A variational non-linear constrained model for the inversion of FDEM data

TL;DR

This paper introduces a novel variational regularization method for inverting Frequency Domain Electromagnetic data to reconstruct soil conductivity, effectively handling the non-linear and ill-posed nature of the problem with promising results on synthetic and real data.

Contribution

It develops a non-linear, sparsity-enforcing regularization model for FDEM data inversion, with proven existence of solutions and noise-dependent regularization properties.

Findings

The method effectively reconstructs soil conductivity from noisy FDEM data.

Numerical examples demonstrate good performance on synthetic and real datasets.

The approach handles non-linearity and ill-posedness of the inverse problem.

Abstract

Reconstructing the structure of the soil using non-invasive techniques is a very relevant problem in many scientific fields, like geophysics and archaeology. This can be done, for instance, with the aid of Frequency Domain Electromagnetic (FDEM) induction devices. Inverting FDEM data is a very challenging inverse problem, as the problem is extremely ill-posed, i.e., sensible to the presence of noise in the measured data, and non-linear. Regularization methods substitute the original ill-posed problem with a well-posed one whose solution is an accurate approximation of the desired one. In this paper we develop a regularization method to invert FDEM data. We propose to determine the electrical conductivity of the ground by solving a variational problem. The minimized functional is made up by the sum of two term: the data fitting term ensures that the recovered solution fits the measured data, while the regularization term enforces sparsity on the Laplacian of the solution. The trade-off between the two terms is determined by the regularization parameter. This is achieved by minimizing an $\ell_2 - \ell_q$ functional with $0 < q \leq 2$. Since the functional we wish to minimize is non-convex, we show that the variational problem admits a solution. Moreover, we prove that, if the regularization parameter is tuned accordingly to the amount of noise present in the data, this model induces a regularization method. Some selected numerical examples on synthetic and real data show the good performances of our proposal.