Christian Gerhards

Christian Gerhards

This paper provides an overview on tools from potential theory on the sphere and some applications in geoscience.

Laurent Baratchart, Christian Gerhards

In Geomagnetism it is of interest to separate the Earth's core magnetic field from the crustal magnetic field. However, measurements by satellites can only sense the sum of the two contributions. In practice, the measured magnetic field is expanded in spherical harmonics and separation into crust and core contribution is achieved empirically, by a sharp cutoff in the spectral domain. In this paper, we derive a mathematical setup in which the two contributions are modeled by harmonic potentials $Φ_0$ and $Φ_1$ generated on two different spheres $\mathbb{S}_{R_0}$ (crust) and $\mathbb{S}_{R_1}$ (core) with radii $R_1<R_0$. Although it is not possible in general to recover $Φ_0$ and $Φ_1$ knowing their superposition $Φ_0+Φ_1$ on a sphere $\mathbb{S}_{R_2}$ with radius $R_2>R_0$, we show that it becomes possible if the magnetization $\mathbf{m}$ generating $Φ_0$ is localized in a strict subregion of $\mathbb{S}_{R_0}$. Beyond unique recoverability, we show in this case how to numerically reconstruct characteristic features of $Φ_0$ (e.g., spherical harmonic Fourier coefficients). An alternative way of phrasing the results is that knowledge of $\mathbf{m}$ on a nonempty open subset of $\mathbb{S}_{R_0}$ allows one to perform separation.

Christian Gerhards

Reconstructing magnetizations from measurements of the generated magnetic potential is generally non-unique. The non-uniqueness still remains if one restricts the magnetization to those induced by an ambient magnetic dipole field (i.e., the magnetization is described by a scalar susceptibility and the dipole direction). Here, we investigate the situation under the additional constraint that the susceptibility is either spatially localized in a subregion of the sphere or that it is band-limited. If the dipole direction is known, then the susceptibility is uniquely determined under the spatial localization constraint while it is only determined up to a constant under the the assumption of band-limitedness. If the dipole direction is not known, uniqueness is lost again. However, we show that all dipole directions that could possibly generate the measured magnetic potential need to be zeros of a certain polynomial which can be computed from the given potential. We provide examples of non-uniqueness of the dipole direction and examples on how to find admissible candidates for the dipole direction under the spatial localization constraint.

Christian Gerhards

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 on a 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 kernel $\tildeΨ_N$. Different from earlier multiscale approaches, it is not the primary goal to obtain an adaptive spatial localization but to simultaneously optimize the related kernels $Φ_N$, $\tildeΨ_N$ in such a way that the former behaves well for the downward continuation while the latter shows a good localization on $Ω_r$ in the region where data is available. The concept is indicated for scalar as well as vector potentials.

Christian Gerhards

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.