DC resistivity modelling in anisotropic media with Gaussian quadrature grids

Abstract

It has been shown that the spectral method (Trefethen 2000) and the spectral element method (Komatitsch and Tromp 1999) have more attractive features than the traditional finite difference (FD) and the finite element (FE) numerical methods used in resistivity modelling. The main advantages lie in the capability to simulate complex physical models and the exponential power convergence. They have been successfully applied to fluid flow dynamic modelling (Boyd 1989), seismic wave simulations (Komatitsch and Tromp 1999) and electromagnetic computations (Martinec 1999). The spectral method uses some global series of orthogonal functions to represent the unknown solution at the irregular collocation points, subject to boundary conditions. The resulting linear system matrix is full. The spectral element method combines the spectral method and the finite element method, and it possesses the main advantages of each. This includes the capability to handle various model shapes, the sparse matrix format of the FEM and the exponential power convergence of the spectral method. In a recent paper (Zhou et al., 2008) we presented the theory for a new resistivity modelling method based on Gaussian Quadrature Grids (GQG). It readily enables calculation of the electric potential in 2.5-D / 3-D heterogeneous, anisotropic models having arbitrary surface topography. The method co-operatively combines the solution of the Variational Principle of the partial differential equation, Gaussian quadrature abscissae and local cardinal functions so that it transforms the 2.5-D / 3-D resistivity modelling problem into a sparse and symmetric linear equation system, which can be solved by an iterative or matrix inversion method.