Skip to main content

Application of GOCE data for regional gravity field modeling


In principle, every component of a disturbing gravity gradient tensor measured in a satellite orbit can be used to obtain gravity anomalies on the Earth’s surface. Consequently, these can be used in combination with ground or marine data for further gravity field modeling or for verification of satellite data. Theoretical relations can be derived both in spectral and spatial forms. In this paper, we focus on the derivation of a spatial integral form in the geocentric spherical coordinates that seems to be the most convenient one for regional gravity field modeling. All of the second partial derivatives of the generalized Stokes’s kernel are derived, and six surface Fredholm integral equations are formulated and discretized. The inverse problems formulated for particular components clearly reveal different behaviors in terms of numerical stability of the solution. Simulated GOCE data disturbed with Gaussian noise are used to study the performance of two regularization methods: truncated singular value decomposition and ridge regression. The optimal ridge regression method shows slightly better results in terms of the root mean squared deviation of the differences from the exact solution.


  • Andrilli, S. and D. Hecker, Elementary Linear Algebra, 644 pp., Elsevier, Amsterdam, 2003.

    Google Scholar 

  • Aster, R. C., B. Borchers, and C. H. Thurber, Parameter Estimation and Inverse Problems, 301 pp., Elsevier, Amsterdam, 2005.

    Google Scholar 

  • Bronshtein, I. N., K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics, 1153 pp., Springer, Berlin, 2004.

    Book  Google Scholar 

  • Drinkwater, M. R., R. Haagmans, D. Muzi, A. Popescu, R. Floberghagen, M. Kern, and M. Fehringer, The GOCE gravity mission: ESA’s first core earth explorer, in Proceedings of the 3rd International GOCE User Workshop, ESA SP-627, 1–8, 2007.

    Google Scholar 

  • European Space Agency, GOCE L1B Products User Handbook, ESA Technical Note GOCE-GSEG-EOPG-TN-06-0137, 90 s., 2006.

    Google Scholar 

  • Heiskanen, W. and H. Moritz, Physical Geodesy, Freeman, San Francisco, 1967.

    Google Scholar 

  • Hoerl, A. E. and R. W. Kennard, Ridge regression: biased estimation for nonorthogonal problems, Technometrics, 12, 55–67, 1970.

    Article  Google Scholar 

  • Kern, M. and R. Haagmans, Determination of gravity gradients from terrestrial gravity data for calibration and validation of gradiometric GOCE data, in Gravity, Geoid and Space Missions, edited by Jekeli, Bastos, and Fernandes, 95–100, Springer, Berlin, 2005.

    Chapter  Google Scholar 

  • Lemoine, F. G., S. C. Kenyon, J. K. Factor, R. G. Trimmer, N. K. Pavlis, D. S. Chinn, C. M. Cox, S. M. Klosko, S. B. Luthcke, M. H. Torrence, Y. M. Wang, R. G. Williamson, E. C. Pavlis, R. H. Rapp, and T. R. Olson, The development of the joint NASA GSFC and NIMA Geopotential Model EGM96, NASA Technical Paper NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt, USA, 1998.

    Google Scholar 

  • Martinec, Z., Green’s function solution to spherical gradiometric boundary-value problems, J. Geod., 77, 41–49, 2003.

    Article  Google Scholar 

  • Novák, P. and E. W. Grafarend, The effect of topographical and atmospheric masses on spaceborne gravimetric and gradiometric data, Stud. Geophys. Geodaet., 50, 549–582, 2006.

    Article  Google Scholar 

  • Pizzetti, P., Sopra il calcolo teorico delle deviazioni del geoide dall’ ellissoide, Atti R. Accad. Sci. Torino, 46, 331, 1911.

    Google Scholar 

  • Reed, G. B., Application of kinematical geodesy for determining the short wave length components of the gravity field by satellite gradiometry, Technical Report No. 201, Department of Geodetic Science, The Ohio State University, Columbus, Ohio, 1973.

    Google Scholar 

  • van Gelderen, M. and R. Rummel, The solution of the general geodetic boundary value problem by least squares, J. Geod., 75, 1–11, 2001.

    Article  Google Scholar 

  • van Gelderen, M. and R. Rummel, Corrections to “The solution of the general geodetic boundary value problem by least squares”, J. Geod., 76, 121–122, 2002.

    Article  Google Scholar 

  • Xu, P. L., Determination of surface gravity anomanlies using gradiometric observables, Geophys. J. Int., 110, 321–332, 1992.

    Article  Google Scholar 

  • Xu, P. L., Truncated SVD methods for discrete linear ill-posed problems, Geophys. J. Int., 135, 505–514, 1998.

    Article  Google Scholar 

  • Xu, P. L. and R. Rummel, Generalized ridge regression with applications in determination of potential fields, Manuscr. Geod., 20, 8–20, 1994.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Juraj Janák.

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit

Reprints and Permissions

About this article

Cite this article

Janák, J., Fukuda, Y. & Xu, P. Application of GOCE data for regional gravity field modeling. Earth Planet Sp 61, 835–843 (2009).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Key words

  • Gradiometry
  • Pizzetti integral formula
  • inverse problem
  • regularization