Abstract. Recovering spherical magnetizations m from magnetic field data in the exterior is a highly non-unique problem. A spherical Hardy-Hodge decomposition supplies information on what contributions of the magnetization m are recoverable but it does not supply geophysically suitable constraints on m that would guarantee uniqueness for the entire magnetization. In this paper, we focus on the case of induced spherical magnetizations and show that uniqueness is guaranteed if one assumes that the magnetization is compactly supported on the sphere. The results are based on ideas presented in [4] for the planar setting.

This paper gives an overview on decompositions of vector fields on the sphere that are of importance in geoscientific modeling. Various versions of the Mie and Helmholtz decomposition are presented. A special emphasis is set to integral representations for the different contributions, which is of interest, e.g., in numerical applications. Furthermore, the decompositions are treated in a global framework on the entire sphere and in a local framework on regular subsurfaces. In the end, an application to the modeling of ionospheric currents is indicated.

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.

Abstract. This paper presents a method for the approximation of harmonic potentials that combines downward continuation of globally available data on a sphere Ω R of radius R (e.g., a satellite's orbit) with locally available data in a subregion Γ r of the sphere Ω r of radius r < R (e.g., the spherical Earth's surface). The approximation is based on a two-step algorithm motivated by spherical multiscale expansions: First, a convolution with a scaling kernel Φ N deals with the downward continuation from Ω R to Ω r , while in a second step, the result is locally refined by a convolution on Ω r with a wavelet kernelΨ N . The kernels Φ N andΨ N are optimized in such a way that the former behaves well for the downward continuation while the latter shows a good localization in Γ r . The concept is indicated for scalar as well as vector potentials.

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