Apart from empirical methods, the methods of interpreting earth-resistivity data may be classified as direct or indirect. In the latter the interpreter attempts to match the resistivity data to one or other of a group of discrete solutions provided by a previously compiled set of tables or curves. In the direct methods, the solution is arrived at by numerical manipulation of the field data.Ideally, a direct method would lead to a complete three-dimensional picture of the resistivity or conductivity distribution for the subsurface volume or model under investigation. Practically, it is necessary to make simplifying assumptions. A common assumption has been that the subsurface or model consists of an array of rectangular blocks embedded in a medium of uniform resistivity. Each block has specified dimensions and a uniform resistivity which is to be determined by the analysis of the field or experimental data.
The best least-squares solution for this assumed subsurface geometry may be expressed in terms of measured potentials or resistivities and a matrix or operator whose elements depend on the subsurface array assumed and the positions taken by the electrodes during the course of the measurements. For each combination of an electrode configuration and a subsurface block a geometric coefficient may be computed. Each coefficient is the algebraic sum of six similar coefficients, one for each surface of the block. The elements of the operators are made up of geometric coefficients or their linear combinations.
Five operators were computed by Ness for an unpublished doctoral thesis (1959) and two more are presented here. In addition, a tabulation is presented of geometric factors for the pole-pole electrode configuration with respect to a large number of buried rectangular horizontal and vertical surfaces. From these factors may be developed the operators for a great variety of subsurface blocks and arrays of blocks, not only for the pole-pole configuration but for any other desired colinear configuration.
The general expression for the geometric coefficient is only an approximation and must undergo certain manipulations to give it the required symmetry properties. As a result the tabulated coefficients display anomalous characteristics which are probably indicative of some degree of error. In addition, the solution to the potential equation on which the work is based is also a first approximation and its development rests on certain simplifying assumptions. Other difficulties arise because there is some uncertainty as to how the contrast between block resistivity and background resistivity should enter into the solution and because it is necessary to assume a value for the background resistivity before the solution can be attempted. Nevertheless, results put forward by other workers suggest that the method is of value. More field and model work is required, however to establish firmly advantages and limitations.
Bukhari, S.A. and Lennox, D.H. (1966): Geometric coefficients for use in numerical resistivity analysis; Research Council of Alberta, RCA/AGS Bulletin 19, 238 p.