## Regularization of Inverse Problems in Satellite Geodesy by Wavelet Methods with Orbital Data Given on Closed Surfaces

• Satellite-to-satellite tracking (SST) and satellite gravity gradiometry (SGG), respectively, are two measurement principles in modern satellite geodesy which yield knowledge of the first and second order radial derivative of the earth's gravitational potential at satellite altitude, respectively. A numerical method to compute the gravitational potential on the earth's surface from those observations should be capable of processing huge amounts of observational data. Moreover, it should yield a reconstruction of the gravitational potential at different levels of detail, and it should be possible to reconstruct the gravitational potential from only locally given data. SST and SGG are modeled as ill-posed linear pseudodifferential operator equations with an injective but non-surjective compact operator, which operates between Sobolev spaces of harmonic functions and such ones consisting of their first and second order radial derivatives, respectively. An immediate discretization of the operator equation is obtained by replacing the signal on its right-hand-side either by an interpolating or a smoothing spline which approximates the observational data. Here the noise level and the spatial distribution of the data determine whether spline-interpolation or spline-smoothing is appropriate. The large full linear equation system with positive definite matrix which occurs in the spline-interplation and spline-smoothing problem, respectively, is efficiently solved with the help of the Schwarz alternating algorithm, a domain decomposition method which allows it to split the large linear equation system into several smaller ones which are then solved alernatingly in an iterative procedure. Strongly space-localizing regularization scaling functions and wavelets are used to obtain a multiscale reconstruction of the gravitational potential on the earth's surface. In a numerical experiment the advocated method is successfully applied to reconstruct the earth's gravitational potential from simulated 'exact' and 'error-affected' SGG data on a spherical orbit, using Tikhonov regularization. The applicability of the numerical method is, however, not restricted to data given on a closed orbit but it can also cope with realistic satellite data.

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Verfasserangaben: Petra Baumann urn:nbn:de:hbz:386-kluedo-12517 Diplomarbeit Englisch 2001 2001 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 02.06.2003 Inverse Problems ; Multiplicative Schwarz Algorithm; Regularization Wavelets ; Splines Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 31-XX POTENTIAL THEORY (For probabilistic potential theory, see 60J45) / 31Bxx Higher-dimensional theory / 31B05 Harmonic, subharmonic, superharmonic functions 42-XX FOURIER ANALYSIS / 42Cxx Nontrigonometric harmonic analysis / 42C15 General harmonic expansions, frames 42-XX FOURIER ANALYSIS / 42Cxx Nontrigonometric harmonic analysis / 42C40 Wavelets and other special systems 47-XX OPERATOR THEORY / 47Axx General theory of linear operators / 47A52 Ill-posed problems, regularization [See also 35R25, 47J06, 65F22, 65J20, 65L08, 65M30, 65R30] 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Mxx Numerical methods [See also 90Cxx, 65Kxx] / 49M27 Decomposition methods 65-XX NUMERICAL ANALYSIS / 65Dxx Numerical approximation and computational geometry (primarily algorithms) (For theory, see 41-XX and 68Uxx) / 65D07 Splines 65-XX NUMERICAL ANALYSIS / 65Jxx Numerical analysis in abstract spaces / 65J22 Inverse problems Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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