Resistivity modeling for arbitrarily shaped two dimensional structures. Part I. Theoretical formulation

TLDR: In this article, a numerical technique was developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution.

Abstract: A numerical technique was developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations, making point as well as area discretization, of the subsurface. Potential distribution at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct, explicit, matrix inversion techniques. An empirical mixed boundary condition is used at the ''infinitely distant'' edges of the lower half-space. Accurate solutions using area discretization method were obtained with significantly less attendant computational costs than with the relaxation, finite-element or network solution techniques, for models of comparable dimensions.