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 de 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.

Constraining 3-D electrical resistance tomography with GPR reflection data for improved aquifer characterization

“…The modeling domain was extended 30 m in all directions to reduce boundary effects. The singularity removal technique of Lowry et al (1989) was used to achieve high accuracy by accounting for the rapid decay of electric potential around each current-source position . Since there is no analytical solution for the primary potentials in the presence of topography, the potential field (for a homogeneous earth) was calculated using a refined mesh around the electrodes before the inversion.…”

A saline tracer test monitored via both surface and cross-borehole electrical resistivity tomography: Comparison of time-lapse results

“…To solve the so-called singularity problem, which typically leads to infinite potential gradients at source positions and to very poor numerical approximations, R2 applies the method illustrated by Coggon (1971) for finite elements structures and Lowry et al (1989) for finite differences schemes: the total measured potential is split up into a primary component and a secondary one, the former representing the non-singular component, the latter representing the response to the singularity; the removal of such singularity component helps reduce the error in the calculation of the apparent resistivities over the model and hence the computational effort.…”

Re-Inversion of Surface Electrical Resistivity Tomography Data from the Hanford Site B-Complex

“…To remove modeling errors and inversion artifacts associated with surface topography (Gunther et al 2006), topographic variations extracted from airborne LiDAR data were incorporated into the mesh as shown in Figure 2. In addition, the mesh was refined around electrode locations to reduce modeling errors resulting from the large potential gradients that occur near current source electrodes (Lowry et al 1989). …”