Journal of Earth Science

, Volume 29, Issue 6, pp 1349–1358 | Cite as

Investigation of the Tikhonov Regularization Method in Regional Gravity Field Modeling by Poisson Wavelets Radial Basis Functions

  • Yihao Wu
  • Bo ZhongEmail author
  • Zhicai Luo
Geophysical Imaging from Subduction Zones to Petroleum Reservoirs


The application of Tikhonov regularization method for dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation (VCE) and minimum standard deviation (MSTD) approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the first-order Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.

Key words

regional gravity field modeling Poisson wavelets radial basis functions Tikhonov regularization method L-curve variance component estimation (VCE) 


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Thanks go to the three anonymous reviewers, who gave constructive comments and beneficial suggestions which help us to improve this manuscript. Thanks also go to Prof. Roland Klees from Delft University of Technology for kindly providing the gravity data. This research was mainly supported by the National Natural Science Foundation of China (Nos. 41374023, 41131067, 41474019), the National 973 Project of China (No. 2013CB733302), the China Postdoctoral Science Foundation (No. 2016M602301), the Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University (No. 15-02-08), and the State Scholarship Fund from Chinese Scholarship Council (No. 201306270014). The final publication is available at Springer via

References Cited

  1. Albertella, A., Sansò, F., Sneeuw, N., 1999. Band-Limited Functions on a Bounded Spherical Domain: The Slepian Problem on the Sphere. Journal of Geodesy, 73(9): 436–447. Scholar
  2. Andersen, O. B., 2010. The DTU10 Gravity Field and Mean Sea Surface. Second International Symposium of the Gravity Field of the Earth (IGFS2). Fairbanks, AlaskaGoogle Scholar
  3. Chambodut, A., Panet, I., Mandea, M., et al., 2005. Wavelet Frames: An Alternative to Spherical Harmonic Representation of Potential Fields. Geophysical Journal International, 163(3): 875–899. Scholar
  4. Girard, A., 1989. A Fast ‘Monte-Carlo Cross-Validation’ Procedure for Large Least Squares Problems with Noisy Data. Numerische Mathematik, 56(1): 1–23. Scholar
  5. Guo, D. M., Bao, L. F., Xu, H. Z., 2015. Tectonic Characteristics of the Tibetan Plateau Based on EIGEN-6C2 Gravity Field Model. Earth Science—Journal of China University of Geosciences, 40(10): 1643–1652. (in Chinese with English Abstract)CrossRefGoogle Scholar
  6. Hansen, P. C., Jensen, T. K., Rodriguez, G., 2007. An Adaptive Pruning Algorithm for the Discrete L-Curve Criterion. Journal of Computational and Applied Mathematics, 198(2): 483–492. Scholar
  7. Hansen, P. C., O’Leary, D. P., 1993. The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems. SIAM Journal on Scientific Computing, 14(6): 1487–1503. Scholar
  8. Hashemi Farahani, H., Ditmar, P., Klees, R., et al., 2013. The Static Gravity Field Model DGM-1S from GRACE and GOCE Data: Computation, Validation and an Analysis of GOCE Mission’s Added Value. Journal of Geodesy, 87(9): 843–867. Scholar
  9. Heck, B., Seitz, K., 2006. A Comparison of the Tesseroid, Prism and Point-Mass Approaches for Mass Reductions in Gravity Field Modelling. Journal of Geodesy, 81(2): 121–136. Scholar
  10. Heiskanen, W. A., Moritz H., 1967. Physical Geodesy, WH Freeman and Co., San FranciscoGoogle Scholar
  11. Hirt, C., 2013. RTM Gravity Forward-Modeling Using Topography/Bathymetry Data to Improve High-Degree Global Geopotential Models in the Coastal Zone. Marine Geodesy, 36(2): 183–202. Scholar
  12. Hoerl, A., Kennard, R., 1970. Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 42(1): 80–86CrossRefGoogle Scholar
  13. Holschneider, M., Iglewska-Nowak, I., 2007. Poisson Wavelets on the Sphere. Journal of Fourier Analysis and Applications, 13(4): 405–419. Scholar
