Scattering matrix evaluation using spatial symmetry in electromagnetic modelling

Abstract

S U M M A R YIn this paper we discuss the possibility of a drastic computational reduction in forming the scattering matrix for electromagnetic modelling of 3-D conductivity structure embedded in a stratified and vertically anisotropic earth using integral equations. This reduction is facilitated by using the lateral homogeneity of the space and the symmetry property of the Green's functions to reduce the redundancy of calculating the scattering matrix by identifying classes of cell pairs which give either identical entries in the scattering matrix or entries that differ only in the sign. It is required that a conductivity structure be discretized into equal-size or equal-sizebased cells. By the latter we mean that the structure is first divided into equal-sized basic cells, and some odd numbers of the basic cells may form secondary, bigger cells where the scattering currents and other field quantities may be assumed to be constant, in order to allow the symmetry reduction while keeping the dimension of the linear system as low as possible. This method of reduction is valid for arbitrary conductivity structure. T h e factor of reduction depends mostly on the number of cells in the lateral direction and can be up to several hundred.

Electromagnetic scattering of large structures in layered earths using integral equations

“…With nonequal-dimensional discretizations, many algorithms are not applicable. Fortunately, we still can use the method of system iteration, and if we utilize the equal-sizebased cells as described by Xiong and Tripp [1993], we still can make use of the spatial symmetry reductions and solve the matrix equation with the method of system iteration without storing the off-diagonal submatrices. The studies presented in this paper are conducted in the low frequency range where electromagnetic waves are diffusive in nature.…”

“…If the structures are discretized into equal-sizebased cells as described by Xiong and Tripp [ 1993], we can still use the method of system iteration and spatial symmetry reduction without storing the offdiagonal submatrices, though the efficiency may be decreased since more computation is involved in regenerating the matrix elements.…”

“…The spatial symmetry of the layered Earth in lateral directions permits a reduction of the computation for forming the impedance matrix [Xiong and Tripp, 1993]. This reduction occurs when a conductivity structure is discretized into equal-size cells, because the lateral homogeneity of the space and the symmetry property of the Green's functions partition the model cells into equivalent classes of cells which give either identical entries in the impedance matrix or entries that differ only in the sign.…”

“…In what follows we shall briefly introduce the method of Xiong [1992b] and Xiong and Tripp [ 1993] and demonstrate the performance of a hybrid of these two methods. We include a convergence analysis of a practical model.…”