Studia Geophysica et Geodaetica

, Volume 49, Issue 4, pp 441–462 | Cite as

GOCE Data Processing: The Spherical Cap Regularization Approach

  • B. Metzler
  • R. Pail
Article

Abstract

Due to the sun-synchronous orbit of the satellite gravity gradiometry mission GOCE, the measurements will not be globally available. As a consequence, using a set of base functions with global support such as spherical harmonics, the matrix of normal equations tends to be ill-conditioned, leading to weakly determined low-order spherical harmonic coefficients. The corresponding geopotential strongly oscillates at the poles.

Considering the special configuration of the GOCE mission, in order to stabilize the normal equations matrix, the Spherical Cap Regularization Approach (SCRA) has been developed. In this approach the geopotential function at the poles is predescribed by an analytical continuous function, which is defined solely in the spatially restricted polar regions. This function could either be based on an existing gravity field model or, alternatively, a low-degree gravity field solution which is adjusted from GOCE observations. Consequently the inversion process is stabilized.

The feasibility of the SCRA is evaluated based on a numerical closed-loop simulation, using a realistic GOCE mission scenario. Compared with standard methods such as Kaula and Tikhonov regularization, the SCRA shows a considerably improved performance.

Keywords

regularization spherical harmonics gravity gradiometry GOCE 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albertella A., Sanso F. and Sneeuw N., 1999. Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere. J. Geodesy, 73, 436–447.Google Scholar
  2. Albertella A., Migliaccio F. and Sanso F., 2001. Data gaps in finite dimensional boundary value problems for satellite gradiometry. J. Geodesy, 75, 641–646.CrossRefGoogle Scholar
  3. Bouman J., 2000. Quality assessment of satellite-based global gravity field models. Publications on Geodesy, New Series, 48, Netherlands Geodetic Commission, Delft, The Netherlands.Google Scholar
  4. Cesare S., 2002. Performance requirements and budgets for the gradiometric mission. Technical Note, GOC-TN-AI-0027, Alenia Spazio, Turin, Italy.Google Scholar
  5. Ditmar P., Kusche J. and Klees R., 2003: Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues. J. Geodesy, 77, 465–477.CrossRefGoogle Scholar
  6. ESA, 1999. Gravity Field and Steady-State Ocean Circulation Mission. Reports for Mission Selection, The Four Candidate Earth Explorer Core Missions, SP-1233(1), European Space Agency, Noordwijk, The Netherlands.Google Scholar
  7. Golub G., Heath M. and Wahba G., 1979. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21, 215–223.Google Scholar
  8. Han S.-C., Jekeli C. and Shum C. K., 2002. Efficient gravity field recovery using in situ disturbing potential observables from CHAMP. Geophys. Res. Lett., 29, Art.No. 1789.Google Scholar
  9. Hansen P., 1992. Analysis of discrete ill-posed problems by means of the L-curve. SIAM Review, 34, 561–580.CrossRefGoogle Scholar
  10. Hoerl A. and Kennard R., 1970. Ridge regression: biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.Google Scholar
  11. Hwang C., 1991. Orthogonal functions over the oceans and applications to the determination of orbit error, geoid, sea surface topography from satellite altimetry. OSU Report, 414, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio, USA.Google Scholar
  12. Ivanov V., 1962. Integral equations of the first kind and an approximate solution for the inverse problem of potential. Soviet Math. Doklady, 3, 210–212.Google Scholar
  13. Kaula W.M., 1966. Theory of Satellite Geodesy: Applications of Satellites to Geodesy. Dover Publications, Mineola, New York, USA.Google Scholar
  14. Klees R., Bouman J., Koop R. and Visser P., 2000. Detailed Scientific Data Processing Approach: Technical Report on Regularization and Downward Continuation. ESA-Project “From Eotvos to mGal”, ESA/ESTEC Contract 13392/98/NL/GD, Final Report, WP 2, European Space Agency, Noordwijk, The Netherlands, 72–103.Google Scholar
  15. Koch K.-R. and Kusche J., 2002. Regularization of geopotential determination from satellite data by variance components. J. Geodesy, 76, 259–268.CrossRefGoogle Scholar
  16. Korte M. and Holme R., 2003. Regularization of spherical cap harmonics. Geophys. J. Int., 153, 253–262.CrossRefGoogle Scholar
  17. Kusche J. and Klees R., 2002. Regularization of gravity field estimation from satellite gravity gradients. J. Geodesy, 76, 359–368.CrossRefGoogle Scholar
  18. Lemoine F.G., Kenyon S.C., Factor J.K., Trimmer R.G., Pavlis N.K., Chinn D.S., Cox C.M., Klosko S.M., Luthcke S.B., Torrence M.H., Wang Y.M., Williamson R.G., Pavlis E.C., Rapp R.H. and Olson T.R., 1998. The Development of the Joint NASE GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96. National Aeronautics and Space Administration, Goddard Space Flight Center, Greenbelt, Maryland, USA.Google Scholar
