Journal of Geodesy

, Volume 82, Issue 4–5, pp 207–221 | Cite as

Efficient GOCE satellite gravity field recovery based on least-squares using QR decomposition

  • Oliver BaurEmail author
  • Gerrit Austen
  • Jürgen Kusche
Original Article


We develop and apply an efficient strategy for Earth gravity field recovery from satellite gravity gradiometry data. Our approach is based upon the Paige-Saunders iterative least-squares method using QR decomposition (LSQR). We modify the original algorithm for space-geodetic applications: firstly, we investigate how convergence can be accelerated by means of both subspace and block-diagonal preconditioning. The efficiency of the latter dominates if the design matrix exhibits block-dominant structure. Secondly, we address Tikhonov-Phillips regularization in general. Thirdly, we demonstrate an effective implementation of the algorithm in a high-performance computing environment. In this context, an important issue is to avoid the twofold computation of the design matrix in each iteration. The computational platform is a 64-processor shared-memory supercomputer. The runtime results prove the successful parallelization of the LSQR solver. The numerical examples are chosen in view of the forthcoming satellite mission GOCE (Gravity field and steady-state Ocean Circulation Explorer). The closed-loop scenario covers 1 month of simulated data with 5 s sampling. We focus exclusively on the analysis of radial components of satellite accelerations and gravity gradients. Our extensions to the basic algorithm enable the method to be competitive with well-established inversion strategies in satellite geodesy, such as conjugate gradient methods or the brute-force approach. In its current development stage, the LSQR method appears ready to deal with real-data applications.


Least-squares Iterative solvers QR decomposition Preconditioning Gravity field recovery GOCE Parallel computing 


