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.

Spherical decompositions in a global and local framework: theory and an application to geomagnetic modeling

“…A second set of vector spherical harmonicsỹ (i) n,k , i = 1, 2, 3, that originates in quantum mechanical considerations on angular momentum (see, e.g., Edmonds 1957), is of great interest for separating the magnetic field with respect to interior and exterior sources (cf. Backus et al 1996;Mayer 2006;Mayer and Maier 2006). This basis system again motivates a spherical decomposition in the style of (1.2), which is treated in some more detail in Sect.…”

“…More information on the second set of vector spherical harmonics can be found, e.g., Edmonds 1957, and with special application to the separation of the geomagnetic field with respect to interior and exterior sources in Backus et al (1996), Mayer (2006) and Mayer and Maier (2006).…”

On the unique reconstruction of induced spherical magnetizations

“…n,k (see, e.g., [1,13,18,19,22]). These vector spherical harmonics can be defined via a suitable connection to the inner harmonics H int n,k and the outer harmonics H ext n,k (i.e., the harmonic extensions of scalar orthonormalized spherical harmonics Y n,k into Ω int and Ω ext , respectively).…”

“…Any function f ∈ l 2 (Ω) can be decomposed into 18) where the scalar functionsF 1 ,F 2 ,F 3 are uniquely determined by the conditions…”

“…By dω we denote the surface element on the sphere Ω R . The non-uniqueness of recovering a vertically integrated magnetization m from the knowledge of V in Ω ext R can be characterized by a fairly well-known decomposition (see, e.g., [1,8,11,12,13,18,19,22]) m =m (1) +m (2) +m (3) , (1.3) which has the property that V ≡ 0 in Ω ext R if and only ifm (2) ≡ 0 (in other words, any magnetization of the form m =m (1) +m (3) produces no magnetic potential in the exterior Ω ext R ). We call such a decomposition a Hardy-Hodge decomposition (cf.…”