Wavelet Decomposition of Spherical Vector Fields with Respect to Sources

Abstract

This article is concerned with an approach of modelling the Earth’s magnetic field as measured by satellites in terms of a special system of vector spherical harmonics and in terms of vector kernel functions, called vector scaling functions and wavelets. The main ingredient is the presentation of a system of vector spherical harmonics which separates a given spherical vector field with respect to its sources, i.e., the spherical vector field is separated into a part which is induced by sources inside the sphere, a part which is induced by sources outside the sphere and a part which is induced by sources on the sphere, which are, for example, currents crossing the sphere. Using this special system of vector spherical harmonics vector scaling functions and wavelets are constructed which keep the advantageous property of separating with respect to sources but which also allow a locally reflected modelling of the respective vector field. At the end of the article, the method is tested on real magnetic field data measured by the German geoscientific research satellite CHAMP.