## Harmonic Spline-Wavelets on the 3-dimensional Ball and their Application to the Reconstruction of the Earth´s Density Distribution from Gravitational Data at Arbitrarily Shaped Satellite Orbits

• We introduce splines for the approximation of harmonic functions on a 3-dimensional ball. Those splines are combined with a multiresolution concept. More precisely, at each step of improving the approximation we add more data and, at the same time, reduce the hat-width of the used spline basis functions. Finally, a convergence theorem is proved. One possible application, that is discussed in detail, is the reconstruction of the Earth´s density distribution from gravitational data obtained at a satellite orbit. This is an exponentially ill-posed problem where only the harmonic part of the density can be recovered since its orthogonal complement has the potential 0. Whereas classical approaches use a truncated singular value decomposition (TSVD) with the well-known disadvantages like the non-localizing character of the used spherical harmonics and the bandlimitedness of the solution, modern regularization techniques use wavelets allowing a localized reconstruction via convolutions with kernels that are only essentially large in the region of interest. The essential remaining drawback of a TSVD and the wavelet approaches is that the integrals (i.e. the inner product in case of a TSVD and the convolution in case of wavelets) are calculated on a spherical orbit, which is not given in reality. Thus, simplifying modelling assumptions, that certainly include a modelling error, have to be made. The splines introduced here have the important advantage, that the given data need not be located on a sphere but may be (almost) arbitrarily distributed in the outer space of the Earth. This includes, in particular, the possibility to mix data from different satellite missions (different orbits, different derivatives of the gravitational potential) in the calculation of the Earth´s density distribution. Moreover, the approximating splines can be calculated at varying resolution scales, where the differences for increasing the resolution can be computed with the introduced spline-wavelet technique.

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Author: Martin J. Fengler, Dominik Michel, Volker Michel urn:nbn:de:hbz:386-kluedo-13714 Schriften zur Funktionalanalysis und Geomathematik (16) Preprint English 2005 2005 Technische Universität Kaiserslautern 2005/03/23 GOCE ; GRACE ; Kugel ; Spline-Wavelets ; reguläre Fläche GOCE ; GRACE ; ball ; regular surface ; spline-wavelets CHAMP ; Harmonische Spline-Funktion ; Inverses Problem ; Kugelfunktion ; Mehrskalenanalyse ; Regularisierung ; Sobolev-Raum Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A15 Spline approximation 42-XX FOURIER ANALYSIS / 42Cxx Nontrigonometric harmonic analysis / 42C40 Wavelets and other special systems 45-XX INTEGRAL EQUATIONS / 45Kxx Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] / 45K05 Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 86-XX GEOPHYSICS [See also 76U05, 76V05] / 86Axx Geophysics [See also 76U05, 76V05] / 86A22 Inverse problems [See also 35R30] Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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