An Integral Equation and its Solution for some Two-and Three-Dimensional Problems in Resistivity and Induced Polarization

Here, U* and σ* are the corresponding quantities for the potential and conductivity without the heterogeneity. The integral is taken over the surface of the heterogeneity, ∂U/∂n is the normal derivative (in the direction of the outward normal) of U, and G is a Green's function. Solutions to this equation can readily be found by using the Galerkin method of solving integral equations. The solutions of this equation when the heterogeneity is a sphere or a cylinder in a uniform ground or beneath a conductive overburden are the most readily found. When the solution of the integral has been found for the potential it is a simple matter to calculate the apparent resistivity or chargeability for any electrode configuration.