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Reconstructing the structure of the soil using non-invasive techniques is avery relevant problem in many scientific fields, like geophysics andarchaeology. This can be done, for instance, with the aid of Frequency DomainElectromagnetic (FDEM) induction devices. Inverting FDEM data is a verychallenging 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 awell-posed one whose solution is an accurate approximation of the desired one.In this paper we develop a regularization method to invert FDEM data. Wepropose to determine the electrical conductivity of the ground by solving avariational problem. The minimized functional is made up by the sum of twoterm: the data fitting term ensures that the recovered solution fits themeasured data, while the regularization term enforces sparsity on the Laplacianof the solution. The trade-off between the two terms is determined by theregularization parameter. This is achieved by minimizing an $\ell_2 - \ell_q$functional with $0 < q \leq 2$. Since the functional we wish to minimize isnon-convex, we show that the variational problem admits a solution. Moreover,we prove that, if the regularization parameter is tuned accordingly to theamount of noise present in the data, this model induces a regularizationmethod. Some selected numerical examples on synthetic and real data show thegood performances of our proposal.

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