## 1. Introduction

[2] Many geoelectrical structures can be modeled by a set of 3-D inhomogeneities embedded in a horizontally layered medium. The integral equation (IE) method is well known as one of the most accurate techniques for modeling these kind of problems [*Weidelt*, 1975; *Hohmann*, 1975; *Wannamaker et al.*, 1984; *Xiong*, 1992]. The IE method consists of several independent steps making it ideal for easy parallelization providing high performance.

[3] The major obstacle in the full integral equation solution is handling the coefficient matrix with a full structure. For models consisting of more than a couple of thousands of cells, the storage of the coefficient matrix is practically impossible, not to mention the horrendous cost of its direct inversion.

[4] There have been several attempts to overcome the storage and computational cost problem for large matrix inversion. *Xiong* [1992] applied a block iterative scheme to solve the electromagnetic forward problem. In that algorithm only a reasonably small submatrix of the original coefficient matrix is stored at a time. *Portniaguine et al.* [1999] applied the compression algorithm, which is based on a special linear transformation of the full coefficient matrix, such that most elements of the resulting matrix become nearly zero. Thus, thresholding those entries to zero results in significant storage reductions.

[5] Iterative schemes form an alternative to the direct matrix inversion. For example, they have been playing a significant role in forward electromagnetic modeling based on differential methods. There are several studies concerning the finite difference or finite element methods dealing with convergence issues of iterative techniques [*Mackie et al.*, 1994; *Alumbaugh and Newman*, 1995; *Smith*, 1996; *Varentsov*, 1999; *Coggon*, 1971; *Jin*, 1993; *Rätz*, 1999; *Druskin et al.*, 1999]. However, only a few iterative methods solving integral equations have been tested [*Samokhin*, 1993; *Habashy et al.*, 1993; *Singer and Fainberg*, 1995; *Zhdanov and Fang*, 1997; *Avdeev et al.*, 1997; *Zhdanov et al.*, 2000].

[6] In papers by *Pankratov et al.* [1995], *Zhdanov and Fang* [1997], *Zhdanov et al.* [2000], and *Avdeev et al.* [2002], an alternative form of the electromagnetic integral equation was used based on the modified Green's operator with a norm less than one. Based on this contraction operator we consider the Contraction Integral Equation (CIE), which can be treated as a preconditioned conventional integral equation. The preconditioners are diagonal operators determined by the conductivity distribution within the geoelectrical model, facilitating inexpensive manipulations. Existing codes based on the solution of the conventional integral equation can be easily improved by applying preconditioning matrices described in this paper.

[7] In contrast to the series representations used by *Pankratov et al.* [1995], *Zhdanov and Fang* [1997], *Zhdanov et al.* [2000], and *Avdeev et al.* [2002], within the framework of CIE formulation any existing iterative solver can be easily applied. Iterative techniques tested for the conventional IE and CIE methods include a) the Successive Iteration (SI) method, b) the Conjugate Gradient Normal Equation Residual (CGNR) method, c) the Biconjugate Gradient (BICG) method, d) the Biconjugate Gradient Stabilized (BICGSTAB) method, e) a quasi-minimal residual variant of the BiCGSTAB (QMRCGSTAB), and f) the Complex Generalized Minimum Residual (CGMRES) algorithms. Also, we examine the effect of an initial model choice for the iterative solution by introducing a simple strategy for multiple frequency modeling.

[8] The model used in the numerical experiments is a relatively complex 3-D structure consisting of several conductive structures of different sizes, conductivity contrasts and depths. This model has been originally selected within the framework of the COMMEMI project [*Zhdanov et al.*, 1997; *Varentsov et al.*, 2000].

[9] As a result of this work we conclude that the convergence rate of the iterative methods applied to the CIE solution is much better than for the conventional IE method. The most effective solvers are the BIGGSTAB, QMRCGSTAB and CGMRES algorithms (equipped with preconditioning based on the CIE method).