the THR1B-files can now be modeled using the estimated scaling factors in the R, S, W-tripod. The results are promising (see Fig. 17.3), except for the W component. This poor quality in W is probably explained by the reduced sensitivity of the accelerometers in cross-track direction (Frommknecht et al. 2003). The accelerations in W are not further considered here, because they have the least influence on the inter-satellite distance and therefore on the gravity field estimation (J€aggi et al. 2011).

17.3

Effect on Satellite Orbits

We are now in a position to study the isolated impact of the thrusters on the satellite orbit and to compare it with the effect of the cleaned surface forces and with the combined effect of the original ACC1Baccelerations. Therefore, we determine orbits in four different ways: (a) Without ACC1B-DATA (b) With the complete, original ACC1B-data (c) Only with the modeled pulses of the thruster firings (d) With the accelerations reduced by the thrusterinduced effects The orbits were determined using the Celestial Mechanics Approach (CMA, Beutler 2005). Piecewise constant accelerations, set up for every 15 min interval, were constrained to zero with a sigma of 3.109 m/s2 and subsequently estimated together with all other parameters. This approach is very flexible, the pseudo-stochastic accelerations are able to absorb the missing non-gravitational accelerations to a large extent. To limit the model error by the static gravity field, AIUB-GRACE02S (J€aggi et al. 2011) was used as background model. AIUB-GRACE02S has been estimated from GRACE GPS and K-Band range-rate data of the years 2006 and 2007. Figure 17.4 shows the resulting K-band range-rate residuals. To separate the effects of interest from errors in the background models, the differences of the residuals from experiments (b) to (d) with respect to those of experiment (a) are shown (the model errors cancel out). The orbit perturbations caused by thruster firings and those caused by the cleaned surface forces are of comparable size. The simulated effect of the thruster fire on the orbit, obtained by direct numerical integration of the modeled accelerations and subsequent high-pass filtering (by subtraction of a moving mean over 3 min) is included in Fig. 17.4. The correlation with the K-band residuals is clearly visible. The differences are explained by the pseudostochastic accelerations. Therefore, we conclude that the thrusters significantly perturb the satellite orbits. Figure 17.5 shows the original range-rate residuals of experiments (b) and (d) and their difference due to the missing thruster accelerations in (d). The systematic effects are at least by a factor of three larger than the thrusters-induced effects. The thrusters (or the thruster

The Impact of Attitude Control on GRACE Accelerometry and Orbits

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mismodeling) are thus not responsible for these systematic effects. In a last orbit experiment we study the interaction between the linear thruster accelerations and the pseudo-stochastic pulses of the CMA using a Keplerian orbit. The Kepler ellipse, disturbed by the modeled thruster accelerations, is then introduced as “observed” orbit (3D-positions at 30 s intervals). These positions are used in an orbit determination, where pseudo-stochastic pulses are set up at 15-min

intervals. The resulting velocity residuals are compared with the range-rate residuals of the real orbit determination process in Fig. 17.6. The simulated and the real residuals are of the same order of magnitude. The jumps in the residuals every 15 min are due to the stochastic pulses. The Kepler orbit is a useful tool to assess the thruster-induced orbit perturbations in a qualitative way. We have not yet answered the question, whether the observed linear accelerations due to the thruster

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actuations are artefacts or real. GRACE orbits are determined according to the experiments (a) to (d) for 60 days of September and October 2007 to answer this question. Figure 17.7 shows daily RMSvalues of the range-rate residuals for all four cases. The results from Fig. 17.4 are confirmed, i.e., the thrusters do significantly perturb the orbits. It becomes clear by now, that their inclusion in the accelerometer files leads to a considerable improvement of the orbit fit. The linear thruster accelerations are thus mainly caused by non-symmetric firings of the thrusters.

17.4

Interpretation of Accelerations and Correlation with K-Band Residuals

Let us now use the cleaned accelerations to study the atmosphere. We limit our analysis to the acceleration differences in the direction of flight S of the satellites, which characterize the spatial or temporal variability of the atmosphere’s density. The state of the atmosphere mainly depends on the insolation, which is why sun-related coordinates are selected to represent the accelerations. The first coordinate is b, the angle

The Impact of Attitude Control on GRACE Accelerometry and Orbits

Fig. 17.7 Daily RMS of the range-rate residuals without using nongravitational accelerations (black), using the complete ACC1B-data (gray), considering only the modeled effect of the thrusters (black, dashed) or the cleaned effect of the surface forces (gray, dashed)

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between the W-axis of the co-rotating tripod and the direction to the sun. The second coordinate is u, the argument of latitude of the satellites on their orbit around the Earth. The Earth orbits the sun within a period of about 365 days and the orbital plane of the satellites rotates due to the precession of the nodes by about 40 per year. Hence the sun passes over the entire u, b-plane within 160 days and b varies by about 180 . The GRACE satellites orbit the Earth about 15 times per day and u therefore completes 15 full 360 cycles per day. Note that the selected coordinates are not strictly sun-fixed, the geocentric latitude of the sun varies between 23 during the year. Figure 17.8 shows acceleration differences for 160 days in the summer of 2007, referring to the same place, but separated 28 s in time. The center of the figure corresponds to the regions with the highest insolation. The heating of the atmosphere causes density fluctuations that result in the prominent signal visible in Fig. 17.8. Moreover, one sees the shadow entry and exit of the satellites, caused by a “discontinuity” in solar radiation pressure. White vertical lines mark arcs, where no orbits are available. Two horizontal lines currently cannot be explained. Figure 17.9, containing the simultaneous accelerations differences (thus corresponding to locations separated by 220 km in the north–south direction) shows much more variations than Fig. 17.8. The shadow entry

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Conclusions

We developed a method to model the linear accelerations on the satellites caused by the frequent thruster pulses. The impact of these pulses on the satellite orbits was studied and it was shown that the observed spikes represent real accelerations, which have to be taken into account for orbit determination. The method can be used to separate the surface forces from the thruster-induced accelerations for the radial and along-track direction. The analysis and interpretation of the cleaned accelerations, which mainly represent the atmospheric conditions at satellite height, is still at an early stage. To study the impact of atmospheric density fluctuations on orbit determination and on subsequent gravity field estimation, we plan to estimate and remove the strong seasonal variations of hydrological origin in the near future to be able to look for correlations between surface forces and K-band residuals. Acknowledgements The authors gratefully acknowledge the generous financial support provided by the Institute for Advanced Study (IAS) of the Technische Universit€at M€unchen.

β Fig. 17.9 Acceleration differences between the two GRACE satellites at the same time (separated 220 km in space), thruster fire removed

and exit are visible extremely well. Inclined, linear structures in the lower part of the figure are not yet explained. Strong atmospherical density variations can be observed close to the Earth’s north pole (around u ¼ +90 ), caused by interactions of the atmosphere with the magnetosphere. The corresponding variations close to the south pole (around u ¼ 90 ) are much less pronounced, because the southern atmosphere gets less insolation during the time period shown. The smoothness of the acceleration differences in Fig. 17.8 indicates that the signals visible in Fig. 17.9 really are atmospheric density fluctuations and not just noise. Currently, we cannot answer the question whether the atmosphere density variations shown in Fig. 17.9 have an impact on the orbit quality. The K-band rangerate residuals show a strong signal related to hydrology, which has to be removed before addressing the issue of the density variations.

References Beutler G (2005) Methods of celestial mechanics. Springer, Berlin. doi:10.1007/b138225 Case K, Kruizinga G, Wu S (2002) GRACE level 1B data product user handbook. D-22027. JPL, Pasadena, CA Flury J, Bettadpur S, Tapley B (2008) Precise accelerometry onboard the GRACE gravity field satellite mission. Adv Space Res 42:1414–1423. doi:10.1016/j.asr.2008.05.004 Frommknecht B, Oberndorfer H, Flechtner F, Schmidt R (2003) Integrated sensor analysis for GRACE – development and validation. Adv Geosci 2003(1):57–63 J€aggi A (2007) Pseudo-stochastic orbit modeling of low earth satellites using the global positioning system. Geod€atischgeophysikalische Arbeiten in der Schweiz, vol 73. doi:9783-908440-17-8 J€aggi A, Beutler G, Meyer U, Prange L, Dach R, Mervart L (2011) AIUB-GRACE02S – status of GRACE gravity field recovery with the celestial mechanics approach. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 30:5683. doi:10.1126/science.1099192 Thomas JB (1999) An analysis of gravity-field estimation based on Intersatellite Dual-1-way biased ranging. JPL, Pasadena, CA, pp 98–115 Wu SC, Kruizinga G, Bertiger W (2006) Algorithm theoretical basis document for GRACE level-1B data processing V1.2. D-27672. JPL, Pasadena, CA

Using Atmospheric Uncertainties for GRACE De-aliasing: First Results

18

L. Zenner, T. Gruber, G. Beutler, A. J€aggi, F. Flechtner, T. Schmidt, J. Wickert, E. Fagiolini, G. Schwarz, and T. Trautmann

Abstract

In standard gravity field processing, short-term mass variations in the atmosphere and the ocean are eliminated in the so-called de-aliasing step. Up to now the background models used for de-aliasing have been assumed to be error-free. As the accuracy assessed prior to launch could not yet be achieved in the analysis of real GRACE data, the de-aliasing process and related geophysical model uncertainties have to be considered as potential error sources in GRACE gravity field determination. The goal of this study is to identify the impact of atmospheric uncertainties on the de-aliasing products and on the resulting GRACE gravity field models. The paper summarizes the standard GRACE de-aliasing process and studies the effect of uncertainties in the atmospheric (temperature, surface pressure, specific humidity, geopotential) input parameters on the gravity field potential coefficients. Finally, the impact of alternative de-aliasing products (with and without atmospheric model errors) on a GRACE gravity field solution is investigated on the level of K-band range-rate residuals. The results indicate that atmospheric model uncertainties are small in terms of the associated spherical harmonic coefficients. The effect in terms of K-band observation residuals is negligible compared to other modeling errors.

18.1 L. Zenner (*) T. Gruber Institut f€ur Astronomische und Physikalische Geod€asie, Technische Universit€at M€ unchen, 80290, M€ unchen, Germany e-mail: [email protected] G. Beutler A. J€aggi Astronomisches Institut, Universit€at Bern, 3012, Bern, Switzerland F. Flechtner T. Schmidt J. Wickert GeoForschungsZentrum, 14473, Potsdam, Germany E. Fagiolini G. Schwarz T. Trautmann Deutsches Zentrum f€ ur Luft- und Raumfahrt, 82234, Weßling, Germany

Introduction

Mass redistributions inside, on, and above the Earth’s surface are responsible for time variable gravity field forces, which directly act on a satellite orbit. If these variations cannot be directly measured by repeated observations within short periods, they have to be removed during the gravity field determination in order to avoid aliasing due to undersampling. Shortterm mass variations in the atmosphere and in the oceans are part of the variations which cannot be measured with GRACE due to the inadequate spacetime sampling of the GRACE mission. To deal with this problem, the high frequency atmospheric and

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oceanic signals are modeled and “removed” prior to or during the GRACE data processing. The “removal” of these high-frequency mass variations is called dealiasing (Flechtner 2007). As the GRACE results are still not meeting the expected pre-launch baseline accuracy, the de-aliasing process and related geophysical model uncertainties are considered here as a possible limiting factor for the GRACE gravity field determination. Therefore we explore ways to improve the de-aliasing process. In a first step, the de-aliasing process and its fundamental formulas, with the main focus on the processing sequence of the atmosphere, is reviewed (Sect. 18.2). As the standard processing scheme assumes error-free atmospheric and oceanic parameters and as it is well known that in areas with sparse observations the atmospheric models are degraded in quality (Salstein et al. 2008), there is reason to assume that one could improve the de-aliasing product and consequently the gravity field solution by taking into account uncertainties of the atmospheric parameters. This is why a mathematical model of error propagation of atmospheric model uncertainties into the gravityfield de-aliasing coefficients was developed. In order to study the impact of atmospheric model uncertainties on intermediate and final gravity field results, a test environment based on a real GRACE data set, data from the European Center for Medium Range Weather Forecast (ECMWF) atmospheric analysis (ECMWF 2009), and from the Ocean Model for Circulation and Tides (OMCT) (Dobslaw and Thomas 2007) was set up. The error scenarios for the atmosphere are defined in Sect. 18.2.2. The ocean is assumed as error-free in this study. For the time being, only atmospheric uncertainties provided by ECMWF were used to get a first insight into the effect of model errors on the de-aliasing coefficients (AOD) and GRACE. In Sect. 18.3.1 the impact of the error assumptions on the estimated de-aliasing coefficients and the geoid is studied. Differences between the error-scenarios are computed and compared to the GRACE error predictions. Finally, the newly computed de-aliasing coefficients are used for a GRACE gravity field determination. In order to identify their impact on the resulting gravity field, observation residuals are investigated in detail in Sect. 18.3.2. Sect. 18.4 summarizes the results and draws conclusions.

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18.2

Theory: The De-aliasing Process

18.2.1 Fundamental Formulas A short overview of the fundamental formulas of the current atmospheric and oceanic de-aliasing processing sequence is provided below. The following formulas outline the computation of the harmonic coefficients Cnm , Snm due to atmospheric and oceanic variations. Pkþ1=2 ¼ akþ1=2 þ bkþ1=2 Ps

(18.1)

Tv ¼ ð1 þ 0:608SÞT

(18.2)

Hkþ1=2 ¼ Hs þ

kmax X Pjþ1=2 RTv ln g Pj1=2 j¼kþ1

Z Z Cnm a2 ð1 þ kn Þ ¼ ½In þ PO ð2n þ 1ÞMg Snm Pnm ðcos yÞ Z0 In ¼ Ps

y

l

cos ml sin ml

a N0 þ a a Hkþ1=2

(18.3)

(18.4)

sin ydydl nþ4 dP

(18.5)

Pk+1/2 ¼ pressure at half-levels ak+1/2, bk+1/2 ¼ model dependent coefficients Tv ¼ virtual temperature R ¼ gas constant for dry air Hk+1/2 ¼ geopotential height at half-levels g ¼ mean gravity acceleration a ¼ semi major axis of reference ellipsoid M ¼ earth mass kn ¼ loading love numbers Po ¼ ocean bottom pressure In ¼ vertically integrated atm. pressure Pnm ¼ assoc. norm. legendre polynomials N’ ¼ mean geoid height above sphere r ¼ a Four input parameters are needed for the determination of the atmospheric potential coefficients: Surface pressure Ps, surface geopotential height Hs, temperature T and specific humidity S at the 91 atmospheric model levels (full-level). After a few steps, the vertical integration of the atmospheric pressure In is performed in (18.5). The atmospheric pressure In is

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Using Atmospheric Uncertainties for GRACE De-aliasing: First Results

then combined with the ocean bottom pressure Po in order to obtain the combined atmospheric and oceanic potential. The combined atmospheric and oceanic pressure is subsequently called PAo (18.6). In a final step (18.4), the numerical integration is performed and the atmosphere and ocean de-aliasing product (AOD) is obtained. A few comments should be made concerning the outlined procedure: Usually, a mean field is subtracted before-hand from the vertically integrated atmospheric pressure In and from the ocean bottom pressure Po, i.e., PAo contains the residual atmospheric and oceanic pressure. The mean fields are subtracted in order to analyze gravitational variations. Since the mean mass distribution of the atmosphere and ocean by definition refers to the static part of the gravity field, only the deviations from the mean value have to be taken into account. In the currently realized GRACE gravity field processing as well as in our investigations, mean fields obtained from the years 2001/2002 are used. For details we refer to Flechtner (2007).

18.2.2 Changing the Current De-aliasing Process: Two Error Scenarios Temperature, specific humidity, surface pressure, and surface geopotential height have so far been assumed to be error-free, although it is known that there are large uncertainties in the atmospheric input parameters, particularly in the surface pressure (Ponte and Dorandeau 2003). The goal of this study was to find out to what extent the atmospheric model uncertainties affect AOD and GRACE results. To answer this question, the atmospheric field errors are propagated into the vertically integrated atmospheric pressure In in a first step. As mentioned previously, for the time-being, the ocean is regarded as error-free. In a second step, in order to take the error of In into account and to further propagate it into the AOD coefficients Cnm , Snm , the present approach of numerical integration (18.4) was replaced by a least-squares adjustment (Gauß-Markov, model). Equation (18.4) was used as an observation equation: Mg X 2n þ 1 X Pnm ðcos yÞ a2 n 1 þ k n m ½Cnm cos ml þ Snm sin ml;

PAO ¼

(18.6)

149

where the combined residual atmospheric and oceanic pressure fields PAO are used as observations and the error of PAO as individual (inverse) weights. The error of PAO equates to the error of as the ocean In is assumed as error-free. In this way we obtain the AOD coefficients as a function of the errors of the input parameters, i.e., as a function of the weighting. In order to see whether the atmospheric model uncertainties significantly affect AOD and GRACE results, two so-called error scenarios were studied: “error-free AOD”: The uncertainties of the four atmospheric parameters are not taken into account. “full-error AOD”: The uncertainties of the four atmospheric parameters are taken into account. The “error-free” experiment thus assumes that all observations PAO have the same weight. The “fullerror” scenario assumes that the observations PAO are weighted individually using the inverse error of PAO on the diagonal of the weighting matrix. After performing the least-squares adjustment for these two scenarios, one obtains two sets of AOD products. By comparing them (Sect. 18.3), one gets insight into the effect of atmospheric uncertainties on the atmospheric and oceanic potential coefficients Cnm , Snm . The following results were achieved using data provided by the ECMWF (Operational archive, Atmospheric model, Analysis and Errors in analysis; ECMWF 2009).

18.3

Results: Impact of Atmospheric Uncertainties on AOD and GRACE

18.3.1 Effect of Model Uncertainties on AOD Figure 18.1 shows the difference between the AOD product for the “full-error” and the “error-free” experiment in terms of geoid heights. The atmospheric model uncertainties affect the geoid in the range of 0.8 to 0.3 mm. The largest differences are visible in regions where the weights for the observations are extreme (high or low). This result seems to be clear: The deviation between the two error-scenarios is only caused by the different weighting of the observations PAO. In the “error-free” case the combined atmospheric and oceanic pressure has the same weights,

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Fig. 18.1 Geoid difference between the “error-free” and “fullerror” scenario. 2007-08-01 00h. Unit (mm)

whereas in the “full-error” scenario the observations are weighted using the inverse error of PAO (which is equal to the vertically integrated atmosphere error since the ocean is assumed as error-free). Therefore, the error of the atmospheric pressure is reflected in the difference between the two error-scenarios. The effect of atmospheric uncertainties on AOD is in the sub-mm domain, which seems to be small. Figure 18.2 shows the difference between the two error-scenarios in terms of spherical harmonic degree variances. The difference between the two error-scenarios is presented in the dotted line marked with circles. Compared to the GRACE baseline (Kim 2000; lowest line in Fig. 18.2) the effect of atmospheric model uncertainties clearly is in the sensitivity range of GRACE. Therefore we conclude that an effect should be visible in the gravity field solution as well. This aspect will be studied in Sect. 18.3.2.

18.3.2 Effect of Model Uncertainties on K-Band-Residuals The K-band range-rate (KBRR) residuals are obtained from an extended orbit-determination problem, where one tries to minimize the differences between the computed K-band range-rate observations based on a priori models and the real observations made by GRACE by only adjusting orbit parameters. In theory, the resulting KBRRresiduals would become zero, if all force models (tides, static gravity field, AOD-model,. . .) as well

as the real observations were error-free. As it is known, this is not the case. Thus differences between the modeled and the real world occur in the KBRRresiduals. The residuals contain all model errors, measurement errors, and unmodeled forces. By applying the new de-aliasing coefficients, which take the atmospheric model uncertainties into account, we hope to reduce these residuals. This would imply that the new (“full-error”) AOD product is more realistic than the standard or “error-free” AOD. For this purpose the Bernese GPS Software (Dach et al. 2007) and the Celestial Mechanics Approach (J€aggi et al. 2011) is used to perform an orbit determination based on kinematic positions and K-band range-rate observations by using the gravity field model AIUB-GRACE02S (J€aggi et al. 2011) as known and using the different sets of AOD coefficients. By leaving all other settings unchanged, one gains insight into the effect of atmospheric model errors. For detailed parametrization information, we refer to J€aggi et al. (2011). In addition to the two error-scenarios described in Sect. 18.2.2, the overall effect of AOD by not applying AOD products for orbit determination was included as well. The impact of the three experiments on the KBRR residuals is shown in Fig. 18.3. Applying (dark grey “area” in Fig. 18.3) or not applying AOD (light grey “area”) clearly affects the KBRR-residuals. Removing the high frequent atmospheric and oceanic mass variations significantly reduces the RMS over 1 year of KBRR-residuals by about 10%. This implies that the inclusion of the short-term atmospheric and oceanic variations for gravity field solutions leads to more realistic results. Figure 18.3 shows that the amplitudes of the prominent peaks are reduced, but not removed, when applying AOD. One reason for that might be due to the fact that AIUB-GRACE02S is a static field where the time-variable gravity field signals are not yet modeled. Thus, the KBRRresiduals contain apart from other model errors mainly the unmodeled time-variable gravity forces (in particular hydrology). However, in general the signal in the residual series is reduced. Is the newly computed full-error AOD product able to further improve de-aliasing, i.e., to further reduce KBRR-residuals? Figure 18.3 (bold black line) shows that this is not the case. This is contrary to our

18

Using Atmospheric Uncertainties for GRACE De-aliasing: First Results

Fig. 18.2 Spherical harmonic degree variance differences between the “error-free” and “full-error” scenario in terms of geoid heights. 2007-08-01 00h. Unit (m)

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expectations at the end of Sect. 18.3.1, where we expressed hope that an effect on GRACE gravity field determination should be visible.

18.4

Conclusion and Discussion

Figure 18.4 is identical to Fig. 18.2, except that it also contains the achieved (as opposed to the expected prelaunch) accuracy of gravity field determination with GRACE (curve with asterisk). As the differences

“error-free” minus “full-error” are clearly below the curve “achieved GRACE baseline”, it is clear that the impact of the refined AOD model studied here is not visible in gravity field determination. It must be pointed out, however, that our results are based on atmospheric error fields taken from the operational analysis of the ECMWF. The results shown are therefore only true for the ECMWF error-fields. The effect of alternative uncertainty estimates from other sources [e.g., NCEP (National Centers for Environmental Prediction)] has not been investigated yet.

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Fig. 18.4 Degree variance differences of error-free AOD and full-error AOD in terms of geoid heights (m) for 2007-08-01 00h, and the predicted and current GRACE error estimates

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Also, one should take into account that the used errorfields might be too optimistic as they result from the assimilation model (ECMWF 2009). More work has to be carried out concerning the determination of representative error parameters (e.g., Schmidt et al. 2008). Previous investigations have shown that the surface pressure error has the most significant effect on the vertically integrated atmospheric pressure. It is therefore essential to determine reliable and reasonable surface pressure values and use them as input for the upcoming investigations. Furthermore, it must be emphasized that so far the ocean has been assumed to be error-free. To determine representative error-fields of ocean bottom pressure and to take them into account during AOD processing will be one of the next steps.The goal of this study was to improve the de-aliasing process by taking atmospheric model uncertainties into account. It was shown that atmospheric model uncertainties are not able to improve the de-aliasing process or GRACE orbit determination, because the actual error of GRACE is above the predicted baseline accuracy. It should be kept in mind, however, that model uncertainties will become more important if the achieved accuracy of GRACE can be further improved towards the assumed baseline. Acknowledgements This study was conducted as part of the IDEAL-GRACE project with the support of the German Research Foundation (Deutsche Forschungsgemeinschaft) and within the SPP1257 priority program “Mass transport and Mass Distribution in the System Earth”. The International Graduate School for Science and Engineering of the Technische Universit€at M€unchen also supported this work.

References Dach R, Hugentobler U, Fridez P, Meindl M (eds) (2007) Bernese GPS software version 5.0. Astronomical Institute, University of Bern, Bern Dobslaw H, Thomas M (2007) Simulation and observation of global ocean mass anomalies. J Geophys Res 112:C05040. doi:10.1029/2006JC004035 ECMWF (2009) MARS user guide. Technical Notes, p 5. http:// www.ecmwf.int/publications/manuals/mars/guide/MarsUser Guide.pdf. Accessed date 17 August 2011 Flechtner F (2007) AOD1B product description document. GRACE project documentation, JPL 327–750, Rev. 1.0. JPL, Pasadena, CA. 17 August 2011 J€aggi A, Beutler G, Meyer U, Prange L, Dach R, Mervart L (2011) AIUB-GRACE02S: status of GRACE gravity field recovery using the celestial mechanics approach. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg Kim J (2000) Simulation study of a low-low satellite-to-satellite tracking mission. Technical report, University of Texas at Austin, Austin, TX Ponte RM, Dorandeau J (2003) Uncertainties in ECMWF surface pressure fields over the ocean in relation to sea level analysis and modeling. J Atm Ocean Technol 20(2):301. doi:10.1175/1520-0426(2003)020 Salstein DA, Ponte RM, Cady-Pereira K (2008) Uncertainties in atmospheric surface pressure fields from global analyses. J Geophys Res 113:D14107. doi:10.1029/ 2007JD009531 Schmidt T, Wickert J, Heise J, Flechtner F, Fagiolini E, Schwarz G, Zenner L, Gruber T (2008) Comparison of ECMWF analyses with GPS radio occultations from CHAMP. Ann Geophys 26:3225–3234. http://www.ann-geophys.net/26/ 3225/2008/. Accessed date 17 August 2011

Challenges in Deriving Trends from GRACE

19

A. Eicker, T. Mayer-Guerr, and E. Kurtenbach

Abstract

The following contribution addresses some of the problems involved with the determination of long-term gravity field variations from GRACE satellite observations. First of all the choice of the time span plays a very important role, especially since it generally is a hard task to derive secular trends from only a few years of satellite data. Another issue, when one is interested in a single trend phenomenon, is the reduction of all other geophysical effects causing long-term gravity field variations. This paper uses the example of trends in continental hydrological water masses for the case of the High Plains aquifer to demonstrate some of the challenges implicated by trend analysis from GRACE.

19.1

Introduction

During the last years, the satellite mission GRACE (Tapley et al. (2004)) has provided significant improvement in the knowledge of the Earth’s gravity field, in the static as well as in the time variable component. The time variabilities derived from GRACE deliver information about short periodic variations such as the seasonal hydrological cycle (e.g. Schmidt et al. 2008), occasional incidents such as large earthquakes (Han et al. 2006), and also long-term trends. The investigations to be presented in this contribution are dedicated to the analysis of these long-term processes from GRACE observations. Examples are the melting of the ice sheets and glaciers in Greenland and Antarctica (Horwath and Dietrich 2009), rebound

A. Eicker (*) T. Mayer-Guerr E. Kurtenbach Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115 Bonn, Germany e-mail: [email protected]

effects due to glacial isostatic adjustment (GIA), see e.g. Steffen et al. (2008), or trends in the continental hydrological water masses due to water withdrawal or climatic changes, as outlined in Rodell et al. (2009). When trying to derive long-term trends from GRACE gravity field observations, one is faced with a number of challenges. First of all, the time span used in the analysis is very important because it is very difficult to derive long-term gravity field variations from only 7 years of GRACE data or even less. The other important aspect that has to be taken into account is that when the focus is on one single phenomenon, all other effects causing long-term gravity field variations have to be removed. This is generally performed by the application of geophysical models, with the drawback that inaccuracies or uncertainties of these models are introduced into the trend analysis. Some confusion might be caused by the fact that these models are not only used for signal separation but also, with a different intention, in the GRACE data analysis (linearization, de-aliasing). The latter point is discussed in

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_19, # Springer-Verlag Berlin Heidelberg 2012

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Sect. 19.2 while Sect. 19.4 is dedicated to the problem of signal separation. The High Plains aquifer (marked in white in Fig. 19.2), located in the central United States with an area of 450,000 km2, serves as an example in this article to demonstrate some of the issues that have to be addressed when deriving long-term gravity field variations from GRACE. This investigation area was chosen because of an assumed mass loss trend due to ground water depletion caused by large-scale irrigation, see Strassberg et al. (2009). It is subject to discussion, whether GRACE is able to detect such human ground water withdrawals, see Rodell (2002) and Strassberg et al. (2009). However, this paper does not intend to give quantifications of potential mass loss or to draw qualitative conclusions whether such investigations are indeed possible with GRACE. The main intention is to point out some of the issues that have be considered when such investigations shall be performed. They are not specific for the example area but are valid for every investigation dealing with the interpretation of GRACE data.