  19. Lerch F., Iz H. and Chan J., 1993. Gravity model solution based upon SLR data using eigenvalue analysis: Alternative methodology. In: D. Smith and D. Turcotte (Eds.), Contributions of Space Geodesy and Geodynamics-Earth Dynamics. Geodynamics Series, 24, American Geophysical Union, 213–219.Google Scholar
  20. Meissl P., 1971. A study of covariance functions related to the Earth's disturbing potential. OSU Report, 151, Department of Geodetic Science, The Ohio State University, Columbus, USA.Google Scholar
  21. Metzler B. and Weimann F., 2003. WP Ia-4.4 Core Solver: Combined SST+SGG solution. GOCE DAPC Final Report, ASAP-CO-008/03, Graz University of Technology, Graz, Austria, 237–264.Google Scholar
  22. Migliaccio F., Reguzzoni M. and Sanso F., 2004. Space wise approach to satellite gravity field determinations in the presence of coloured noise. J. Geodesy, 78, 304–313.CrossRefGoogle Scholar
  23. Moritz H., 1980. Advanced Physical Geodesy. Herbert Wichmann Vlg., Karlsruhe, Germany.Google Scholar
  24. Morozov V., 1984. Methods for Solving Incorrectly Posed Problems. Springer, Berlin, Heidelberg, New York.Google Scholar
  25. Pail R., 2002. In-orbit calibration and local gravity field continuation problem. ESA-Project “From Eotvos to mGal”, Final Report, ESA/ESTEC Contract 14287/00/NL/DC, WP 1, European Space Agency, Noordwijk, The Netherlands, 9–112.Google Scholar
  26. Pail R., 2004. GOCE Quick-Look Gravity Field Analysis: Treatment of gravity gradients defined in the Gradiometer Reference Frame. Proc. Second International GOCE User Workshop, Frascati, March 2004, European Space Agency, Noordwijk, The Netherlands.Google Scholar
  27. Pail R., Plank G. and Schuh W.-D., 2001. Spatially restricted data distribution on the sphere: the method of orthonormalized functions and applications. J. Geodesy, 75, 44–56.CrossRefGoogle Scholar
  28. Pail R. and Plank G., 2002. Assessment of three numerical solution strategies for gravity field recovery from GOCE satellite gravity gradiometry implemented on a parallel platform. J. Geodesy, 76, 462–474.CrossRefGoogle Scholar
  29. Pail R., Lackner B. and Preimesberger T., 2003. WP Ia-4.1 Quick-Look Gravity Field Analysis (QL-GFA). GOCE DAPC Final Report, ASAP-CO-008/03, Graz University of Technology, Graz, Austria, 107–161.Google Scholar
  30. Pail R. and Plank G., 2004. GOCE gravity field processing strategy. Stud. Geophys. Geod., 48, 289–308.CrossRefGoogle Scholar
  31. Pail R. and Wermuth M., 2003. GOCE SGG and SST quick-look gravity field analysis. Advances in Geosciences, 1, 5–9.Google Scholar
  32. Preimesberger T. and Pail R., 2003. GOCE quick-look gravity solution: application of the semianalytic approach in the case of data gaps and nonrepeat orbits. Stud. Geophy. Geod., 47, 435–453.CrossRefGoogle Scholar
  33. Rapp R., Wang Y. and Pavlis N., 1991. The Ohio state 1991 geopotential and sea surface topography harmonic coefficient models. OSU Report, 410, Department of Geodetic Science and Surveying, The Ohio State University, Columbus, USA.Google Scholar
  34. Rummel R., van Gelderen M., Koop R., Schrama E., Sanso F., Brovelli M., Miggliaccio F. and Sacerdote F., 1993. Spherical harmonic analysis of satellite gradiometry. Publications on Geodesy, 39, Neth. Geod. Comm., Delft, The Netherlands.Google Scholar
  35. Rummel R., Gruber T. and Koop R., 2004. High level processing facility for GOCE: Products and processing strategy. Proc. Second International GOCE User Workshop, Frascati, March 2004, European Space Agency, Noordwijk, The Netherlands.Google Scholar
  36. Schuh W.-D., 1996. Tailored numerical solution strategies for the global determination of the Earth's gravity field. Mitteilungen geod. Inst. TU Graz, No. 81, Graz Univ. of Technology, Graz, Austria.Google Scholar
  37. Smit J., Koop R., Visser P., van den Ijssel J., Sneeuw N., Miller J., Oberndorfer H., 2000. GOCE end to end performance analysis. ESTEC Contract No. 12735/98/NL/GD, European Space Agency, Noordwijk, The Netherlands.Google Scholar
  38. Sneeuw N., 2002. A Semi-Analytical Approach to Gravity Field Analysis from Satellite Observations. Dissertation, DGK, Reihe C, Munich, 527 pp., Bayerische Akademie d. Wissenschaften, Munich, Germany.Google Scholar
  39. Sneeuw N. and Bun R., 1996. Global spherical harmonic computation by two-dimensional Fourier methods. J. Geodesy, 70, 224–232.Google Scholar
  40. Sneeuw N. and van Gelderen M., 1997. The polar gap. In: F. Sanso and R. Rummel (Eds), Geodetic Boundary Value Problems in View of the One Centimeter Geoid. Lecture Notes in Earth Sciences, 65, Springer, 559–568.Google Scholar
  41. Tikhonov A., 1963. Regularization of incorrectly posed problems. Sov. Math. Dokl., 4, 1035–1038.Google Scholar
  42. Tikhonov A. and Arsenin V, 1977. Solutions of Ill-Posed Problems. Wiley and Sons, New York, USA.Google Scholar
  43. Xu P., 1992. Determination of surface gravity anomalies using gradiometric observables. Geophys. J. Int., 110, 321–332.Google Scholar

Copyright information

© StudiaGeo s.r.o. 2005

Authors and Affiliations

  • B. Metzler
    • 1
  • R. Pail
    • 1
  1. 1.Institute of Navigation and Satellite GeodesyGraz University of TechnologyGrazAustria

Personalised recommendations