Existence of an Optimal Conductivity and Fréchet Differentiability In 2-D Electromagnetic Inversion

Abstract

SUMMARY A 2-D electromagnetic inverse problem relevant for geophysical exploration is considered. an exact theory is described which proves Frechet differentiability of the data and the existence of optimal, or best-fitting, solutions. the earth is represented by a 2-D electrical conductivity structure with a known, horizontal strike direction and the data are surface measurements of electric field due to telluric earth currents flowing along-strike. First, the TE-mode equations governing the forward problem are cast in variational form. It is then shown that there exists at least one admissible conductivity that minimizes the discrepancy between the observations and variational solutions to the forward problem. Admissible conductivities are drawn from any compact subset of the class of bounded, positive functions. Standard techniques from elliptic partial differential equations, non-linear functional analysis and optimization play key roles in establishing the result. In particular, the Lax-Milgram lemma assures that a unique solution to the forward problem exists. an implicit function theorem approach is used to establish the regularity of the mapping from the conductivity to the data. the existence of optimal models follows from the well-known fact that a continuous functional attains its minimum when varied over a compact set. Previously, Frechet differentiability and existence theories for geo-electromagnetic inverse problems have been limited to induction in 1-D media. However, characterization of the optimal models in two dimensions, analogous to Parker's D+ conductivity models in one dimension, is impeded by the lack of closed-form solutions to the forward problem.