The inversion of magnetic field data for the underlying magnetization is a frequent problem in geoscience. It occurs at planetary scales, inverting satellite magnetic field information for lithospheric sources, as well as at microscopic scales, inverting for the sources in thin slices of rock samples. All scales have in common that the inverse problem is nonunique and highly instable. Here, we want to provide an overview on this topic and indicate various scenarios for which additional assumptions may ameliorate some of the issues of ill-posedness. This ranges from the assumption of an (infinitely) thin lithosphere (where the Hardy-Hodge decomposition can be used for the characterization of uniqueness) to a priori knowledge about the location or shape of magnetic inclusions within a rock sample (where the Helmholtz decomposition plays a role for the uniqueness aspect).

The sources of geophysical signals are often spatially localized. Thus, adequate basis functions are required to model such properties. Slepian functions have proven to be a very successful tool.

Here, we consider theoretical properties of the Slepian spatial-spectral concentration problem for the space of multi-variate polynomials on the unit ball in $\mathbb{R}^d$ normalized under Jacobi weights. In particular, we show the phenomena of the step-like shape of the eigenvalue distribution of concentration operators, and characterize the transition by the Jacobi weight $W_{0}$, which serves as an analogue of the $2\Omega T$ rule in the classical Slepian theory. A numerical demonstration is performed for the 3-D ball with Lebesgue weights.

We consider challenging inverse problems from the geosciences: the downward continuation of satellite data for the approximation of the gravitational potential as well as the travel time tomography using earthquake data to model the interior of the Earth. Thus, we are able to monitor certain influences on the system Earth, in particular the mass transport of the Earth or its interior anomalies.

For the approximation of these linear(ized) inverse problems, different basis systems can be utilized. Traditionally, we a-priori either choose a global, e.g. spherical harmonics on the sphere or polynomials on the ball, or a local one, e.g. radial basis functions and wavelets or finite elements.

The Learning Inverse Problem Matching Pursuits (LIPMPs), however, have the unique characteristic to enable the combination of global and local trial functions for the approximation of inverse problems. The latter is obtained iteratively from an intentionally overcomplete set of trial functions, the dictionary, such that the Tikhonov functional is reduced. Moreover, the learning add-on allows the dictionary to be infinite such that an a-priori choice of a finite number of trial functions is negligible. Further, it increases the efficiency of the methods.

In this talk, we give details on the LIPMPs and show some current numerical results.

Earth exploration is in many cases connected to inverse problems, since often regions of interest cannot be accessed sufficiently. This is the case for the recovery of structures in the Earth's interior. However, it is also present in the investigation of processes at the Earth's surface, e.g. if a sufficient global or regional coverage is required or if remote areas are of interest.

Many of these problems are associated to an instability of the inverse problems, which is why a variety of regularization methods for their stabilization has been developed so far. However, a notable number of the problems is also ill-posed because of a non-unique solution. Phantom anomalies and other artefacts might be possible consequences. In some cases, the mathematical structure of the underlying null spaces is entirely understood (e.g. for a certain class of Fredholm integral equations of the first kind). In other cases, such a theory is still missing. Nevertheless, also for mathematically well described cases, numerical methods often ignore what can be visible and what can be invisible in available data.

The purpose of this talk is to create some more sensitivity regarding the challenges of inverse problems with non-unique solutions.

[1] S. Leweke, V. Michel, R. Telschow. On the non-uniqueness of gravitational and magnetic field data inversion (survey article), in: Handbook of Mathematical Geodesy (W. Freeden, M.Z. Nashed, eds.), Birkhäuser, Basel, 883-919, 2018.

[2] V. Michel. Geomathematics - Modelling and Solving Mathematical Problems in Geodesy and Geophysics. Cambridge University Press, Cambridge, 2022.

[3] V. Michel, A.S. Fokas. A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods, Inverse Problems 24: 25pp, 2008.

[4] V. Michel, S. Orzlowski. On the null space of a class of Fredholm integral equations of the first kind, Journal of Inverse and Ill-Posed Problems 24: 687-710, 2016.