Scattering matrix evaluation using spatial symmetry in electromagnetic modelling

Abstract

SUMMARY In 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-size- . based ce11s. 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. The factor of reduction depends mostly on the number of cells in the lateral direction and can be up to several hundred. the Green's tensors which are merely scalar and cannot be Electromagnetic modelling of 3-D conductivity structures vectorized. Thus, in the integral equation method the using integral equations has been widely accepted in formation of the scattering matrix can dominate the geophysics since its appearance in the early 1970s computation time. (Hohmann 1971, 1975; Weidelt 1975). Yet the quadratically With the introduction of the method of system iteration growing requirements on computer storage and on (Xiong 1992a), matrix factorization time and storage computation time in forming the scattering matrix, and the requirements were markedly reduced. In the method of cubic growth of matrix factorization time with the number of system iteration a conductivity structure is divided into discretization cells have remained the major obstacles for its many substructures. The direct matrix inversion is applied to application for about two decades. each substructure only, whereas the mutual interactions The computation of the scattering matrix elements usually among the substructures are treated as being due to consumes about 80 per cent of the total CPU time for equivalent sources. This method not only solves the problem models of moderate size. The cubically growing time of storage requirements, but also greatly reduces the consumption of the matrix factorization is dominant only for computation time needed for the solution of the matrix models with a fairly large number of cells, say over 500 cells. equation. Furthermore, the method of system iteration also With the optimization efforts of Xiong (1992a) where he allows vector and/or parallel processing. Hence, the approached the integration of the secondary parts of the scattering matrix formation time dominates the computation Green's tensors in the vertical direction analytically in the for large structures, too. Hankel transform domain and thus increased the accuracy We can use geometric model symmetries, if present, to and efficiency of the numerical integrations, the computation mitigate these computational problems. Among others, time required by matrix formation was greatly reduced. Tripp & Hohmann (1984) (see also Tripp 1990) put forward Even so, forming the scattering matrix still takes about 50 a block diagonalization method which reduces the scattering per cent of the total CPU time for structures with moderate matrix to four block diagonal matrices and reduced the discretization. Also, the computation of the scattering computation time for matrix formation by a factor of 4. The 459