A numerical technique is 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 by using point‐ as well as area‐discretization of the subsurface. Potential distributions 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 are obtained with significantly less attendant computational costs than with the relaxation, finite‐element, or network solution techniques for models of comparable dimensions.

A numerical technique has been developed to solve the three‐dimensional (3-D) potential distribution about a point source of current located in or on the surface of a half‐space containing an arbitrary 3-D conductivity distribution. Self‐adjoint difference equations are obtained for Poisson’s equation using finite‐difference approximations in conjunction with an elemental volume discretization of the lower half‐space. Potential distribution at all points in the set defining the subsurface are simultaneously solved for multiple point sources of current. Accurate and stable solutions are obtained using full, banded, Cholesky decomposition of the capacitance matrix as well as the recently developed incomplete Cholesky‐conjugate gradient iterative method. A comparison of the 2-D and 3-D simple block‐shaped models, for the collinear dipole‐dipole array, indicates substantially lower anomaly indices for inhomogeneities of finite strike‐extent. In general, the strike‐extents of inhomogeneities have to be approximately 10 times the dipole lengths before the response becomes 2-D. The saturation effect with increasing conductivity contrasts appears sooner for the 3-D conductive inhomogeneities than for corresponding models with infinite strike‐lengths. A downhole‐to‐surface configuration of electrodes produces diagnostic total field apparent resistivity maps for 3-D buried inhomogeneities. Experiments with various lateral and depth locations of the current pole indicate that mise‐à‐la‐masse surveys give the largest anomaly if a current pole is located asymmetrically and, preferably, near the top surface of the burried conductor.

Electromagnetic coupling responses in frequency and time‐domain induced‐polarization measurements over a multilayered earth are evaluated. For collinear dipole‐dipole and pole‐dipole configurations over a dissipative layered subsurface, the percent frequency effects of electromagnetic coupling are seen to be as high as 60 percent for large [Formula: see text] values, where L is the length of the receiving dipole, [Formula: see text] is the conductivity of the top layer of the half‐space, and f is the higher frequency of excitation used. In both frequency and time‐domain analyses, the distinctive effects of layering compared to that of a homogeneous half‐space response are shown for different electrode configurations, layer geometry, and electrical parameters of the subsurface. The pole‐dipole configuration of electrodes, in general, exhibits higher coupling compared to the dipole‐dipole configuration. In time‐domain measurements, the late off‐time transient decays reflect almost entirely the normal polarizability of the layered subsurface, in that the coupling responses are significant only during the early off‐time of the transient. The mutual impedance between grounded dipoles of arbitrary length is computed by extension of the complete solution of the boundary‐value problem of a horizontal electric dipole situated over a multilayered half‐space. A number of nomograms are presented for various layered structures to eliminate the electromagnetic coupling response in the induced‐polarization measurements in order to obtain the true polarization effect of the subsurface.

A complete solution of the boundary value problem of a horizontal magnetic dipole over homogeneous and n‐layered half‐spaces is outlined. Quasi‐static expressions for the electric and magnetic fields have been obtained and a comparison of the complete solution with the quasi‐static approximation in practical frequency ranges is made. An analysis of the phases and amplitudes of the magnetic field components and of the polarization parameters of the magnetic field reveals that the phase of the vertical component of the magnetic field and the ellipticity of the magnetic field polarization ellipse are the most sensitive indicators of layering. Amplitude measurements are, in general, less effective than phase measurements for resolution of layered earth structures. Results from both parametric and geometric modes of sounding have been studied in detail for a number of two‐ and three‐layered models of varying thicknesses and conductivity contrasts. Deduction of layering for different thicknesses of the top layer from the measurements of [Formula: see text] and polarization parameters, seems relatively easier when the underlying layer is more conductive than the top layer. For models in which the underlying layer is less conductive than the top layer, the phases of both [Formula: see text] and wave tilt are more diagnostic of changes in layer parameters.