Computation of potentials from current electrodes in cylindrically stratified media: A stable, rescaled semi-analytical formulation

TL;DR

This paper introduces a stable, rescaled semi-analytical method for calculating electric potentials in cylindrically layered media, effectively handling numerical issues caused by large variations in layer properties.

Contribution

It develops a novel rescaling of modified-Bessel functions and compares extrapolation techniques to improve the robustness and efficiency of potential computations in stratified media.

Findings

Rescaled Bessel functions prevent numerical underflow and overflow.

Extrapolation methods accelerate convergence of integrals.

Algorithm verified in geophysical exploration scenarios.

Abstract

We present an efficient and robust semi-analytical formulation to compute the electric potential due to arbitrary-located point electrodes in three-dimensional cylindrically stratified media, where the radial thickness and the medium resistivity of each cylindrical layer can vary by many orders of magnitude. A basic roadblock for robust potential computations in such scenarios is the poor scaling of modified-Bessel functions used for computation of the semi-analytical solution, for extreme arguments and/or orders. To accommodate this, we construct a set of rescaled versions of modified-Bessel functions, which avoids underflows and overflows in finite precision arithmetic, and minimizes round-off errors. In addition, several extrapolation methods are applied and compared to expedite the numerical evaluation of the (otherwise slowly convergent) associated Sommerfeld-type integrals. The proposed algorithm is verified in a number of scenarios relevant to geophysical exploration, but the general formulation presented is also applicable to other problems governed by Poisson equation such as Newtonian gravity, heat flow, and potential flow in fluid mechanics, involving cylindrically stratified environments.