Conjugate direction relaxation solutions for 3-D magnetotelluric modeling

In recent years, there has been a tremendous amount of progress made in three‐dimensional (3-D) magnetotelluric modeling algorithms. Much of this work has been devoted to the integral equation technique (e.g., Hohmann, 1975; Weidelt, 1975; Wannamaker et al., 1984; Wannamaker, 1991). This method has contributed significantly to our understanding of electromagnetic field behavior in 3-D models. However, some of the very earliest work in 3-D modeling concentrated on differential methods (e.g., Jones and Pascoe, 1972; Reddy et al., 1977). It is generally recognized that differential methods are better suited than integral equation methods to model arbitrarily complex geometries, and consequently this area has recently been receiving a great deal of attention (e.g., Madden and Mackie, 1989; Xinghua et al., 1991; Mackie et al., 1993; Smith, 1992, personnal communication). Differential methods lead to large sparse systems of equations to be solved for the unknown field values. It is possible to use relaxation algorithms to quickly obtain approximate solutions to these systems of equations without resorting to standard matrix inversion routines or sparse matrix solvers.