A goal-oriented adaptive finite-element method for 3D MT anisotropic modeling with topography

Traditional 3D MT data interpretations are based on an isotropic model that is sometimes inappropriate, because it has been well established that electrical anisotropy is widely present in the deep earth. The MT anisotropic modelling is generally worked on structured meshes that has a limited accuracy but cannot model complex geology. We present a goal-oriented adaptive unstructured finite-element method for accurately and efficiently simulating 3D anisotropic MT responses. This is largely different from the traditional 3D MT modelling, like finite-difference or integral equation method that uses artificially refined grids. The accuracy of the latter methods are severely influenced by the quality of mesh, especially for complex geology and topography. We use Galerkin method to discretize the electric field vector wave equation for obtaining the final finite-element equation. Once the electric field is solved, the magnetic field can be calculated via Faraday's law. By solving two polarization modes with source parallel to the x- and y-axes respectively, we can get the impedance tensor. For the adaptive strategy, we use the continuity of normal current to evaluate the posterior error, while the weighting coefficient of the posterior error is obtained by solving a dual problem of the forward problem. Besides, we use a convergence rule to determine which receiver will be used in the next refinement iteration. We check the accuracy of our algorithm against the analytical solutions for a layered anisotropic earth. Further, we study the anisotropic effect on the adaptive meshes and MT responses by models with anisotropic bodies embedded in a half-space. At last, we study the MT responses for an anisotropic body located under a trapezoid hill. The numerical experiments show that our algorithm is an effective tool for modelling 3D anisotropic MT soundings with topography. The numerical results demonstrate that: 1) Anisotropy influences the adaptive meshes differently based on the resistivity contrasts in different directions; 2) The influences of topography on MT responses depends on the polarization mode, while the anisotropy influences the MT responses depending on the direction of anisotropic principal axes and polarization; 3) Anisotropy is resolvable from the distribution of the apparent resistivities by polar plots. Our study aims to provide foundations for the interpretation of 3D MT data on the conditions of topography and anisotropy.