19.2

Gravity Field Processing

In order to interpret the GRACE solutions and to derive, for example, trend information out of the GRACE products it is necessary to understand some details about the processing procedures. During the processing of the original GRACE data (level 1B), the observations y are expressed as non-linear functionals y ¼ f (x) of the unknown gravity field parameters x. In the examples described below, these parameters are given as a set of spherical harmonics coefficients for each monthly solution. After a linearization process the coefficients can be estimated by means of a least-squares adjustment. The linearization is given by: y ¼ f ðx0 Þ þ

@f ðx x0 Þ: @x x0

(19.1)

Here x0 describes the approximate solution, functionals of which are reduced from the original observations. This means that the least squares adjustment delivers a solution for Dx ¼ x x0. In order to keep the linearization error small, the approximate

solution should be as close as possible to the real gravity field signal in the corresponding month as possible. Therefore, the following models are taken into account by the processing centers: a static gravity field solution, a trend for the lower spherical harmonics, tides (direct tides, Earth and ocean tides), periodic changes of the centrifugal potential due to polar motion (pole tide and ocean pole tide), and mass variations of the atmosphere and the ocean. Apart from the linearization, these models serve also for the reduction of short-term mass variations that cannot be modelled by monthly representations (dealiasing). To obtain the complete gravity field signal for a monthly solution, the components removed by these models should be restored according to: x ¼ x0 þ Dx:

(19.2)

But in case of the official GRACE products, this is not carried out in a consistent manner. Only the static solution is re-added, but the trend and all other models are not present in the solutions. The atmosphere and ocean model, however, is delivered as an additional product in terms of monthly mean values. The consideration of these background models differs slightly from one analysis center to the next, for example by a different treatment of the permanent tides and different trend models for the lower harmonic coefficients. Details can be found in Flechtner (2007), Bettadpur (2007), and Watkins and Yuan (2007). Another aspect that has to be taken into account when dealing with GRACE solutions is the fact that the coefficients of degree n ¼ 1 cannot be determined from GRACE observations, and are therefore set to zero. This corresponds to the realization of a coordinate system in the center of mass (CM) of the complete Earth system (solid Earth including ocean and atmosphere and all other subsystems). The CM varies in comparison to the center of the solid Earth (CE) or the center of figure (CF). A good discussion of this topic can be found in Blewitt (2003). While there exist models to account for the seasonal variations of these changes, the longterm variations are still under investigation (Klemann and Martinec 2009). It is commonly known that the accuracy of the GRACE monthly solutions decreases with increasing spherical harmonic degree, i.e. with increasing resolution. Therefore, the solutions have to be filtered in order to allow a meaningful interpretation of the

19

Challenges in Deriving Trends from GRACE

results. The choice of the filtering technique has a significant influence of the solution. As there have been a number of investigations on this topic in the past, see for example Steffen et al. (2009) and Werth et al. (2009), we will not go into further detail at this point.

19.3

Time Span

Some of the geophysical phenomena causing longterm gravity field variations have been going on for time spans of decades to millenia and even longer. In contrast to this, the measurement period of GRACE with only a few years of observations is very short. Taking into account the strong inter-annual gravity field variations, it is very difficult to derive representative trends from such a short time series. It has been shown, that especially when only parts of the GRACE period are evaluated, the estimated trend changes significantly, an example given by Horwath and Dietrich (2009) for the case of ice mass loss in Western Antarctica. In the study at hand, we perform a similar analysis for the temporal evolution in the High Plains aquifer, using the GFZ-RL04 monthly gravity field series (Flechtner et al. 2009) for the time span January 2004 to December 2008. The monthly gravity field solutions provided by the CSR (Bettadpur 2007), the JPL (Watkins and Yuan 2007) and the ITG-Grace03s (Mayer-G€urr et al. 2007) time series were evaluated as well, all providing similar results to those presented below. But for reasons of clarity and because the ITGGrace solutions are so far only available until April 2007, only the results for the GFZ solutions will be displayed. The monthly gravity field models are smoothed by a 500 km Gaussian filter and integrated over the area of the High Plains aquifer. Figure 19.1 displays the mass variations of the GFZ-RL04 monthly gravity field solutions in terms of geoid heights. If only the time span until the end of December 2006 is considered, the linear trend (black line on the left) shows a strongly negative behavior suggesting an interpretation as mass loss in the area. This is the time span under investigation in Strassberg et al. (2009), but the examination of the complete time series up to today shows that statements about such trends are critical from only a few years. When introducing the next 2 years of GRACE data,

155

the trend for 2007–2008 (black line on the right) shows a different behavior by being slightly positive. The overall linear trend for the whole time span (indicated by the gray line) is therefore slightly positive as well. Of course the results raise the question, whether a linear function is the correct representation of the long-term variability in this specific area. Alternatively, one could think about the use of basis functions changing slowly in time or applying low pass filters to remove short-term variations.

19.4

Separation of Trend Signals

GRACE can only observe the integrated mass signal without being able to distinguish between different mass sources. In order to separate different mass signals external information is required, which is commonly obtained from geophysical models. More precisely this implies that if one is interested in only one specific geophysical phenomenon (e.g. continental water variations), the GRACE solutions must be reduced by models of all other mass variations. At this point it has to be emphasized that this task of signal separation is in principle independent of the linearization and dealiasing process described above. This is especially important when considering the different error sources. During the linearization process, the introduction of approximate values does not lead to any additional model errors, apart from linearization errors which are negligible in case of GRACE. Therefore, after readding all models that are used in the processing, the result is the complete mass signal which is only influenced by the observation errors of GRACE. This solution should be independent of the inaccuracies in the models as they are only used as approximation values. In contrast to this, the model errors play a very important role for the task of signal separation, as the quality of the separation can only be as good as the accuracy of the models. In case of GRACE it is difficult to distinguish between signal separation and linearization as the restore step is not carried out in a consistent manner, as described in Sect. 19.2. In the context of our example of mass changes in the High Plains aquifer, the main focus is on the determination of changes in the continental water budget. Therefore, all other effects have to be removed from the GRACE signal. There is a wide variety of different mass variations that have to be considered.

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Fig. 19.1 Monthly solutions as provided by the GFZ-RL04 time series averaged over the High Plains aquifer region. One overall trend has been calculated for the entire time span (gray

line) and individual trends for the years 2004–2006 and 2007–2008 (black lines)

The first ones are the variabilities of the atmospheric masses and the effects on the ocean caused by these variabilities. These short- and long-periodic changes are included in the so-called atmosphere and ocean dealiasing product (AOD1B; see Flechtner 2005) which is reduced during the analysis process especially to prevent the solutions from aliasing errors caused by short periodic signals and is not restored afterwards. This means that the separation of the atmosphere masses has already been performed during the data processing step. Nevertheless, if a more suitable atmospheric model (e.g. a local model) is available, it might be a good choice to re-add the global model and remove a more accurate local model which is better tailored to the specific area. Another source for long-term variations might be the long-periodic part of the tidal forces (e.g. the 18.6year cycle), of the Earth tides and the ocean tides which appear as trends when only a few years of data are analyzed. However, those tidal effects are also taken into account by models during the GRACE gravity field analysis process from all of the different analysis centers and can therefore be regarded as nonexistent in the GRACE monthly solutions. These assumptions are only valid, of course, to the extent

of the accuracy of the given models, as errors in the modeling of these long-term variations might be wrongly interpreted as originating from a different source. It has to be emphasized that the AOD1B and tide models are used twice, each time with a different objective: for linearization but also for signal separation. A different issue is the handling of the trend which is removed during the GRACE processing. Most analysis centers use rates given for the lower spherical harmonic coefficients (dot coefficients) as proposed by the IERS conventions (McCarthy and Petit 2004). While these rates might be sufficient as approximation values, they are not suitable for signal separation. They were derived during the development of the EGM96 gravity field model by a combination of models and observations. The intention of these rates was to account for the integrated mass trend. It is not possible to distinguish which subsystems contribute to the dot coefficients. There is certainly some part of the glacial isostatic adjustment effect present, but a complete GIA model cannot be described by only a few coefficients. Furthermore, we do not know for certain whether there are also some hydrological effects comprised in those coefficients. Therefore, we

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Challenges in Deriving Trends from GRACE

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Fig. 19.2 Spatial distribution of different phenomena that cause long-term trends in GRACE data, displayed in terms of geoid heights per year in mm. Top: dot coefficients, middle: glacial isostatic adjustment, bottom: secular variations in pole tide model. The region of the High Plains aquifer in the Central United States is illustrated in white

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the rates given by the IERS are displayed in the top part of Fig. 19.2 as changes in geoid heights in mm per year. A considerably large negative rate is visible for our example of the area containing the High Plains aquifer.

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After re-adding the mass trend proposed by the IERS dot coefficients, the next step is to remove the known long-term mass change phenomena. As a first aspect in this context the glacial isostatic adjustment has to be mentioned, causing a mass redistribution due to the uplift of the land mass as a reaction to the deglaciation after the last ice age. This phenomenon can be observed especially in the Scandinavian and Northern Canadian area, but the effects can be witnessed further south as well. The spatial distribution of the GIA is displayed in the middle part of Fig. 19.2 as a rate in terms of millimeter of geoid height per year, the corresponding model was provided by Klemann and Martinec (2009). In the area of the High Plains aquifer positive as well as negative rates can be found in the GIA model, with the positive trend prevailing. The second aspect that has to be considered is the effect of the change in the centrifugal potential caused by polar motion (pole tide). While all of the analysis centers reduce a pole tide model during their gravity field estimation process, this model only contains the short-term variations of the pole tides, but not the secular mass changes. The spatial distribution of these secular variations, calculated according to the IERS conventions, can be found in the bottom part of Fig. 19.2. This effect is rather small globally, but in the investigation area in the central United States a comparably large positive trend can be observed nevertheless. The influence that the effects described above have on the estimation of the trend in the investigation area can be observed in Fig. 19.3. The original GFZ-RL04 time series is plotted in gray, where only the effects given by the de-aliasing product and the different tidal models have been removed. The remaining effects have been integrated over the High Plains region and the corresponding rates are shown as well. The effect caused by GIA is indicated by the orange line, here the positive trend can be noticed. The pole tide trend is given in green, even though this effect is quite small globally, in the region at interest a significant positive trend is present. The IERS rates are plotted in the black line and as already concluded from the spatial distribution, a large negative trend is visible when integrating over the High Plains region. Re-adding these rates and subtracting the GIA and pole tide model results in the reduced time series illustrated in red with the corresponding linear trend. In contrast to the original time series, the overall trend

A. Eicker et al.

is now negative. After having taken into account the variations of atmosphere and ocean, of tides and solid Earth mass transports, the remaining variations can be interpreted as changes in the hydrological budget. Here it has to be kept in mind that the changes in the gravitational potential (expressed in terms of geoid heights) does not only contain the changes in the water masses but also the reaction of the Earth’s crust due to loading. These effects can be separated using a loading theory. Specifically this implies the multiplication of the spherical harmonic coefficients with the load love numbers, see Wahr et al. (1998). Within this step it is reasonable to convert the resulting trend signal from potential to the generating water masses in terms of equivalent water heights. This conversion would have been mathematically possible at an earlier stage as well, but it is difficult to interpret effects such as GIA in terms of water heights. The resulting trend in terms of equivalent water heights of the original and the reduced time series is shown in Fig. 19.4. In contrast to the analysis in terms of geoid height, the trend is slightly positive. It is doubtful whether this trend can be regarded as significant. It is extremely difficult to make a reliable statement in this matter, as not only the currently achievable accuracy of GRACE has to be taken into account, but also errors in the background models used for signal separation. Especially for the geophysical models plausible error estimations are hardly available.

19.5

Conclusions

This article shows some of the pitfalls occurring when analyzing and interpreting long-term trends derived from GRACE data. The first important conclusion is the need of long time series to derive trends especially in areas with strong inter-annual variability. In order to give reliable statements concerning secular mass changes longer time series are inevitable, which emphasizes the importance of a GRACE follow-on mission without any gap. The second issue deals with the separation of signals originating from different geophysical processes. For example, if one is interested in changes in the hydrological water masses, the contributions of atmosphere, oceans, tides, and solid Earth have to be taken into account. These effects are usually removed by models. It has to be

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Challenges in Deriving Trends from GRACE

Fig. 19.3 Influence of the different background models on the trend in the High Plains aquifer: original GFZ-RL04 time series (gray), GIA model (orange), pole tide trend (green), rates

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proposed by the IERS (black). Reduced time series with corresponding linear trend: red

Fig. 19.4 The original (gray) and the reduced (red) time series expressed in terms of water heights in mm

kept in mind that in case of GRACE some part of the separation step has already taken place during the data analysis, while further models have to be reduced by the user. Other models are only used for linearization or de-aliasing purposes and are not suitable for signal

separation. This article summarizes the correct use of the different models and demonstrates their importance for the interpretation of trend signals from GRACE data using the example of the High Plains aquifer region.

160 Acknowledgments The authors would like to thank Volker Klemann from the GFZ for providing the GIA model. The support by the DFG (Deutsche Forschungsgemeinschaft) within the frame of the special priority program SPP1257 “Mass transport and mass distribution in the Earth system” is gratefully acknowledged.

References Bettadpur S (2007) UTCSR level-2 processing standards document for level-2 product release 0004. GR-03-03. CSR, Austin, TX Blewitt G (2003) Self-consistency in reference frames, geocenter definition, and surface loading of the solid Earth. J Geophys Res 108(B2):2103. doi:10.1029/2002JB002082 Flechtner F (2005) GRACE AOD1B product description document (Rev. 2.1) Flechtner F (2007) GFZ level-2 processing standards document for level-2 product release 0004. GR-GFZ-STD-001. GFZ, Potsdam Flechtner F, Dahle Ch, Neumayer KH, Koenig R, Foerste Ch (2009) The release 04 CHAMP and GRACE EIGEN gravity field models. In: Flechtner F, Gruber T, Guentner A, Mandea M, Rothacher M, Wickert J (eds) Satellite geodesy and earth system science – observation of the earth from space. Springer, Berlin (in preparation) Han S-C, Shum CK, Bevis M, Ji C, Kuo C-Y (2006) Crustal dilatation observed by GRACE After the 2004 SumatraAndaman Earthquake. Science 313:658–662. doi:10.1126/ science.1128661 Horwath M, Dietrich R (2009) Signal and error in mass change inferences from GRACE: the case of Antarctica. Geophys J Int 177(3):849–864. doi:10.1111/j.1365-246X.2009.04139.x Klemann V, Martinec Z (2009) Contribution of glacial-isostatic adjustment to the geocenter motion. Tectonophysics. doi:10.1016/j.tecto.2009.08.031 Mayer-G€urr T, Eicker A, Ilk KH (2007) ITG-Grace03 gravity field model. http://www.geod.uni-bonn.de/itg-grace03.html

A. Eicker et al. McCarthy DD, Petit G (2004) IERS conventions 2003. IERS technical notes, 32 Verlag des Bundesamts fuer Kartographie und Geod€asie, Frankfurt am Main Rodell M (2002) The potential for satellite-based monitoring of groundwater storage changes using GRACE: the High Plains aquifer, Central US. J Hydrol 63:245–256. doi:10.1016/ S0022-1694(02)00060-4 Rodell M, Velicogna I, Famiglietti JS (2009) Satellite-based estimates of groundwater depletion in India. Nature 460: 999–1002. doi:10.1038/nature08238 Schmidt R, Petrovic S, G€untner A, Barthelmes F, W€unsch J, Kusche J (2008) Periodic components of water storage changes from GRACE and global hydrology models. J Geophys Res 113:B08419. doi:10.1029/2007JB005363 Steffen H, Denker H, M€uller J (2008) Glacial isostatic adjustment in Fennoscandia from GRACE data and comparison with geodynamic models. J Geodyn 46(3–5):155–164. doi:10.1016/j.jog.2008.03.002 Steffen H, Petrovic S, M€uller J, Schmidt R, W€unsch J, Barthelmes F, Kusche J (2009) Significance of secular trends of mass variations determined from GRACE solution. J Geodyn 48(3–5):157–165. doi:10.1016/j.jog.2009.09.029 Strassberg G, Scanlon BR, Chambers D (2009) Evaluation of groundwater storage monitoring with the GRACE satellite: case study of the High Plains aquifer, central United States. Water Res Int 45:W05410. doi:10.1029/2008WR006892 Tapley BD, Bettadpur S, Watkins M, Reigber C (2004) The gravity recovery and climate experiment: mission overview and early results. Geophys Res Lett 31:L09607 Wahr J, Molenaar M, Bryan F (1998) Time variability of the Earth’s gravity field: hydrological and oceanic effects and their possible detection using GRACE. J Geophys Res 103(B12):30, 20530, 229 Watkins M, Yuan D-N (2007) JPL level-2 processing standards document for level-2 product release 04. ftp://podaac.jpl. nasa.gov/pub/grace/doc/ Werth S, G€untner A, Schmidt R, Kusche J (2009) Evaluation of GRACE filter tools from a hydrological perspective. Geophys J Int 179(3):14991515

AIUB-GRACE02S: Status of GRACE Gravity Field Recovery Using the Celestial Mechanics Approach

20

A. J€aggi, G. Beutler, U. Meyer, L. Prange, R. Dach, and L. Mervart

Abstract

The gravity field model AIUB-GRACE02S is the second release of a model generated with the Celestial Mechanics Approach using GRACE data. Intersatellite K-band range-rate measurements and GPS-derived kinematic positions serve as observations to solve for the Earth’s static gravity field in a generalized orbit determination problem. Apart from the normalized spherical harmonic coefficients up to degree 150, arc-specific parameters like initial conditions and pseudo-stochastic parameters are solved for in a rigorous least-squares adjustment based on both types of observations. The quality of AIUB-GRACE02S has significantly improved with respect to the earlier release 01 due to a refined orbit parametrization and the implementation of all relevant background models. AIUB-GRACE02S is based on 2 years of data and was derived in one iteration step from EGM96, which served as a priori gravity field model. Comparisons with levelling data and models from other groups are used to assess the suitability of the Celestial Mechanics Approach for GRACE gravity field determination.

20.1

Introduction

The Gravity Recovery And Climate Experiment (GRACE) mission, launched on March 17, 2002, has significantly improved our knowledge of the Earth’s gravity field in terms of accuracy, spatial and temporal resolution. The time-varying part of the Earth’s gravity field has been inferred with unprecedented accuracy from space by analyzing GPS, accelerometer, and

A. J€aggi (*) G. Beutler U. Meyer L. Prange R. Dach Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland e-mail: [email protected] L. Mervart Institute of Advanced Geodesy, Czech Technical University, Thakurova 7, 16629, Prague, Czech Republic

inter-satellite K-band observations (Tapley et al. 2004). For the static part GRACE models report an accuracy of about 3 cm in terms of geoid heights at a spatial resolution of 200 km (F€orste et al. 2008a), which is an improvement by about one order of magnitude compared to the high-resolution model EGM96 (Lemoine et al. 1997). Apart from the official GRACE models derived by the Center for Space Research (CSR) of the University of Texas at Austin, the Jet Propulsion Laboratories (JPL), and the GeoForschungsZentrum Potsdam/Groupe de Recherche de Ge´ode´sie Spatiale (GFZ /GRGS), alternative state-of-the-art GRACE gravity field models have been computed, e.g., at the Institut f€ur Geod€asie und Geoinformation of the University of Bonn (Mayer-G€urr 2008).

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_20, # Springer-Verlag Berlin Heidelberg 2012

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Gravity field recovery from satellite data has also been initiated at the Astronomical Institute of the University of Bern (AIUB). The so-called Celestial Mechanics Approach (CMA) has been applied to both the high-low satellite-to-satellite tracking (hlSST) data of CHAMP and the low-low (ll) SST data of GRACE. Consolidated results for CHAMP and first results for GRACE may be found in Prange et al. (2009) and J€aggi et al. (2010), respectively. This article focuses on updates of the CMA, which were implemented to improve the combined processing of GRACE hl-SST and ll-SST data of the K-band ranging system (Sect. 20.2). Various gravity field recovery experiments illustrate the sensitivity of the CMA on, e.g., the orbit parametrization or the background modeling (Sect. 20.3). A significantly improved gravity field model AIUB-GRACE02S has been derived from GRACE data covering the years 2006 and 2007 (Sect. 20.4). The quality of this 2-year solution is assessed and perspectives for future developments are presented.

20.2

Celestial Mechanics Approach

GRACE gravity field determination using the CMA is based on the analysis of Level 1B inter-satellite Kband measurements (Case et al. 2002) and GPSderived kinematic positions (J€aggi et al. 2009). The static part of the gravity field is modeled with a series of normalized spherical harmonic (SH) coefficients (Heiskanen and Moritz 1967) from degree 2 up to a maximum degree of 150. The time variable part is currently not yet modeled. Based on a priori orbits derived from the kinematic positions of both GRACE satellites and the inter-satellite measurements (Sect. 20.1) normal equations for both types of observations are set up on a daily basis for the unknown gravity field coefficients and for additional arc-specific parameters, i.e., two normal equation systems based on the kinematic positions of GRACE A and B (Sect. 20.2), and one normal equation system based on the ll-SST data of the K-band ranging system (Sect. 20.3). The resulting daily normal equations are then combined into one system for each daily arc. Finally, arc-specific parameters are pre-eliminated and the combined daily normal equations are accumulated into monthly, annual, and multi-annual systems, which are eventually inverted to solve for the SH coefficients

without applying any regularization. Details about the general procedure may be found in (J€aggi et al. 2010).

20.2.1 A Priori Orbit Generation As opposed to the processing scheme described by J€aggi et al. (2010), improved a priori orbits are used here for gravity field recovery: Kinematic positions and K-band measurements are already taken into account for the a priori orbit generation. Using a specific force model (a priori gravity field model, accelerometer data, ocean tide model,. . .), only the weighted kinematic positions of both GRACE satellites are fitted in a first step by numerically integrating the corresponding equations of motion (Beutler 2005) and by adjusting the arc-specific orbit parameters. Efficient numerical integration techniques are applied to solve the so-called variational equations (Beutler 2005) in order to obtain the needed partial derivatives with respect to the orbit parameters. The normal equations of this first step are stored as they are subsequently needed for the generation of the “final” a priori orbits. Apart from the initial osculating elements constant and once-per-revolution empirical accelerations are set up, which act over the entire arc in the radial, along-track, and cross-track directions, and constrained piecewise constant accelerations acting over 15 min intervals in the same three directions. Additional polynomial coefficients up to degree 3 are set up per arc in the along-track direction in order to compensate for instrument drifts when using accelerometer data. For numerical reasons the arc-specific parameters are not set up separately for the two GRACE satellites. Transformed parameters representing the mean values of the corresponding parameters of both GRACE satellites, which are mainly determined by the kinematic positions, and half of the differences, which are mainly determined by the K-band observations are used. Based on the orbits of GRACE A and B from the first step, K-band measurements are used to set up the K-band normal equations for the same arc-specific parameters. The three normal equation contributions are then combined into one system for each daily arc, which is eventually inverted to solve for the arcspecific orbit parameters. The a priori orbits used for gravity field recovery are finally obtained by propagating the improved state vectors by numerical integration. As these orbits represent the K-band

20

AIUB-GRACE02S: Status of GRACE Gravity Field Recovery

observations already at a rather comfortable level even when using an a priori gravity field model from the pre-GRACE era, the gravity field solution should not be harmed by linearization effects (see Sect. 20.3).

20.2.2 Daily Normal Equations from Positions Based on the a priori orbits of GRACE A and B gravity field recovery from orbit positions is set up as a generalized orbit improvement process for each GRACE satellite as described by J€aggi et al. (2010). The actual orbit of one satellite is expressed as a truncated Taylor series with respect to the unknown arc-specific orbit parameters and the unknown SH coefficients about the a priori orbit. In analogy to Sect. 20.2.1 the partial derivatives of the a priori orbit are computed with respect to all parameters. These partials eventually allow it to set up the daily normal equations based on kinematic positions for all parameters according to a standard least-squares adjustment. In analogy to Sect. 20.2.1 the arc-specific parameters are set up as the sum and the difference of the original parameters in order to avoid numerical problems.

20.2.3 Daily Normal Equations from II-SST Based on the a priori orbits of GRACE A and B gravity field recovery from K-band measurements is set up as a differential orbit improvement process as described by J€aggi et al. (2010). The actual orbit difference is expressed as a truncated Taylor series with respect to the unknown parameters about the a priori orbit difference. The same techniques as in Sect. 20.2.2 are applied to solve the variational equations separately for both GRACE satellites. As K-band observations only contain information about the line-of-sight orbit difference, a projection of the partial derivatives on the line-of-sight direction is required. These projected partials eventually allow it to set up the daily normal equations based on K-band measurements for all parameters. The transformed arc-specific parameters are set up for both GRACE satellites, which implies that the daily normal equation matrices would be singular if only K-band data were used.

163

20.3

Gravity Field Recovery Studies

Gravity field models from GRACE kinematic positions and K-band range-rate data of the year 2007 were derived using the methodology outlined in Sect. 20.2. Due to the computational burden of gravity field recovery, only static fields up to degree 120 were estimated. EGM96 served as a priori model up to the same degree. A spacing of 15 min for the piecewise constant accelerations turned out to be sufficient to account for various model shortcomings. In order to illustrate the sensitivity of the CMA on orbit parametrization (Sect. 20.3.1), data weighting (Sect. 20.3.2), linearization effects (Sect. 20.3.3), and background models (Sect. 20.3.4), different solutions were made, which are subsequently compared with the gravity field model ITG-GRACE03S (Mayer-G€urr 2008) in terms of the square-roots of the degree difference variances up to degree 120.

20.3.1 Orbit Parametrization As the two GRACE satellites are only separated by about 30 s on the same orbital trajectory, they experience almost the same perturbations. An orbit and gravity field determination tailored to the GRACE configuration may therefore use this information and introduce additional, relative constraints between the estimated piecewise constant accelerations of both GRACE satellites. A series of gravity field models was thus generated to study the impact of relative constraints, which have been chosen to be 100 times tighter than the absolute constraints imposed on the piecewise constant accelerations. Figure 20.1 shows the impact of such constraints. The largest part of the improvement w.r.t. ITG-GRACE03S is already obtained by applying relative constraints only for the along-track direction. Additional constraints for the radial direction have a small positive impact, as well, whereas relative constraints for the cross-track direction do not significantly change the solution. The agreement (consistency) with the reference model for degree 2 is, however, degraded when relative constraints are applied for the radial and the along-track directions. Comparisons with EIGEN-GL04C (F€orste et al. 2008a) even revealed a degradation when relative constraints are only applied for the along-track

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164 Fig. 20.1 Square-roots of degree difference variances of annual recoveries when using different constraining options

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direction (not shown). Further investigations are necessary to better understand this mechanism.

20.3.2 Data Weighting According to J€aggi et al. (2010) a nominal weighting ratio of 1:108 between the GPS (L1) carrier phase

observations (error propagation taken into account by epoch-wise covariance information) and the K-band range-rate observations has been used for the determination of the gravity field solutions shown in Fig. 20.1. It turned out, however, that the agreement with ITG-GRACE03S may be slightly improved if stronger ratios are used, e.g., 1:109 or even 1:1010 (not shown). Despite the fact that a weighting ratio of 1:1010 is

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AIUB-GRACE02S: Status of GRACE Gravity Field Recovery

165

unrealistic in view of the a posteriori RMS errors, which are about 1 mm for the (L1) GPS carrier phase observations, and about 0.23 mm/s for the K-band range-rate observations (see Fig. 20.4), the latter was chosen for all subsequent solutions.

piecewise constant accelerations are able to compensate for the unmodeled non-gravitational accelerations to a great extent. Note that the cross-track component of the accelerometer data has not been taken into account for the solutions presented in this article, as almost no impact could be detected. The differences between the nominal and the experimental solutions are visible only up to about degree 90 because of an inadequate implementation of the ocean tide model used in the CMA at that time (see Sect. 20.3.4.3), and because of the omission errors due to the limited maximum degree of 120.

20.3.3 Linearization Gravity field recovery may depend on the a priori gravity field model used to set up the (linearized) observation equations. In order to rule out such dependencies for the CMA, EGM96 served as a priori gravity field model for all solutions presented in this article. Figure 20.2 shows the agreement of this baseline solution (iteration 1) with ITG-GRACE03S. A full iteration cycle was performed in order to assess the magnitude of linearization effects. For that purpose the previously obtained gravity field model (iteration 1) was used as a priori model for an additional iteration step (iteration 2). Figure 20.2 confirms that the agreement with ITG-GRACE03S is almost identical for both iteration steps. For most degrees the differences between the two iterations are about one order of magnitude smaller than the differences w.r.t. ITGGRACE03S. Gravity field recovery with the CMA is thus considered as highly independent of the a priori gravity field.

20.3.4 Background Modeling Since the results presented by J€aggi et al. (2010) all relevant background models are now implemented in the CMA. Their impact on static GRACE gravity field solutions is studied in the following subsections.

