Separating inner and outer Earth's magnetic field from CHAMP satellite measurements by means of vector scaling functions and wavelets

This article is concerned with an approach of modelling the Earth's magnetic field, as measured by satellites, in terms of vector kernel functions, which we will call vector scaling functions and wavelets. The major ingredient is the presentation of a system of vector spherical harmonics which separates a given spherical vector field with respect to its sources, that is, the spherical vector field is separated into a part which is generated by sources inside the sphere, a part which is generated by sources outside the sphere and a part which is generated 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 source but which also allow a locally adapted modelling of the respective vector field. At the end of the article, the method is tested on magnetic field data measured by the German geoscientific research satellite CHAMP.