Joint Application of Secondary Field and Coupled Potential Formulations to Unstructured Meshes for 3-D CSEM Forward Modeling

We developed a 3-D joint modeling algorithm of controlled source electromagnetic (CSEM) responses with unstructured meshes. The crucial component is to construct an approximating matrix from the stiffness matrix based on coupled potential formulation. Through an incomplete Cholesky decomposition of the approximating matrix and application of the mapping matrix connecting the secondary electric fields and coupled potentials, we obtain an efficient preconditioner for iterative solution of the linear system based on the secondary field formulation. We first demonstrate the accuracy of the algorithm by comparison with analytic solutions for a three-layered earth model. The linear system of equations can be solved efficiently by a quasi-minimum residual (QMR) method with the newly developed preconditioner, while the iterative solution process converges much slower if we use a preconditioner constructed from the original stiffness matrix. Besides, the divergence correction (DC) technique extensively applied for modeling with rectangular meshes is inefficient for modeling with unstructured meshes. We further verify the reliability of the new algorithm by comparison with published algorithms for complex earth models. The new modeling algorithm could provide accurate numerical solutions for complex 3-D earth model within affordable computation resources. We also compare the numerical solutions of the joint modeling algorithms for unstructured meshes and structured meshes. There are observable large differences for the numerical solutions of complex models, which may confirm the privilege of the numerical modeling algorithms with unstructured meshes.