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

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Abstract

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.

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References Cited

  • 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. https://doi.org/10.1007/pl00003999

    Article  Google Scholar 

  • Andersen, O. B., 2010. The DTU10 Gravity Field and Mean Sea Surface. Second International Symposium of the Gravity Field of the Earth (IGFS2). Fairbanks, Alaska

    Google Scholar 

  • 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. https://doi.org/10.1111/j.1365-246x.2005.02754.x

    Article  Google Scholar 

  • Girard, A., 1989. A Fast ‘Monte-Carlo Cross-Validation’ Procedure for Large Least Squares Problems with Noisy Data. Numerische Mathematik, 56(1): 1–23. https://doi.org/10.1007/bf01395775

    Article  Google Scholar 

  • 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. https://doi.org/10.3799/dqkx.2015.148 (in Chinese with English Abstract)

    Article  Google Scholar 

  • 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. https://doi.org/10.1016/j.cam.2005.09.026

    Article  Google Scholar 

  • 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. https://doi.org/10.1137/0914086

    Article  Google Scholar 

  • 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. https://doi.org/10.1007/s00190-013-0650-3

    Article  Google Scholar 

  • 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. https://doi.org/10.1007/s00190-006-0094-0

    Article  Google Scholar 

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

    Google Scholar 

  • 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. https://doi.org/10.1080/01490419.2013.779334

    Article  Google Scholar 

  • Hoerl, A., Kennard, R., 1970. Ridge Regression: Biased Estimation for Nonorthogonal Problems. Technometrics, 42(1): 80–86

    Article  Google Scholar 

  • Holschneider, M., Iglewska-Nowak, I., 2007. Poisson Wavelets on the Sphere. Journal of Fourier Analysis and Applications, 13(4): 405–419. https://doi.org/10.1007/s00041-006-6909-9

    Article  Google Scholar 

  • 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. https://doi.org/10.1109/10.867966

    Article  Google Scholar 

  • 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. https://doi.org/10.1007/s00190-007-0196-3

    Article  Google Scholar 

  • Koch, K. R., 1987. Bayesian Inference for Variance Components. Manuscr. Geod., 12: 309–313

    Google Scholar 

  • Koch, K. R., Kusche, J., 2002. Regularization of Geopotential Determination from Satellite Data by Variance Components. Journal of Geodesy, 76(5): 259–268. https://doi.org/10.1007/s00190-002-0245-x

    Article  Google Scholar 

  • Kusche, J., Klees, R., 2002. Regularization of Gravity Field Estimation from Satellite Gravity Gradients. Journal of Geodesy, 76(6/7): 359–368. https://doi.org/10.1007/s00190-002-0257-6

    Article  Google Scholar 

  • 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. https://doi.org/10.3189/2013jog12j147

    Article  Google Scholar 

  • Rummel, R., Schwarz, K. P., Gerstl, M., 1979. Least Squares Collocation and Regularization. Bulletin Géodésique, 53(4): 343–361. https://doi.org/10.1007/bf02522276

    Article  Google Scholar 

  • Simons, F. J., Dahlen, F. A., 2006. Spherical Slepian Functions and the Polar Gap in Geodesy. Geophysical Journal International, 166(3): 1039–1061. https://doi.org/10.1111/j.1365-246x.2006.03065.x

    Article  Google Scholar 

  • 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. https://doi.org/10.1007/s11200-008-0022-2

    Article  Google Scholar 

  • Tikhonov, A. N., 1963. Regularization of Incorrectly Posed Problems. Sov. Math. Dokl., 4(1): 1624–1627

    Google Scholar 

  • Wittwer, T., 2010. Regional Gravity Field Modelling with Radial Basis Functions: [Dissertation]. Delft University of Technology, Delft

    Google Scholar 

  • 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. https://doi.org/10.6038/cjg20160505 (in Chinese with English Abstract)

    Google Scholar 

  • 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. https://doi.org/10.1186/s40623-017-0618-2

    Google Scholar 

  • 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. https://doi.org/10.6038/cjg20160308 (in Chinese with English Abstract)

    Google Scholar 

  • Xu, P. L., 1992. The Value of Minimum Norm Estimation of Geopotential Fields. Geophysical Journal International, 111(1): 170–178. https://doi.org/10.1111/j.1365-246x.1992.tb00563.x

    Article  Google Scholar 

  • Xu, P. L., 1998. Truncated SVD Methods for Discrete Linear Ill-Posed Problems. Geophysical Journal International, 135(2): 505–514. https://doi.org/10.1046/j.1365-246x.1998.00652.x

    Article  Google Scholar 

  • Xu, P. L., 2009. Iterative Generalized Cross-Validation for Fusing Heteroscedastic Data of Inverse Ill-Posed Problems. Geophysical Journal International, 179(1): 182–200. https://doi.org/10.1111/j.1365-246x.2009.04280.x

    Article  Google Scholar 

  • Xu, P. L., Rummel, R., 1994. Generalized Ridge Regression with Applications in Determination of Potential Fields. Manuscr. Geod., 20: 8–20

    Google Scholar 

  • 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. https://doi.org/10.1007/s00190-006-0032-1

    Article  Google Scholar 

  • 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. https://doi.org/10.3799/dqkx.2015.140 (in Chinese with English Abstract)

    Article  Google Scholar 

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Acknowledgments

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 https://doi.org/10.1007/s12583-017-0771-3.

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Authors and Affiliations

  1. MOE Key Laboratory of Fundamental Physical Quantities Measurement, School of Physics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Yihao Wu & Zhicai Luo

  2. School of Geodesy and Geomatics, Wuhan University, Wuhan, 430079, China

    Bo Zhong

  3. Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University, Wuhan, 430079, China

    Bo Zhong

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  1. Yihao Wu
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  3. Zhicai Luo
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Correspondence to Bo Zhong.

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Wu, Y., Zhong, B. & Luo, Z. Investigation of the Tikhonov Regularization Method in Regional Gravity Field Modeling by Poisson Wavelets Radial Basis Functions. J. Earth Sci. 29, 1349–1358 (2018). https://doi.org/10.1007/s12583-017-0771-3

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  • DOI: https://doi.org/10.1007/s12583-017-0771-3

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