A 3‐D perturbation solution for the EM induction problem in a spherical earth—the forward problem

Abstract

A solution of the three‐dimensional forward problem for electrical conductivity in a spherical earth is considered. This work is based upon the standard decomposition which separates the magnetic field into toroidal magnetic (TM) mode and poloidal magnetic (PM) mode, and the assumption that the lateral inhomogeneity of the conductivity structure of the earth at mid‐mantle depths is small. Under this restriction, we apply a perturbation method. The conductivity may be written as the summation of its major part, which is radially symmetric, and a perturbation term which is a function of all three variables, i.e. σ=σ0(r) +σ1 (r, Θ, ϕ). It can be shown that the zeroth‐order approximation of the problem is just the one‐dimensional case in which the two modes TM and PM can be totally separated. Higher order solutions introduce lateral heterogeneity into the system and make the two modes couple into each other. A finite difference method is used to solve the zeroth‐order equation. The first‐order perturbation solution depends on the solution to the zeroth‐order problem. The three‐dimensional perturbation to the initial one‐dimensional solution is obtained by expanding a function of conductivity in terms of spherical harmonics. The three‐dimensional solution is calculated from the coefficients of this expansion by numerical integration. Preliminary results are compared with Wannamaker's flat earth two dimensional finite element model for a restricted set of conductivity models, and a reasonable agreement is obtained. A three‐dimensional model has also been calculated up to second‐order approximation. Copyright © 1992, Wiley Blackwell. All rights reserved