We have developed an edge-based finite-element (FE) modeling algorithm with a divergence correction for calculating the controlled-source electromagnetic (CSEM) responses of a 3D conductivity earth model. We solve a curl-curl equation to directly calculate the secondary electric field to eliminate the source singularity. The choice of the edge-based FE method enables us to properly handle the discontinuity of the normal component of electric fields across conductivity boundaries. Although we can solve the resulting complex-symmetric linear system of equations efficiently by a quasiminimal residual (QMR) method preconditioned with an incomplete Cholesky decomposition for the high-frequency band, the iterative solution process encounters a common problem in the field formulation and does not converge within a practically feasible number of iterations for low frequencies. To overcome this difficulty and to accelerate the iterative solution process in general, we combine a divergence correction technique with the secondary field solution using the QMR solver. We have found that applying the divergence correction intermittently during the iterative solution process ensures the calculation of sufficiently accurate electric and magnetic fields and can significantly speed up the solution process by more than an order of magnitude. We have tested the efficiency and accuracy of our algorithm with 1D and 3D models, and we have found that the divergence correction technique is able to guide the electric field to satisfy the boundary conditions across conductivity interfaces. Although there is a computational overhead required for applying the divergence correction, that cost is significantly offset by the substantial gains in the solution accuracy and speed-up. The work makes the field-based curl-curl formulation using edge elements an efficient and practical method for CSEM simulations.

The galvanic distortion induced by the electric charge buildup across near-surface inhomogeneities can severely affect the interpretation of magnetotelluric (MT) data for deeper structures. In addition to the methods already available, we have developed an alternative approach for processing MT impedance data with such distortions using an equivalent source technique. One prerequisite for the method is that all data are acquired on the surface, which is nearly always the case in land-based MT surveys. The method works with the electric field scaled from the impedance data and constructs an equivalent electrical polarization layer that attempts to reproduce the signal in the data while misfitting the galvanic distortion. Because of the uncorrelated characteristics of galvanic distortions across multiple stations at the same frequency, they can be distinguished and removed by constructing an equivalent source layer of electrical polarization using a regularized inverse formulation. The tradeoff between the signal and distortion is achieved through the use of generalized cross-validation method during the equivalent source construction, whereas the choice of equivalent source parameters also affects the separation. Numerical tests indicate that good results are obtained when the depth of the equivalent source layer is slightly greater than 10 times the nominal data spacing, and the lateral extent is twice that of the data area. The simultaneous processing with multiple frequencies yields more stable apparent resistivity curves than the separate single-frequency processing. The method has performed well in removing the galvanic distortions in the synthetic- and field-data examples.

In the traditional framework of 3D controlled-source electromagnetic (CSEM) modeling, either the field formulation or the coupled potential formulation is implemented. We have developed an improved algorithm that can calculate the 3D CSEM responses based on a joint application of the secondary field and secondary coupled potential formulations. The vector finite-element method is implemented for the discretization of the curl-curl equation of the secondary field. The vector and nodal elements are implemented simultaneously for the discretization of the coupled potential formulation. We use the stiffness matrix resulting from the coupled potential formulation to construct an efficient preconditioner, which is used for the solution of the linear system of equations obtained based on the secondary field formulation. The linear system of the equations could be solved efficiently by the quasiminimum residual method within several hundred iterations with the constructed preconditioner. We compare different modeling schemes to calculate the CSEM responses for a horizontally layered earth model excited by an electric dipole and find that the iteration process of the improved algorithm converges faster on average than other modeling schemes for the frequency band considered. Thus, the least computation time can be obtained over the frequency band for the improved algorithm compared with other modeling schemes, with the same level of accuracy reached. We further verify the efficiency of the algorithm through a small-scale 3D earth model excited by a magnetic dipole. Finally, we illustrate the performance of the algorithm with two 3D subsurface models, including a land-based model and a marine disk model. By comparison with published work, it is illustrated that our algorithm can efficiently calculate accurate land-based and marine CSEM responses with the new constructed preconditioner implemented.

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.

The controlled-source electromagnetic (CSEM) forward modeling with vector and scalar potentials could provide insights into the inductive and galvanic effects. We have developed a finite-element (FE) forward modeling algorithm for the CSEM with vector and scalar potentials. The first-order vector and nodal shape functions are used for the vector and scalar potentials, respectively. To mitigate the null space and thus the nonuniqueness of the potentials caused by the curl operator, we introduce a new preconditioner constructed from an incomplete Cholesky decomposition with zero fill-ins of the Laplacian approximation to the resulted matrix after the FE discretization. We implemented a preconditioned quasiminimal residual method to iteratively solve the resulting linear system with the new preconditioner. We first verified the accuracy of the algorithm through comparison against the analytic solutions obtained for a three-layered earth model. The new iterative modeling algorithm converges much faster compared with the modeling algorithm with the preconditioner constructed from the original stiffness matrix. The efficiency and accuracy are further illustrated by comparison with the modeling scheme based on coupled potentials with explicit enforcement of the Coulomb gauge condition for the vector potential by modeling three 3D models. In addition, the accuracy of our algorithm is demonstrated via a comparison of our numerical solution for a 3D model with the numerical solution obtained from the integral equation method. We also present the inductive and galvanic components of the horizontal secondary electric field for numerical solutions obtained with direct and iterative solvers. The numerical test demonstrated that, aside from the efficiency improved with the new iterative solver, the unstable and nonuniqueness problem for the potentials is eliminated by the new preconditioner with the Laplacian approximation involved.