A two-dimensional integral model of the first-kind for LIN electromagnetic data inversion

In this paper we introduce a two-dimensional first-kind integral model to describe the interaction between the soil and an electromagnetic device. This model is used to reconstruct the electrical conductivity of the soil from electromagnetic data. The definition of the two-dimensional model is derived, and a numerical study of the forward model based on Gauss–Legendre quadrature formulae is presented. To solve the inverse problem, a linear system obtained from the discretization of the integral equation in the model is considered. The main difficulty is the severe ill-conditioning of the system, so the Tikhonov regularization method is applied and different regularization matrices and choice-rules for the regularization parameter are proposed. Several numerical tests show the effectiveness of the proposed approach.