20.3.4.1 Non-gravitational Accelerations Accelerometer data are taken into account for GRACE gravity field recovery to separate the non-gravitational accelerations from the gravitational signal. Figure 20.3 compares the agreement of this nominal solution and of an experimental solution obtained without accelerometer data with ITG-GRACE03S. As expected, GRACE gravity field recovery clearly benefits from using accelerometer data. It is, however, remarkable that solutions of quite good quality may be obtained with the CMA even if accelerometer data are left out from the processing. Obviously, the estimated

20.3.4.2 Atmospheric and Oceanic Dealiasing Figure 20.3 also shows the impact of the non-tidal atmosphere and ocean short-term mass variations (Flechtner et al. 2006) on a static gravity field solution. Although the atmospheric and oceanic dealiasing (AOD) products are important for the generation of GRACE monthly solutions and clearly improve the K-band range-rate residuals (see Fig. 20.4 or Zenner et al. (2011) for more details), only a relatively small effect is visible in a static 1-year solution. 20.3.4.3 Ocean Tides The implementation of the ocean tide model in the CMA has been revised. Figure 20.3 shows that the improved implementation mainly had a positive effect on the high degrees above 90. The analysis of ocean tides also included the implementation of the models FES2004 (Lyard et al. 2006) and EOT08a (Savcenko and Bosch 2008) in the CMA. As no large differences on a static one-year solution have been found when using FES2004 (not shown) or EOT08A, the more recent EOT08a ocean tide model has been selected for the generation of the AIUB-GRACE02S model.

20.4

AIUB-GRACE02S

The AIUB-GRACE02S model was derived from kinematic positions and K-band range-rate data covering the years 2006 and 2007 using the methods from Sects. 20.2 to 20.3. As opposed to Sect. 20.3, the maximum degree was increased from 120 to 150. EGM96 thus also served as a priori model up to degree 150.

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166 Fig. 20.3 Square-roots of degree difference variances of annual recoveries when using different processing options

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20.4.1 Comparison with Other Models Figure 20.5 shows the agreement of the two solutions obtained from the 2006 and 2007 data sets and of the combined 2-year solution AIUB-GRACE02S w.r.t. ITG-GRACE03S. Compared to the figures from the previous section, clear improvement results for the higher degrees, but also the medium to high degrees

50

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show a better agreement with ITG-GRACE03 due to the increased maximum degree. Figure 20.5 even suggests that a maximum degree larger than 150 might have been chosen for the 2-year solution. The quality of the lower degrees improved slightly as well. Figure 20.6 shows the agreement of the three satellite-only models AIUB-GRACE02S, EIGEN-5S (F€orste et al. 2008b), and GGM03S (Tapley et al.

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AIUB-GRACE02S: Status of GRACE Gravity Field Recovery

Fig. 20.5 Square-roots of degree difference variances of annual recoveries and the combined 2-year solution

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2007) with ITG-GRACE03S. Although AIUBGRACE02S is based on only about half of the data volume of the other models, the agreement with ITGGRACE03S is comparable to that of EIGEN-5S and GGM03S for most degrees. Deficiencies in the AIUB-GRACE02S solution are only visible for degrees below 20. Longer data spans and, in particular, the estimation of time variable signals will further improve the low degree SH coefficients. Figure 20.4

ITG−GRACE03S GGM03S EIGEN−5S AIUB−GRACE02S

50 100 Degree of spherical harmonics

150

indicates that the time variability of the Earth’s gravity field has a clear impact on the residuals of gravity field recovery.

20.4.2 Validation with External Data T. Gruber from the Institut f€ur Astronomische und Physikalische Geod€asie of the Technische Universit€at

A. J€aggi et al.

168 Table 20.1 RMS of differences (cm) between levelling and model geoid heights up to a maximum degree of 120 Levelling data EUREF GPS BRD EUVN BRD GPS Canada GPS 1998 Canada GPS 2007 Australia GPS Japan GPS USA GPS

2007 Solution 22.2 04.4 05.0 20.9 15.3 25.0 11.2 34.2

AIUB-GRACE02S 22.2 05.3 05.7 20.2 14.5 24.6 10.8 33.6

M€unchen compared the geoid heights derived from AIUB-GRACE02S, ITG-GRACE03S, EIGEN-5S, and GGM03S with geoid heights derived from different sets of recent levelling data using the method described in (Gruber 2004). Table 20.1 shows the RMS values of the differences around the mean values between the filtered levelling geoid heights and geoid heights from the different models up to degree 120. The solution AIUB-GRACE02S is of the same quality as the other satellite-only models. More pronounced differences between the models are only visible for some of the high-quality data sets, e.g., the BRD EUVN or BRD GPS data sets. Compared to the solution obtained from the GRACE data of the year 2007, AIUB-GRACE02S has improved for all data sets except for BRD EUVN and BRD GPS. The reason is yet unclear.

20.5

Conclusions

We used the CMA for gravity field determination with GPS-derived kinematic GRACE positions and Level 1B K-band range-rate data to generate the static gravity field model AIUB-GRACE02S with data covering the years 2006 and 2007. We showed that it is possible to start the recovery with EGM96 as a priori model and that the solution may be obtained in one iteration step. Accelerometer data are important for the quality of the resulting gravity field model, but models of remarkable quality may also be obtained with the CMA without accelerometer data. The overall quality of AIUB-GRACE02S was found to be comparable to that of other well-known satellite-only gravity fields, which is also confirmed by the validation with terrestrial measurements.

ITG-GRACE03S 21.9 03.7 04.4 19.6 15.0 24.3 10.3 33.4

EIGEN-5S 22.6 05.9 06.3 19.9 15.5 24.5 11.8 33.3

GGM03S 22.5 05.4 06.2 19.7 14.8 24.2 10.6 33.3

The quality of the very low degrees of AIUBGRACE02S is not yet optimal and should be improved in a future release. Part of the degradation might be due to too strong relative constraints applied to the piecewise constant accelerations. This aspect has to be further studied. The unmodeled time variability of the Earth’s gravity field, which clearly shows up in the a posteriori K-band range-rate residuals, could also contribute to the not yet optimally estimated low degree SH coefficients. Further refinements of the CMA will include the estimation of time variable signals for the low degree SH coefficients. This and the use of a longer data span will further improve GRACE gravity field recovery with the CMA. Acknowledgements The authors gratefully acknowledge the generous financial support provided by the Swiss National Science Foundation and the Institute for Advanced Study (IAS) of the Technische Universit€at M€unchen.

References Beutler G (2005) Methods of celestial mechanics. Springer, Berlin Case K, Kruizinga G, Wu S (2002) GRACE level 1B data product user handbook. D-22027. JPL, Pasadena, CA Flechtner F, Schmidt R, Meyer U (2006) De-aliasing of shortterm atmospheric and oceanic mass variations for GRACE. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the earth system from space. Springer, Heidelberg, pp 83–97 F€orste C, Schmidt R, Stubenvoll R, Flechtner F, Meyer U, K€onig R, Neumayer H, Biancale R, Lemoine JM, Bruinsma S, Loyer S, Barthelmes F, Esselborn S (2008a) The GeoForschungsZentrum Potsdam/Groupe de Recherche de Ge´ode´sie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C. J Geod 82:331–346 F€orste C, Flechtner F, Schmidt R, Stubenvoll R, Rothacher M, Kusche J, Neumayer KH, Biancale R, Lemoine JM,

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Barthelmes F, Bruinsma S, K€ onig R, Meyer U (2008b) EIGEN-GL05C – a new global combined high-resolution GRACE-based gravity field model of the GFZ-GRGS cooperation. Geophysical Research Abstracts, vol. 10. EGU, Vienna Gruber T (2004) Validation concepts for gravity field models from satellite missions. In: Proceedings of second international GOCE user workshop “GOCE, The Geoid and Oceanography”, ESA-ESRIN, Frascati Heiskanen WA, Moritz H (1967) Physical geodesy. Freeman, San Francisco, CA J€aggi A, Dach R, Montenbruck O, Hugentobler U, Bock H, Beutler G (2009) Phase center modeling for LEO GPS receiver antennas and its impact on precise orbit determination. J Geod 83:1145–1162 J€aggi A, Beutler G, Mervart L (2010) GRACE gravity field determination using the celestial mechanics approach – first results. In: Mertikas S (ed) Gravity, geoid and earth observation. Springer, Berlin, pp 177–184 Lemoine FG, Smith DE, Kunz L, Smith R, Pavlis EC, Pavlis NK, Klosko SM, Chinn DS, Torrence MH, Williamson RG, Cox CM, Rachlin KE, Wang YM, Kenyon SC, Salman R, Trimmer R, Rapp RH, Nerem RS (1997) The development of the NASA GSFC and NIMA joint geopotential model. In: Segawa J, Fujimoto H, Okubo S (eds) IAG symposia: gravity, geoid and marine geodesy. Springer, Berlin, pp 461–469

Lyard F, Lefevre F, Letellier T, Francis O (2006) Modelling the global ocean tides: insight from FES2004. Ocean Dyn 56:394–415 Mayer-G€urr T (2008) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb€ogen am Beispiel der Satellitenmissionen CHAMP und GRACE. Schriftenreihe 9, Institut f€ur Geod€asie und Geoinformation, University of Bonn, Bonn Prange L, J€aggi A, Dach R, Bock H, Beutler G, Mervart L (2009) AIUB-CHAMP02S: the influence of GNSS model changes on gravity field recovery using spaceborne GPS. Adv Space Res 45:215–224 Savcenko R, Bosch W (2008) EOT08a – empirical ocean tide model from multi-mission satellite altimetry. DGFI report 81, Deutsches Geod€atisches Forschungsinstitut, Munich Tapley BD, Bettadpur S, Ries JC, Thompson PF, Watkins M (2004) GRACE measurements of mass variability in the Earth system. Science 305(5683):503–505 Tapley B, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Poole S (2007) The GGM03 mean earth gravity model from GRACE. Eos Trans AGU 88(52) Zenner L, Gruber T, Beutler G, J€aggi A, Flechtner F, Schmidt T, Wickert J, Fagiolini E, Schwarz G, Trautmann T (2011) Using atmospheric uncertainties for GRACE de-aliasing – first results. In: Kenyon S et al (eds) Geodesy for planet earth. Springer, Heidelberg

.

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes

21

B.C. Gunter, T. Wittwer, W. Stolk, R. Klees, and P. Ditmar

Abstract

In this study we address the question of whether regional gravity field modeling techniques of GRACE data can offer improved resolution over traditional global spherical harmonic solutions. Earlier studies into large, equatorial river basins such as the Amazon, Zambezi and others showed no obvious distinction between regional and global techniques, but this may have been limited by the fact that these equatorial regions are at the latitudes where GRACE errors are known to be largest (due to the sparse groundtrack coverage). This study will focus on regions of higher latitude, specifically Greenland and Antarctica, where the density of GRACE measurements is much higher. The regional modeling technique employed made use of spherical radial basis functions (SRBF), complete with an optimal filtering algorithm. Comparisons of these regional solutions were made to a range of other publicly available global spherical harmonic solutions, and validated using ICESat laser altimetry. The timeframe considered was a 3 year period spanning from October 2003 to September 2006.

21.1

Introduction

The launch of the Gravity Recovery and Climate Experiment (GRACE) in 2002 started a new wave of research into the Earth’s mass transport processes. The measurements from the mission’s twin satellites have enabled the multi-year tracking of many large scale processes, such as continental hydrology and ice mass changes in the cryosphere. While these first studies have produced some truly excellent results, there is always the desire to push the boundaries of what

B.C. Gunter (*) T. Wittwer W. Stolk R. Klees P. Ditmar Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands e-mail: [email protected]

GRACE can observe, in terms of spatial and temporal resolution. Previous studies have demonstrated that the current processing standards of GRACE data provide mass change accuracies on the order of 2 cm of equivalent water height (EWH) over spatial scales of 400 km and time intervals of 1 month (Klees et al. 2008a). This analysis was done by comparing the performance of a range of different GRACE processing strategies, including both regional and global methods, over selected river basins and other “dry” regions where little to no hydrological signal is expected. The global methods tested primarily involved traditional spherical harmonic solutions from various processing centers (CSR, GFZ, JPL, CNES, DEOS), but with various spatial filters applied. The regional solutions examined included the “mascon” approach (Luthcke et al. 2006) as well as

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solutions computed using spherical radial basis functions. In short, the overall conclusion of this earlier study was that there was no clear advantage to using regional techniques over global methods for the river basins studied. In fact, it turned out that the choice of spatial filter was the most important aspect in the comparisons; however, one of the limitations of this particular study was that most of the regions studied were at relatively low latitudes, where the density of GRACE measurements is the lowest. For higher latitude regions, it is possible that the increased data density might offer a higher signal-to-noise ratio that regional techniques might be able to better exploit. As a result, a follow-on study was conducted (Stolk 2009) to perform a similar analysis over regions such as Greenland and Antarctica, to see if the conclusions would be the same. This paper will provide an overview of the methods and conclusions of this follow-on study.

21.2

Spherical Radial Basis Functions

The focus of the regional techniques for the highlatitude regions involved the application of spherical radial basis functions (SRBF). The general concept behind this approach is to use a distribution of spacelocalizing functions to represent any complex spherical shape, such as the Earth’s gravity field. The functions can be constructed using a number of different methods, although the kernel adopted for the current study makes use of Poisson wavelets of order three (Holschneider et al. 2003; Wittwer 2009). The shape and spatial distribution of the SRBFs are determined by the depth (i.e., bandwidth) and the level (i.e., spacing on a Reuter grid) assigned to each SRBF, as illustrated in Fig. 21.1. As with spherical harmonic solutions, SRBF solutions suffer from north-south error patterns (i.e., “stripes”), which require the application of a suitable filter. The anisotropic, non-symmetric (ANS) filter developed by Klees et al. (2008b) offers a number of benefits over other traditional filtering techniques, such as destriping or Gaussian smoothing, primarily because use is made of the full statistical information of the solution (i.e., signal and noise variance-covariance matrices are used). For example, if spherical harmonics are used to parameterize the time-variable gravity field, and we let N^ x ¼ b represent the normal

equations for a monthly GRACE solution, the ANS filter W can be applied as follows: ^xw ¼ W^x ¼ ðN þ D1 Þ1 b

(21.1)

where N is the normal matrix, D the signal covariance matrix (i.e., the auto-covariance matrix of the vector ^x), ^x the estimated parameter vector, and b the rightside vector. The matrix N is determined from the partial derivatives of the system dynamics; however, the auto-covariance matrix, D, must be determined empirically. This is done through an iterative process whereby the (time-independent) variances of the signal from the actual time series of monthly solutions (e.g., 36 months for this study) are computed at the nodes of an equal-angular grid and then transformed back to the spherical harmonic domain to form D. This signal covariance information has the effect of suppressing spurious noise in regions that typically do not have much mass variations (e.g., oceans and deserts), while also allowing the solution to adjust more freely in areas where the mass change signal has larger variations (e.g., river basins). Since this signal covariance matrix is computed from the time series of GRACE solutions, it is particular to the solution technique. A straightforward generalization of this concept to a SRBF parameterization is obtained when the relationship between spherical harmonic coefficients, x, and SRBF coefficients, a, is exploited. This relationship can be written as x ¼ Qa

(21.2)

Hence, given the auto-covariance matrix in the spherical harmonic domain, D, we can obtain the corresponding auto-covariance matrix in the SRBF domain according to a ¼ Qþ x ) Da ¼ Qþ DðQþ ÞT

(21.3)

where Q+ ¼ (QTQ)1QT is the pseudo-inverse of the Q matrix. This approach, however, fails because the spectrum of a given SRBF parameterization comprises spherical harmonic degrees, which may exceed the maximum degree of a given GRACE monthly solution (the number of harmonic coefficients in x is often much larger than the number of SRBF coefficients in a).

21

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes

173

Fig. 21.1 Example spherical radial basis functions

Therefore, the optimal filter needs to be designed directly in the SRBF domain. If f is the time-variable gravity signal in terms of equivalent water heights, and a comprises the SRBF coefficients, we write the SRBF synthesis as f ¼ Ba

(21.4)

Using the pseudo-inverse of B, B+ ¼ (BTB)1BT, we write a ¼ Bþ f

(21.5)

and obtain the auto-covariance matrix in the SRBF domain, Da, as Da ¼ Bþ D ðBþ ÞT

(21.6)

Hence, if Na a ¼ ba is the system of normal equations in terms of SRBFs, the equivalent expression of (21.1) is aW ¼ Wa a ¼ ðNa þ Da Þ1 ba

(21.7)

With these relationships, the signal covariance matrix now can be computed, and the ANS filter applied to the SRBF coefficients. Note that since the computation of the signal covariance matrix is done iteratively, an initial set of values must first be chosen. The standard deviations chosen for this initial signal variance covariance matrix are essentially arbitrary, although proper choices might reduce the number of iterations needed. For the current study, the initial standard deviations were set to 50 mm globally. This initial standard deviation is propagated from the spatial domain to the frequency domain using (21.6), then a new signal variance matrix is created from the

filtered solution. Iteration is halted when the difference in equivalent water height between two consecutive iterations for each grid point is less than 35 mm (chosen experimentally to balance convergence speed and the determination of accurate signal variability). The determination of the optimal values for the level and depth of the SRBF solutions depends on the spatial variations, and noise content, of the data involved. Placing a dense grid of functions at a relatively shallow depth (i.e., small bandwidth) may result in noisy solutions, especially for GRACE data. The general approach used here was to employ a level high enough to represent what was believed to be the signal content in the data, and to place these functions as deep as possible in an attempt to smooth out the noise in the data. Many combinations of level and depth were evaluated, with the determination that a level 90 (i.e., ~220 km Reuter grid spacing), depth 900 km parameterization offers the highest quality solutions for Greenland and Antarctica.

21.3

Comparisons

Having finalized the optimal parameterization and filtering of the SRBF solutions, the next step was to compare the results of the mass change estimates derived from these solutions to those derived from other techniques, over Greenland and Antarctica. Since the goal of the study was simply to compare global versus regional techniques, only a limited set of spherical harmonic solutions were involved, and included those from the Center for Space Research (CSR) and the Delft Institute of Earth Observation and Space Systems (DEOS), who now produce a set of publically available monthly gravity solutions called the DEOS Mass Transport (DMT-1) models

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Table 21.1 Descriptions of the various global and regional solutions used for comparison Model name CSR DS400

Solution type Global

CSR DS0 DMT-1

Global Global

SRBF global

Regional

SRBF regional

Regional

Description Spherical harmonic solution to 60x60 derived from CSR RL04 data (see http://podaac.jpl.nasa.gov/ grace) ; destriped; 400km Gaussian smoothing applied; SLR C20 values; degree 1 coefficients taken from Swenson et al. (2008) Similar to above, except without Gaussian smoothing applied DEOS Mass Transport models (see http://www.lr.tudelft.nl/psg/grace); spherical harmonic solutions to 120x120 generated from KBR L1B data using the range-combination approach (Liu et al. 2009); anisotropic non-symmetric (ANS) filter applied (Klees et al. 2008b) Spherical radial basis function approach using Poisson wavelets, in which a global distribution of nodes with a Reuter grid spacing of level 90 and depth of 900km is used. Low level data derived from the same KBR L1B data as the DMT-1 solution, with a similar anisotropic non-symmetric filter applied (adapted for use with SRBF’s) Similar to the SRBF global approach, but using only regional data (i.e., within a 30 extended boundary from the target region)

(Liu et al. 2009). A summary of the solutions used in the comparisons is provided in Table 21.1. In short, the CSR solutions used both un-filtered and Gaussian filtered solutions, with an additional destriping filter applied similar to that of Swenson et al. (2008) (see Gunter et al. (2009) for further details of the CSR solution processing). The DMT-1 solutions are global spherical harmonic solutions computed using the acceleration approach (Ditmar and van Eck van der Sluijs 2004; Liu 2008), and with the ANS filter applied. Two types of SRBF solutions were tested, one using a global distribution of functions (SRBF global), and one using a more regional distribution (SRBF regional) in which a latitudinal buffer of 30 was used to reduce edge effects. For each solution, both the long-term linear trends (with bias and annual/ semi-annual terms included) and monthly variations in the signals were examined. The timeframe considered was a 3 year period spanning from October 2003 to September 2006. Some selected results from the comparisons are shown in Figs. 21.2 and 21.3. Figure 21.2 shows the geographical plot of the linear trends for the CSR400, DMT-1 and SRBF regional solutions over Greenland and Antarctica. Figure 21.3 is a plot of the maximum amplitude of the annual signal variation which is coestimated along with the linear trend parameter. This is useful to visualize where the largest fluctuations in mass change exist. The first observation that can be made from looking at these two figures is that the resolution for the ANS filtered solutions is much higher than those of the CSR DS400 solution, particularly for Greenland. The

DMT-1 and SRBF solutions are quite similar, but differences do exist. It is also interesting to note that the amplitude plots for the DMT-1 and SRBF regional solutions show subtle difference as well. For example, the SRBF solution shows a noticeable variation at the tip of the Antarctic Peninsula, where the DMT-1 solution does not. Similarly, in the Amundsen Sea sector (SW Antarctica), the SRBF solutions show two distinctive peaks, where the DMT-1 solutions show only one.

21.4

Validation

The determination of whether the differences seen in Figs. 21.2 and 21.3 represent genuine improvements in the signal recovered by the SRBF solutions is a difficult question to answer, and is a topic of current and future research efforts. One attempt made in this study to do this utilized surface elevation change data from the Ice, Cloud and Land Elevation Satellite (ICESat), a laser altimetry mission launched in 2003. ICESat observes the volume changes due to ice mass changes, which are naturally correlated to the mass changes observed by GRACE. The spatial resolution of ICESat is also much higher than that of GRACE, so a test was developed whereby the ICESat data was smoothed using a full-width Gaussian filter [as opposed to the traditional half-width filter normally used in geodesy, e.g., Jekeli (1981)] at intervals ranging from 0 to 2,500 km. Trend maps over the 3-year time period for both GRACE and ICESat were computed and each map was individually normalized. The normalization was needed because the ICESat

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes DMT–1

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Fig. 21.2 Geographical plot of the 3-year trend computed from selected global and regional solutions, in units of equivalent water height CSR DS400

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Fig. 21.3 Geographical plot of the estimated annual amplitude variations computed from selected global and regional solutions, in units of equivalent water height

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map represents physical height changes (dh/dt, in cm/ yr), whereas GRACE maps represent annual changes in EWH (also in cm/yr). As these are not the same quantities, the normalization allows a more direct comparison of the two data types under the assumption that a strong change in volume directly corresponds to a strong change in mass (and vice-versa). For each smoothing increment, correlations were computed between the smoothed ICESat map and the corresponding GRACE map. A peak in the resulting correlation curve would give an indication of the spatial resolution of the GRACE solution tested. The results of this test for all of the GRACE solutions mentioned in Table 21.1 are provided in Fig. 21.4. For Greenland, the correlations with ICESat for the ANS filtered solutions (DMT-1 and the SRBF solutions) peak at around 1,300 km (full-width) 1

Correlation

0.8

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ICESat Gaussian smoothing full width [km] Fig. 21.4 Results of the spatial correlation test to ICESat data for Greenland (top) and Antarctica (bottom), for the various GRACE solutions

Gaussian smoothing, where the CSR solutions peak in the 2,200–2,500 km range. This implies that the ANS filter is the driving force for the accuracy levels in Greenland, and not necessarily the solution technique. For Antarctica, the situation is slightly different. Here, the correlation peak of the SRBF regional solution is approximately 5–10% higher than the SRBF global solution and the unfiltered CSR solution. This would suggest that the SRBF regional approach is achieving slightly better spatial resolution than the other global approaches. Conclusions

The results of the analysis for this study supports the earlier conclusions by Klees et al. (2008a) that the choice of the spatial filter used in the GRACE processing has the largest impact on the comparisons. When compared to the standard destriping and Gaussian filter approach (i.e., DS400), the anisotropic, non-symmetric (ANS) filter offers many benefits in terms of improved spatial resolution. That said, there were other indications that the choice of solution method may also offer some improvements, although to a much smaller degree. For Antarctica, the SRBF regional solution had the best spatial correlation when compared to the corresponding height change data from ICESat (Fig. 21.4), and was the only solution to observe annual variations in the Antarctic Peninsula (Fig. 21.3). For Greenland, all ANS filtered solutions (global and regional) performed essentially the same, with all of them offering substantial improvements over the corresponding CSR fields (DS400 and DS0). This is primarily due to the fact that the CSR fields have inherently lower resolution (with maximum degree and order 60), and because a Gaussian filter was applied (equivalent ANS filtered CSR solutions were not possible since the monthly noise covariance matrices are not publicly available). Regardless, the results suggest that, at a minimum, the regional SRBF techniques are equivalent to other global spherical harmonic solutions (i. e., DMT-1), but that there is also the possibility that a 5–10% improvement might be gained, depending on the region. Future studies will attempt to verify these results with more extensive comparisons with independent data sets, such as in-situ glaciological measurements or other satellite measurements.

21

Comparison of Regional and Global GRACE Gravity Field Models at High Latitudes

Acknowledgements The authors would like to thank Tim Urban at the UT-Austin Center for Space Research for providing the ICESat crossover height change data.

References Ditmar P, van Eck van der Sluijs AA (2004) A technique for modeling the Earth’s gravity field on the basis of satellite accelerations. J Geodesy 78:12–33 Gunter BC, Urban T, Riva REM, Helsen M, Harpold R, Poole S, Nagel P, Schutz B, Tapley B (2009) A comparison of coincident GRACE and ICESat data over Antarctica. J Geodesy 83(11):1051–1060. doi:10.1007/s00190-009-0323-4 Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys Earth Planet Inter 135:107–124 Jekeli C (1981) Alternative methods to smooth the Earth’s gravity field. Technical Report 327, Ohio State University, Department of Geodetic Science and Surveying, December 1981 Klees R, Liu X, Wittwer T, Gunter BC, Revtova EA, Tenzer R, Ditmar P, Winsemius HC, Savenije HHG (2008a) A comparison of global and regional GRACE models for land hydrology. Surv Geophys 29(4–5):335–359

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Klees R, Revtova EA, Gunter BC, Ditmar P, Oudman E, Winsemius HC, Savenije HHG (2008b) The design of an optimal filter for monthly GRACE gravity models. Geophys J Int 175:417–432 Liu X (2008) Global gravity field recovery from satelliteto-satellite tracking data with the acceleration approach. Ph.D. Thesis, Netherlands Geodetic Commision, Publication on Geodesy 68, Delft, The Netherlands Liu X, Ditmar P, Siemes C, Slobbe DC, Revtova EA, Klees R, Riva R, Zhao Q (2009) DEOS Mass Transport model (DMT1) based on GRACE satellite data: methodology and validation. Geophys J Int 181(2):769–788 Luthcke S, Rowlands D, Lemoine F, Klosko S, Chinn D, McCarthy J (2006) Monthly spherical harmonic gravity field solutions determined from GRACE inter-satellite range-rate data alone. Geophys Res Lett 33:L02402 Stolk W (2009) An evaluation of the use of radial basis functions for mass change estimates at high latitudes. Msc Thesis, Delft University of Technology, The Netherlands Swenson S, Chambers D, Wahr J (2008) Estimating geocenter variations from a combination of GRACE and ocean model output. J Geophys Res 113, B08410, doi:10.1029/ 2007JB005338 Wittwer T (2009) Regional gravity field modeling with radial basis functions. Ph.D. Thesis, Netherlands Geodetic Commision, Publication on Geodesy 72, Delft, The Netherlands

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A New Approach for Pure Kinematical and Reduced-Kinematical Determination of LEO Orbit Based on GNSS Observations

22

A. Shabanloui and K.H. Ilk

Abstract

The geometrical point-wise satellite positions of a Low Earth Orbiter (LEO) equipped with a Global Navigation Satellite System (GNSS) receiver can be derived by GNSS analysis techniques based on hl-SST (high-low Satellite to Satellite Tracking) observations. In the geometrically determined LEO orbit, there is no connection between subsequent positions, and consequently, no information about the velocity and the acceleration or in general kinematical information of the satellite is available. If the kinematical parameters which consistently connects positions, velocities and accelerations are determined by a best fitting process based on the observations, we perform a pure Kinematical Precise Orbit Determination (KPOD). In addition, the proposed approach has a capability to use certain dynamical constraints based on the dynamical force function model. In this case, we introduce a Reduced-Kinematical Precise Orbit Determination (RKPOD) of a specific level depending on the strength of the “dynamical constraints”. The various possibilities and the corresponding results of CHAMP orbits based on GNSS observations are presented.