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  1. Alkhatib H, Schuh W-D (2007) Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for large-scale least squares problems. J Geod 81:53–66 doi:10.1007/s00190-006-0034-zCrossRefGoogle Scholar
  2. Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J, Du Croz J, Greenbaum A, Hammarling S, McKenney A, Sorensen D (1999) LAPACK Users’ guide, 3rd edn. SIAM, PhiladelphiaGoogle Scholar
  3. Baur O, Austen G (2005) A parallel iterative algorithm for large-scale problems of type potential field recovery from satellite data. In: Proceedings joint CHAMP/GRACE science meeting, Geoforschungszentrum Potsdam, online publication ( Scholar
  4. Baur O, Grafarend EW (2006) High-performance GOCE gravity field recovery from gravity gradients tensor invariants and kinematic orbit information. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the Earth system from space. Springer Heidelberg, pp 239–253Google Scholar
  5. Baur O, Sneeuw N, Grafarend EW (accepted) Methodology and use of tensor invariants for satellite gravity gradiometry. J Geod (accepted)Google Scholar
  6. Björck A (1996) Numerical methods for least squares problems. SIAM, PhiladelphiaGoogle Scholar
  7. Boxhammer Ch (2006) Effiziente numerische Verfahren zur sphärischen harmonischen Analyse von Satellitendaten. Dissertation, University of Bonn, 95ppGoogle Scholar
  8. Boxhammer Ch, Schuh W-D (2006) GOCE gravity field modeling: computational aspects—free kite numbering scheme. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the Earth system from space. Springer Heidelberg, pp 209–224Google Scholar
  9. Chandra R, Menon R, Dagum L, Kohr D, Maydan D, McDonald J (2001) Parallel programming in OpenMP. Academic, New YorkGoogle Scholar
  10. Colombo OL (1984) The global mapping of gravity with two satellites. Netherlands Geodetic Commission, New Series, 7(3)Google Scholar
  11. Ditmar P, Klees R, Kostenko F (2003a) Fast and accurate computation of spherical harmonic coefficients from satellite gravity gradiometry data. J Geod 76:690–705 doi:10.1007/s00190-002-0298-xCrossRefGoogle Scholar
  12. Ditmar P, Kusche J, Klees R (2003b) Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues. J Geod 77:465–477 doi:10.1007/s00190-003-0349-1CrossRefGoogle Scholar
  13. ESA SP-1233 (1999) The four candidate Earth explorer core missions - gravity field and steady-state ocean circulation mission. European Space Agency Report SP-1233(1), GranadaGoogle Scholar
  14. van Gelderen M, Koop R (1997) The use of degree variances in satellite gradiometry. J Geod 71:337–343 doi:10.1007/s001900050101CrossRefGoogle Scholar
  15. Golub GH, Kahan W (1965) Calculating the singular values and pseudoinverse of a matrix. SIAM J Numer Anal 2:205–224Google Scholar
  16. Gundlich B, Koch K-R, Kusche J (2003) Gibbs sampler for computing and propagating large covariance matrices. J Geod 77:514–528 doi:10.1007/s00190-003-0350-5CrossRefGoogle Scholar
  17. Hanke M, Hansen PC (1993) Regularization methods for large-scale problems. Surv Math Ind 3:253–315Google Scholar
  18. Hanke M, Vogel CR (1999) Two-level preconditioners for regularized inverse problems I: theory. Numer Math 83:385–402CrossRefGoogle Scholar
  19. Hansen PC (1987) The truncated SVD as a method for regularization. Numer Math 27:534–553CrossRefGoogle Scholar
  20. Hestenes MR, Stiefel E (1952) Methods of conjugate gradients for solving linear systems. J Res Nat Bur Stand 49:409–436Google Scholar
  21. Ilk KH, Visser P, Kusche J (2003) Satellite Gravity Field Missions. Final Report Special Commission 7, vol. 32, General and technical reports 1999–2003Google Scholar
  22. Jacobsen M, Hansen PC, Saunders MA (2003) Subspace preconditioned LSQR for discrete ill-posed problems. Numer Math 43:975–989CrossRefGoogle Scholar
  23. Keller W (2002) A wavelet approach for the construction of multi-grid Solvers for large linear systems. In: Adam J, Schwarz K-P (eds) Vistas for geodesy in the new millennium. Springer, Berlin, pp 265–270Google Scholar
  24. Klees R, Koop R, Visser P, van den IJssel J (2000) Efficient gravity field recovery from GOCE gravity gradient observations. J Geod 74:561–571 doi:10.1007/s001900000118CrossRefGoogle Scholar
  25. Klees R, Ditmar P, Broersen P (2003) How to handle coloured observation noise in large-scale least-squares problems. J Geod 76:629–640 doi:10.1007/s00190-002-0291-4CrossRefGoogle Scholar
  26. Kusche J (2001) Implementation of multigrid solvers for satellite gravity anomaly recovery. J Geod 74:773–782 doi:10.1007/s001900000140CrossRefGoogle Scholar
  27. Kusche J, Mayer-Gürr T (2001) Iterative solution of ill-conditioned normal equations by Lanczos methods. In: Adam J, Schwarz K-P (eds) Vistas for geodesy in the new millennium. Springer, Berlin, pp 248–252Google Scholar
  28. Kusche J, Klees R (2002) Regularization of gravity field estimation from satellite gravity gradients. J Geod 76:359–368 doi:10.1007/s00190-002-0257-6CrossRefGoogle Scholar
  29. Kusche J (2003) A Monte-Carlo technique for weight estimation in satellite geodesy. J Geod 76:641–652 doi:10.1007/s00190-002-0302-5CrossRefGoogle Scholar
  30. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and NIMA geopotential model EGM96. NASA Goddard Space Flight Center, Greenbelt, 575ppGoogle Scholar
  31. Nolet G (1993) Solving large linearized tomographic problems. In: Iyer HM, Hirahara K (eds) Seismic tomography: theory and practice. Chapman & Hall, London, pp 248–264Google Scholar
  32. Paige CC, Saunders MA (1982a) LSQR: an algorithm for sparse linear equations and sparse least squares. ACM T Math Softw 8:43–71CrossRefGoogle Scholar
  33. Paige CC, Saunders MA (1982b) LSQR: sparse linear equations and least squares problems. ACM T Math Softw 8:195–209CrossRefGoogle Scholar
  34. Pail R, Plank G (2002) Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform. J Geod 76:462–474 doi:10.1007/s00190-002-0277-2CrossRefGoogle Scholar
  35. Pail R (2005) A parametric study on the impact of satellite attitude errors on GOCE gravity field recovery. J Geod 79:231–241 doi:10.1007/s00190-005-0464-zCrossRefGoogle Scholar
  36. Phillips DL (1962) A technique for the numerical solution of certain integral equations of the first kind. J Assoc Comput Mach 9:54–97Google Scholar
  37. Rapp RH, Cruz JY (1986) Spherical harmonic expansions of the Earth’s gravitational potential to degree 360 using 30’ mean anomalies. Technical Report 376, Department of Geodetic Sciences and Surveying, Ohio Sate University, ColumbusGoogle Scholar
  38. Reubelt T, Götzelmann M, Grafarend EW (2005) Harmonic analysis of the Earth gravitational field from kinematic CHAMP orbits based on numerically derived satellite accelerations. In: Flury J, Rummel R, Reigber C, Rothacher M, Boedecker G, Schreiber U (eds) Observation of the Earth system from space. Springer, Heidelberg, pp 27–42Google Scholar
  39. Rummel R (1986) Satellite gradiometry. In: Sünkel H (ed) Mathematical and numerical techniques in physical geodesy. Lect Notes Earth Sci, vol 7. Springer, Berlin, pp 317–363Google Scholar
  40. Rummel R, Sansò F, van Gelderen M, Brovelli M, Koop R, Miggliaccio F, Schrama E, Scerdote F (1993) Spherical harmonic analysis of satellite gradiometry. Netherlands Geodetic Commission, New Series, 39Google Scholar
  41. Schmidt M, Fengler M, Mayer-Gürr T, Eicker A, Kusche J, Sánchez L, Han S-C (2007) Regional gravity modeling in terms of spherical base functions. J Geod 81:17–38 doi:10.1007/s00190-006-0101-5CrossRefGoogle Scholar
  42. Schuh W-D (1996) Tailored numerical solutions strategies for the global determination of the Earth’s gravity field. Mitteilungen der Universität Graz 81Google Scholar
  43. Sneeuw N (2000) A semi-analytical approach to gravity field analysis from satellite observations. Deutsche Geodätische Kommission, Series C 527, MunichGoogle Scholar
  44. Tikhonov AN (1963) Regularization of incorrectly posed problems. Sov Mat Dokl 4:1035–1038Google Scholar
  45. Van der Sluis A, Van der Vorst HA (1987) Numerical solution of large, sparse linear algebraic systems arising from tomographic problems. In: Nolet G (ed) Seismic tomography. Reidel Publications, pp 49–84Google Scholar
  46. Yao ZS, Roberts RG, Tryggvason A (1999) Calculating resolution and covariance matrices for seismic tomography with the LSQR method. Geophys J Int 138:886–894CrossRefGoogle Scholar
  47. Xu J (1997) An introduction to multilevel methods. In: Ainsworth M, Levesley K, Marietta M, Light W (eds) Wavelets, multilevel methods and elliptic PDEs. Numerical Mathematics and Scientific Computation, Clarendon Press, pp 213–302Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of GeodesyUniversität StuttgartStuttgartGermany
  2. 2.Delft Institute of Earth Observation and Space Systems (DEOS)Delft University of TechnologyDelftThe Netherlands
  3. 3.Department of Geodesy and Remote SensingGFZ PotsdamTelegrafenbergGermany

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