  14. Johnston, P. R., Gulrajani, R. M., 2000. Selecting the Corner in the L-Curve Approach to Tikhonov Regularization. IEEE Transactions on Biomedical Engineering, 47(9): 1293–1296. Scholar
  15. Klees, R., Tenzer, R., Prutkin, I., et al., 2008. A Data-Driven Approach to Local Gravity Field Modelling Using Spherical Radial Basis Functions. Journal of Geodesy, 82(8): 457–471. Scholar
  16. Koch, K. R., 1987. Bayesian Inference for Variance Components. Manuscr. Geod., 12: 309–313Google Scholar
  17. Koch, K. R., Kusche, J., 2002. Regularization of Geopotential Determination from Satellite Data by Variance Components. Journal of Geodesy, 76(5): 259–268. Scholar
  18. Kusche, J., Klees, R., 2002. Regularization of Gravity Field Estimation from Satellite Gravity Gradients. Journal of Geodesy, 76(6/7): 359–368. Scholar
  19. Luthcke, S. B., Sabaka, T. J., Loomis, B. D., et al., 2013. Antarctica, Greenland and Gulf of Alaska Land-Ice Evolution from an Iterated GRACE Global Mascon Solution. Journal of Glaciology, 59(216): 613–631. Scholar
  20. Rummel, R., Schwarz, K. P., Gerstl, M., 1979. Least Squares Collocation and Regularization. Bulletin Géodésique, 53(4): 343–361. Scholar
  21. Simons, F. J., Dahlen, F. A., 2006. Spherical Slepian Functions and the Polar Gap in Geodesy. Geophysical Journal International, 166(3): 1039–1061. Scholar
  22. Tenzer, R., Klees, R., 2008. The Choice of the Spherical Radial Basis Functions in Local Gravity Field Modeling. Studia Geophysica et Geodaetica, 52(3): 287–304. Scholar
  23. Tikhonov, A. N., 1963. Regularization of Incorrectly Posed Problems. Sov. Math. Dokl., 4(1): 1624–1627Google Scholar
  24. Wittwer, T., 2010. Regional Gravity Field Modelling with Radial Basis Functions: [Dissertation]. Delft University of Technology, DelftGoogle Scholar
  25. Wu, Y. H., Luo, Z. C., 2016. The Approach of Regional Geoid Refinement Based on Combining Multi-Satellite Altimetry Observations and Heterogeneous Gravity Data Sets. Chinese J. Geophys., 59(5): 1596–1607. (in Chinese with English Abstract)Google Scholar
  26. Wu, Y. H., Luo, Z. C., Chen, W., et al., 2017. High-Resolution Regional Gravity Field Recovery from Poisson Wavelets Using Heterogeneous Observational Techniques. Earth, Planets and Space, 69(34): 1–15. Scholar
  27. Wu, Y. H., Luo, Z. C., Zhou, B. Y., 2016. Regional Gravity Modelling Based on Heterogeneous Data Sets by Using Poisson Wavelets Radial Basis Functions. Chinese J. Geophys., 59(3): 852–864. (in Chinese with English Abstract)Google Scholar
  28. Xu, P. L., 1992. The Value of Minimum Norm Estimation of Geopotential Fields. Geophysical Journal International, 111(1): 170–178. Scholar
  29. Xu, P. L., 1998. Truncated SVD Methods for Discrete Linear Ill-Posed Problems. Geophysical Journal International, 135(2): 505–514. Scholar
  30. Xu, P. L., 2009. Iterative Generalized Cross-Validation for Fusing Heteroscedastic Data of Inverse Ill-Posed Problems. Geophysical Journal International, 179(1): 182–200. Scholar
  31. Xu, P. L., Rummel, R., 1994. Generalized Ridge Regression with Applications in Determination of Potential Fields. Manuscr. Geod., 20: 8–20Google Scholar
  32. Xu, P. L., Shen, Y. Z., Fukuda, Y., et al., 2006. Variance Component Estimation in Linear Inverse Ill-Posed Models. Journal of Geodesy, 80(2): 69–81. Scholar
  33. Xu, S. F., Chen, C., Du, J. S., et al., 2015. Characteristics and Tectonic Implications of Lithospheric Density Structures beneath Western Junggar and Its Surroundings. Earth Science—Journal of China University of Geosciences, 40(9): 1556–1565. (in Chinese with English Abstract)CrossRefGoogle Scholar

Copyright information

© China University of Geosciences and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.MOE Key Laboratory of Fundamental Physical Quantities Measurement, School of PhysicsHuazhong University of Science and TechnologyWuhanChina
  2. 2.School of Geodesy and GeomaticsWuhan UniversityWuhanChina
  3. 3.Key Laboratory of Geospace Environment and Geodesy, Ministry of EducationWuhan UniversityWuhanChina

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