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

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