Generalization Kernel for Geopotential Approximation by Harmonic Splines

Authors: Elena Kotevska

Abstract:

This paper presents a generalization kernel for gravitational potential determination by harmonic splines. It was shown in [10] that the gravitational potential can be approximated using a kernel represented as a Newton integral over the real Earth body. On the other side, the theory of geopotential approximation by harmonic splines uses spherically oriented kernels. The purpose of this paper is to show that in the spherical case both kernels have the same type of representation, which leads us to conclusion that it is possible to consider the kernel represented as a Newton integral over the real Earth body as a kind of generalization of spherically harmonic kernels to real geometries.

Keywords: Geopotential, Reproducing Kernel, Approximation, Regular Surface

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1071031

References:

[1] W. Freeden, On Approximation by Harmonic Splines, Manuscripta Geodetica 6, 1981, pp. 193-244.
[2] W. Freeden, R. Reuter, Spherical Harmonic Splines: Theoretical and Computational Aspects, em Meth. u. Verf. d. Math. Physik 27, 1983, pp. 79-103
[3] W. Freeden, A Spline Interpolation Method for Solving Boundary Value Problems of Potential Theory from Discretely Given Data Numerical Methods of Partial Differential Equations 3, 1987, pp. 375-398
[4] W. Freeden, Multiscale Modelling of Spaceborne Geodata, B.G.Teubner, Leipzig, Stuttgart, 1999.
[5] W. Freeden, V. Michel, Multiscale Potential Theory (With Applications to Geoscience). Birkh¨auser Verlag, Boston, Basel, Berlin, 2004
[6] W. Freeden, M. Schreiner, Spherical Functions of Mathematical Geosciences, Springer, Berlin, Heidelberg, 2009
[7] E. W. Grafarend, Six Lectures on Geodesy and Global Geodynamics, Mitt. Geod¨at. Inst. Techn. Univ. Graz, 41, 1982, pp. 531-685
[8] M. Gutting, Fast Multipole Methods for Oblique Derivative Problems, PhD thesis, University of Kaiserslautern, Geomathematics Group, Shaker, Aachen, 2008
[9] K. Hesse, Domain Decomposition Methods in Multiscale Geopotential Determination from SST and SGG, PhD thesis, University of Kaiserslautern, Geomathematics Group, Shaker, Aachen, 2002
[10] E. Kotevska, Real Earth Oriented Gravitational Potential Determination, PhD Thesis, TU Kaiserslautern, Geomathematics group, online publication
[11] T. Krarup, A Contribution to the Mathematical Foundation of Physical Geodesy, Geodaetisk institut, Kobenhavn, 1969.