Symmetry properties of the scattering matrix in 3-D electromagnetic modeling using the integral equation method

Abstract

Modeling large three‐dimensional (3-D) earth conductivity structures continues to pose challenges. Although the theories of electromagnetic modeling are well understood, the basic computational problems are practical, involving the quadratically growing requirements on computer storage and cubically growing computation time with the number of cells required to discretize the modeling body.

Scattering matrix evaluation using spatial symmetry in electromagnetic modelling

“…The method also reduces the storage requirement by a factor of 16 a n d the matrix factorization time by a factor of 16. About the same efficiency was reached by Xiong (1992b) for plane wave excitations where only the scattering current in one quarter needs to be computed. Those reductions, however, are limited to symmetrical models only.…”

“…If we ignore the 3-D conductivity structure and look vertically down at the stratified earth, the earth will be homogeneous in all lateral directions along each ( x -y ) plane. This lateral homogeneity of the space was used by Xiong (1992b) to study the symmetry relations of the scattering matrix to reduce the modelling problem to a quarter of a symmetrical 3-D structure only. Following the same lines as Xiong (1992b), however, we can show in what follows that most of the elements of the scattering matrix for a 3-D structure, which is discretized into equal-sue cells, are related by some simple symmetry relations, which facilitate drastic reductions in computing the scattering matrix elements.…”

“…This lateral homogeneity of the space was used by Xiong (1992b) to study the symmetry relations of the scattering matrix to reduce the modelling problem to a quarter of a symmetrical 3-D structure only. Following the same lines as Xiong (1992b), however, we can show in what follows that most of the elements of the scattering matrix for a 3-D structure, which is discretized into equal-sue cells, are related by some simple symmetry relations, which facilitate drastic reductions in computing the scattering matrix elements. For simplicity of discussion, and without losing any generality, we consider a block structure discretized into N, X N, X N, equal-size rectangular prismatic cells, as shown by the solid lines in Fig.…”

“…The spatial homogeneity and the linearity of the Green's tensors with the directions of the coordinate axes allow us to deduce the symmetry relations of the components of the Green's tensors G(r;l I r:li)j), G(ril I rzli)$-jl), and G(r;l I r:('-,)) with the components of G(rf, 1 r;). This can be readily done by taking into account the XIand Y'-axis directions as well as the directions of components of the Green's tensors, as discussed in Xiong (1992b). If we write down all the components of the Green's tensors G(r;l I rf'),…”

Electromagnetic scattering of large structures in layered earths using integral equations

“…For scatterers with two planes of symmetry Tripp and Hohmann [1984] and Tripp [1990] presented a group theoretic block diagonalization method for arbitrary sources of excitation which reduces the scattering impedance matrix to a block diagonal matrix, with subsequent reduction of storage by a factor of 16 and the matrix factorization time by a factor of 16. Ting and Hohmann [1981], Havedata et al [1987], and Xiong [1992a] computer time and appears to be a bottleneck for computers with fast processors. While the method of system iteration is applicable regardless of the discretizations of the substructures, equally discretized structures give new vitality to this method.…”