Singularity removal: A refinement of resistivity modeling techniques

Abstract

Modeling techniques commonly exhibit errors of 3 to 10 percent or more in the calculation of apparent resistivities over earth models for which analytic solutions are easily available. A singularity occurs in the solution of any elliptic partial differential equation for which the forcing function is not smooth. The inability to adequately represent in discrete space a discontinuous function (in this case, the delta function describing the introduction of current at a point) commonly results in numerical error near the source of a modeled singularity. Inspection of an integrated finite‐difference method for modeling the dc resistivity geophysical technique indicates much of the error encountered is of singular origin. A procedure is herein detailed by which the singularity is mathematically removed from the modeling process and reintroduced as a last step, thus preventing it from contributing to the numerical error. Using this procedure, the average error in apparent resistivity values for a model of a polar‐dipole traverse over a nonconducting sphere is reduced by 40 percent. For a dipole‐dipole traverse of a two‐layer model the error decreases by 75 percent, and in the case of a Wenner profile of a model of a vertically faulted earth, the average error is diminished by 90 percent.