ISBN 9783868539301

978-3-86853-930-1, Reihe Mathematik

Christian Gerhards
Spherical Multiscale Methods in Terms of Locally Supported Wavelets: Theory and Application to Geomagnetic Modeling

202 Seiten, Dissertation Technische Universität Kaiserslautern (2011), Softcover, A5

Zusammenfassung / Abstract

In this thesis, we have treated spherical multiscale methods with locally supported wavelets that are motivated by problems of geomagnetic modeling. These methods prove useful whenever the modeled quantity is of strongly localized nature or if the data sets are not equally distributed or of strongly varying quality, so that the classical globally oriented approaches via spherical harmonics do not reflect the given data situation. Our special focus has been set to the construction of locally supported wavelets for crustal field modeling (more precisely, the separation of the Earth’s magnetic field with respect to interior and exterior contributions) and for the reconstruction of ionospheric currents. Sometimes motivated by frequency oriented vector spherical harmonics, we have transferred the mentioned problems into a spatially oriented setting involving the operators ∆∗, ∇∗, L∗ and D.

A major ingredient to this thesis is the treatment of spherical decompositions, especially the derivation of integral representations for the Helmholtz and Mie scalars, and adaptations to the problems under consideration. This is described in detail in Chapter 3 and forms the foundation of further considerations. The regularization of the weakly singular convolution kernels occurring in these integral representations is the crucial step for the multiscale representation. The difference of two of these regularized kernels for different scaling parameters vanishes outside a scale-dependent spherical cap, thus implying the desired locally supported wavelet kernels. Furthermore, by use of Green’s function for the Beltrami operator and the single layer kernel, these kernels can be expressed explicitly by elementary functions, simplifying their numerical evaluation and implementation.