22.1

Introduction

Estimated positions based on GNSS techniques are purely geometric and there is no connection between subsequent positions, consequently no information about the velocity and the acceleration of the LEO is available. Therefore, to describe the time dependency of the satellite’s motion, it is necessary to provide a properly constructed function which consistently connects positions, velocities and accelerations. Such

A. Shabanloui (*) K.H. Ilk Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115, Bonn, Germany e-mail: [email protected]

a continuous function can be provided by a least squares adjustment process based on a table of geometrical positions. It should be pointed out that the orbit determination in this investigation is restricted to short arcs. In order to keep the accumulated effects of the disturbing forces on LEO as small as possible, satellite arcs can be divided into short arcs. Especially, in case of the measurement of LEO’s surface forces (e.g. CHAMP or twin-GRACE) or its compensation (e.g. GOCE), it is preferable to reduce the residual effects of an incomplete compensation by selecting short arcs. In this paper, the kinematical orbit is represented by a sufficient number of approximated parameters, including the boundary values of the short arc. These parameters are determined in such a way that

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180

the hl-SST observations (or geometrical positions as pseudo-observations) are approximated in the best possible way w.r.t. a properly selected norm. The approach is based on approximated parameters, which have also a clearly defined relation to the dynamical model of the satellite’s motion. If the kinematical representation parameters are determined by a best fitting process based on the hl-SST observations or geometrical positions, then a pure kinematical POD is realized. If instead all orbit representation parameters are determined by a model of the forces acting on the LEOs, then a Dynamical Precise Orbit Determination (DPOD) is introduced. In addition, there is also a possibility to use certain dynamical constraints based on the dynamical force function models. Therefore, a reduced-kinematical precise orbit determination of a specific level depending on the strength and contribution of the “dynamical restrictions” can be realized. It should be pointed out that this formulation of a POD problem allows a smooth transition from pure KPOD to RKPOD and finally fully DPOD of a LEO (Shabanloui 2008). In addition to the kinematic and the dynamic orbit determination technique, J€aggi (2007) also introduced a further type of methods called reduced dynamic methods which is located somewhere in between the kinematic and purely dynamic methods. This group of methods is characterized by the fact that additionally to the parameters of the dynamic methods, stochastic parameters of various types are introduced at intervals of a few minutes along the orbit. In this way the total orbit is divided into short arcs with different continuity properties at the arc boundaries depending on the type of the stochastic parameters. There are various articles which demonstrate the advantages of this method (e.g. J€aggi 2007). On the other hand, Sˇvehla and Rothacher (2003) proposed a reduced-kinematic POD which is here defined as a GPOD with a-priori low accuracy dynamical information between successive epochs in radial, cross and along-track directions. Their proposed reduced-kinematic method is only used to improve the characteristics of the purely GPOD by a considerable reduction of spikes and jumps. It should be mentioned that the proposed reduced-kinematical (geometrical) POD by Sˇvehla and Rothacher (2003) reduces geometrical information, while our new proposed reduced-kinematical POD in this investigation

A. Shabanloui and K.H. Ilk

reduces kinematical information in the LEO orbit (Shabanloui 2008).

22.2

Methodology

The precise orbit determination procedure is formulated as a Boundary Value Problem (BVP) of Newton-Euler’s equation of motion in the form of an integral equation of Fredholm type (Sect. 22.2.1). The solution of a Fredholm integral can be formulated in a semi-analytical way, either as a series of Fourier coefficients (Sect. 22.2.2), or as a series of Euler and Bernoulli polynomials (Sect. 22.2.3). The kinematical LEO orbit can be represented based on the combination of Euler–Bernoulli polynomials and Fourier series (Sect. 22.2.5). With introducing “dynamical restrictions” to the Fourier coefficients, the reduced kinematical POD is realized (Sect. 22.2.6).

22.2.1 Equation of Motion Based on the Linear Extended Newton Operator The basic idea was proposed as a general method for artificial satellite orbit determination by Schneider in 1968 (Schneider 1968). Before applying this method in the orbit determination, it was modified for the Earth gravity field determination by Reigber (1969). This orbit determination technique has been further developed and modified to various applications in satellite geodesy, especially to recover the Earth’s gravity field based on POD methods (Ilk 1977). A technique for the numerical solution of two BVP of Newton–Euler’s equation of motion as well as of Lagrange’s equation of motion based on Schneider’s proposed method was developed in the same publication. Later, this technique has been successfully applied to recover the Earth gravity field based on high–low SST and low–low SST observations (Mayer-G€urr 2006), but was not applied to produce the kinematical precise orbit of a LEO. At that time, orbit observations were sparse and not suited for precise kinematical orbit computation. The availability of GNSS changed the situation dramatically. The mathematical–physical model for densely tracked LEO orbits is based on the formulation of the equation of motion as (Ilk 1977)

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination

€rðtÞ ¼ aðt; r; r_ Þ

(22.1)

where the vectors r, r_ and €r denote the position, velocity and acceleration of the LEO, respectively. The function a indicates the specific (mass-related) force function acting on the LEO. The solution of Newtons equation of motion formulated as a BVP with the boundary values rA :¼ rðtA Þ;

rB :¼ rðtB Þ

ð1 rðtÞ ¼ rðtÞ T

K II ðt; t0 ÞaII ðt0 ; r; r_ Þdt0 (22.2)

t0 ¼0

with the starting time tA, the end time tB, the normalized time variable t and the arc length T t tA t :¼ ; T

with t 2 ½tA ; tB ;

T :¼ tB tA

K II ðt; t0 Þ ¼

0 1 sinmð1tÞsinmt ; sinm m 0 1 sinmð1t Þsinmt ; sinm m

sinðupt0 ÞaII ðt0 ; r; r_ Þdt0 :

t0 ¼0

(22.6)

It can be shown (Ilk (1977)), that the solution series (22.5) contains a generalized Fourier series of the difference function dðtÞ : ¼ rðtÞ rðtÞ 1 X ¼ du sinðuptÞ :¼ d1 F ð tÞ

(22.7)

u¼1

du ¼ 2

0

tt:

sinmð1 tÞ sinmt (22.3) rA þ rB sinm sinm qﬃﬃﬃﬃﬃﬃ with the mean anamoly m ¼ GM a3 T, where a is the semi-major axis of the LEO orbit and GM denotes the standard Earth’s gravitational constant. The reduced force function acting on the satellite reads m2 rðtÞ: T2

(22.4)

The solution of the BVP (Fredholm’s integral equation in (22.2)) reads rðtÞ ¼ rðtÞ þ dðtÞ 1 X du sinðuptÞ ¼ r ð tÞ þ

dðt0 Þsinðupt0 Þdt0 :

(22.8)

t0 ¼0

r ð tÞ ¼

u¼1

ð1

ð1

t0 t

The Keplerian orbit as the reference motion reads

aII ðt; r; r_ Þ ¼ aðt; r; r_ Þ

2T 2 du ¼ 2 2 u p m2

with the Fourier coefficient

as well as the the integral kernel (

with the sine coefficient

22.2.2 Interpretation of the Solution of Fredholm’s Integral Equation as Fourier Series

reads according to Schneider (1968)

2

181

(22.5)

If the difference function d (t) is continued to an odd periodic function with the period 2T or to the normalized interval [–1, 1], then we get because of property of sine function dðtÞ ¼ dðtÞ

(22.9)

and all cosine terms vanish (i.e. special case of Fourier series). Therefore, all properties of the Fourier series hold for the solution of the differential function. If we restrict the upper summation indices of the Fourier series to finite number n then the Fourier series in (22.7) taking the Fourier remainder function (RF (t)) into account reads n d1 F ðtÞ ¼ dF ðtÞ þ RF ðtÞ n 1 X X ¼ du sinðuptÞ þ du sinðuptÞ: u¼1

u¼nþ1

(22.10)

182

A. Shabanloui and K.H. Ilk

When neglecting the Fourier remaining term RF (t) in (22.10), then Fourier coefficients from discrete not necessary equidistant positions can be estimated by a least square adjustment (Shabanloui 2008).

J X 2 ð1Þjþ1

du ¼

2jþ1

ðupÞ

j¼1

ð1Þu d½2j ð1Þ d½2j ð0Þ

þ zðb; J; uÞ: (22.13)

22.2.3 From Fourier Series to Series of Euler–Bernoulli Polynomials

Introducing (22.13) into (22.7) results in

If (22.8) is partially integrated, then the first integration by parts reads

d(tÞ ¼

ð1

2jþ1

ð1Þu dj2jj ð1Þ

ð u pÞ j2jj d ð0Þ þBðb; J; uÞ sin ðuptÞ: u¼1

j¼1

(22.14)

dðt0 Þsinðupt0 Þdt0

du ¼ 2 t0 ¼0

¼ 2dðt0 Þ þ2

1 J X X 2ð1Þjþ1

1 up

1 1 cosðupt0 Þj0 up ð1 d½1 ðt0 Þcosðupt0 Þdt0 :

(22.11)

dðtÞ ¼

t0 ¼0

1 X

du sinðuptÞ

u¼1

By inserting the limits and considering that the difference function at the boundaries is zero, then we get 1 du ¼ 2 up

If we separate the inner sum of (22.14) in terms of even and odd indices u we get,

¼

½1

0

0

2jþ1

j¼1

þ

ð1

J X 2 ð1Þjþ1

J X 2 ð1Þj j¼1

0

d ðt Þcosðupt Þdt : þ

t0 ¼0

ð2pÞ

1 X

ðpÞ2jþ1

d½2j ð1Þ d½2j ð0Þ

d½2j ð1Þ þ d½2j ð0Þ

1 X sinð2uptÞ u2jþ1 u¼1

1 X sinð2u 1Þpt u¼1

ð2u 1Þ2jþ1

zðb; J; uÞsinðuptÞ:

u¼1

(22.15)

After 2J + 2 integrations by parts (Klose 1985), the general expression reads du ¼

J X

d½2j ðt0 Þ

j¼1

þb

The terms in (22.15) can be replaced by the absolutely and uniformly continuous series expansions of the Euler polynomials (Abramowitz and Stegun 1972)

2 ð1Þjþ1

1 cosðupt0 Þj0 2jþ1

ðupÞ ð1

2 ðupÞ2Jþ2

d½2Jþ2 ðt0 Þsinðupt0 Þdt0

t0 ¼0

E2j ðtÞ ¼

1 4 ð1Þj ð2jÞ! X sinð2u 1Þpt 2jþ1 2jþ1 p u¼1 ð2u 1Þ

(22.16)

(22.12) and the Bernoulli polynomials

with b¼

þ1 1

J ¼ 1; 4; 5; 8; 9; 12; . . . ; J ¼ 2; 3; 6; 7; 10; 11; . . . :

Evaluating the integration limits in the first term of (22.12) and denoting the remaining terms as x (b, J, u) we get

B2jþ1 ðtÞ ¼

1 2 ð1Þjþ1 ð2j þ 1Þ! X sinð2uptÞ : (22.17) u2jþ1 ð2pÞ2jþ1 u¼1

If (22.16) and (22.17) are inserted into (22.15), then we get:

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination

d ð tÞ ¼

1 X u¼1

¼

J X j¼1

þ

22.2.5 Kinematical Precise Orbit Determination

du sinðuptÞ ¼ d1 F ð tÞ e2j E2j ðtÞ þ

J X

b2jþ1 B2jþ1 ðtÞ

j¼1

1 X u¼1

zðb; J; uÞsinðuptÞ ¼ dJP ðtÞ þ RP ðtÞ (22.18)

with the term RP (t) as the Euler–Bernoulli truncation error term from degree J + 1 to infinity and the coefficients of the Euler polynomials (Shabanloui 2008) 1 ½2j d ð1Þ þ d½2j ð0Þ 2ð2jÞ!

e2j ¼

(22.19)

and the coefficients of the Bernoulli polynomials b2jþ1 ¼

1 d½2j ð1Þ d½2j ð0Þ : ð2j þ 1Þ!

183

(22.20)

In Shabanloui (2008), it was demonstrated that the upper degree J of the series in terms of Euler– Bernoulli polynomials should be very high to achieve an approximation accuracy at the sub-millimeter level. Also, it was shown that a fit of a series in terms of Euler–Bernoulli polynomials to the geometrically determined LEO’s short arc at the reasonable upper degree Jmax ¼ 4 should guarantee a sufficient determination of the Euler–Bernoulli coefficients corresponding to sufficiently precise arc derivatives at the boundaries (refer to (22.19) and (22.20)). Therefore, the resulting residual sine series after removing the Euler–Bernoulli terms should show a fast convergence and small residuals when compared to the true ephemerides (Shabanloui 2008). The satellite arc is presented kinematically by the selected reference motion, the Euler–Bernoulli polynomial up to degree Jmax and the residual Fourier series up to index n as rðtÞ ¼ rðtÞ þ

Jmax X

e2j E2j ðtÞ

j¼1

22.2.4 Determination of the Euler–Bernoulli Polynomials Coefficients If we restrict the upper summation indices of the Euler–Bernoulli polynomials to finite number J, then the LEO short arc can be represented based on GNSS observations according to (22.18) by omitting RP (t) as dðtÞ ’

J X j¼1

e2j E2j ðtÞ þ

J X

b2jþ1 B2jþ1 ðtÞ: (22.21)

j¼1

It should be mentioned that an Euler polynomial of degree 2j and a Bernoulli polynomial of degree 2j + 1 belong together as a pair. Based on the selected reference motion (refer to (22.3)), the LEO’s short arc can be approximated with the Euler–Bernoulli polynomial of degree Jmax in the space or spectral domain. We may note that the approximation quality increases with increasing upper degree J and the convergence of the Euler–Bernoulli polynomials requires a large upper index J (Shabanloui 2008).

þ

Jmax X j¼1

b2jþ1 B2jþ1 ðtÞ þ

n X

u sinðuptÞ: d

u¼1

(22.22) u denotes the residual Fourier coefwhere the vector d ficient. Based on the (22.22), in the first step, Euler–Bernoulli coefficients up to degree Jmax ¼ 4 are estimated from discrete GNSS observations. In the second step, the short arc boundary values as well as the residual Fourier coefficients are determined based on GNSS observations which are reduced w.r.t. the Euler–Bernoulli term up to degree Jmax ¼ 4.

22.2.6 Reduced-Kinematical Precise Orbit Determination The kinematical orbit parameters contain no dynamical information of the force function model; they are only based on observations taken at discrete epochs of the satellite motion along the orbit. In the following, we will extend the hybrid case of the kinematical orbit determination procedures as treated in Sect. 22.2.5 in

184

A. Shabanloui and K.H. Ilk

such a way that dynamical restrictions can be introduced in the orbit determination procedure. This dynamical ~u information is contained in the orbit coefficients d which are related to the force function according to ~j (i and j as ~ i to d (22.6). If the dynamical quantities d start and end indices) from (22.6) are considered as a-priori dynamical information with the corresponding ~i Þ to Cðd ~j Þ, then the observavariance-covariance Cðd tion equation taking dynamical restrictions into account reads as ¼

A1 0

A2 I

x1 x2

(22.23)

with (pseudo) observations,

Diff. (m)

l1 ¼ ð rðt1 Þ . . . rðtK Þ ÞT ~ T ~ l2 ¼ d . . . dj i where l1 and l2 denote geometrically determined LEO ~i to d ~j which positions and dynamical restrictions d are derived from (22.6), respectively. The terms A1 and A2 are design matrices w.r.t. the boundary values and the Fourier series coefficients which are computed based on (22.3) and (22.5), and x1 and x2 are the unknown boundary values and the Fourier series coefficients, respectively. The a-priori variancecovariance information reads C¼

C1

0

d

x

dy

10

dz

20

30

20

30

20

30

Minute 0.0004 0 -0.0004

0

10 Minute

0 C2 C1 ¼ diagðCðt1 Þ CðtK ÞÞ ~ ~ C2 ¼ diag Cðd i Þ Cðdj Þ

0.02 0.01 0 -0.01 -0.02 0

Diff. (m/s)

Numerical Tests

The orbit determination approach is tested based on simulated GNSS observations for a 30 min arc of a simulated CHAMP orbit with a sampling rate of 30 s. The geometrical positions of CHAMP are estimated based on the simulated hl-SST carrier phase observations which are contaminated with a white noise of 2 cm.

(22.24)

where C1 and C2 denote LEO position variancecovariance matrix derived from GPOD procedure and variance-covariance of dynamical restriction, respectively. Based on the estimated boundary values and the Fourier coefficients from (22.23), the Euler–Bernoulli polynomials up to degree Jmax ¼ 4 by a least squares adjustment are estimated (for more details refer to Shabanloui 2008). The coefficients of the residual Fourier series up to degree n from (22.22) are determined based on the estimated boundary

Diff. (m/s2)

l1 l2

22.3

2E-005 0 -2E-005

0

10 Minute

Coef. (m)

values, the Euler–Bernoulli coefficients and the Fourier coefficients. Therefore, the orbit parameters are determined in such a way that the kinematical parameters are constrained according to the specified level of “dynamical restrictions”.

0.08 0.04 0 -0.04 -0.08 -0.12

dv,x

dv,y

20

dv,z

40

60

Index

Fig. 22.1 Position differences, velocity differences, acceleration differences and the coefficients of the residual Fourier series for the kinematical POD, respectively

Diff. (m)

22

A New Approach for Pure Kinematical and Reduced-Kinematical Determination 0.02 0.01 0 -0.01 -0.02

d

0

x

dy

10

dz

20

30

20

30

20

30

Diff. (m/s)

Minute 0.0008 0.0004 0 -0.0004 -0.0008

0

10

Diff. (m/s2)

Minute 4E-005 2E-005 0 -2E-005 -4E-005

0

10

Coef. (m)

Minute dv,x

0.08

dv,y

dv,z

185

To verify results, kinematical ephemerides are determined based on estimated orbit representation parameters at the interval of 10 s and compared with the true ephemerides. The kinematical position, velocity and acceleration differences as well as the Fourier coefficients are shown in Fig. 22.1. It should be pointed out that the kinematical position differences are in range of the given white noise (2 cm). To test the proposed RKPOD procedure, the same short arc of CHAMP as used in the kinematical case, ~ 1 to d ~5 has been selected. The Fourier coefficients d are determined based on EGM96 and have been introduced to the observation equations as dynamical restrictions. The reduced-kinematical position, velocity and acceleration differences at the interval 10 s as well as the residual Fourier coefficients for the reduced-kinematical case are shown in Fig. 22.2. The comparison of the RKPOD with the KPOD shows some improvements in the results, because of introducing dynamical restrictions to estimation procedure (refer to Table 22.1).

0 -0.08

Conclusions

Fig. 22.2 Position differences, velocity differences, acceleration differences and the coefficients of the residual Fourier series for the reduced-kinematical POD case, respectively

The new proposed kinematical and reducedkinematical POD procedures open a wide window to represent low-flying orbits. The method is very flexible with the possibility of smooth transition from pure kinematical to fully dynamical POD.

Table 22.1 3D RMS of positions, velocities and accelerations for kinematical and reduced-kinematical POD numerical tests

Acknowledgment We gratefully acknowledge the financial support of the BMBF under the project ”REAL-GOCE”.

20

40

60

Index

Method KPOD RKPOD

Pos.(m) 0.011339 0.008873

Vel.(m/s) 0.000351 0.000337

Acc.(m/s2) 0.000015 0.000016

References The geometrically estimated positions are used as pseudo-observations in case of the kinematical and reduced kinematical orbit determination procedures. To demonstrate the kinematical POD procedure, the Keplerian orbit is used as reference motion. A series of Euler–Bernoulli polynomials up to degree Jmax ¼ 4 has been fitted to the difference function d(t). Based on geometrical determined LEO positions, the Keplerian orbit and the estimated Euler–Bernoulli polynomials coefficients, the residual Fourier coefficients up to the index n ¼ 30 are estimated.

Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York Klose U (1985). Beitr€age zur L€osung einer Integralgleichung vom HAMMERSTEINschen Typ. Diploma Thesis, TUM, Germany Ilk KH (1977) Berechnung von Referenzbahnen durch L€osung selbstadjungierter Randwertaufgaben, DGK, Reihe C, Heft 228, Munich, Germany J€aggi A (2007) Pseudo-stochastic orbit modeling of low earth satellites using the GPS. Ph.D. Thesis, University of Bern, Switzerland Mayer-G€urr T (2006) Gravitationsfeldbestimmung aus der Analyse kurzer Bahnb€ogen am Beispiel der Satellitenmission

186 CHAMP und GRACE. Ph.D. Thesis, IGG, University of Bonn, Germany Reigber C (1969) Zur Bestimmung des Gravitationsfeldes der Erde aus Satellitenbeobachtungen, DGK, Reihe C, Heft Nr. 137, Munich, Germany Schneider M (1968) A general method of orbit determination. Ph.D. Thesis, Ministry of Technology, Farnborough, England

A. Shabanloui and K.H. Ilk Shabanloui A (2008) A new approach for a kinematic-dynamic determination of low satellite orbits based on GNSS observations. Ph.D. Thesis, IGG, University of Bonn, Germany Sˇvehla D, Rothacher M (2003) Kinematic, reduced-kinematic, dynamic and reduced-dynamic precise orbit determination in the LEO orbit, 2nd CHAMP Science Meeting, Potsdam, Germany

Pure Geometrical Precise Orbit Determination of a LEO Based on GNSS Carrier Phase Observations

23

A. Shabanloui and K.H. Ilk

Abstract

The interest in a precise orbit determination of Low Earth Orbiters (LEOs) especially in pure geometrical mode using Global Navigation Satellite System (GNSS) observations has been rapidly grown. Conventional GNSS-based strategies rely on the GNSS observations from a terrestrial network of ground receivers (IGS network) as well as the GNSS receiver on-board LEO in double difference (DD) or in triple difference (TD) data processing modes. With the advent of precise orbit and clock products at centimeter level accuracy provided by the IGS centers, the two errors associated with broadcast orbits and clocks can be significantly reduced. Therefore, higher positioning accuracy can be expected even when only a single GNSS receiver is used in a zero difference (ZD) procedure. Along with improvements in the International GNSS Services (IGS) products in terms of Global Position System (GPS) satellite orbits and clock offsets, the Precise Point Positioning (PPP) technique based on zero (un-) differenced carrier phase observations has been developed in recent years. In this paper, the zero difference procedure has been applied to the CHAllenging Minisatellite Payload (CHAMP) high–low GPS Satellite to Satellite Tracking (hl-SST) observations, then the solution has been denoted as Geometrical Precise Orbit Determination (GPOD). The estimated GPOD CHAMP results are comparable with results of other groups e.g. Sˇvehla at TUM (Sˇvehla D, Rothacher M (Sˇvehla and Rothacher 2002) and Bock at Bern (Bock 2003) but because of different outliers detection and data processing strategies, the GPOD results presented here are more or less different than the other groups’ results. The estimated geometrical orbit of CHAMP is point-wise and its accuracy relies on the geometrical status of the GNSS satellites and on the number of the tracked GNSS satellites as well as on the GNSS measurement accuracy in the data

A. Shabanloui (*) K.H. Ilk Institute of Geodesy and Geoinformation, University of Bonn, Nussallee 17, 53115, Bonn, Germany e-mail: [email protected] S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_23, # Springer-Verlag Berlin Heidelberg 2012

187

188

A. Shabanloui and K.H. Ilk

processing. The position accuracy of 2–5 cm of CHAMP based on high–low GPS carrier phase observations with zero difference procedure has been achieved. These point-wise absolute positions can be used to estimate kinematical orbit of the LEOs.

23.1

Introduction

Among many possibilities to determine precise orbit of LEOs based on GNSS observations, zero differenced of high–low GPS-SST observations makes possible to be independent of the ground GNSS stations. In other words, geometrical LEO orbit can be estimated point-wise at the tracked epochs with only high–low observations which connect the GPS satellites to LEO with high precision. With the proposed method, only table of absolute positions at the desired epochs are estimated and subsequently no velocities and no accelerations and other kinematical parameters can be directly estimated in this procedure. In order to estimate kinematical parameters (e.g. velocities) specified functions or polynomials have to be fitted to the determined geometrical absolute positions of a LEO based on high–low GPS-SST observations. In this paper, methodology, numerical tests for the CHAMP satellite and some conclusions for the proposed geometrical precise orbit determination are presented.

the outliers in the high–low GPS-SST observations have to be flagged in the pre-processing process and excluded from the main GPS data processing. The detection methods are described in Sect. 23.2.2 in more detail.

23.2.1 Zero Differenced High–Low GPS-SST Observations In the zero (un-)differenced observations (refer to Fig. 23.1), always only the geometrical connection between GPS satellites (as sender) and GPS receiver on-board LEO (as receiver) is used to estimate the LEO geometrical orbit (Shabanloui 2008). The zero differenced carrier phase GPS-SST observations between the GPS satellite s and the LEO satellite r at frequency i with respect to the ambiguity parameter and all the error terms can be written as (Shabanloui 2008) si

23.2

E e3

Methodology

Zero differenced procedure of GPS-SST observations is used to estimate the geometrical absolute positions and the clock offset of a LEO. The approximated LEO absolute positions and the clock offsets can be determined epoch-wise based on code pseudo-range GPS-SST observations. The geometrical LEO absolute positions are improved with the accurate screened carrier phase observations. It has to be mentioned that the estimator, based on the least squares assuming standard normal distribution is not robust. Therefore, any outliers in the high–low GPS-SST observations, prior variances or other kinds of random parameters could dramatically affect the estimates of the unknown parameters and the variance components. Therefore,

Geocenter

rsi(t – –r tsi)

r

r(t)

E e2 E e1

Fig. 23.1 Zero differenced geometrical orbit determination

23

Pure Geometrical Precise Orbit Determination of a LEO s Fsr;i ðtÞ ¼ li fsr;i ðtÞ ¼ rsr ðtÞ cdtr ðtÞ þ li Nr;i

þ xsr;Fi ðtÞ þ eFi ðtÞ

(23.1)

s s where at epoch t, Fr,j and fr,j are the observed carrier phase between the GPS satellite s and the LEO GPS receiver r at frequency of i in unit of length and cycle respectively, rr,s is the true geometrical distance between the corresponding GPS and LEO, cdtr is the s LEO GPS receiver clock offset, Nr,j is the ambiguity parameter, li is the wavelength of the given GPS signal at frequency of i. The terms xsr,Фi and eФi represent the summation of all error effects on the LEO and the GPS satellites and the remaining error that cannot be modeled in the carrier phase observations, respectively (Shabanloui 2008). In the zero differenced principal, the error terms either have to be modeled with accurate specified models or have to be eliminated in the data processing procedure. For example, to eliminate the largest part of ionospheric effect on the carrier phase observations, the linear combination (L3) of the carrier phase observation at two frequencies (L1) and (L2) has been used. Consequently, the ionosphere free carrier phase observations at the epochs t can be written as (Shabanloui 2008):

s þ xsr;F3 ðtÞ þ eF3 ðtÞ Fsr;3 ðtÞ ¼ rsr ðtÞ cdtr ðtÞ þ l3 Nr;3

(23.2) with considering the GPS signal sending time and the Sagnac effect on the carrier phase observation, the observation equation can be rewritten as (Shabanloui 2008), Fsr;3 ðtÞ ¼ RZ ðoe tsr Þrs ðt tsr Þ rr ðtÞ s þ l3 Nr;3 cdtr ðtÞ þ xsr;F3 ðtÞ þ eF3 ðtÞ

(23.3) or the observation equation reads as (Leick 1995), s Fsr;3 ðtÞ ¼ rsm ðt tsr Þ rr ðtÞ þ l3 Nr;3 cdtr ðtÞ þ xsr;F3 ðtÞ þ eF3 ðtÞ

(23.4)

where oe, tsr and Rz(oetsr) represent rotation rate of the Earth, GPS signal travel time between sender and receiver and rotation matrix of the ITRF around the Z axis by the angle oetsr, respectively. The term rs(ttsr) represents the absolute position of the GPS

189 s satellite s at the sending time, rm (ttrs) is the GPS satellite position after applying the GPS signal travel time and the Sagnac effect and finally rr(t) is LEO receiver r at the receiving time, respectively.

