Abstract
The choice of the optimal spherical radial basis function (SRBF) in local gravity field modelling from terrestrial gravity data is investigated. Various types of SRBFs are considered: the point-mass kernel, radial multipoles, Poisson wavelets, and the Poisson kernel. The analytical expressions for the Poisson kernel, the point-mass kernel and the radial multipoles are well known, while for the Poisson wavelet new closed analytical expressions are derived for arbitrary orders using recursions. The performance of each SRBF in local gravity field modelling is analyzed using real data. A penalized least-squares technique is applied to estimate the gravity field parameters. As follows from the analysis, almost the same accuracy of gravity field modelling can be achieved for different types of the SRBFs, provided that the depth of the SRBFs is chosen properly. Generalized cross validation is shown to be a suitable technique for the choice of the depth. As a good alternative to generalized cross validation, we propose the minimization of the RMS differences between predicted and observed values at a set of control points. The optimal regularization parameter is determined using variance component estimation techniques. The relation between the depth and the correlation length of the SRBFs is established. It is shown that the optimal depth depends on the type of the SRBF. However, the gravity field solution does not change significantly if the depth is changed by several km. The size of the data area (which is always larger than the target area) depends on the type of the SRBF. The point-mass kernel requires the largest data area.
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References
Blaha G., Blessette R.P. and Hadgigeorge G., 1986. Global point-mass adjustment of the oceanic geoid based on satellite altimetry. Marine Geodesy, 10, 97–129.
Chambodut A., Panet I., Mandea M., Diament M., Holschneider M. and Jamet O., 2005. Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys. J. Int., 163, 875–899.
Förstner W., 1979. Ein Varfahren zur Schätzung von Variaz-und Kovarianzkomponenten. Allgemeine Vermessungs-Nachrichten, 86, 446–453.
Freeden W., Gervens T. and Schreiner M., 1998. Constructive Approximation of the Sphere (with Applications to Geomathematics). Oxford Science Publication, Clarendon Press.
Hardy R.L. and Göpfert W.M., 1975. Least squares prediction of gravity anomalies, geoidal undulations, and deflections of the vertical with multiquadric harmonic functions. Geophys. Res. Lett., 2, 423–426.
Heikkinen M., 1981. Solving the Shape of the Earth by Using Digital Density Models. Report 81(2), Finnish Geodetic Institute, Helsinki, Finland.
Holschneider M., Chambodut A. and Mandea M., 2003. From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys. Earth Planet. Inter., 135, 107–124.
Klees R. and Wittwer T., 2007. A data-adaptive design of a spherical basis function network for gravity field modeling. In: Tregoning P. and Rizos C. (Eds.), Dynamic Planet-Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools. IAG Symposia, 130, Springer-Verlag, Berlin, Heidelberg, 303–328.
Klees R., Tenzer R., Prutkin I. and Wittwer T., 2008. A data-driven approach to local gravity field modeling using spherical radial basis functions. J. Geodesy, 82, 457–471, doi: 10.1007/s00190-007-0196-3.
Koch K.R. and Kusche J., 2002. Regularization of geopotential determination from satellite data by variance components. J. Geodesy, 76, 259–268.
Krarup T., 1969. A Contribution to the Mathematical Foundation of Physical Geodesy. Report 44, Danish Geodetic Institute, Copenhagen, Denmark.
Kusche J., 2003. A Monte-Carlo technique for weight estimation in satellite geodesy. J. Geodesy, 76, 641–652.
Lehmann R., 1993. The method of free-positioned point masses-geoid studies on the Gulf of Bothnia. Bulletin Géodésique, 67, 31–40.
Lehmann R., 1995. Gravity field approximation using point masses in free depths. International Association of Geodesy Buleltin, Section IV, No.1, 129–140.
Lelgemann D., 1981. On numerical properties of interpolation with harmonic kernel functions. Manuscripta Geodaetica, 6, 157–191.
Lelgemann D. and Marchenko A.N., 2001. On Concepts for Modeling the Earth’s Gravity Field. Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften, Reihe A, Heft Nr. 117, München, Germany.
Marchenko A.N., 1998. Parameterization of the Earth’s Gravity Field: Point and Line Singularities. Lviv Astronomical and Geodetical Society, Lviv, Ukraine.
Panet I., Chambodut A., Panet I., Diament M., Holschneider M. and Jamet O., 2006. New insights on intraplate volcanism in French Polynesia from wavelet analysis of GRACE, CHAMP, and sea surface data. J. Geophys. Res., 111, B09403, doi: 10.1029/2005JB004141.
Reilly J.P. and Herbrechtsmeier E.H., 1978. A systematic approach to modeling the geopotential with point mass anomalies. J. Geophys. Res., 83, 841–844.
Sansò F. and Tscherning C.C., 2003. Fast spherical collocation: theory and examples. J. Geodesy, 77, 101–112.
Schmidt M., Kusche J., van Loon J., Shum C.K., Han S.C. and Fabert O., 2005. Multi-resolution representation of regional gravity data. In: Jekeli C., Bastos L. and Fernandes J. (Eds.), Gravity, Geoid and Space Missions. Springer-Verlag, Berlin, Heiderlberg, New York, 167–172.
Schmidt M., Fengler M., Mayer-Gürr T., Eicker A., Kusche J., Sanchez J. and Han S.C., 2007. Regional gravity modelling in terms of spherical base functions. J. Geodesy, 8, 17–38, doi: 10.1007/s00190-006-0101-5.
Sünkel H., 1981. Point Mass Models and the Anomalous Gravitational Field. Report No. 328, The Ohio State University, Dept of Geod Sciences, Columbus, Ohio.
Sünkel H., 1983. The Generation of a Mass Point Model from Surface Gravity Data. Report No. 353, The Ohio State University, Dept of Geod Sciences, Columbus, Ohio.
Tscherning C.C., 1986. Functional methods for gravity field approximation. In: Sünkel H. (Ed.), Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, 7, Springer-Verlag, Berlin, Heidelberg, 3–47.
Vermeer M., 1992. Geoid determination with mass point frequency domain inversion in the Mediterranean. Mare Nostrum, GEOMED Report 2, Madrid, 109–119.
Vermeer M., 1995. Mass point geopotential modeling using fast spectral techniques; historical overview, toolbox description, numerical experiment. Manuscripta Geodaetica, 20, 362–378.
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Tenzer, R., Klees, R. The choice of the spherical radial basis functions in local gravity field modeling. Stud Geophys Geod 52, 287–304 (2008). https://doi.org/10.1007/s11200-008-0022-2
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DOI: https://doi.org/10.1007/s11200-008-0022-2