Locally Supported Wavelets for the Separation of Spherical Vector Fields With Respect to Their Sources

Abstract

We provide a space domain oriented separation of magnetic fields into parts generated by sources in the exterior and sources in the interior of a given sphere. The separation itself is well-known in geomagnetic modeling, usually in terms of a spherical harmonic analysis or a wavelet analysis that is spherical harmonic based. In contrast to these frequency oriented methods, we use a more spatially oriented approach in this paper. We derive integral representations with explicitly known convolution kernels. Regularizing these singular kernels allows a multiscale representation of the internal and external contributions to the magnetic field with locally supported wavelets. This representation is applied to a set of CHAMP data for crustal field modeling.

On the reconstruction of inducing dipole directions and susceptibilities from knowledge of the magnetic field on a sphere

“…L 2 (S) and H k (S) denote the corresponding scalar valued function spaces. For the rest of this section, we briefly recapitulate some notations and results from [2,3,6,[14][15][16][17]. In particular, proofs of the main results of this section, i.e.…”

On the unique reconstruction of induced spherical magnetizations

“…In order to be able to obtain a spherical version of Theorem 1.4(c), we reformulate the decomposition of Theorem 2.1 in terms of a set of pseudo-differential operatorsõ (1) ,õ (2) ,õ (3) as indicated in [8,10,11,12]. More precisely, 12) where D denotes the pseudo-differential operator…”

“…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.…”

Multiscale Modeling of the Geomagnetic Field and Ionospheric Currents

“…In geomagnetic applications they have been used in Freeden and Gerhards (2010) and Gerhards (2011aGerhards ( , 2012.…”

“…Spherical versions have already been introduced in Dahlke et al (1995), Schröder and Swelden (1995), Windheuser (1996), Holschneider (1996), and Freeden et al (1998). The application to geomagnetic problems, however, is rather recent (see, e.g., Bayer et al 2001;Holschneider et al 2003;Maier and Mayer 2003;Chambodut et al 2005;Mayer and Maier 2006;Freeden and Gerhards 2010;Gerhards 2012).…”