Evaluation of electrical conductivity and magnetic permeability variations with depth from surface voltage measurements

Measurements of alternating current potential drop at the surface of a conductor can be predicted based on knowledge of the electromagnetic material properties of the conductor and their variation with position. Here we consider the inverse problem of finding the variations of the material properties with depth from multi-frequency potential drop data, which are generated by injecting alternating currents into the conductor. Measurements are carried out using a four-point probe. A possible application of this inverse problem is that of determining the depth dependence of case-hardening in steel. In this article, we define a mathematical framework for the inverse problem and propose a linearization method for solving it. The linearization is based on the assumption that the variations of the material properties with depth are small. The proposed algorithm is tested against both simulated and real experimental data.