23.2.2 Data Pre-processing For all applications of GPS, an efficient pre-processing and data screening of GNSS observations are necessary. It is particularly an important issue for the processing of the high–low GPS-SST observations of space-borne GNSS receivers on-board LEOs to determine precise orbit of the LEOs. In other words, any orbit determination procedure based on high–low GPS-SST observations depends crucially on the ability to remove invalid or degraded observations from the estimation procedures. It is clear that code pseudo-range GPS-SST observations are only used to determine initial geometrical absolute positions of a LEO. The procedure applied to detect outliers in the code observations is based on the “Majority voting”, which used the estimated LEO clock offset at every epoch (Bock 2003). In other words, the Majority voting is based on an epoch-wise processing of the GPS-SST observations. Therefore, at the first step, the receiver clock is synchronized with the GPS time, then based on the Majority voting algorithm, the bad observations are flagged to be excluded from GPS data processing. Based on the iterative least squares data processing, these flagged observations may be detected and subsequently be excluded from the main GPS data processing. The same Majority algorithm can be applied to the subsequent time differenced carrier phase observations between two epochs to detect outliers in the high–low carrier phase observations (Bock 2003). But the key factor limiting the performance of the pre-screening procedure of the carrier phase observations is the quality of the a-priori LEO absolute positions. Therefore, the rejection threshold should be set not too small. Consequently, a second screening procedure has to be applied to the carrier phase observations to find all bad observations which deteriorate the solution. The second issue, in case of carrier phase observation, is the cycle slips in the high–low GPS-SST observations. In order to obtain high precision orbit based on the carrier phase observations, cycle slips in the carrier phase observations have to be detected,

190

A. Shabanloui and K.H. Ilk

identified and repaired at the pre-processing stage. A slip of only few cycles can influence the measurements such that a geometrical precise orbit cannot be achieved. Therefore, the detection of the cycle slips in the carrier observations is the major issue in the geometrical precise orbit determination procedure. Repairing process in this case needs to determine the size of cycle slips at the every carrier phase frequency, but sometimes it is very difficult to identify the size of an every frequency part. Therefore, after identification of the cycle slips, to avoid the repairing of the carrier phase, as an alternative, the new ambiguity parameters can be introduced to detected carrier phases in the GPS main processing. In order to detect the cycle slips in carrier phase observation, the observations or combination of observations have to be used sensitively to the cycle slips. Therefore, the detection of cycle slips in this case can be performed based on the quantities which based on the geometrical free combination of carrier phase observation at Li frequencies (i.e. fi, i ¼ 1,2) and code pseudo-range observations. The geometrical free combination (frs, GF(t) of carrier phase observation can be formulated as (Hofmann-Wellenhof et al. 2001):

GPS ambiguity terms, but is non-linear with respect to LEO absolute position. Therefore, to estimate geometrical absolute position, the observation equation has to be linearized. The linearized ionosphere-free GPSSST carrier phase observation reads (Xu 2007),

f1 s f ðtÞ f2 r;2 f1 s bion ðtÞ f2 s ¼ Nr;1 Nr;2 ð1 12 Þ þ ef ðtÞ f2 f1 f2 (23.5)

which xr(t), yr(t) and zr(t) represents the coordinates of the LEO absolute position at time t. The linearized observation can be rewritten as (Shabanloui 2008),

fsr;GF ðtÞ ¼ fsr;1 ðtÞ

the left side of the geometrical free carrier phase observations at two frequencies shows that only time-varying quantity is the ionosphere term bion. This equation shows that the ionosphere effect has been reduced with respect to the original carrier phase observations (Shabanloui 2008). Now, if there are no cycle slips, the temporal variations of the geometrical free of carrier phase observations would be small for normal ionosphere conditions. The geometrical free observations have disadvantages to determine the cycle slips on L1 and L2 or both, but with introducing new ambiguities, this problem can be solved.

23.2.3 Solution Obviously, the carrier phase GPS-SST observation is linear with respect to the LEO clock offset and the

Fsr;3 ðtÞ ¼ Fsr;3;0 ðtÞ þ

@Fsr;3 ðtÞ ðx x0 Þ @x

(23.6)

where x is a vector of the unknown parameters which contains the LEO absolute position, the LEO clock offset and GPS ambiguity terms at the time t respectively; and x0 is a vector of the initial values of the unknown parameters as follows, 0

1 xr ðtÞ B yr ðtÞ C B C B zr ðtÞ C B C B C x :¼ B cdtr ðtÞ s1 C; B l3 Nr;3 C B C B .. C @ . A sn l3 Nr;3

x0 :¼ xj0

(23.7)

DFsr;3 ðtÞ ¼ asr ðtÞDxðtÞ; wsr ðtÞ ¼

s20 cosðzsr ðtÞÞ: s2Fs

(23.8)

r;3

The terms wrs(t), s2Fs and zrs(t) represent the obserr;3 vation weight, carrier phase observation variance and zenith distance of the GPS satellite s from the GPS receiver r, respectively. If we assume that, a number n of GPS satellites si, i¼1, . . ., n are available at the time t. The Gauss-Markov model for all observed carrier phase observations corresponding to weight matrix with considering the unit vector between the GPS satellite s and LEO on-board receiver r reads (Shabanloui 2008), rsm ðt tsr Þ rr ðtÞ esri ðtÞ ¼ rs ðt ts Þ rr ðtÞ m r i i esr;y esr;zi ; ¼ esr;x

(23.9)

23

Pure Geometrical Precise Orbit Determination of a LEO

1 1 DFsr;3 ðtÞ C B .. A @ . n ðtÞ DFsr;3 1 esr;y ðtÞ esr;z1 ðtÞ 1 .. .. .. . . . n esr;y ðtÞ esr;zn ðtÞ 1

23.3

0

0

1 esr;x ðtÞ B .. ¼@ . n ðtÞ esr;x

191

1 .. . 0

1 0 0 .. .. C: . .A 0 1

0

1 Dxr ðtÞ B Dyr ðtÞ C B C B Dzr ðtÞ C B C B C :B DcdtrsðtÞ C þ «ðtÞ; 1 B l3 Nr;3 C B C B C .. @ A . sn l3 Nr;3 Wl ðtÞ ¼ diagðwsr1 ðtÞ wsrn ðtÞÞ:

(23.10)

Obviously, at the first epoch, the number of observations are smaller than the number of the unknowns (under determined problem). Therefore, the carrier phase observations of the other epochs have to be summed up and the observation equation has to be solved in a batch processing for all desired epochs. The observation equations for all observed epochs can be written as, « þ Dl ¼ ADx; Wl

(23.11)

by a least square adjustment, the unknowns read 1 T D^x ¼ AT Wl A A Wl Dl:

Numerical Tests

To verify the proposed geometrical orbit determination procedure, four 30 min arcs (short arcs) of CHAMP have been selected. As a pre-processing step, the outliers and cycle slips in carrier phase observations are removed with the “Majority voting” technique and different combination form of the observations. Finally, to minimize or mitigate the effect of multi-path effect nearby weighting the carrier phase observations with respect to the GPS zenith distance, a 15 cut-off angle has been applied to the observations. The ground tracks of four CHAMP short arcs are shown in Fig. 23.2. Because of the fact that the accuracy of GPS-SST carrier phase observations are in the range of some millimeters, one can expect that the millimeter level of accuracy for the GPS receiver is achievable. In case of GPS receivers at the ground stations, the problematic modeling of the atmosphere parameters can reduce the accuracy of GPS estimated positions. The situation is less critical for GPS receiver on-board LEO. There is no tropospheric effect in the altitude of LEOs (e.g. 430 km); but at this altitude, the ionosphere can dramatically affect the GPS-SST observations. The multipath effect is another major problem in the data processing of LEOs. To validate the estimated geometrical CHAMP orbit externally, the CHAMP PSO dynamical orbits provided by GFZ-Potsdam have been used. In Fig. 23.3, the differences between estimated CHAMP absolute positions and PSO dynamical orbit for the four short arcs (cases a–d) are shown. Figure 23.4 represents the carrier phase GPS-SST observations for four short arcs. It should be pointed

(23.12)

The estimated unknowns are the corrections to the LEO absolute positions, the LEO clock offsets and the GPS ambiguity terms. Because of the non-linearity of the observations, the convergence is achieved after few iterations e.g. after i iterations; subsequently the unknowns vector and variance-covariance read,

90°N

d

60°N 30°N

c

0° 30°S

b

a

60°S

^xi ¼ ^xi1 þD ^ xi ;

1 C^xi ¼ AT Wl A

(23.13)

the estimated unknowns are the LEO absolute positions, the LEO clock offsets and the float GPS ambiguity terms at all desired epochs.

90°S 180°W

120°W

60°W

0°

60°E

120°E

180°

Fig. 23.2 The ground tracks of four 30 min short arcs for the time: (a) 2002 03 21 13h 30m 0.0s – 14h 00m 0.0s, (b) 2002 07 20 12h 48m 0.0s – 13h 18m 0.0s, (c) 2003 03 21 17h 20m 0.0s – 17h 50m 0.0s, (d) 2003 03 31 17h 00m 0.0s – 17h 30m 0.0s

a

A. Shabanloui and K.H. Ilk 0.12

dx

dy

dz

Diff. (m)

0.08 0.04 0 -0.04

a

0.02

Residuals (m)

192

0.01 0 -0.01 -0.02

-0.08

52354.565

52354.565 52354.57 52354.575 52354.58 MJD (days) 0.08

dx

dy

dz

Diff. (m)

0.04 0

52354.575

52354.58

MJD (days)

b

0.02

Residuals (m)

b

52354.57

0.01 0 -0.01

-0.04 -0.02

-0.08

52475.535

MJD (days)

c

0.2

dx

dy

dz

Diff. (m)

0.1 0

c

0.012

Residuals (m)

52475.535 52475.54 52475.545 52475.55 52475.555

0.008

-0.1

0 -0.004 -0.008 52719.725 52719.73 52719.735 52719.74 MJD (days)

MJD (days) dx

dy

dz

Diff. (m)

0.1 0

d

0.012

Residuals (m)

52719.725 52719.73 52719.735 52719.74

d

52475.55

0.004

-0.2

0.2

52475.54 52475.545 MJD (days)

0.008 0.004 0 -0.004 -0.008 -0.012

-0.1

52729.71 52729.715 52729.72 52729.725 52729.73 MJD (days)

-0.2 52729.71 52729.715 52729.72 52729.725 52729.73 MJD (days)

Fig. 23.3 Absolute position differences between estimated geometrically orbit and PSO dynamical CHAMP orbit for the case (a), case (b), case (c) and case (d)

out that the residuals for all four short arcs are in the range of 2 cm. The geometrical absolute positions of CHAMP can be estimated with an accuracy of 2–5 cm. In the cases (c) and (d), the residuals in some epochs show zero values which means availability of minimum four GPS satellites at these epochs. To show real precision of estimated CHAMP orbits, polynomial of degree five seems to be a proper degree to fit to the differenced ephemeris. Therefore, a polynomial of degree five is fitted to the differences between the

Fig. 23.4 Carrier phase GPS-SST observation residuals for case (a), case (b), case (c) and case (d)

estimated CHAMP orbit and the PSO dynamical orbit. After removing trend with the polynomial of degree five, RMS values of orbits is shown in Table 23.1. Table 23.1 RMS values of the estimated CHAMP orbits w.r.t CHAMP PSO dynamical orbit after removing trend with polynomial degree five Case a b c d

X(m) 0.0047 0.0024 0.0140 0.0087

Y(m) 0.0096 0.0062 0.0171 0.0073

Z(m) 0.0122 0.0075 0.0085 0.0105

23

Pure Geometrical Precise Orbit Determination of a LEO

23.4

Conclusions

Conventional GNSS-based POD strategies rely on the GNSS observations from a terrestrial network of ground receivers (IGS network) as well as the GNSS receiver on-board LEO in different difference data processing modes. Advent of GNSS orbits and clock offset, the geometrical precise orbit of LEO has been realized based on high–low GPS-SST observations. Zero differenced procedure provides an efficient possibility to estimate geometrical orbit of a LEO. The proposed geometrical LEO POD strategy could be characterized as pure geometrical type, point-wise and three-dimensional. In other words, only geometrical parameters (e.g. absolute positions) can be estimated based on the geometrical observations. An efficient data pre-processing and data screening procedure are very important to achieve precise geometrical LEO orbits based on GPS-SST observations. Specially, the detection, identification and removing (repairing) procedure are the most important issues in the main data processing. In this paper, real case results demonstrate that an accuracy of centimeter has been achieved for the geometrical orbit of CHAMP. It has to be mentioned that the estimated geometric precise LEO orbits can be used for further investigations e.g. determination of kinematical (continuous) LEO orbit for space borne recovery of the Earth gravity field based on GPS-SST observations. It is clear that,

193

geometrical or kinematical POD of a LEO have not been affected by the dynamical information from the Earth gravity field. Acknowledgment We gratefully acknowledge the financial support of the “German Federal Ministry for Education and Research” (BMBF) under the project “LOTSE-CHAMP/ GRACE”.

References Bock H (2003) Efficient methods for determining precise orbits for low earth orbiters using the global positioning system (GPS). PhD thesis, Astronomical Institute, University of Bern, Switzerland Hofmann-Wellenhof B, Lichtenegger H, Collins J (2001) GPS, theory and practice. Springer, New York Leick A (1995) GPS satellite surveying, 3rd edn. Wiley, New York Shabanloui A (2008) A new approach for a kinematic-dynamic determination of low satellite orbits based on GNSS observations. PhD thesis, Department of Astronomical, Physical and Mathematical Geodesy, Institute of Geodesy and Geo-information, University of Bonn, Germany Sˇvehla D, Rothacher M (2002) Kinematic orbit determination of LEOs based on zero or double difference algorithms using simulated and real SST data. In: Adam J, Schwarz K-P (eds) Vistas for geodesy in the new millennium, vol 125. Springer, Berlin, pp 322–328 Xu G (2007) GPS, theory, algorithms and applications, 2nd edn. Springer, Berlin

.

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field: Boundary Problems and a Target Function

24

P. Holota and O. Nesvadba

Abstract

The purpose of the presentation is to discuss the combination of terrestrial and satellite gravity field data and to show its spectral and space domain interpretation. Potential theory is of key importance in this field. However, the problems discussed are overdetermined by nature. Therefore, methods typical for the solution of boundary-value problems are used together with an optimization concept, target functions and regularization techniques. Two cases are treated. They are motivated by the use of a satellite-only model of the gravity field of the Earth or by data coming from satellite missions (especially GOCE) in common with terrestrial gravity measurements. For the results reached in the spectral domain summation techniques are applied in order to find the interpretation of the results in terms of kernel (Green’s) functions related to the particular combination scheme. This enables to show the tie between the global and the local modelling of the gravity field. In order to demonstrate the efficiency of the procedure results of numerical simulations are added. Differences of the closed loop simulation are very promising.

24.1

Introduction

In this paper a concept is discussed enabling combination of terrestrial gravity measurements with a satellite-only model of the Earth’s gravity field or with data coming from satellite missions, in particular GOCE [treated in terms of the space-wise approach, see e.g.

P. Holota (*) Research Institute of Geodesy, Topography and Cartography, 250 66 Zdiby 98, Praha-vy´chod, Zdiby, Czech Republic e-mail: [email protected] O. Nesvadba Land Survey Office, Pod Sı´dlisˇteˇm 9, 182 11, Praha 8, Czech Republic e-mail: [email protected]

Migliaccio et al. (2004)]. The paper ties to Holota (2007) and Holota and Nesvadba (2006, 2007, 2009). In the sequel OTOPO means a domain bounded by two surfaces, from above by a geocentric sphere of radius Re and from below by a surface GTOPO , which is star-shaped at the origin and about as irregular as the surface of the Earth. Moreover, let h be the height of the boundary point x 2 GTOPO above a geocentric sphere of radius Ri . We will consider the following two problems D T ¼ 0 in jxj

@T þ 2T ¼ jxjDg @jxj

O TOPO for

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_24, # Springer-Verlag Berlin Heidelberg 2012

x 2 GTOPO

(24.1) (24.2)

195

196

P. Holota and O. Nesvadba

T ¼ t for

jxj ¼ Re

(24.3)

where t means the input from an available satelliteonly model of the Earth’s gravity field and D T ¼ 0 in jxj

@T þ 2T ¼ jxjDg @jxj @2 T @ j x j2

¼G

O TOPO for

for

x 2 GTOPO jxj ¼ Re

(24.4) (24.5)

(24.6)

where G represents the input from satellite gradiometry. Note also that DT means Laplace’s operator applied on T, Dg is the gravity anomaly, x ðx1 ; x2 ; x3 Þ, jxj ¼ ðS3i¼1 x2i Þ1=2 and x1 ; x2 ; x3 are the usual rectangular Cartesian co-ordinates in threedimensional Euclidean space R3 . Our starting point will be the transform of the domain O TOPO given by x1 ¼ ½ r þ oðrÞ hð’; lÞ cos ’ cos l

(24.7)

x2 ¼ ½ r þ oðrÞ hð’; lÞ cos ’ sin l

(24.8)

x3 ¼ ½ r þ oðrÞ hð’; lÞ sin ’

(24.9)

where ’ and l are the spherical coordinates of the point x 2 O TOPO (geocentric latitude and longitude, respectively), r is defined by jxj ¼ r þ oðrÞ hð’; lÞ and oðrÞ is a suitably chosen and twice differentiable auxiliary function of r 2 ½Ri ; Re . The transformation, as defined in Holota and Nesvadba (2009), is a oneto-one mapping between O TOPO and a domain O, O f x; x 2 R3 ; Ri < jxj Re is not regular at infinity, for r ! 1 it does not decrease as c=r (c is a constant) or faster. This is

198

P. Holota and O. Nesvadba

caused by measurement errors. The data given for r ¼ Ri are sufficient to determine a harmonic function in Oext fx 2 R3 ; r>Ri g and thus in O Oext . The data for r ¼ Re are excess data and give rise to “interðeÞ nal” terms ðr=Re Þn T~n that are not regular at infinity. In solving this overdetermined problem, we will look for a harmonic function f , which is regular at infinity and minimizes the functional ð Fðf Þ ¼

ð ðf ; gÞ1 ¼

hgrad f ; grad gi dx

(24.36)

Oext ð1Þ

We look for a function f 2 H2 ðOext Þ that minimizes the functional ð Cðf Þ ¼ jgradðf T~ Þj2 dx

(24.37)

O 2

ðf T~ Þ dx

(24.31)

O

In particular we suppose that f 2 H2 ðOext Þ, where H2 ðOext Þ is a space of harmonic functions with inner product ð ðf ; gÞ

In this case the function f is defined by the following integral identity ð

ð

hgrad f ; gradvidx ¼ hgrad T~ ; gradvidx (24.38)

O

O ð1Þ

r 2 fg dx

(24.32)

valid for all v 2 H2 ðOext Þ. In consequence f is given by (24.34), but now with an ¼ 0 for all n.

Oext

Hence, assuming F has its minimum at a point f 2 H2 ðOext Þ, we know that Gaˆteaux’ differentials of F equals zero at the point f . This yields ð

ð fv dx ¼ T~ v dx for all v 2 H2 ðOext Þ

O

(24.33)

O

24.6

ð1Þ

Optimization in H2 -Traces ð1Þ

Considering the fact that functions from H2 ðOext Þ have precisely defined traces on the boundary @O of ð1Þ

O, we can also look for f 2 H2 ðOext Þ that minimizes the functional ð

and we easily obtain that

Yðf Þ ¼

1 nþ1 h i X Ri T~nðiÞ þ an T~nðeÞ f ¼ r n¼0

2 ð f T~ Þ dS

(24.34)

Again, Y attains its minimum and ð

with

ð fv dS ¼

an ¼

ð2n 1Þð 1 q2 Þ n2 q 2ð 1 q2n1 Þ

@O

(24.35)

see Fig. 24.1. In particular a0 ¼ ð1 þ qÞ=2q and lim an ¼ 0 as n ! 1, see Holota and Nesvadba (2006, 2007).

Optimization in ð1Þ

ð1Þ H2

Let H2 ðOext Þ be the space of harmonic functions on Oext equipped with inner product

T~ v dS

(24.40)

@O

ð1Þ

valid for all v 2 H2 ðOext Þ is the respective condition for Y to have a minimum at the point f . Hence, the function f can be represented by (24.34) too, but with an ¼ bn , n ¼ 1; 2; . . . 1 and bn ¼

24.5

(24.39)

@O

1 þ q n1 q 1 þ qn1

(24.41)

see Fig. 24.1. One can show that b0 ¼ 1 and that lim bn ¼ 0 as n ! 1.

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field

Fig. 24.1 Values of an and bn for Ri ¼ 6; 378 km and two cases of Re : Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km, i.e., for q ¼ 0:96228 and q ¼ 0:94099, respectively

199

1,2

· a n for q = 0, 96228

1,0

a n for q = 0, 94099

0,8

+ b n for q = 0, 96228

0,6

× b n for q = 0, 94099

0,4 0,2 degree n 0,0 0

24.7

AðiÞ n ¼

Optimized Solution: Target Function Fðf Þ

In order to show more clearly how the optimization works we have to return to the structure of the ðiÞ ðeÞ harmonics T~n and T~n . For the combination of gravimetry and a satellite-only model they are given by (24.25) and (24.26). Thus the optimization leads to f ¼

1 nþ1 X Ri n¼0

r

AðiÞ n

Ri f Dg þ AðeÞ n tn n1 n

25

(24.42)

with nþ1 Þ=Dn AðiÞ n ¼ ðn 1Þð 1 an q

(24.43)

n AðeÞ n ¼ ½ðn þ 2Þq þ an ðn 1Þ =Dn

(24.44)

f¼

100

125

AðiÞ n

r

Ri f R2e Dgn þ þAðeÞ Gn n n1 ðn þ 1Þðn þ 2Þ

AðeÞ n ¼

ðn þ 1Þðn þ 2Þ ½ðn þ 2Þqn þ an ðn 1Þ Dgn (24.47)

illustrated in (comparative) Fig. 24.3. For a greater range of degree n the plots shown in Figs. 24.2 and 24.3 can be seen in Holota and Nesvadba (2006, 2007).

24.8

(24.45)

150

n1 nðn 1Þ an ðn þ 1Þðn þ 2Þqnþ1 g Dn (24.46)

Optimization – Comparizon: Target Functions Fðf Þ, Cðf Þ and Yðf Þ

Similarly as for the optimization based on the target function Fðf Þ we can proceed in case of target functions Cðf Þ and Yðf Þ. The optimized solution is again given by (24.42) for the first problem and ðiÞ ðeÞ (24.45) for the second problem, but with An and An illustrated in Figs. 24.2 and 24.3.

1 nþ1 X Ri n¼0

with

75

and

and

illustrated in (comparative) Fig. 24.2. For the combination of gravimetry and satellite ðiÞ ðeÞ gradiometry T~n and T~n are given by (24.29) and (24.30). Thus the optimization leads to

50

24.9

Space Domain Interpretation: Terrestrial Term

Our aim is now to sum the series

200

P. Holota and O. Nesvadba

Φ(f)

Ψ (f)

Θ (f )

2,0

2,0

2,0

• q = 0, 96228

q = 0, 94099 1,5

1,5

1,5 (i)

(i)

1,0

(i)

An

An

0,5

An

1,0

1,0

0,5

0,5

(e)

An

0,0

0,0

0,0 (e)

(e)

An

An

degree n -0,5

-0,5 0

2

-0,5 0

50

25

50

0

25

50

ðeÞ Fig. 24.2 Combination of gravimetry and a satellite-only model: Coefficients AðiÞ n and An for target functions Fðf Þ, Cðf Þ, Yðf Þ and for q ¼ 0:96228 and q ¼ 0:94099, i.e., for Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km

Φ( f )

Ψ (f)

2,0

Θ (f)

2,0

2,0

1,5

1,5

• q = 0, 96228

q = 0, 94099 1,5 (i)

1,0

0,5

(i)

(i)

An

An

An 1,0

1,0

0,5

0,5

(e)

An

0,0

0,0

0,0 (e)

-0,5

-0,5 0

25

(e)

An

degree n 50

An -0,5

0

25

50

0

25

50

ðeÞ Fig. 24.3 Combination of gravimetry and satellite gradiometry: Coefficients AðiÞ n and An for target functions Fðf Þ, Cðf Þ, Yðf Þ and for q ¼ 0:96228 and q ¼ 0:94099, i.e., for Re ¼ Ri þ 250 km and Re ¼ Ri þ 400 km

fterr ¼

1 nþ1 X Ri n¼0

r

AðiÞ n

Ri f Dg n1 n

(24.48)

representing the terrestrial term in (24.42) or (24.45) for the target function Fðf Þ. To our opinion the function combines the effect of boundary data and the smoothness in constructing the optimized solution optimally. For the first problem we obtain

R2i 1q f R3 2 ð1 þ qÞq f Dg1 Dg0 þ 2i r 2ð2q 1Þ r 6q2 ð Ri f ds þ S ðr; cÞ Dg 4p s (24.49) 1 X 2n þ 1 Ri nþ1 AðiÞ Pn ðcos cÞ S ðr; cÞ ¼ n r n1 n¼2

fterr ¼

(24.50)

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field

201

15 Classical Stokes' function S(psi) = S(1,psi) 12 S* Kernel, terrestrial data and a satellite-only model, q = 0.94099 9

S* Kernel, terrestrial data and a satellite-only model, q = 0.96228 S* Kernel, gravimetry and satellite gradiometry, q = 0,94099

6 S* Kernel, gravimetry and satellite gradiometry, q = 0.96228 3 0 -3 -6 psi ® -9 0°

30°

60°

90°

120°

150°

180°

Fig. 24.4 Kernel functions in the integral representation of the terrestrial term (Stokes’ function drawn for comparison) ðiÞ

with An given by (24.43). Note that c is the angle between the computation and the running point of the integration. Similarly, for the second problem we have fterr ¼

R2i 1 þ q f R3 1 þ q f Dg0 2i Dg1 r 4q r 6q2 ð Ri f ds þ S ðr; cÞ Dg 4p s

ð1Þ

fsat ¼

1 nþ1 X Ri n¼0

r

AðeÞ n tn

(24.52)

and (24.51)

where the kernel S ðr; cÞ appears again, but with ðiÞ coefficients An given by (24.46). It is clear that for the space domain interpretation of the terrestrial term the dependence of S ðr; cÞ on the angle c is of considerable instructive value. For r ¼ Ri the diagrams were derived (indirectly) from summation of about 300–450 first terms of the difference between the series expression of Stokes’ function and the series in (24.50), see Fig. 24.4. The figure shows that in fterr the influence of distant zones is considerably suppressed.

24.10 Space Domain Interpretation: Satellite Term Of course, our aim also is to sum the satellite term

ð2Þ fsat

¼

1 nþ1 X Ri n¼0

r

AðeÞ n

R2e Gn ðn þ 1Þðn þ 2Þ

(24.53)

in (24.42) and (24.45), respectively. After some algebra we have ( ) ð ð1Þ ð1Þ S ðr; cÞ 0 t 1 fsat ¼ ds ð2Þ ð2Þ S ðr; cÞ G 4p s 0 fsat (24.54) 1 X Ri nþ1 AðeÞ Pn ðcos cÞ Sð1Þ ðr; cÞ ¼ n ð2n þ 1Þ r n¼0 (24.55) ðeÞ

with An given by (24.44) and Sð2Þ ðr; cÞ ¼

1 X n¼0

ðeÞ

AðeÞ n

nþ1 2n þ 1 Ri Pn ðcos cÞ ðn þ 1Þðn þ 2Þ r

with An given by (24.47). The kernels Sð1Þ and Sð2Þ are illustrated in the following figure (Fig. 24.5).

202 Fig. 24.5 Kernel functions in the integral representation of the satellite terms

P. Holota and O. Nesvadba 8000

10

7000

9

S^(1) Kernel: gravimetry and a satellite only model, Re = Ri + 400km

6000 5000

8 7

S^(1) Kernel: gravimetry and a satellite only model, Re = Ri + 250km

4000

S^(2) Kernel: gravimetry and satellite gradiometry, Re = Ri + 400km S^(2) Kernel: gravimetry and satellite gradiometry, Re = Ri + 250km

6 5

3000 4 2000

3

1000

2

0

1

psi →

-1000

psi →

0 0°

30°

60°

90°

0°

30°

60°

90°

Fig. 24.6 (Above) fterr and (below) fsat for r ¼ Ri in case of the combination of gravimetry and a satellite-only model (r ¼ 6; 378 km and Re ¼ Ri þ 250 km, i.e. q ¼ 0:96228)

24

On a Combined Use of Satellite and Terrestrial Data in Refined Studies on Earth Gravity Field

203

Fig. 24.7 (Above) fterr and (below) fsat for r ¼ Ri in case of the combination of gravimetry and a satellite gradiometry (r ¼ 6; 378 km and Re ¼ Ri þ 250 km, i.e. q ¼ 0:96228)

24.11 Numerical Simulations and Final Comments The procedure discussed here was tested by means of simulated data. For W we took the EGM08 potential from Pavlis et al. (2008) and U was the potential of the Somigliana-Pizzetti normal gravity field with parameters of the GRS1980 system. f was simulated at more than The anomaly Dg 10,000,000 points of a “regular” grid on the sphere of radius Ri ¼ 6; 378 km given by the tenth level of the icosahedron refinement. The hierarchically created grids were also exploited for Romberg’s integration method in calculating the integrals in (24.49), (24.51) and (24.54), see Nesvadba et al. (2007).

