Multi-order vector finite element modeling of 3D magnetotelluric data including complex geometry and anisotropy.

Abstract

This thesis presents the development of a computational algorithm in Fortran, to model 3D magnetotelluric (MT) data using a Multi-order Vector Finite Element Method (MoVFEM) to include complex geometry (such as topography, and subsurface interfaces). All the modules in MoVFEM have been programmed from the beginning, unless specified by referencing the libraries used. The governing equations to be solved are the decoupled electromagnetic (EM) partial differential equations for the secondary electric field, or the secondary magnetic field, with a symmetric conductivity tensor to include anisotropy. The primary fields are the solution of a plane-wave within the air domain. Two boundary conditions are implemented, namely the Generalized Perfect Matched Layers method (GPML) and Dirichlet boundary conditions. Three Dirichlet boundary schemes are applied, first considering zero EM fields at the boundaries of the computational domain; secondly, considering the boundaries as homogeneous Earth; and finally, considering the boundaries as a layered Earth. Two formulations of GPML are implemented in this algorithm, firstly the original GPML formulation and secondly, the GPML parameters are modified for the MT and Controlled Source Electromagnetic (CSEM) problem. High-order edge-elements are defined based on covariant projections, and mixed-order edge-elements for hexahedra. The vector basis functions are defined for linear elements (12 edge-elements), quadratic elements (36 edge-elements), and Lagrangian elements (54 edge-elements). By this definition, the vector basis will have zero divergence in the case of rectangular elements and relatively small divergence in the case of distorted elements. The validation of this computational algorithm is performed with a homogeneous Earth, where the analytic solution of the MT problem is known. In the validation, the convergence of the solution is analyzed for different grid spacing and for different element-orders with Dirichlet boundary conditions. High-order elements produce accurate solutions with larger spacing than the fine grid needed for linear-order elements. After the convergence analysis, the solution obtained with all the proposed boundary conditions, and edge-element orders are compared for one frequency, and for a frequency range. In the homogeneous Earth, Dirichlet boundary condition presents backward reflections from the boundaries of the computational domain to the center of the model. Both GPML formulations produce more stable solutions, where no boundary reflections are present. However the MT responses fluctuate within a small range close to the values for the homogeneous Earth. The GPML formulation for MT and CSEM produce more accurate results and stabilize the MT responses over a frequency range. This algorithm is applied to synthetic examples with complex conductivity structures. Some of these synthetic examples have been published previously, thus the results of this algorithm are compared qualitatively. In the case of anisotropy and complex geometry, the proposed synthetic examples have not been published, and a discussion of how the MT responses behave for these Earth examples is presented. This computational algorithm could be extended with the use of an adaptive method, and it could be implemented in an algorithm for 3D inversion of MT data.