Three-dimensional DC resistivity modelling using EXCMG on Unstructured Tetrahedral Grids

Abstract

In geophysical applications, direct current (DC) resistivity surveys usually face the challenge of mountainous topography and electrical anisotropy. And the speed of solving the forward modeling problems greatly impacts the efficiency of imaging/inverting and interpreting the DC data, especially under the above conditions. We present an extrapolation cascadic multigrid method (EXCMG) on unstructured-block tetrahedral grids to solve the large linear system resulting from finite element discretization of three-dimensional DC resistivity modelling with topography. By using a truncated half-sphere computational domain, we simplify the Robin boundary condition for the regular secondary potential in 3-D media. The EXCMG method begins with a relatively coarse unstructured-block tetrahedral mesh, which can capture the model's basic features. Then, a regular refinement strategy is applied to generate a series of nested tetrahedral meshes. We design a novel multigrid prolongation operator, which combines Richardson extrapolation and second-order interpolation on tetrahedra. The elaborately designed prolongation operator can generate quite a good initial guess for the multigrid smoothing operator on the next refined mesh, thereby greatly enhancing the convergence speed and computational efficiency. A variety of models, including the two-layer anisotropic model and the model with terrain, are tested to demonstrate that the proposed EXCMG method can be used to calculate problems with complex electrical structures accurately and efficiently. The solving time of the proposed method is much less than that of the traditional iterative method (e.g., SSOR-BiCGStab) and the algebraic multigrid (AMG) method. Thus, it provides an efficient tool for developing a 3-D DC inversion method that can deal with large-scale problems involving topography and anisotropic media, and it can also be applied to other large-scale problems encountered in geophysics.