Similarly, also the data t and G were simulated at more than 10,000,000 points of a “regular” grid, but on the sphere of radius Re ¼ Ri þ 250 km. The restrictions of T ¼ W U and @ 2 [email protected] 2 were used for this purpose. The case of gravimetry and a satellite-only model was treated first. The terrestrial term represented by (24.49) and the satellite term given by the first of (24.54) were computed for r ¼ Ri and Re ¼ Ri þ 250 km. They are shown in Fig. 24.6. Figure 24.7 concerns the combination of gravimetry and satellite gradiometry. The fterr term is given by (24.51) and fsat by the second of (24.54). The computation was done for r ¼ Ri and Re as above. In both the cases the composition (optimized solution) f ¼ fterr þ fsat was compared with T as obtained

204

directly from W and U, and restricted to r ¼ Ri . The results of the comparison are nearly the same in both the cases considered. Globally the RMS of the differences expressed in GeoPotential Units (1 GPU 1 m2 s2 ) does not exceed 0.06 GPU (max. difference smaller than 0.9 GPU) and excellently conforms to an apriori estimate of the error obtained from Romberg’s integration method. The results well confirm our earlier conclusions derived from numerical simulations based on the gravity field model EGM96, see Holota and Nesvadba (2006, 2007). In addition the dependence of the kernels Sð1Þ and Sð2Þ on the angle c, shown in Fig. 24.5, offers a suitable springboard for investigations on polar cap problem in using satellite data. Acknowledgements The work on this paper was done within ESA GOCE Project ID: 4311. The presentation of the paper at the IAG 2009 Scientific Assembly “Geodesy for Planet Earth”, Buenos Aires, Argentina, August 31 to September 4, 2009, was sponsored by the Ministry of Education, Youth and Sports of the Czech Republic through Projects No. LC506. All this support is gratefully acknowledged.

References Holota P (2007) On the combination of terrestrial gravity data with satellite gradiometry and airborne gravimetry treated in terms of boundary-value problems. In: Tregoning P and

P. Holota and O. Nesvadba Rizos C (eds) Dynamic Planet. IAG Symp., Cairns, Australia, 22–26 Aug 2005. IAG Symposia, vol 130, Springer, Berlin etc., Chap 53, pp 362–369 Holota P and Nesvadba O (2006) Optimized solution and a numerical treatment of two-boundary problems in combining terrestrial and satellite data. Proc. 1st Intl. Symp. of the IGFS, 28 Aug-1 Sept, 2006, Istanbul, Turkey. Spec. Issue: 18, Genl. Command of Mapping, Ankara, 2007, pp 25–30 Holota P and Nesvadba O (2007) A regularized solution of boundary problems in combining terrestrial and satellite gravity field data [CD-ROM]. Proc. ‘The 3rd International GOCE User Workshop’, ESA-ESRIN Frascati, Italy, 6–8 Nov 2006 (ESA SP-627, Jan 2007), pp 121–126 Holota P and Nesvadba O (2009) Domain transformation, boundary problems and optimization concepts in the combination of terrestrial and satellite gravity field data. In: Sideris M (ed) Observing our changing earth, Proc. 2007 IAG Gen. Assembly, Perugia, Italy, July 2–13, 2007. IAG Symposia, vol 133, Springer, Berlin etc. pp 219–228 Migliaccio F, Raguzzoni M, Sanso` F (2004) Space-wise approach to satellite gravity field determination in the presence of coloured noise. J Geod 78:304–313 Nesvadba O, Holota P and Klees R (2007) A direct method and its numerical interpretation in the determination of the Earth’s gravity field from terrestrial data. In: Tregoning P and Rizos C (eds) Dynamic Planet. IAG Symp., Cairns, Australia, 22–26 Aug 2005. IAG Symposia, vol 130, Springer, Berlin etc., Chap. 54, pp 370–376 Pavlis NK, Holmes SA, Kenyon SC and Factor JK (2008) An Earth Gravitational Model to Degree 2160: EGM2008. Presented at the 2008 General Assembly of the European Geosciences Union, Vienna, Austria, April 13–18, 2008 Sokolnikoff IS (1971) Tensor analysis, theory and applications to geometry and mechanics of continua. Nauka Publishers, Moscow (in Russian)

Moho Estimation Using GOCE Data: A Numerical Simulation

25

Mirko Reguzzoni and Daniele Sampietro

Abstract

The GOCE mission, exploiting for the first time the concept of satellite gradiometry, promises to estimate the Earth’s gravitational field from space with unprecedented accuracy and spatial resolution. Also inverse gravimetric problems can get benefit from GOCE observations. In this work the general problem of estimating the discontinuity surface between two layers of different density is investigated. A possible solution based on a local Fourier analysis and Wiener deconvolution of satellite data (such as gravitational potential and its second radial derivative) is proposed. Moreover a numerical method to combine in an efficient way gridded satellite data with sparse ground data, like gravity anomalies, has been implemented. Numerical simulations on different synthetic Moho profiles have been carried out. Finally a two-dimensional simulation on realistic data over the Alps has been set up. The results confirm that GOCE data can significantly contribute to the detection of geophysical structures, leading to a much better determination of the signal long wavelengths (up to about 200 km). The use of local ground data improves the satellite-only estimate, making possible the recovery of higher resolution details.

25.1

Introduction

The crust-mantle boundary, the so called Moho discontinuity, has been studied with profitable results by means of seismic profiles and ground gravity anomalies

M. Reguzzoni Department of Geophysics of the Lithosphere, OGS, c/o Politecnico di Milano, Polo Regionale di Como, Via Valleggio 11, 22100 Como, Italy e-mail: [email protected] D. Sampietro (*) DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio 11, 22100 Como, Italy e-mail: [email protected]

(e.g. Grad and Tiira 2009). However there are areas where, due to the lack of observations, the crustal structure has not been yet determined by geophysical sounding. In such areas the Moho surface can be estimated using isostatic theories, but especially young orogenic regions are not necessarily in isostatic equilibrium. Since satellite data are collected worldwide, they can be profitably used to overstep these problems and correct isostatic models even where ground observations are not available. Studies on this subject have been done using GRACE data (see e.g. Shin et al. 2007) or simulated GOCE gradiometric data (see e.g. Benedek and Papp 2009). In our solution, the gravitational potential and its second radial derivative at

S. Kenyon et al. (eds.), Geodesy for Planet Earth, International Association of Geodesy Symposia 136, DOI 10.1007/978-3-642-20338-1_25, # Springer-Verlag Berlin Heidelberg 2012

205

206

M. Reguzzoni and D. Sampietro

satellite altitude, as observed from a GOCE like mission, are combined with point-wise observations at ground level (e.g. gravity anomalies) to model the Moho surface. This inverse gravimetric problem is in general ill-posed: in fact the solution is not unique (i.e. the space of all solutions corresponding to a fixed potential is infinite-dimensional) and not stable because the Newton operator is a smoother and its inverse is unbounded (Sanso´ 1980). In this work the non-uniqueness has been treated by taking peculiar hypotheses on the shape of the density discontinuity. In fact it has been proved that to guarantee the uniqueness of the solution it is possible to consider a two layer density discontinuity, in planar or spherical approximation, where the two different densities are known (see e.g. Sampietro and Sanso´ 2009 and the references therein). Note that for a lot of geophysical problems, included the one considered in this work, these kinds of approximations are commonly adopted and in general well satisfied (see e.g. Gangui 1998). In the first part of the paper the proposed algorithm is theoretically illustrated (Sect. 25.2) and numerically assessed (Sect. 25.3) on the basis of one-dimensional Moho profiles in planar approximation. Then the algorithm is generalized to the two-dimensional case and tested on a realistic simulation in the Alpine area (Sect. 25.4). The results obtained in both cases are presented in the paper, drawing conclusions and discussing future perspectives.

ð ð ð z0 þDðxÞ

TðxÞ ¼

Tz ðxÞ ¼

z0 hðxÞ

@ @z

ð ð ð z0 þDðxÞ z0 hðxÞ

Methodology

The analysis is developed accordingly to the inversion algorithm described in Reguzzoni and Sampietro 2008 and to the theory of partitioned (or stepwise) collocation solution (see Tscherning 1974). Let us recall here the main concepts. We consider a Cartesian coordinate system, where the (x, y) plane is at satellite level, x is oriented along track and z is pointing downward. The topography profile h(x) and the Moho profile D(x) are both defined with respect to the ground level z ¼ z0 ¼ 250 km. In order to solve the gravimetric inverse problem the following deterministic models have to be inverted (Heiskanen and Moritz 1967):

(25.1)

G r dxd dz qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 ðx xÞ þ þ ðz zÞ

z¼z0

(25.2)

Tzz ðxÞ ¼

@ @z2 2

ð ð ð z0 þDðxÞ z0 hðxÞ

G r dx d dz qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðx xÞ2 þ 2 þ ðz zÞ2

z¼0

(25.3) where T is the anomalous potential, G is the gravitational constant, r the density contrast and x, , z are integration variables in the x, y, z directions respectively (planar approximation). Modelling the Moho as DðxÞ ¼ D þ dDðxÞ, linearizing (25.1)–(25.3) around D in the z direction and finally integrating with respect to over a bounded interval, we get:

dTðxÞ ¼

dTz ðxÞ ¼

25.2

G r dx d dz qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðx xÞ2 þ 2 þ z2

dTzz ðxÞ ¼

ð þ1 1

ð þ1 1

ð þ1 1

kT ðx xÞ dDðxÞ dx

(25.4)

kTz ðx xÞ dDðxÞ dx

(25.5)

kTzz ðx xÞdDðxÞ dx

(25.6)

where dT(x), dTz(x) and dTzz(x) are the residual gravitational effects, i.e. the actual effect minus the effects due to the topography and to the mean Moho D, while kT, kTz and kTzz are convolution kernels (for details on the form of these kernels we refer again to Reguzzoni and Sampietro 2008). To estimate the Moho we have to solve the system obtained by inverting (25.4)–(25.6) degraded with the corresponding observation noise. As for the noise in (25.4) and (25.6), none of the two quantities is a direct observation of the GOCE

25

Moho Estimation Using GOCE Data: A Numerical Simulation

mission: in fact the potential T is derived from GPS tracking data, for example by applying the so called energy integral approach (Jekeli 1999; Visser et al. 2003), while the second radial derivatives Tzz are obtained by preprocessing the gradiometer observations taken in the instrumental reference frame (Cesare 2002; Pail 2005). However the resulting potential is known to have an almost white error, while the second radial derivatives have a time-correlated error with spectral characteristics almost identical to the original observations (Migliaccio et al. 2004a). As for the ground gravity anomalies Tz, a hypothesis of white noise is generally adopted. At this point a remark is due: the three different observations have complementary spectral characteristics and spatial distribution. In particular T and Tzz contain the low and medium frequencies of the signal because they are observed at satellite altitude; they are regularly sampled along the orbit and therefore are given on a regular grid when inverting a Moho profile; note that, in the two-dimensional case, T and Tzz can be anyway predicted on a grid (Migliaccio et al. 2007) thanks to the good spatial coverage of the GOCE data. On the other hand, Tz contains the highest frequencies of the gravimetric signal because it is observed at ground level; Tz observations are generally available at sparse points and cannot be reasonably gridded due to their typical dishomogeneous spatial distribution, e.g. in mountain areas. For these reasons it is important to combine, in an optimal way, the point-wise gravity data with the gridded satellite data. This can be achieved by means of collocation, treating dD as a random signal: dD^ ¼ CTy;dD C1 y;y y

(25.7)

207

the covariance matrix between a and b plus the error covariance matrix Cna ;nb . Since we consider the noise of the three observations uncorrelated, the error covariances are present only on the diagonal blocks of the matrix Cy,y. It can be noticed that all the needed covariance matrices, with the exception of the error covariance matrices, can be computed by propagating CdD,dD through the corresponding convolution kernels. In this way the resulting covariance matrices describe the different spectral contents of the observables. The solution of (25.7) can be obtained by using a partitioned method separating the gridded observations from the sparse ones:

dD^ ¼ mT

dTzz þ lT dTz dT

(25.8)

where: 2 6 m¼4

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T

31 02 B6 @4

7 5

3 CTzz ;dD

7 6 54

CT;dD

2 3T 2 C C 6 Tzz ;Tz 7 6 Tzz ;Tzz B 5 4 l ¼ @CTz ;Tz 4 CT;Tz CT;Tzz 2

6 B @CTz ;dD 4

CTzz ;Tz

3T 2 7 6 5 4

CT;Tz

CTzz ;Tz

7 C 5lA;

CT;Tz

0

0

3 1

2

CTzz ;T

31 2

311 C 6 Tzz ;Tz 7C 4 5A CT;Tz

7 5

CT;T

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T

31 2 6 4

7 5

31 CTzz ;dD

7C 5A:

CT;dD

The solution can be rearranged in the following way: T T dTzz T dTzz ^ þ l dTz m2 dD ¼ m 1 dT dT

(25.9)

with: where: 2

3

2

3

CTzz ;dD CTzz ;Tzz CTzz ;T CTzz ;Tz 7 7 6 6 Cy;dD ¼ 4 CT;dD 5; Cy;y ¼ 4 CT;Tzz CT;T CT;Tz 5; CTz ;dD CTz ;Tzz CTz ;T CTz ;Tz 3 2 dTzz 7 6 y ¼ 4 dT 5; dTz dD is the unknown Moho depth, y are the observations, Ca,b is the covariance matrix between a and b and Ca;b

2 m1 ¼ 4 2 m2 ¼ 4 0

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T

CTzz ;Tzz

CTzz ;T

CT;Tzz

CT;T 2

B l ¼ @CTz ;Tz 4

31 2 5

31 2 5

CTzz ;Tz CT;Tz

4

3T

4

CTzz ;dD CT;dD CTzz ;Tz

3 5;

3

5; CT;Tz 11 0

5 m C 2A

2

B @CTz ;dD 4

CTzz ;Tz CT;Tz

3T

1

5 m C 1 A:

The matrices m1 and m2 are the two heaviest terms to be computed, because there are in general much

208

more satellite data than ground observation points and, therefore, the covariance matrix of the satellite data is much larger than the one of the ground data. In other words, after computing m1 and m2, the determination of l is easy from the numerical point of view. Note that the first term of the sum in (25.9) is exactly the collocation estimate of the Moho when considering as observations T and Tzz only. Since both these observations are regularly sampled, and the Moho is estimated on the same gridded points, the matrix m1 can be efficiently computed in terms of Multiple Input Single Output (MISO) Wiener filter in the frequency domain (Papoulis 1977; Sideris 1996). The other onerous term, namely m2, can be seen as the collocation transfer operator from satellite to ground data. Naturally, if the ground data were regularly sampled too, this term could be also efficiently computed in the frequency domain. In reality, as we explained before, ground data are generally sparse and cannot be conveniently gridded. Our idea is first to compute the term m2 as if the ground data were available on a regular grid, thus exploiting the properties of the Fourier transform; this leads to a discrete convolution kernel, which is interpolated by splines to derive a continuous model. By applying a principle of stationarity (invariance by translations), this continuous kernel is shifted to each ground observation point and then resampled over the satellite data grid to get the corresponding row of the matrix m2. Of course this

M. Reguzzoni and D. Sampietro

matrix is not Toeplitz even if it is computed according to the stationarity principle, because ground data are not regularly distributed. Note that the harmonicity of Tz predicted by m2 is in general not guaranteed when splines are used to model the continuous kernel; however this is not critical here, because all covariances and cross-covariances are anyway consistent with one another through the Newton integral and the use of a unique covariance of dD.

25.3

Tests on Synthetic Models

The proposed methodology has been assessed on different synthetic models. The aim of the first model is to illustrate the complementary behaviour of the three kinds of observations used. The reference Moho is taken as a simple profile with non-zero mean, a linear trend and a spike modelled by a Gaussian function, see Fig. 25.1. The Moho covariance function and the corresponding spectrum, which are computed disregarding the trend in the data, have a Gaussian shape as shown in Fig. 25.2. Even if it is quite simplistic, this model can be thought as the Moho counterpart of a mountain loading the crust. The depth of the root extends from 32 to 38 km (we fixed D ¼ 30 km) and the density contrast between the root and the underlying mantle is 0.6 g/cm3. The simulation is performed on nearly 150

Fig. 25.1 Estimated Moho using only T, only Tzz or combining T and Tzz. Reference Moho in black solid line

25

Moho Estimation Using GOCE Data: A Numerical Simulation

209

Fig. 25.2 Error spectra using only T, only Tzz or combining T and Tzz. Moho signal spectrum in black solid line. To convert timefrequency into space-frequency divide by the mean satellite velocity of about 7 km/s

Fig. 25.3 Error spectra comparing a single-track solution versus a ten-track solution. Error spectrum using only gridded ground Tz. Moho signal spectrum in black solid line. To convert time-frequency into space-frequency divide by the mean satellite velocity of about 7 km/s

points, which are regularly spaced at a distance of 7 km, consistently with the sampling rate of the GOCE observations. Concerning the data noise along the satellite orbit, the potential is degraded by a white noise with a standard deviation of 0.3 m2/s2, while the second radial derivatives have a coloured noise with a spectrum

derived from a realistic simulation of the GOCE mission (Catastini et al. 2007). The result of the inversion is shown in Figs. 25.1 and 25.2. It can be seen that low frequencies (such as biases and trends) are completely lost when using Tzz observations, but are recovered from T observations. On the other hand, the high resolution details

210

M. Reguzzoni and D. Sampietro

Fig. 25.4 Estimated Moho using satellite T and Tzz or combining satellite data with sparse ground Tz observations. Reference Moho in black solid line

(the spike in our simulation) are caught by Tzz observations. Obviously, combining the two solutions it is possible to estimate both the low and the high frequencies of the signal. This analysis is confirmed by the comparison between the signal and error spectra shown in Fig. 25.2. Here we consider as “well reconstructed” those frequencies for which the error spectrum is lower than 20% of the signal spectrum. According to this threshold, it can be stated that the maximum resolvable resolution with GOCE data only is about 250 km. This result can be improved by considering more than one satellite orbit flying over the Moho profile. Assuming independent observations for each orbit and considering for example ten orbits (corresponding to a reasonable repeat period of two months for the GOCE mission), the maximum resolution becomes 200 km, see Fig. 25.3. Ground gravity data add information at higher frequencies: a numerical simulation shows that if also ground gravity anomalies were regularly sampled as the GOCE observations, then the corresponding error spectrum could be computed (see Fig. 25.3), making possible to reconstruct Moho details up to about 70 km. A second simulation scenario is set up in order to investigate the improvement in the result due to the combination of gridded satellite data and sparse ground gravity data. The algorithm described in Sect. 25.2 is used. A high resolution detail, i.e. a

smaller Gaussian spike, is added to the previously simulated Moho model and we assume to have few (40) gravity anomaly observations at random points at ground level concentrated over this new feature (see Fig. 25.4). A white noise with a standard deviation of 1 mGal is used for these observations. As one can see, these additional data locally improve the Moho estimation coming from satellite-only data; in particular the estimation error decreases from 0.36 to 0.16 km. On the other hand it has been tested that, if the ground data are far from the higher resolution detail, they have no impact on the final solution, thus showing the numerical stability of the method. Finally a test on a realistic Moho profile over the Alpine region has been studied. In this case the reference Moho has been computed starting from the actual topography and applying the Airy isostasy theory (see Watts 2001). The result ranges from 30 to 46 km and clearly shows the presence of the root due to the Alps (see Fig. 25.5). Since gravity anomaly observations are usually placed in flat areas, we simulate a set of 40 ground observations in the Po valley. Again we can notice that local ground data contribute to improve high frequencies: in fact where the point-wise ground observations are present the error r.m.s. significantly decreases (from 0.72 to 0.36 km), while where they are not available the satellite-only solution and the combined solution have practically the same smooth behaviour.

25

Moho Estimation Using GOCE Data: A Numerical Simulation

211

Fig. 25.5 Estimated Moho profile over the Alps using satellite T and Tzz or combining satellite data with sparse ground Tz observations. Reference Moho in black solid line

25.4

A Two-Dimensional Numerical Example in the Alpine Area

In the previous section, some synthetic Moho profiles have been used to test the performances of the proposed inversion algorithm. However the final goal is to estimate the Moho discontinuity surface and not only a profile along the satellite orbit. For this reason the method has been generalized to the two-dimensional case (Sampietro 2009) and has been tested on a realistic numerical example in a region of 10 7 in the centre of Europe (latitude between 44 and 51 North and longitude between 5 and 15 East). This area presents a complex topography characterized by the presence of the Alps. Again, the Airy isostasy theory is used to simulate a reference Moho over the region of interest. Starting from this surface, the gravitational potential and its second radial derivatives have been numerically computed at satellite altitude on a grid of 0.2 0.2 by using (25.1)–(25.3), see Table 25.1. Note that these simulated observables are realistic as for the Moho contribution, but do not represent the full gravity signal. In this experiment, the noise is derived by applying the so called space-wise approach (Migliaccio et al. 2004b; Reguzzoni and Tselfes 2009) to realistic alongtrack simulated GOCE observations (Catastini et al.

Table 25.1 Statistics on simulated signal and corresponding errors on a grid at satellite altitude T signal [m2/s2] T error [m2/s2] Tzz signal [mE] Tzz error [mE]

Min 0.57 0.19 12.62 1.70

Max 1.24 0.29 17.68 1.86

Mean 0.17 0.02 0.14 0.10

Std 0.43 0.10 7.57 0.70

2007), obtaining grids of potential and second radial derivatives as intermediate results. Some statistics of the error of these grids are reported in Table 25.1. Note that the GOCE signal generated by the simulated Moho over the selected Alpine area is about one order of magnitude larger than the grid error, especially in the case of the second radial derivatives. In order to apply a Fourier analysis, it is required that the error covariance matrices of the gridded data have a Toeplitz–Toeplitz structure (Grenander and Szeg€o 1958). This is not the case because the gridding is performed by least squares collocation in spherical approximation (Tscherning 2004). For this reason the error covariance matrix has been approximated by averaging covariances along diagonals for each block of the Toeplitz–Toeplitz structure, namely: Cout i;iþk ¼

N k 1 X Cin N k j¼1 j;jþk

i ¼ 1; 2; :::; N k k ¼ 0; 1; :::; N 1 (25.10)

212

where Cin is the original error covariance matrix, Cout is the approximated Toeplitz–Toeplitz error covariance matrix and N is the block dimension depending on the grid definition. The differences between the original and the approximated matrix are about one order of magnitude smaller than the original covariances, numerically justifying this approximation.

M. Reguzzoni and D. Sampietro

The estimated Moho with and without the aid of additional ground gravity anomalies is shown in Fig. 25.6. These ground observations are located in the centre of the study area, i.e. close to the Alps and along the valleys in order to make the simulation more realistic. Note that the region covered by satellite data is generally larger due to the smoothing effect of

Fig. 25.6 Estimated Moho surface over the Alps using satellite data only (top) or combining satellite data and ground gravity data (bottom)

25

Moho Estimation Using GOCE Data: A Numerical Simulation

the gravity field with increasing altitude. As expected, the satellite data fix the long wavelengths of the Moho, while the ground data improve the details, In particular, the error r.m.s. is reduced from 0.95 to 0.86 km in the area where ground data are available.

25.5

Conclusions and Perspectives

The inverse gravimetric problem of reconstructing the shape of the Moho using GOCE satellite observations has been studied, as well as the integration of additional ground data in a numerically efficient way. In this work we assume a simple two layer model (to guarantee the uniqueness of the solution) with constant densities of mantle and crust in planar approximation; under these hypotheses, positive results have been obtained. In particular the long wavelengths are well recovered thanks to the use of the gravitational potential, while higher resolution details come from gradiometric observations. Sparse gravity anomaly observations at ground level further improve the resolution of the final model. All in all, the estimated Moho over the Alpine area presents errors of less than 1 km for a grid of 0.2 0.2 resolution. However, it has to be stressed that the GOCE-only solution, without integrating any ground observations, is just 10% worse than the combined solution. This shows that reasonable Moho models can be computed at lower resolution also in areas where ground observations are not available, just using GOCE data. Beyond the positive results there are still open issues that need to be solved. First of all, the hypothesis of planar approximation has to be replaced by the spherical one in order to apply the inversion algorithm over larger areas. This implies that the whole theory, including filter design, has to be revisited. Furthermore, it seems interesting to proceed with the research by including additional geophysical information, such as geological models and bounds. This should contribute to get a more reliable solution from the physical point of view. Last but not least, the application of the method to real GOCE data, being the mission in its operational phase. However this requires to face the main problem of every inversion algorithm, that is how to disentangle the different gravimetric signals mixed up into the observed data. The results presented in this paper

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are based on the assumption that the only source of the gravimetric signal was the Moho discontinuity. As a matter of fact modelling crustal dishomogeneities, as well as unwrapping the contributions of large deep features from those closer to the surface, is a problem that can be solved only with additional geological information. Acknowledgements The present research has been partially funded by the Italian Space Agency (ASI) through the GOCEITALY project.

References Benedek J, Papp G (2009) Geophysical inversion of on board satellite gradiometer data – a feasibility study in the ALPACA region, Central Europe. Acta Geodaetica et Geophysica Hungarica 44(2):179–190 Catastini G, Cesare S, De Sanctis S, Dumontel M, Parisch M, Sechi G (2007) Predictions of the GOCE in-flight performances with the End-to-End System Simulator. In: Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Italy, pp 9–16 Cesare S (2002) Performance requirements and budgets for the gradiometric mission, Technical Note, GOC-TN-AI-0027, Alenia Spazio, Turin, Italy Gangui AH (1998) A combined structural interpretation based on seismic data and 3-D gravity modeling in the Northern Puna, Eastern Cordillera, Argentina. Dissertation FU Berlin, Berliner Geowissenschaftliche Abhandlungen, Reihe B, Band 27, Berlin, Germany Grad M, Tiira T (2009) The Moho depth map of the European Plate. Geophys J Int 176(1):279–292 Grenander U, Szeg€o G (1958) Toeplitz forms and their applications. University of California Press, Berkeley, Los Angeles Heiskanen WA, Moritz H (1967) Physical geodesy. Springer, Wien, Austria Jekeli C (1999) The determination of gravitational potential differences from satellite to satellite tracking. Celestial Mech Dyn Astron 75:85–101 Migliaccio F, Reguzzoni M, Sanso´ F, Zatelli P (2004a) GOCE: dealing with large attitude variations in the conceptual structure of the space-wise approach. In: Proc. of the 2nd International GOCE User Workshop 8–10 March 2004, Frascati, Italy Migliaccio F, Reguzzoni M, Sanso´ F (2004b) Space-wise approach to satellite gravity field determination in the presence of coloured noise. J Geod 78(4–5):304–313 Migliaccio F, Reguzzoni M, Sanso´ F, Tselfes N (2007). On the use of gridded data to estimate potential coefficients. Proc. of the 3rd International GOCE User Workshop, 6–8 November 2006, Frascati, Rome, Italy, pp 311–318 Pail R (2005) A parametric study on the impact of satellite attitude errors on GOCE gravity field recovery. J Geod 79 (4–5):231–241 Papoulis A (1977) Signal analysis. McGraw-Hill, New York

214 Reguzzoni M, Sampietro D (2008) An inverse gravimetric problem with GOCE data. In: IAG Symposia “Gravity, Geoid and Earth Observation”, Mertikas SP (ed), vol 135, Springer, Berlin, pp 451–456 Reguzzoni M, Tselfes N (2009) Optimal multi-step collocation: application to the space-wise approach for GOCE data analysis. J Geod 83(1):13–29 Sampietro D (2009) An inverse gravimetric problem with GOCE data. PhD Thesis, Doctorate in Geodesy and Geomatics, Politecnico di Milano, Italy Sampietro D, Sanso´ F (2009) Uniqueness theorems for inverse gravimetric problems. Proc. of the VII Hotine-Marussi Symposium, 6–10 June 2009, Rome, Italy (in press) Sanso´ F (1980) Internal collocation. Memorie dell’Accademia dei Lincei. vol XVI, N. 1 Shin YH, Xu H, Braitenberg C, Fang J, Wang Y (2007) Moho undulations beneath Tibet from GRACE-integrated gravity data. Geophys J Int 170(3):971–985

M. Reguzzoni and D. Sampietro Sideris MG (1996) On the use of heterogeneous noisy data in spectral gravity field modeling methods. J Geod 70 (8):470–479 Tscherning CC (1974) A FORTRAN IV Program for the Determination of the Anomalous Potential Using Stepwise Least Squares Collocation. Reports of the Department of Geodetic Science, No. 212, The Ohio State University, Columbus, Ohio Tscherning CC (2004) Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares Collocation using simulated data. In: IAG Symposia “A Window on the Future of Geodesy”, Sanso´ F (ed), vol 128, Springer, Berlin, pp 277–282 Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geod 77(3–4):207–216 Watts AB (2001) Isostasy and flexure of the lithosphere. Cambridge University Press, Cambridge

CHAMP, GRACE, GOCE Instruments and Beyond

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P. Touboul, B. Foulon, B. Christophe, and J.P. Marque

Abstract

The electrostatic accelerometers of the CHAMP satellite as well as of the GRACE two ones have provided the necessary information to distinguish the satellite actual trajectories from the pure gravitational orbits. By providing the measurements of the satellite non-gravitational forces, one can distinguish the position or velocity fluctuations of the satellite due to the Earth gravity anomalies from those due to the drag fluctuations. In-orbit calibration and validation of onboard instruments, bandwidth, bias stability and resolution proof the success of the mission scientific geodesic return. The basic principle of these sensors stays on the servo-control of one solid mass, maintained motionless from the instrument highly stable structure. Care is paid for the mass motion detection, down to tenth of Angstrom, and to the fine measurement of the servo-controlled forces applied on the mass through electrostatic pressures. With the same concept and technologies, the GOCE inertial sensors have been designed, produced and tested to reach even better performances in order to deal with the milli-E€otv€os gradiometer objectives. The performance of the instrument and the interest of the obtained measurements do not only depend on the sensor accuracy itself but also on the onboard environment (magnetic, thermal, vibrational. . .), on the satellite attitude motions and on the in-orbit configuration and aliasing aspects. Future missions will have also to consider these aspects, especially when envisaging cryogenic electrostatic sensors which can exhibit better self accuracy or when considering satellite to satellite laser tracking.

