Divergence Problem in Solution of Direct Method for Three-Dimensional Magnetotelluric Forward Modeling

Abstract

The direct solver is widely utilized in three-dimensional (3D) magnetotelluric (MT) forward modeling, primarily due to its robustness and ease of implementation. However, divergence issues may arise in the MT direct solution under certain conditions, such as very long periods, high electrical contrasts, or non-uniform discretization, which can result in significant inaccuracies in MT responses. A novel application of the divergence correction procedure, originally developed to accelerate the convergence rate of iterative solvers, is introduced to MT direct solver solutions. It effectively eliminates residual divergence and improves the accuracy of MT responses. Using quasi-analytical solutions from the infinite fault model as a benchmark, a thorough and systematic analysis of the contributing factors reveals that a longer period, significant conductivity variations (especially in high-resistivity environments), or non-uniform discretization can increase the residual divergence in the direct solutions, ultimately leading to inaccuracies in MT responses. The numerical examples of the COMMEMI (comparison of modelling methods for electromagnetic induction problems) 3D-1A model further validate the results of the previous systematic analysis. In addition, a numerical example involving a more complex model with randomly assigned resistivity values highlights the practical significance of the issue. To ensure the numerical reliability of direct solver solutions, it is recommended that the total divergence value (ψall) should be closely monitored, and corrections should be applied whenever ψall exceeds 10-8. These findings offer valuable insights and recommendations for improving the accuracy of MT responses computed using direct solver solutions.