IAG 150 Years pp 233-239 | Cite as

Regional Gravity Field Modeling by Radially Optimized Point Masses: Case Studies with Synthetic Data

  • Miao LinEmail author
  • Heiner Denker
  • Jürgen Müller
Conference paper
Part of the International Association of Geodesy Symposia book series (IAG SYMPOSIA, volume 143)


A two-step point mass method with free depths is presented for regional gravity field modeling based on the remove-compute-restore (RCR) technique. Three numerical test cases were studied using synthetic data with different noise levels. The point masses are searched one by one in the first step with a simultaneous determination of the depth and magnitude by the Quasi-Newton algorithm (L-BFGS-B). In the second step, the magnitudes of all searched point masses are readjusted with known positions by solving a linear least-squares problem. Tikhonov regularization with an identity regularization matrix is employed if ill-posedness exists. One empirical and two heuristic methods for choosing proper regularization parameters are compared. In addition, the solutions computed from standard and regularized least-squares collocation are presented as references.


Free depths Least-squares collocation Point mass method Regional gravity field modeling Tikhonov regularization 



Three anonymous reviewers are acknowledged for their valuable comments which improved the original manuscript. The first author is financially supported by China Scholarship Council (CSC) for his PhD study in Germany.


  1. Ameti P (2006) Downward continuation of Geopotential in Switzerland. PhD Thesis, TU Darmstadt, Darmstadt, GermanyGoogle Scholar
  2. Antunes C, Pail R, Catalão J (2003) Point mass method applied to the regional gravimetric determination of the geoid. Stud Geophys Geod 47:495–509CrossRefGoogle Scholar
  3. Barthelmes F (1986) Untersuchungen zur approximation des äußeren Schwerefeldes der Erde durch Punktmassen mit optimierten Positionen. Report Nr. 92, Veröffentlichungen des Zentralinstitut Physik der Erde, Potsdam, GermanyGoogle Scholar
  4. Bentel K, Schmidt M, Gerlach C (2013) Different radial basis functions and their applicability for regional gravity field representation on the sphere. GEM Int J Geomath 4:67–96CrossRefGoogle Scholar
  5. Bjerhammar A (1986) Megatrend solutions in physical geodesy. NOAA Technical Report No. 116 NGS 34, National Oceanic and Atmospheric Administration, USAGoogle Scholar
  6. Bouman J (1998) Quality of regularization method. DEOS Report Nr. 98.2, Delft Institute for Earth-Orient Space Research, Delft University of Technology, Delft, NetherlandsGoogle Scholar
  7. Bowin C (1983) Depth of principal mass anomalies contributing to the earth’s geoidal undulations and gravity anonalies. Mar Geod 7:61–100CrossRefGoogle Scholar
  8. Byrd RH, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16:1190–1208CrossRefGoogle Scholar
  9. Claessens SJ, Featherstone WE, Barthelmes F (2001) Experiences with point-mass gravity field modeling in the Perth Region, Western Australia. Geomet Res Aust 75:53–86Google Scholar
  10. Eicker A (2008) Gravity field refinement by radial basis functions from in-situ satellite data. PhD Thesis, University of Bonn, Bonn, GermanyGoogle Scholar
  11. Klees R, Tenzer R, Prutkin I, Wittwer T (2008) A data-driven approach to local gravity field modeling using spherical radial basis functions. J Geod 82:457–471CrossRefGoogle Scholar
  12. Koch K-R, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. J Geod 76:259–268CrossRefGoogle Scholar
  13. Kusche J, Klees R (2002) Regularization of gravity field estimation from satellite gravity gradients. J Geod 76:359–368CrossRefGoogle Scholar
  14. Lehmann R (1993) The method of free-positioned point masses-geoid studies on the Gulf of Bothnia. Bull Géod 67:31–40CrossRefGoogle Scholar
  15. Lin M, Denker H, Müller J (2014) Regional gravity field modeling using free-positioned point masses. Stud Geophys Geod. doi: 10.1007/s11200-013-1145-7 Google Scholar
  16. Marchenko AN, Tartachynska ZR (2003) Gravity anomalies in the Black sea area derived from the inversion of GEOSAT, TOPEX/POSEIDON and ERS-2 altimetry. Bull Geod Sci Affini LXII:50–62Google Scholar
  17. Marchenko AN, Barthelmes F, Mayer U, Schwintzer P (2001) Regional geoid determination: an application to airborne gravity data in the Skagerrak. Scientific Technical Report No. 01/07, GFZ, Potsdam, GermanyGoogle Scholar
  18. Mayer-Gürr T, Rieser D, Höck E, Brockmann JM, Schuh W, Krasbutter I, Kusche J, Maier A, Krauss S, Hausleitner W, Baur O, Jäggi A, Meyer U, Prange L, Pail R, Fecher T, Gruber T (2012) The new combined satellite only model GOCO03s. International Symposium on Gravity, Geoid and Height Systems, Venice, Italy, October 9–12 (oral presentation)Google Scholar
  19. Moritz H (1980) Advanced physical geodesy. Herbert Wichman, KarlsruheGoogle Scholar
  20. Nocedal J, Wright SJ (1999) Numerical optimization. Springer, New YorkCrossRefGoogle Scholar
  21. Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2012) The development and evaluation of the earth gravitational model 2008 (EGM2008). J Geophys Res 117:B04406. doi: 10.1029/2011JB008916 CrossRefGoogle Scholar
  22. Schmidt M, Fengler M, Mayer-Guerr T, Eicker A, Kusche J, Sanchez L, Han S (2007) Regional gravity field modeling in terms of spherical base functions. J Geod 81:17–38CrossRefGoogle Scholar
  23. Tenzer R, Klees R (2008) The choice of the spherical radial basis functions in local gravity field modeling. Stud Geophys Geod 52:287–304CrossRefGoogle Scholar
  24. Tikhonov AN (1963) Solution of incorrectly formulated problems and the regularization method. Soviet Math Dokl 4:1035–1038Google Scholar
  25. Tscherning CC, Rapp RH (1974) Closed covariance expressions for gravity anomalies, geoid undulations and deflections of the vertical implied by anomaly degree variance models. OSU Report 208, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio, USAGoogle Scholar
  26. Zhu C, Byrd RH, Lu P, Nocedal J (1994) LBFGS-B: fortran subroutines for large-scale bound constrained optimization. Report NAM-11, EECS Department, Northwestern University, Evanston, IL, USAGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für Erdmessung (IfE), Leibniz Universität HannoverHannoverGermany

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