26.1

Introduction

Space gradiometry missions have been already envisaged in the 1980s with different types of instruments demonstrating the interest of low altitude

P. Touboul (*) B. Foulon B. Christophe J.P. Marque ONERA - The French Aerospace Lab, F-91761 Palaiseau, France e-mail: [email protected]

satellites with drag compensation systems to eliminate the effects of the non-gravitational forces on the satellite orbital motion [1, 2]. To that aim the concept and the technology of a three axis linear electrostatic accelerometer have been developed in view of providing both the measurement of the satellite drag and the Earth gravity gradient effect [3]. From the experience acquired with the three flights of the ASTRE instrument, developed for the on-board shuttle microgravity environment survey [4], the

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STAR ultra-sensitive space accelerometer has been designed, based on the accurate electrostatic levitation of a parallelepiped proof-mass [5]. This accelerometer has been integrated at the center of mass of the CHAMP satellite, first geodesic mission of the 2000 decade. For now quite 10 years, STAR performs the measurement of the non-gravitational accelerations of the satellite which perturb its low altitude trajectory at less than 500 km. The satellite orbit fluctuations are finely determined taking advantage of the on-board GPS receiver, aiming at the accurate recovery of the low spherical harmonics of the Earth’s gravity field. The concept of operation, the design and the technology of the accelerometer have been especially selected for space applications with the potential to optimize the definition for different mission performance and on-board accommodation requirements. So, the Super-STAR instrument has been designed in view of the two GRACE satellites with a reduced full scale range and a resolution ten times better leading to 1010 ms2 over 1 Hz bandwidth. The launch of the GRACE satellite in 2002 has allowed for the first time to compare the in orbit outputs of two accelerometers quite in same operational conditions: both spacecrafts are identical, flying on the same orbit with a 200 km distance from each other, corresponding to less than half a minute, the radiation pressure and the atmospheric drag being thus similar. The difference of velocities between the two satellites is provided with an accuracy of about 1 mm/s by a microwave Doppler device which links the two satellites and is finely analyzed in conjunction with the accelerometer outputs. This leads to the determination of the Earth’s gravity potential, every month and even more, allowing us to evaluate the secular and seasonal fluctuations of large mass distribution like the Antarctic and Greenland ices or the hydrologic basins of large rivers like Mississippi or Amazon [6]. Another way of improving the accuracy of the static Earth field with a much better geographical resolution is the accommodation of a space gradiometer on-board a drag-free satellite at an altitude lower than 300 km, like in the recently launched GOCE mission [7]. The gravity gradiometer is composed of several identical sensors mounted on a rigid and stable structure. In addition to the linear and angular acceleration of the satellite, it provides the fine measurements of the three diagonal components of the Earth’s gravity gradient tensor for the determination of the higher

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harmonics of the potential. Taking advantage of the fine active thermal control of the instrument case and of the drag-free compensation system of the satellite, the full range of the sensors is limited to a few 106 ms2 to the benefit of the resolution of better than 2 1012 ms2Hz1/2 in the frequency measurement bandwidth, from 5 103 to 0.1 Hz. A resolution of a few milli-E€otv€os is thus expected at an altitude as low as 260 km. After the in-orbit switch on of the six accelerometers in April 2009, first gradiometric measurements have been recently obtained showing new space signatures of the Earth gravity field that will provide absolute reference for ocean circulation studies, geophysics, water or atmospheric changes [8]. With nine electrostatic space accelerometers, now in orbit on board four satellites at altitudes between 450 and 260 km, the performance of the concept, its robustness and flexibility to different functioning environment is clearly demonstrated to the benefit of the observation of the Earth gravity field, which is not only the static reference for many other disciplines but also the variation of mass distribution and transportation. Furthermore, improvements of the instrument can be expected as well as improvements of its accommodation on board future satellites and of involved space measurement techniques. So, beyond these three missions, other solid Earth missions are envisaged but also applications in fundamental physics or planetology.

26.2

The STAR and Super STAR Accelerometer for the CHAMP and GRACE Mission

The German CHAMP mission (CHAllenging Microsatellite Payload for geophysical research and application) was dedicated to Earth’s observation: global magnetic and gravity fields mapping. The satellite has been launched on July 15th, 2000 from the cosmodrome Plesetsk by a Russian COSMOS rocket at an altitude of 454 km in a circular orbit with a 87.3 inclination. CHAMP performed for the first time the combination of uninterrupted three dimensional high low tracking of its low orbit perturbations by the satellites of the GPS constellation and a high-precision three-axes measurement of the satellite surface forces: residual drag, solar and Earth radiation pressures and attitude manoeuvre thrusts are measured by the STAR

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(Space Three-axis Accelerometer for Research) accelerometer integrated at the centre of mass of the satellite. A by-product of the accelerometer measurements is the determination of the atmospheric density variations during the decade of the mission. STAR is a six-axis accelerometer providing the three linear accelerations along the instrument sensitive axes and the three angular accelerations about these axes (see Fig. 26.1). STAR presents a measurement range of 104 ms2 and exhibits a resolution of better than 3 109 ms2 for the y and z axes and 3 108 ms2 for the x axis within the measurement bandwidth from 104 to 101 Hz. The measurements are integrated over 1 s before delivery to the satellite data bus. The configuration of the instrument is compatible with ground tests which demand specific characteristics of the less accurate x-axis for the operation under 1 g gravity field. For example, measurements of residual low frequency vibrations on a specific testing pendulum platform have been performed along the y and z axes of the accelerometer controlled horizontal with a resolution of 108 ms2 Hz1/2 at frequencies lower than 0.1 Hz corresponding to the requirement of 3 109 ms2 rms in the measurement bandwidth. The accelerometer low level of bias (less than 105 ms2) of the same y and z sensitive axes has also been verified in free fall in the Bremen drop tower [9]. The following GRACE mission put in evidence the temporal variability of the Earth’s gravity field. In

addition to the High-Low GPS satellite tracking and to the accelerometer surface force measurement, the GRACE mission is based on low–low K-band tracking between two identical satellites separated by about 220 km on the same quasi circular orbit. The twin GRACE satellites were launched on March 17th 2002 from Plesetsk by Rockot launch vehicle at a initial altitude of 500 km on a quite polar orbit (89.0 inclination). The configuration of the two SuperSTAR accelerometers is quasi identical to STAR and takes advantage of the CHAMP mission experience. The concept of electrostatic servo-controlled accelerometer is well suited for space applications: the electrostatic forces give the possibility to generate very weak but accurate accelerations while the capacitive sensing offers a high position resolution with negligible backaction. The accelerometer proof-mass is fully suspended with six servo-control loops acting along its six degrees of freedom, suppressing any mechanical contact to the benefit of the resolution and yielding to a six-axis accelerometer. The internal cage of the accelerometer, constituted by three electrode-plates made of silica glass, surrounds the solid parallelepiped proof-mass, of 4 4 1 cm sides and 72 g weight, made in Chromium coated Titanium alloy (see Fig. 26.2). The accelerometer performance depends mainly on the geometrical accuracy and the stability of the core mechanical assembly. In operation, the mass is maintained motionless at the centre of the cage with

Fig. 26.1 STAR sensor unit before integration in the CHAMP satellite

Fig. 26.2 GRACE accelerometer mechanical core

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stability up to a few 1012 m/Hz1/2. The fine measurements of the applied voltages on the electrodes provide the accelerometer outputs. Independent analyses performed on the in orbit data confirm that the instrument specifications are achieved for both GRACE accelerometers in particular, along the sensitive axes Y and Z, the bias being less than 2 106 ms2 [10] and the noise PSD: S(f) ¼ 1020 (1 + 5 103/f) m2s4/Hz [11]. In fact, the instrument is so sensitive that the on board environment has to be precisely decorrelated from the self behavior of the instrument to exploit the sensor limit of performance.

26.3

From GRACE Accelerometer to GOCE Gradiometer

The aim of ESA’s Gravity Field and Steady-State Ocean Circulation Explorer mission is also to generate accurate global mapping of the Earth’s gravity field for oceanographic, geophysical, hydrological or climatologic applications. Compare with previously launched missions, GOCE seeks particularly to better understand the highest spherical harmonics of the gravity field model. Launched on March 17th 2009 from Plesetsk, the GOCE mission exploits for the first time a three-axis gradiometer consisting of six electrostatic accelerometers offering an outstanding resolution of 2 1012 ms2Hz1/2. The accelerometers were successfully activated on April 6th 2009. The satellite is on a sun-synchronous, quasi circular and quasi polar (96.5 ) orbit at an altitude near 260 km. Weighting 1 ton with a length of 5 m, the spacecraft has a very rigid structure without moving parts and exhibiting a limited cross section (1 m diameter) in order to reduce its drag. Its drag compensation system counteracts all non- gravitational forces acting on the spacecraft along its velocity vector through two ion thrusters. The electrostatic gravity gradiometer is composed of three pairs of accelerometer sensor head including the Platinum test masses, on one Carbon–Carbon ultra stable structure, three front end electronics units for the masses sensing and actuations, one gradiometer interface unit for the experiment control and the mass control laws, contained in three accurately thermal controlled stages. The outputs delivered by each pair of accelerometers are combined to provide the gravity

gradient along the gradiometer axis: this is the main science data, to be separated from residual angular accelerations of the satellite. This angular acceleration about the three axes, and by integration the angular rate, can be deduced from the transverse axes of the accelerometers, symmetrically distributed around the gradiometer centre which is also the satellite centre of inertia. In addition the common measured acceleration provides the external forces applied on the satellite to be cancelled along the velocity vector by the drag compensation system. The GOCE sensors are similar to the CHAMP and GRACE ones except the very dense proof-mass made in PtRh10 alloy, the eight pairs of electrodes instead of six for redundancy, plus the digital control loops instead of the previous analogue circuits. The main characteristics of the three accelerometers are compared in Table 26.1. On April 6th 2009, the six accelerometers were switched on and the proof-masses were immediately all levitated at the centre of their cage. The next day, the six accelerometers passed in Science Mode with a more limited full range and better resolution. On May 29th 2009, the GOCE satellite was successfully commanded in Drag-Free mode demonstrating the performing association of the accelerometer measurements and the electric ion propulsion system [12]. The residual along-track external acceleration (red spectrum on Fig. 26.3) is around 109 ms2Hz1/2 in the measurement bandwidth, more than one order of magnitude less than the specified value. Estimation of the accelerometer noise can be deduced from redundant sensor measurement (blue spectrum of Fig. 26.3) and is perfectly in line with the predicted noise computed from on ground tests and evaluations (black curve): the coherence of the two curves between 1 and 5 Hz, where the position sensing source is the main contributor confirms the right operation of the capacitive sensors; similarly, the coherence of the curves between 10 and 100 mHz where the electrostatic actuation is the main contributor, confirms the right operation of the electrode voltage driving amplifiers. This result, associated to the coherence of the data provided by the 36 channels of the six accelerometers is a very good clue for the success of the future data processing of the mission. The very low satellite altitude, facilitated by the Sun’s exceptionally weak activity and associated in consequence to a longer

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Table 26.1 Comparison between the mains characteristics of the CHAMP, GRACE and GOCE accelerometers Mission PM material PM mass Gap X Gap Y,Z Vd Vp Ge Acceleration sensitivity Y,Z Position sensitivity Y,Z Measurement range Y,Z Specified resolution Y,Z Specified freq. bandwidth

CHAMP TA6V 0.072 60 75 5 20 1.8 104 0.4 104 6 years) with high spatial resolution (comparable to that provided by GOCE) and high temporal resolution (weekly or better, so to reduce the level of aliasing of the high frequency phenomena found in the time series of the Earth’s gravity field variation provided by GRACE), and it must improve the separability of the observed geophysical signals (Koop and Rummel 2007). In one of the preparatory studies carried out for ESA by Thales Alenia Space I (TAS-I) (Laser Doppler 2005), a resolution of

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0.1 mm/year of the geoid height variation rate at spherical harmonic degree ¼ 200 (or ~100 km spatial resolution) was preliminarily identified satisfying the most stringent geophysical performance requirement.

27.2

FD2

D2 g2

Satellite 1

FD1

D1

Δd = ΔdG + ΔdD

g1

Earth

Measurement Techniques and Mission Scenario

Fig. 27.1 Principle of the LL-SST technique

The monitoring of the temporal variations of the Earth’s gravity field over a long period of time requires flying at an altitude higher than that of GOCE, in order to compensate the drag forces with an affordable amount of propellant. Since the gravitational potential U of degree rapidly decreases with the orbit radius r, i.e. U‘ / r ð‘þ1Þ ;

d

ZJ2000 S2

ZO 1

XO 1 YO 1

S1

(27.1)

it would be necessary to increase proportionally the gradiometer baseline (and/or the sensitivity of the accelerometers) in order to maintain the same signalto-noise ratio (see Cesare et al. 2009). Considering also that the signal of the temporal variations of the gravity field is at least one order of magnitude weaker than that of the static geopotential, for altitudes above ~300 km at the moment the most consolidated technique to measure the Earth’s gravity field is the “LowLow Satellite-to-Satellite Tracking” (LL-SST), which exploits the satellites themselves as the “proof masses” immersed in the Earth’s gravity field, as in the GRACE mission. Assuming a loose formation of two satellites, the distance variation between their centres of mass (Dd, produced by both gravitational and non-gravitational forces together) is gauged via length metrology. The drag accelerations (D1, D2) experienced by the two satellites are separately measured by means of accelerometers. From of Dd€D ¼ D1 D2 , the non-gravitational component RR the distance variation is obtained ðDdD ¼ Dd€D Þ and subtracted from Dd to isolate the component (DdG) produced by Earth’s gravity field only (see Fig. 27.1). Thanks to the separation between the satellites, this “measurement instrument”, which can be regarded as a kind of one-dimensional gradiometer with a very long baseline, has a higher sensitivity for the phenomena being targeted than a gradiometer embarked on a single satellite.

sun

i

γ = 90°

XJ2000

line of nodes

YJ2000

orbit

Fig. 27.2 Basic mission scenario

The simplest scenario implementing LL-SST consists of a loose formation of two co-orbiting satellites. In particular, the main parameters of the mission scenario defined in Laser Doppler study (2005) are (see Fig. 27.2): • Circular orbit with mean altitude of 325 km • Orbit inclination is 96.78 (sun-synchronous) • Local time of the ascending node: 6 am (dawn-dusk orbit) or 6 pm (dusk-dawn orbit) • Inter-satellite distance d ¼ 10 km These parameters and the requirements on the fundamental observables utilized for the determination of the Earth’s gravity field (Dd and Dd€D , as shown in Fig. 27.3), have been so chosen to fulfill the requirements on the geoid variation rate (0.1 mm/ year) at 100 km spatial resolution. They have been derived using analytical models and numerical simulations of the gravity field determination through the LL-SST technique, with the support of geodesy institutes (Laser Doppler 2005; Cesare et al. 2006).

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The Future of the Satellite Gravimetry After the GOCE Mission

Fig. 27.3 Requirements on Dd€D (above) and inter-satellite (or “relative”) Dd (below) measurement error spectral density. These requirements are strictly applicable to the frequency

Although this basic mission scenario still does not meet all the objectives of a NGGM, in particular with respect to the improvement of the temporal resolution and the separability of the geophysical signals, it can provide the “building block” for the realization of more complex scenarios that can potentially fulfill these objectives. For instance, a dense and uniform coverage of the Earth surface that allows generating gravity field solutions in less than 1 week can be obtained by flying two pairs of co-orbiting satellites

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range [1100 mHz], the identified measurement bandwidth (MBW) of the NGGM

in two circular orbits with altitude at 312 km and with 90 and 62.7 inclination respectively, as in Bender et al. (2008).

27.2.1 Laser Metrology System The metrology system designed for the NGGM (Laser Doppler 2005; Laser Interferometry 2008) is based on a Michelson-type heterodyne laser interferometer (see

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Fig. 27.4 Functional scheme of the laser metrology

Fig. 27.4). A measurement laser beam produced by a frequency-stabilized Nd:YAG source (wavelength ¼ 1,064 nm, output optical power ¼ 750 mW) is emitted by satellite 1 (follower) towards satellite 2 (leader). Here it is retro-reflected by a corner-cube optical system towards satellite 1, where it is superimposed to a

reference laser beam generated by the same source but frequency shifted relatively to the measurement beam. The measurement beam is amplitude modulated in on-off mode with a period corresponding to the roundtrip time to avoid the occurrence of spurious signals and non-linearity caused by the unbalance between the

27

The Future of the Satellite Gravimetry After the GOCE Mission

optical power of the strong reference beam and of the weak returning measurement beam. The distance variation is obtained from the phase variation of the interference signal formed by the measurement and the reference laser beams. A breadboard of such an interferometer has been realized (as in Fig. 27.5) and subject to functional and performance tests up to ~90 m distance. The test results shown in Fig. 27.6 prove that the requirement (as depicted in Fig. 27.3) can be met in the MBW. The laser metrology system is completed by: • An Angle Metrology for measuring the angles of the two satellites relative to the laser beam

Fig. 27.5 Breadboard of the laser interferometer Fig. 27.6 Results of the performance test (distance measurement error spectral density)

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• A Beam Steering Mechanism (BSM), i.e. a device in charge of the acquisition and maintenance of the optical link between the satellites and of the laser beam fine pointing during the measurement phase • A Lateral Displacement Metrology for measuring the lateral displacement of the satellite 2 relative to the beam sent by the satellite 1 and driving in closed loop the BSM The Angle Metrology is merged with the Lateral Displacement Metrology and is realized by three small telescopes focusing the collected light on three Position Sensing Detectors (PSDs). Each PSD measures the position and the energy of the laser beam spot focused by the optics on the detector plane. The orientation and the lateral shift of satellite 2 relative to the laser beam are derived from the spot position and from the optical power measured by the three PSDs respectively. Breadboards of the Angle and Lateral Displacement Metrology and of the BSM have been realized (Fig. 27.7) and tested by TAS-I (Laser Doppler 2005; System Support 2008), including a test of the whole laser beam pointing system (BSM driven in closedloop by the Lateral Metrology). The elements of the laser metrology system described above are arranged on an optical bench, shared with the accelerometers utilized for the drag accelerations measurement (as shown in Fig. 27.8).

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satellite 2 can take its role, after the two satellites exchange positions.

27.2.2 Control Requirements

Fig. 27.7 Breadboard of the BSM

Angle Metrology

Beam Steering Mechanism Retro Reflectors

Interferometer core Accelerometers Angle/Lateral Metrology

Fig. 27.8 Configuration of the optical bench installed on each satellite

Two accelerometers, which can be similar to those of GOCE, are used on each satellite, arranged symmetrically about the centre of mass (COM). So doing, it is possible to measure the drag acceleration of the satellite COM, leaving the space in proximity of the COM free for the accommodation of the laser retro-reflector. The same optical bench is replicated on both satellites even if the laser is emitted only by satellite 1. Then in case of failure of the interferometer on satellite 1,

A complete assessment of the control requirements was done in Laser Doppler (2005), and revised and updated along the following studies: the control boundaries were conceived for the baseline configuration of two satellites chasing each other in a drag-free environment and flying in loose formation flying. The modeled metrology is the consolidated LL-SST technique, enhanced by the BSM. More in detail, in order to operate the accelerometers at the performance level required for the NGGM, each satellite must be endowed with a drag-free control system to reduce the level of the non-gravitational 2 accelerations below 106m/s and below a spectral 2 pﬃﬃﬃﬃﬃﬃ 8 density level of 10 m s Hz between 1 and 10 mHz, along each axis. The drag compensation is realized using ion thrusters in the same way as in GOCE. The formation control is in charge of keeping satellite 2 in the “control box” established by the working range of the optical metrology system: DdX ¼ 500 m (with respect to the nominal distance, d ¼ 10 km); DdY, DdZ ¼ 50 m along the Y, Z axes of the satellite 1 Local Orbital Reference Frame. The challenge of the formation control design for this mission consists in keeping the relative motion within these boundaries without interfering with the scientific measurements, which require that the satellites must be “free” to move under the effect of the gravity field over time scales of 1,000 s, and also without spoiling the drag-free environment (formation control accelerations must fulfil the drag-compensation requirements too), while minimizing thrusters use in terms of dynamic range, propellant consumption. Control architecture and recommended actuator technology is elaborated in Cesare et al. (2009).

27.3

Parallel Studies

27.3.1 NGGM All the illustrated studies have constituted a solid background for the preparation of a new pre-Phase A study, so called “Assessment of a Next Generation

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The Future of the Satellite Gravimetry After the GOCE Mission

Gravity Mission to Monitor the Variations of the Earth’s Gravity Field” (ESA ITT 1-5914/09/NL/CT). Thus, the ESA Future Missions Division has started two parallel studies led by two competitive consortia (Astrium D and ThalesAlenia Space I) in the second quarter of 2009, with the main objective of establishing mission architecture aimed at the optimal recovery of the Earth variable gravity field, through satellite-to-satellite tracking observation techniques.

27.3.2 Mass Transport Study The “Mass transport study” (Monitoring and Modelling Individual 2008) aimed at monitoring and modeling individual sources of mass distribution and transport in the Earth mass model (11 years, 0.5 0.5) was compiled including mass fields stemming from atmosphere, oceans, continental water, ice (ice sheets and glaciers) and the solid Earth. This model was used to optimize the ability to observe changes in mass over the continents and oceans. Detailed and realistic close-loop simulations were performed to study separability and aliasing characteristics inherent in different satellite mission concepts. One of the conclusions was that temporal aliasing is intrinsic to observing gravity field changes by satellites, but e.g. leads to relatively smaller distortions for hydrology than for oceanography. Furthermore, the combination of LL-SST and orbit observables does not allow the precise determination of the spherical harmonic degree 1 terms or geo-centre variations. Finally, flying more than one pair of LLSST satellites can significantly reduce the gravity field retrieval errors.

27.4

Parallel Technology Developments

27.4.1 Atomic Clocks (ESA studies) The European Space Agency is currently carrying on two parallel activities on atomic clocks. The ACES (Atomic Clock Ensemble in Space) mission on the International Space Station fosters research on coldatom sensor technology for space, in view of applications in different domains, e.g. fundamental physics tests (violation of special relativity, search for drifts of fundamental constants) and relativistic

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geodesy (till 10 cm accuracy) with a time and frequency metrology at 1016 stability and accuracy level. ACES will be ready for launch in 2013. Secondarily, a Pre-phase A study of an atomic clock ensemble in space based on the optical transitions of strontium and ytterbium atoms has started in 2006. The so-called SOC (Space Optical Clocks, ESA-AO-2004-100) will take advantage of the ACES heritage and will push stability and accuracy of atomic frequency standards down to the 1018 regime.

27.4.2 Atom Interferometers The European Space Agency has been funding a number of technology development activities and studies in the area of atom interferometry in view of the everincreasing requirements for future inertial satellite platforms. Recently, a study has been initiated to explore the “Applications and Implementations of Atom-Based Inertial Quantum Sensors” in future space missions, in the fields of Space Science, Earth Observation, Fundamental Physics, Microgravity, and Navigation. As part of this ongoing study, the fundamental limits of atom interferometry were translated into the expected performance of a realistic gravity gradiometer payload for a future gravity mission. Gravity gradients can be measured by comparing the time evolution of two clouds of atoms (106 to 108 or more atoms), suitably cooled and prepared by means of lasers as well as other (time varying) electromagnetic fields, and separated by a given baseline. The time evolution of each cloud is measured by sequentially inducing suitable atomic transitions by means of lasers. This constitutes the atom interferometric measurement (see Peters et al. 1999). Comparison of the two atomic clouds is also achieved optically by ensuring the optical phase locking of the interferometric measurements effected on the two clouds. A major advantage compared to a macroscopic gravity gradiometer, such as the one onboard GOCE, is that the absolute local value of gravity is measured for each atomic cloud. Therefore, the atom interferometry gravity gradiometer does not suffer from drift but exhibits constant performance across the measurement frequency band. The acceleration sensitivity of an atom interferometer increases, amongst others, with the square of the interrogation time. This time is effectively limited on

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the ground to the achievable free fall time of an atomic cloud, which amounts to a few 100 ms in a typical vacuum chamber. By contrast, space operation, characterized by the intrinsic microgravity environment, would enable interrogation times of several tens of seconds leading to a sensitivity improvement of four orders of magnitude. Assuming a baseline of 1 m, this would amount to sensitivities of the order of 1 mE/Hz in space. This is one order of magnitude better than the GOCE specifications for measurements at frequencies above ~1 mHz, and, thanks to the absence of drift, well below, by several orders of magnitude, for frequencies below 1 mHz. The above estimate, however, does not yet represent a fundamental limit. Performance also increases with the magnitude of the probing laser wave vector and the signal-tonoise ratio, which is a function of the number of atoms in the cloud. Both these quantities can be effectively increased using quantum entanglement techniques. Although these still need to be demonstrated, the potential sensitivity for the same 1 m baseline atom interferometer gravity gradiometer would then be in the order of 1mE/Hz. In order to become suitable for future space mission, atom interferometry technology requires further development in the area of miniaturization and integration of both optical and laser components, as well as vacuum chambers. Furthermore, space worthiness also needs testing. These developments and investigations are currently underway in a number of European institutes (see Schmidt et al. 2009; Stern et al. 2009).

27.5

Conclusions

Since 2003 a number of studies have been initiated by ESA focusing on various technological, scientific and mission aspects of a potential future gravity field mission. It is important that the technology, not only related to the payload but also to satellite and/or constellation control, reaches the right readiness level. In addition, tools need to be developed to allow full mission performance analysis. Besides LL-SST other interesting technologies are under development but at different levels of maturity. The road towards availability of these techniques for future missions is still long and full of challenges.

Acknowledgements The authors gratefully acknowledge Prof. Achim Peters, Head, and Malte Schmidt, of the Humboldt University of Berlin, for their expert advice on atom interferometry.

