arXiv subjects

C. Gerhards

Publications and source records attributed to C. Gerhards.

4 recordsLinked to original sources

arXiv2026

Global versus regional internal--external potential field separation

Internal--external field separation is crucial for many aspects of geomagnetism, aiming at distinguishing contributions of the magnetic field generated within a given observation surface from those generated in the exterior. When data are available on a full spherical observation surface, this separation is a standard, stable, and widely used procedure dating back to Gauss. However, when data are only available in a subdomain of the observation surface (as is the case for aeromagnetic and ground-based surveys), the situation drastically changes. Here we show that, without prior assumptions, an internal--external field separation is not uniquely possible. Given the geophysically reasonable assumption that the exterior sources, e.g., magnetospheric and ionospheric current systems, are located above a source-free spherical shell, we show that a unique separation becomes possible but that it is highly unstable. The results are based on the spherical Hardy--Hodge decomposition and explain the intrinsic difficulties of regional data-based internal--external potential field separation.

math.FA↗

arXiv2021

Unique reconstruction of simple magnetizations from their magnetic potential

Inverse problems arising in (geo)magnetism are typically ill-posed, in particular {they exhibit non-uniqueness}. Nevertheless, there exist nontrivial model spaces on which the problem is uniquely solvable. Our goal is here to describe such spaces that accommodate constraints suited for applications. In this paper we treat the inverse magnetization problem on a Lipschitz domain with fairly general topology. We characterize the subspace of $L^{2}$-vector fields that causes non-uniqueness, and identify a subspace of harmonic gradients on which the inversion becomes unique. This classification has consequences for applications and we present some of them in the context of geo-sciences. In the second part of the paper, we discuss the space of piecewise constant vector fields. This vector space is too large to make the inversion unique. But as we show, it contains a dense subspace in $L^2$ on which the problem becomes uniquely solvable, i.e., magnetizations from this subspace are uniquely determined by their magnetic potential.

math.AP↗

arXiv2020

Decomposition of $L^{2}$-vector fields on Lipschitz surfaces: characterization via null-spaces of the scalar potential

For $\partial Ω$ the boundary of a bounded and connected strongly Lipschitz domain in $\mathbb{R}^{d}$ with $d\geq3$, we prove that any field $f\in L^{2} (\partial Ω; \mathbb{R}^{d})$ decomposes, in an unique way, as the sum of three silent vector fields---fields whose magnetic potential vanishes in one or both components of $\mathbb{R}^d\setminus\partial Ω$. Moreover, this decomposition is orthogonal if and only if $\partial Ω$ is a sphere. We also show that any $f$ in $L^{2} (\partial Ω; \mathbb{R}^{d})$ is uniquely the sum of two silent fields and a Hardy function, in which case the sum is orthogonal regardless of $\partial Ω$; we express the corresponding orthogonal projections in terms of layer potentials. When $\partial Ω$ is a sphere, both decompositions coincide and match what has been called the Hardy-Hodge decomposition in the literature.

math.AP↗

arXiv2015

A Parameter Choice Strategy for the Inversion of Multiple Observations

In many geoscientific applications, multiple noisy observations of different origin need to be combined to improve the reconstruction of a common underlying quantity. This naturally leads to multi-parameter models for which adequate strategies are required to choose a set of 'good' parameters. In this study, we present a fairly general method for choosing such a set of parameters, provided that discrete direct, but maybe noisy, measurements of the underlying quantity are included in the observation data, and the inner product of the reconstruction space can be accurately estimated by the inner product of the discretization space. Then the proposed parameter choice method gives an accuracy that only by an absolute constant multiplier differs from the noise level and the accuracy of the best approximant in the reconstruction and in the discretization spaces.

math.NA↗