On the inverse conductivity problem in the half space

Abstract

We present an analytic treatment of the inverse problem of reconstructing the electrical conductivity of the lower half space from electrical measurements performed on its surface. As the domain under consideration is infinite, the inversion requires the knowledge of data up to infinite distances. One way of overcoming this problem is to approximate the half space by a large cylinder and to use an asymptotic estimate for data at large distances.We have transformed the governing differential equation into an integral equation and regularized it by the use of a priori information. In this way we obtain a stable Fredholm integral equation of the second kind for a regularized conductivity distribution. This equation can be solved either numerically or by using its eigenfunctions which we have computed explicitly.