References Bender PL, Wiese DN, and Nerem RS (2008) “A Possible DualGRACE Mission with 90 Degree and 63 Degree Inclination Orbits”, In: Proceedings of the 3 rd International Symposium on Formation Flying, Missions and Technologies, Noordwijk (NL) Cesare S, Sechi G, Bonino L, Sabadini R, Marotta AM, Migliaccio F, Reguzzoni M, Sanso` F, Milani A, Pisani M, Leone B, Silvestrin P (2006) “Satellite-to-satellite laser tracking mission for gravity field measurement”. In: Proceedings of the 1st International Symposium of IGFS, 28 Aug–1 Sept. 2006, Istanbul, Turkey. Harita Dergisi, Special Issue 18, pp 205–210 Cesare S, Allasio A, Aguirre M, Leone B, Massotti L, Muzi D, and Silvestrin P (2009) The Measurement of Earth’s Gravity Field after the GOCE Mission. In: Proceedings of 60th International Astronautical Congress, Daejeon, Korea, 2009, IAC-09.B1.2.7 Enabling Observation Techniques for Future Solid Earth Missions (2004) ESA Contract No: 16668/02/NL/MM, Final Report, Issue 2, 15 July 2004 Koop R, Rummel R (2007) The Future of Satellite Gravimetry, Final Report of the Future Gravity Mission Workshop, 12–13 April 2007 ESA/ESTEC, Noordwiik, Netherlands Laser Doppler Interferometry Mission for determination of the Earth’s Gravity Field (2005), ESA Contract 18456/04/NL/ CP, Final Report, Issue 1, 19 December 2005 Laser Interferometry High Precision Tracking for LEO (2008), ESA Contract No. 0512/06/NL/IA, Final Report, July 2008 Monitoring and Modelling Individual Sources of Mass Distribution and Transport in the Earth System by Means of Satellites (2008), ESA Contract No. 20406/06/NL/HE, Final report, November 2008 Peters A, Chung KY, Chu S (1999) Measurement of gravitational acceleration by dropping atoms. Nature 400:849–852 Schmidt M, Senger A, Gorkhover T, Grede S, Kovalchuk E, and Peters A (2009) “A Mobile Atom Interferometer for High Precision Measurements of Local Gravity”, In: Frequency Standards and Metrology – Proceedings of the 7th Symposium pp 511–516 Stern G, Battelier B, Geiger R, Varoquaux G, Villing A, Moron F, Carraz O, Zahzam N, Bidel Y, Chaibi W, Pereira Dos Santos F, Bresson A, Landragin A, Bouyer P (2009) Lightpulse atom interferometry in microgravity. Eur Phys J D 53:353–357 System Support to Laser Interferometry Tracking Technology Development for Gravity Field Monitoring (2008), ESA Contract No. 20846/07/NL/FF, Final report, September 2008

Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

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R. Mayrhofer and R. Pail

Abstract

Modern satellite gravity field recovery missions use accelerometric, intersatellite tracking or gradiometric observables for deducing gravity field related data. In this study an alternative observable type for gravity field recovery, the relativistic frequency shift, is investigated. As Einstein stated in his general theory of relativity, gravity can be considered as attribute of space-time. In this view mass alters the geometric shape of the metric tensor. Moreover mass, respectively gravity, has effects on electromagnetic wave propagation [Einstein (Annalen der Physik 35:898–908 1911)]. Although these relativistic effects are quite small and difficult to measure, with upcoming atomic clocks which have sufficient accuracy and short-term stability it will be possible to derive meaningful gravity related information. Since relativistic effects are used this method is called Post-Newtonian method. The main target of this paper is to demonstrate the validity of the derived relativistic equations. The scientific quality of the relativistic frequency shift observed by means of highly accurate atomic clocks is investigated. In our basic scenario a low earth orbit (LEO) sends an electromagnetic wave to a receiver. The reference station determines the frequency shift of the signal, which is connected to the time dilatation between the atomic clock of the satellite and an identical atomic clock nearby the receiver. A simplified, mathematical model for numerical simulations of this configuration is presented. The effect of different error sources are investigated by numerical closed-loop simulations. Thus, the performance requirements of atomic clocks, position and velocity determination and limiting factors for deducing earth’s gravity field can be derived.

28.1

R. Mayrhofer (*) R. Pail Institute of Navigation and Satellite Geodesy, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria e-mail: [email protected]

Introduction

In this study the principle of the Post-Newtonian method for earth gravity field determination is presented. The observable for gravity field reconstruction is the frequency shift of an electromagnetic signal transmitted from a satellite to a receiver station. This frequency shift is caused by relativistic time dilatation

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which is related to the gravity potential. By numerical simulations the spatial and spectral performance of a LEO satellite mission equipped with atomic clocks, which will be available within the next tow decades, is investigated. There have been done some studies which are related to future satellite gravity field missions and general relativistic effects. There is for example the Einstein Gravity Explorer mission proposal (Schiller et al. 2009) in which testing methods of relativistic effects and physical constants based on atomic clock measurements are investigated. M€ uller et al. (2007) explored the impact of relativity on various geodetic topics like the geoid, reference systems, geodynamics, Global Positioning System (GPS), Satellite Laser Ranging (SLR), and Very Long Base Interferometry (VLBI). Gulkett (2003) investigated relativistic effects on GPS and LEO and showed how to implement them correctly within a relativistic framework. The IAU already included relativistic effects for frame transformations (Soffel et al. 2003). For being able to realise a satellite mission as proposed in this paper, atomic clocks with sufficient quality will be needed. Actual atomic clocks achieve a short term stability of 1016 s (between two measurement epochs) on earth (Schiller 2007) and are expected to achieve 1018 s within the next 15 years. According to Cacciapuoti (2006), actual space-borne atomic clocks achieve an accuracy of 1015 s. Thus, a satellite mission with an atomic clock with 1018 s stability should be possible within the next 30 years. In this study, the equations for the Post-Newtonian method are derived in Chaps. 28.2 and 28.3. In Chap. 28.4 the simulation setup is described in more detail. Chapter 28.5 shows the simulation results, a performance analysis, and the analysis of the spectral and spatial error behaviour. Finally a conclusion and outlook is given in Chaps. 28.6 and 28.7.

28.2

Relativistic Time Dilatation

The metric tensor gmn describes the curvature of spacetime. Thus, it can be used for deducing a description of the relativistic frequency shift. The line element ds can be described by ds2 ¼ gmn ðxÞdxm dxn

(28.1)

Here Einstein’s tensor convention has been applied (Einstein 1916). Double upper and lower indices describe a summation. The gradient dxk ¼ ½c dt dx1 dx2 dx3 T of the scalar vector field x ¼ ðxk Þ contains position and time information. c is the speed of light in vacuum, dt the time element of a chosen time system and dxi describe the three dimensional coordinate elements. The metric tensor gmn ðxÞ is a function of xk ðm; n; k ¼ ½0; 1; 2; 3Þ, which means that the curvature of space-time depends on time and position. A solution of Einstein’s field equations delivers the elements of the metric tensor. A series expansion representation of the tensor elements is (Soffel et al. 2000) 2 FðxÞ 2 F2 ðxÞ þ Oðc5 Þ c2 c4 4 Fi ðxÞ g0i ¼ þ Oðc5 Þ c3 2 FðxÞ 2 F2 ðxÞ þ Oðc6 Þ gij ¼ dij 1 þ c2 c4 (28.2)

g00 ¼ 1 þ

where FðxÞ is the gravity potential and i; j ¼ ½1; 2; 3. The earth’s static gravity field potential is usually expressed by a spherical harmonic series expansion (Heiskanen and Moritz 1967): FE ðr; y; lÞ ¼

1 lþ1 GM X R R l¼0 r

l X

ðClm cos ml þ Slm sin mlÞ Plm ðcos yÞ

m¼0

(28.3) Here r, y and l are spherical coordinates, R is the earth’s reference radius, GM the gravitational constant times mass of the earth, l and m are the degree and order of the fully normalized spherical coefficients Clm ; Slm , and Plm ðcos yÞ represents the fully normalized Legendre function. Equation (28.3) describes the functional model for setting up the design matrix for least squares adjustment in our simulation environment. The spherical coefficients Clm ; Slm are the system parameters, while the gravity potential F, which is derived from relativistic frequency shifts is the observable of our system.

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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

The local line element dsclock of a clock moving within a gravity field affects the displayed time dtclock and frequency uclock of the clock. dtclock ¼

1 uclock

¼

dsclock 1 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ gmn ðxÞdxm dxn (28.4) c c

x ¼ ðxk Þ and dxk are related to the coordinates of the clock. The coordinate elements of a moving clock can be described by dxk ¼ ½ c dt

v1 dt

v2 dt

v3 dt

(28.5)

Here, vi ¼ vi ðxÞ denotes the velocity of the clock and is related to its coordinates, too. Merging (28.2), (28.4) and (28.5) and omitting elements smaller then c3 leads to a description of the inherent time of a moving body: 2 FðxÞ v2 ðxÞ dt2 dt2 ¼ 1 þ þ c2 c2

(28.6)

v ðxÞ is the local scalar velocity of the clock. The asterisk ‘*’ is used for underlining that v is scalar and preventing to mix it up with the velocities from (28.5). dt represents a virtual time of an non-moving, gravityfree (inertial) body located at infinite distance.

28.3

Functional Model

A LEO satellite transmits an electromagnetic signal via microwave link to a receiver station. This receiver station could be a geostationary satellite or a reference station located on earth’s surface. By comparing the local frequency of the transmitted signal with the local frequency of the received signal the time dilation between receiver and transmitter is defined. By using (28.6) and again omitting elements smaller then c3 the ratio of receiver and transmitter frequency is obtained by uR dtT ¼ uT dtR vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 2 u1 þ 2F2 T þ vT2 c c t ¼ þ Oðc4 Þ v2R 2FR 1 þ c2 þ c2

DuRT ¼

(28.7)

The lower indices R and T describe a receiver, respectively a transmitter related variable. The gravity

233

potentials are related to the receiver or transmitter position FR ¼ FðxkR Þ; FT ¼ FðxkT Þ. This equation is used for synthesizing relativistic frequency shifts in our simulation environment. Moreover it is the fundamental equation for the Post-Newtonian approach. As the relativistic time dilation, which is not modeled by the Newtonian framework, is taken into account, the nomenclature ‘Post-Newtonian’ has bee chosen to describe this method. The gravity potential FT deduced from the frequency shift DuRT is the prime observable for the Post-Newtonian method. A general description of (28.7) is DuRT

uR dtT ¼ ¼ ¼ uT dtR

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ gmn ðxT ÞdxmT dxnT gmn ðxR ÞdxmR dxnR

(28.8)

By combining (28.8) with the description of the metric tensor elements in (28.2), (28.7) can be achieved. Based on this function the gravity potential FT at the satellite position can be calculated from a measured frequency shift DuRT . After some reformulations a function, which is used in our simulation environment as functional model for deducing the gravity potential at transmitter position from the frequency shift DuRT , is achieved: 2 v2T c 2 FT ¼ DuRT OR 2 þ 1 c 2

(28.9)

Here a support variable OR has been introduced. It contains all position, velocity and gravity potential information of the receiver station: OR ¼ 1 þ

2 FR v2R þ 2 : c2 c

Equations (28.9) and (28.10) describe two relativistic effects. The first one is time dilatation caused by relative movement of receiver and transmitter, the second one time dilatation caused by the gravity potential difference between transmitter and receiver location. The gravity potential FðxÞ is composed of all occurring gravity potentials. As a first approximation in our simulations, all non-earth gravity potentials and tide signals have been neglected. Beside the special relativistic and general relativistic effects the Doppler shift is the third large effect which influences the

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frequency observations of the receiver station. As a satellite in a LEO achieves large velocities, the radial velocity between receiver and transmitter cause a Doppler frequency shift which has to be modeled. The Doppler shift (Doppler 1842) is defined by sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ c vTR uR ¼ uT c þ vTR

(28.10)

where vTR is the scalar radial velocity between transmitter and receiver. The Doppler shift is applied to the measured frequency at the receiver station. It has to be mentioned that as we are working in a relativistic framework, the radial velocity has to be derived in a coordinate system located at the center of the receiver station.

28.4

Simulation Setup

Figure 28.1 shows the schematic set-up of our simulation software. In a configuration file the orbit properties, computation switches and observation noise types are defined. The software computes based on this configuration, the orbit positions and the frequency related effects by using (28.7) and (28.10). In our simulation environment (28.9) has been used to calculate the gravity potential at the satellite position from the synthesized frequency shifts. The simulation environment has been designed to determine the influence of data noise on the reproduced gravity field model. Therefore it is possible to manually add realistic noise on frequency and velocity measurements. Following effects on the frequency shift observable have been simulated:

• Special relativistic frequency shift caused by relative velocity of receiver and transmitter clock (28.7). • General relativistic frequency shift caused by relative potential difference at receiver and transmitter clock positions (28.7). • Doppler Effect caused by relative radial velocity of receiver and transmitter clock (28.10). For every effect a realistic stochastic noise signal was added on noise-free frequency and velocity measurements. Shin et al. (2008) suppose a coloured noise for H-maser clocks with increasing amplitudes at low and high frequencies and linear behaviour in-between. Figure 28.2 shows the power spectrum density function of the atomic clock noise with amplitude 1017 and 1018 s generated for our simulations based on this information. For the velocity error, white noise has been assumed. In the frame of gravity field adjustment, the stochastic models, which define the metric of the normal equation system, have been consistently incorporated by correspondingly designed digital filters applied to both, the observation time series and the columns of the design matrix (Schuh, 2001). A nearly polar (89.5 inclination), circular repeat orbit with 25 days and 403 cycles at 300 km mean height with 10 s sampling interval has been chosen for all simulations.

28.5

28.5.1 Performance Analysis The main observable of the Post-Newtonian method is the frequency shift. Equation (28.7) shows that beside the atomic clock noise the velocity determination noise

Orbit Definition

Simulation Definition

Fig. 28.1 Schematic presentation of the simulation environment used in this study

Simulation Results

Orbit Synthesis

Signal Synthesis

Noise Definition

Gravity Field Coefficients equation (28.3) least squares adjustment

Gravity Field Analysis equation (28.9)

equation (28.7), (28.10)

Potential Reconstruction

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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

235

Fig. 28.2 Smoothed noise amplitude spectrum of coloured clock noise with amplitude 1017 s and 1018 s

Fig. 28.3 Degree error median plot of simulations with frequency noise with 1017 s and 1018 s amplitude and velocity noise with 105 m/s and 106 m/s white noise

of the transmitter is a stochastic variable, too and is the second limiting factor for the Post-Newtonian method. Based on the frequency shift the transmitter potential can be determined by using (28.10). Thus, first a noise free data set has been defined, and the calculation method has been verified by using closed-loop computations. Next, realistic coloured clock noise (Fig. 28.2) with amplitudes of 1016 up to 1018 s has been applied to the synthetic observations. Finally, signals including white velocity noise at amplitudes of 104 to 106 m/s have been used instead. Figure 28.3 shows the degree error median of the by least squares adjustment reproduced gravity field

coefficients up to d/o (degree and order) 150. It can be seen that the velocity noise of 105 m/s has an effect on the gravity field reconstruction error which is comparable to 1017 s clock noise, while 106 m/s velocity noise has a similar influence as 1018 s clock noise. Moreover it can be seen that future atomic clocks (Cacciapuoti 2006; Schiller 2007) with an expected short term stability of 1018 s clock noise, and positioning precision of 106 m/s velocity noise it would be possible to deduce earth’s gravity field up do d/o 120. It has to be mentioned that following effects, which will in practice have additional contributions on the total error budget, have been neglected in our simulations:

236

• • • • • •

Tidal and non-tidal temporal variable effects. Non-conservative potentials Relativistic effects on frame transformations. Non-uniformly rotating earth. Non-inertial potentials (satellite rotation). Receiver position related errors (position, velocity and the gravity potential at receiver position are assumed to be error-free). • Dispersive atmosphere related effects. However, it can be expected that the error terms included in this study are the dominant ones.

Fig. 28.4 Geoid height error of simulation with 1018 s clock noise applied on frequency observations. A homogeneous and isotropic error structure is provided

Fig. 28.5 Degree error median plot of Doppler velocity error with 104 m/s noise simulation. Compared to the simulations shown in Fig. 28.3 the influence of Doppler noise is negligibly small

R. Mayrhofer and R. Pail

28.5.2 Spatial and Spectral Error Structure As the gravity potential at the transmitter position, which is deduced from frequency measurements, is the observable for the least squares adjustment, the error structure of the recovered earth gravity field corresponds to the error structure of direct potential observations. Thus, in the case of white noise, the error amplitude increases with higher degree and order of deduced spherical coefficients and the slope of the degree error median is related to the chosen orbit height. Figure 28.4

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Future Satellite Gravity Field Missions: Feasibility Study of Post-Newtonian Method

shows the homogeneous and isotropic spatial error distribution of a simulation configuration with 1018 s clock noise. Simulations with white velocity noise superposed show similar spatial error structures.

28.5.3 Doppler Shift A simulation setup has been defined, where the Doppler shift has been applied on synthetic frequency observations based on noise-free data with which relativistic influences were modeled before. White noise with amplitude of 104 m/s has been superposed on the noise-free velocities (28.10). Figure 28.5 shows that the Doppler shift can be modeled very well even at high velocity noise amplitudes. Moreover, the influence on the recovered gravity field is negligibly small up to d/o 250. So the Doppler shift is no limiting factor. The reason for this is because compared to (28.9), the radial velocity in (28.10) is not scaled by c2 .

28.6

Conclusions

It has been shown that the Post-Newtonian method is a feasible method for reproducing earth’s gravity field for lower and medium frequencies, provided that the technological development of space-borne atomic clocks proceeds in the future. Additionally it has been shown that the derived equations are valid within the defined mission scenario. The two dominant error contributions of this method are the atomic clock noise and the satellite velocity error of the precise orbit determination. The Doppler-effect, which also influences the frequency measurements, can be modeled with sufficient precision, so its error does not leak into the recovered gravity coefficients. It has to be mentioned that the simulations done here should be seen as a concept study and some more realistic simulations will be done to provide more information about the behaviour of the method. A velocity determination precision up to 106 m/s and an atomic clock short term stability of 1018 s is required to resolve the gravity field up to d/o 120. The main advantage of this method is its homogeneous and isotropic spatial error structure.

28.7

237

Outlook

With upcoming atomic clocks below 1016 s short term stability and improving positioning and velocity determination methods, the presented method can be an additional piece for a global earth gravity field monitoring framework in a not too far future. Beside the single-satellite mission presented in this study, various satellite constellations and formations can be designed. In what extent satellite formations like Pendulum, Cartwheel or LISA-like lead to improved precision still has to be investigated by numerical simulations. There is no doubt that multi-satellite missions would underline the possible power of the Post-Newtonian method. One, two or three geostationary satellites could be used as reference stations. Additional rover satellites could be placed in different orbit types. A dense network of satellites could improve the time resolution, so the time variable gravity field could be optimally mapped. These satellites could be equipped with two or more RF-antennas, so one satellite could establish a connection to two or more other satellites, which would further increase the measurement density of the network. As the equations used in this study are strongly simplified, the influences of other effects have to be further investigated. First real orbits and non-conservative forces have to be modeled. Next, the influence of satellite rotation has to be investigated. Additional attitude information from star-tracker measurements and its noise behaviour have to be simulated. All observations and calculations have to be done in a relativistic framework. So the influences of frame transformations in this relativistic framework have to be investigated. All other effects listed in Chap. 28.5.1 will be further investigated in upcoming simulations. This will lead to a much more complex mathematical description which will be harder to linearize, but will also be closer to reality. The main target of upcoming simulations will be to set up a more realistic environment and to design multi-satellite missions that support the advantages of the Post-Newtonian method. Moreover there will be done simulations concerning the time variable gravity field. It will be investigated how different mission design affects the quality of the static and time variable gravity field and if there is

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a possibility to deduce models with higher spatial and temporal resolution. Acknowledgements We would like to thank L. Vitushkin and an anonymous reviewer for their valuable comments which helped to improve the manuscript.

References Cacciapuoti L (2006) Atomic Clocks in Space. ESA-ESTEC (SCI-SP), Frascati Doppler C (1842) Ueber das farbige Licht der Doppelsterne und einiger anderer Gestirnen des Himmels. Abhandlungen der k. b€ ohm. Gesellschaft der Wissenshaften Folge V Band 2, In Commision bei Borrosch & Andre´, Prag € Einstein A (1911) Uber den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes. Annalen der Physik 35, Verlag von Johann Ambrosius Barth, Leipzig pp 898–908 Einstein A (1916) Die Grundlage der allgemeinen Relativit€atstheorie. Annalen der Physik 49, Verlag von Johann Ambrosius Barth, Leipzig pp 769–821 Gulkett M (2003) Relativistic effects in GPS and LEO. Rapport for the Cand. Scient. degree, University of Copenhagen, Department of Geophysics, The Niels Bohr Institute for Physics, Astronomy and Geophysics, Denmark Heiskanen WA, and Moritz H (1967). Physical Geodesy. W. H. Freeman & Co Ltd, San Francisco, ISBN-13 978–0716702337

R. Mayrhofer and R. Pail M€uller J, Soffel M and Klioner SA (2007) Geodesy and relativity. Journal of Geodesy 82 Number 3, Springer, Berlin, ISSN 0949–7714, pp 133–145 Schiller S (2007) Gravimetry with optical clocks. Workshop on The Future of Satellite Gravimetry 12–13 April 2007, ESTEC Schiller S, Tino GM, Gill P, Salomon C, Sterr U, Peik E, Nevsky A, G€orlitz A, Svehla D, Ferrari G, Poli N, Lusanna L, Klein H, Margolis H, Lemonde P, Laurent P, Santarelli G, Clairon A, Ertmer W, Rasel E, M€uller J, Iorio L, L€ammerzahl C, Dittus H, Gill E, Rothacher M, Flechner F, Schreiber U, Flambaum V, Ni, Wei-Tou, Liu, Liang, Chen, Xuzong, Chen, Jingbiao, Gao, Kelin, Cacciapuoti L, Holzwarth R, Heß MP, Sch€afer W (2009) Einstein gravity explorer – a medium class fundamental physics mission. Exp Astron 23 (2):573–610 Shin MY, Park C and Lee SJ (2008) Atomic Clock Error Modeling for GNSS Software Platform. Position, Location and Navigation Symposium, IEEE/ION Plans 2008, pp 71–76, 1-4244-1537-03/08 Schuh W-D (2001) Improved modeling of SGG-data sets by advanced filter strategies. ESA-Project (Hg.): From E€otv€os to mGal+, WP2, Midterm-Report, pp 113–181 Soffel M, Klioner A, Petit G, Wolf P, Kopeikin SM, Bretagnon P, Brumberg VA, Capitaine N, Damour T, Fukushima T, Guinot B, Huang T-Y, Lindegren L, Ma C, Nordtvedt K, Ries JC, Seidelmann PK, Vokrouhlicky D, Will CM, Xu C (2003) The IAU 2000 resolution for astrometry, celestial mechanics, and metrology in the framework: explanatory supplement. Astron J 126:2687–2706, The American Astronomical Society. USA

Local and Regional Comparisons of Gravity and Magnetic Fields

29

C. Jekeli, O. Huang, and T.L. Abt

Abstract

A long recognized connection between the gravitational gradients of the Earth’s crust and its magnetic anomalies, known as Poisson’s relationship, is the object of investigation in this paper. We develop the mathematical and theoretical basis of this relationship in both the space and frequency domains. Anomalies of the magnetic field thus implied by the gravitational gradients (or other derivatives of the gravitational potential) are called pseudo-magnetic anomalies; and, they assume a linear relationship between the mass density of the source material and its magnetization induced by the Earth’s main magnetic field. Tests in several regions of the U.S. that compare gravitational gradients derived from the highresolution model, EGM08, and a continental magnetic anomaly data base reveal that the correlation implied by Poisson’s relationship is not consistent. Some areas exhibit high positive correlation at various frequencies, while others have even strong negative correlation. Therefore, useful applications of Poisson’s relationship depend on the validity of the underlying assumptions that, conversely, may also be investigated and studied using a combination of gradiometric and magnetic data.

29.1

Introduction

The gravitational and magnetic fields of the Earth are connected by the fact that both are generated in part by the same source material, gravitationally because this material has mass density, and magnetically because it is magnetized by the Earth’s main magnetic field due to its liquid outer core. At the microscopic level, the atomic structure of the material has both mass and

C. Jekeli (*) O. Huang T.L. Abt Division of Geodetic Science, School of Earth Sciences, The Ohio State University, 125 South Oval Mall, Columbus, OH 43210, USA e-mail: [email protected].edu

electric dipoles that respond to an applied magnetic field and, in turn, generate an induced field. Mathematically, both fields satisfy Laplace’s differential equation and the solutions are similar, one involving a Green’s function for a distribution of monopoles, the other a Green’s function for a distribution of dipoles. The putative relationship between gravity and magnetic fields was developed already by Poisson (1826), was popularized by Baranov (1957), and essentially posits that magnetic anomalies are gravitational gradients, that is, higher-frequency signals. Mathematical elaborations were carried out by Gunn (1975) and Klingele et al. (1991), and others. The idea that gravitational gradients are intimately connected to the magnetization of the Earth’s crust has motivated

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geophysicists to use more easily obtained magnetic data to make gravimetric-type interpretations (Fedi 1989), and to combine (e.g., Dindi and Swain 1988; Ates and Keary 1995) or contrast (Briden et al. 1982) magnetic and gravimetric data for an improved characterization of subsurface geologic structures. While most previous investigations used gravimetric data to model the gravitational gradients, the correlation between gravitation and magnetism can now also take advantage of various gradiometric data sets (e.g., Jekeli et al. 2008). The purpose of this paper, however, is to investigate this correlation on the basis of the new very-high-degree gravitational model, EGM08 (Pavlis et al. 2008), from which gravitational gradients are readily computed.

29.2

Basic Theory

The gravitational potential, V, at some point, x, due to a mass point (monopole) with mass, m, located at x0 , is well known and given by V ðxÞ ¼ G

m ; jx x0 j

(29.1)

where G ¼ 6:674 1011 m3 /(kg s2 Þ is Newton’s gravitational constant. For the (anomalous) magnetic field, the most elemental source is a dipole, modeled as arising from a microscopic loop current. It can be shown (Telford et al. 1990) that the corresponding magnetic potential, A, is AðxÞ ¼

m0 p e ; 4p jx x0 j2

(29.2)

where m0 ¼ 4p 107 kg m/ ðamp sÞ2 is the magnetic permeability of free space, p is the magnetic dipole moment (vector), and e is the unit vector from x0 to x. For a continuous volume distribution of mass monopoles, we define the mass density as rðx0 Þ ¼ dm=dv; and, analogously, for a continuous distribution of magnetic dipole moments, we define the magnetic dipole moment density as z ðx0 Þ ¼ dp=dv. This density is also called the magnetic polarization or the magnetization intensity. The field is given in each case above, according to the law of superposition, by

ððð V ð xÞ ¼ G v

m A ð xÞ ¼ 0 4p

ððð v

rðx0 Þ dv; jx x0 j z ðx 0 Þ e jx x0 j 2

dv;

(29.3)

(29.4)

where dv ¼ dx1 0 dx2 0 dx3 0 . It is easily verified that the scalar magnetic potential is also expressed as A ð xÞ ¼

m0 4p

ðð ð

z ðx0 Þ:rx

v

1 dv jx x0 j

(29.5)

The magnetization, z, of the crust material, in part, is induced by the main field flux density, H0 , that is generated by the Earth’s outer liquid core, and includes a remnant (remanent) magnetization left from the time of rock formation: z ðx0 Þ ¼ wðx0 ÞH0 ðx0 Þ þ z R ðx0 Þ;

(29.6)

where w is the (presumably scalar) magnetic susceptibility. For present purposes, we assume that the Koenigsberger ratio, Q ¼ z R =wH 0 , is sufficiently small so as to make z R negligible. Furthermore, locally, H0 is almost constant in magnitude and direction. For a given volume of material (x0 2 v), we assume that the mass density and the magnetic susceptibility are linearly related: wðx0 Þ ¼

w0 rðx0 Þ; r0

(29.7)

where r0 and w0 are constants; then z ðx0 Þ ¼ k0

z0 rðx0 Þ; r0

(29.8)

where z 0 ¼ w0 H0 , H0 ¼ H0 k0 , and k0 ¼ ð cos cos x sin cos x

sin x ÞT ;

(29.9)

and where x is the dip angle (inclination) with respect to the local horizon, and is the strike angle (azimuth, or declination) with respect to local geodetic north. With (29.8), the magnetic potential becomes the pseudo-magnetic potential:

29

Local and Regional Comparisons of Gravity and Magnetic Fields

m z AðxÞ ¼ 0 0 k0 rx 4p r0

ððð v

rðx0 Þ dv; jx x0 j

(29.10)

where k0 rx is the directional derivative in the (constant) direction of the magnetization of the material. Combining (29.3) and (29.10), we thus have Poisson’s relation: A ð xÞ ¼

m0 z0 k0 rx V ðxÞ; 4pGr0

(29.11)

or, in terms of the force fields, g ¼ rV, F ¼ rA (the signs are conventional, and the subscript on the gradient operator is now redundant): F ð xÞ ¼

m0 z0 rgT ðxÞk0 : 4pGr0

(29.12)

241

Therefore, from (29.13), we have DB ¼

m0 z 0 T k GðxÞk0 ; 4pGr0 0

which directly relates magnetometry and gravity gradiometry. Technically, the right side of (29.16) is called the pseudo-magnetic anomaly. Under the assumption of a flat Earth, it is possible to relate the gravitational and magnetic fields locally also in the spatial frequency domain. Consider a potential that satisfies Laplace’s equation in the exterior space, z0 G ; > > < tGD1 p tG r D1 Ii3 þ D1 Ii3 ; > r > Ii3 ð1 ð1 pÞ tGD1 Þ; > D2 > : r pIi3 ;

0 t

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