Abstract
Global gravity field solutions are commonly modelled in spherical harmonic basis functions. Additionally, radial basis functions on the sphere with quasi-local support are used to model regional refinements of gravity fields. However, these functions are usually not orthogonal on a sphere, which makes the modelling process more complex. In this paper we study and compare different radial basis functions and their performance in regional gravity field modelling on the sphere by making use of simulated data. In addition to the type of radial basis function also the size of the study area on the sphere, the point grid, the margins and the method which is used to solve the singular system have to be taken into account. The synthetic signal, which we use in our simulation, is a residual signal in a bandwidth which corresponds to the bandwidth of GOCE satellite gravity observations.
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References
Antoni, M., Keller, W., Weigelt, M.: Representation of regional gravity fields by radial base functions. In: Observing our Changing Earth: Proceedings of the 2007 IAG General Assembly (International Association of Geodesy Symposia), vol. 133, pp. 293–299 (2009)
Chambodut, A., Panet, I., Mandea, M., Diament, M., Holschneider, M., Jamet, O.: Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys. J. Int. 163(3), 875–899 (2005). doi:10.1111/j.1365-246X.2005.02754.x
ESA: Gravity field and steady-state ocean circulation mission. Reports for mission selection: the four candidate earth explorer core missions, ESA SP-1233(1) (1999)
Fehlinger, T.: Multiscale formulations for the disturbing potential and the deflections of the vertical in locally reflected physical geodesy. Dissertation, University of Kaiserslautern, Department of Mathematics, Verlag Dr. Hut. ISBN 978-3-86853-213-5 (2009)
Fehlinger, T., Freeden, W., Gramsch, S., Mayer, C., Michel, D., Schreiner, M.: Local modelling of sea surface topography from (geostrophic) ocean flow. ZAMM 87(11–12), 775–791 (2007)
Fischer, D.: Sparse regularization of a joint inversion of gravitational data and normal mode anomalies. University of Siegen, Dissertation (2011)
Freeden, W., Michel, V.: Multiscale Potential Theory With Applications to Geoscience. Springer Basel AG (2004). ISBN 081764105X
Freeden, W., Schreiner, M.: Spaceborne gravitational field determination by means of locally supported wavelets. J. Geod. 79, 431–446 (2005). doi:10.1007/s00190-005-0482-x
Freeden, W., Schreiner, M.: Local multiscale modelling of geoid undulations from deflections of the vertical. J. Geod. 79, 641–651 (2006)
Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere With Applications to Geoscience. Oxford Science Publications (1998). ISBN 019853682
Freeden, W., Fehlinger, T., Klug, M., Mathar, D., Wolf, K.: Classical globally reflected gravity field determination in modern locally oriented multiscale framework. J. Geod. 83, 1171–1191 (2009). doi:10.1007/s00190-009-0335-0
Gerhards, C.: Spherical multiscale methods in terms of locally supported wavelets: theory and application to geomagnetic modeling. University of Kaiserslautern, Department of Mathematics, Dissertation (2012)
Hofmann-Wellenhof, B., Moritz, H.: Physical Geodesy. Springer Wien, New York (2005)
Holschneider, M., Chambodut, A., Mandea, M.: From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys. Earth Planet. Inter. 135(2–3), 107–124 (2003). doi:10.1016/S0031-9201(02)00210-8
Klees, R., Tenzer, R., Prutkin, I., Wittwer, T.: A data-driven approach to local gravity field modelling using spherical radial basis functions. J. Geod. 82(8) (2008). doi:10.1007/s00190-007-0196-3
Koch, K.R.: Parameter estimation and hypothesis testing in linear models. Springer, Berlin (1999)
Koch, K.R., Kusche, J.: Regularization of geopotential determination from satellite data by variance components. J. Geod. 76, 259–268 (2002). doi:10.1007/s00190-002-0245-x
Moritz, H.: Advanced Physical Geodesy, 2nd edn. Wichmann Verlag, Karlsruhe (1989)
Panet, I., Jamet, O., Diament, M., Chambodut, A.: Modelling the earth’s gravity field using wavelet frames. In: Jekeli, C., Bastos, L., Fernandes, J., Sansò, F. (eds.) Gravity, Geoid and Space Missions, International Association of Geodesy Symposia, vol. 129. Springer, Berlin, pp. 48–53 (2005)
Panet, I., Kuroishi, Y., Holschneider, M.: Wavelet modelling of the gravity field by domain decomposition methods: an example over Japan. Geophys. J. Int. 184(1), 203–219 (2010). doi:10.1111/j.1365-246X.2010.04840.x
Press, W.H., Flanner, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical recipes: the art of scientific computing. Cambridge University Press, Cambridge (2007)
Schmidt, M., Fengler, M., Mayer-Gürr, T., Eicker, A., Kusche, J., Sánchez, L., Han, S.C.: Regional gravity modeling in terms of spherical base functions. J. Geod. 81(1), 17–38 (2007). doi:10.1007/s00190-006-0101-5
Tenzer, R., Klees, R.: The choice of the spherical radial basis funcitons in local gravity field modeling. Stud. Geophys. Geod. 1–18 (2007)
Wolf, K.: Multiscale modeling of classical boundary value problems in physical geodesy by locally supported wavelets. Dissertation, University of Kaiserslautern, Department of Mathematics, Verlag Dr. Hut, ISBN 978-3-86853-249-4 (2009)
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Bentel, K., Schmidt, M. & Gerlach, C. Different radial basis functions and their applicability for regional gravity field representation on the sphere. Int J Geomath 4, 67–96 (2013). https://doi.org/10.1007/s13137-012-0046-1
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DOI: https://doi.org/10.1007/s13137-012-0046-1