Stable evaluation of Green's functions in cylindrically stratified regions with uniaxial anisotropic layers

TL;DR

This paper introduces a numerically stable algorithm for computing electromagnetic Green's functions in cylindrically layered media with uniaxial anisotropy, addressing computational instability issues in extreme parameter regimes.

Contribution

A new rescaling-based method for stable evaluation of Green's functions in anisotropic layered media, extending spectral representations for practical geophysical applications.

Findings

Algorithm remains stable across all parameter ranges.

Numerical results demonstrate robustness in geophysical exploration scenarios.

Addresses issues of overflow, underflow, and round-off errors.

Abstract

We present a robust algorithm for the computation of electromagnetic fields radiated by point sources (Hertzian dipoles) in cylindrically stratified media where each layer may exhibit material properties (permittivity, permeability, and conductivity) with uniaxial anisotropy. Analytical expressions are obtained based on the spectral representation of the tensor Green's function based on cylindrical Bessel and Hankel eigenfunctions, and extended for layered uniaxial media. Due to the poor scaling of these eigenfunctions for extreme arguments and/or orders, direct numerical evaluation of such expressions can produce numerical instability, i.e., underflow, overflow, and/or round-off errors under finite precision arithmetics. To circumvent these problems, we develop a numerically stable formulation through suitable rescaling of various expressions involved in the computational chain, to yield a robust algorithm for all parameter ranges. Numerical results are presented to illustrate the robustness of the formulation including cases of practical interest to geophysical exploration.