For large-scale magnetotelluric (MT) forward modeling using vector finite-element methods, iterative solvers are commonly applied. However, as frequency decreases close to zero, the iterative solution process struggles to converge. This is mainly caused by the fact that the weak conductivity term in the curl-curl equation governing the electromagnetic (EM) diffusion in the earth fades during the iterative solution for EM fields. This can lead to violation of the divergence-free condition for current and false jumps in the calculated fields. We develop a regularization technique for vector finite-element MT forward modeling, in which a scaled grad-div operator for electrical fields is included in the curl-curl equation to enforce the divergence-free condition explicitly. Because of its use of basis functions, the direct edge element discretization of the scaled term can lead to zero coefficients. To address this, an equivalent substitute of the scaled grad-div operator based on potential representation of electrical fields is used, discretized with node elements. For one specific element, this is finally reduced to a scaled gradient of the surface integral of current across all the surfaces of the element, resulting in better connectivity of the linear system of equations. The correctness of our algorithm is verified with two synthetic models and an inversion model from real data. The numerical performance is compared to the traditional iterative divergence correction technique (and without the application of the divergence correction for one case) for each of the three models. The results indicate that our algorithm is generally more efficient and stable compared to the traditional technique for all models at all periods considered, with significant improvement at long periods.

Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.

The integral equation method (IEM) and differential equation methods have been widely applied to provide numerical solutions of the electromagnetic (EM) fields caused by inhomogeneity for the controlled-source EM method. IEM has a bounded computational domain and has been well-known for its efficiency, whereas differential equation methods are commonly used for complex geologic models. To use the advantages of the two types of approaches, a hybrid method is developed based on the combination of IEM and the edge-based finite-element method (vector FEM). In the hybrid scheme, Maxwell’s differential equations of the secondary electric fields in the frequency domain are derived for a volume with boundary placed slightly away from the inhomogeneity. The vector FEM is applied to solve Maxwell’s differential equations, and a system of linear equations for the secondary electric fields can be derived by the minimum theorem. The secondary electric fields on the boundary are represented by IEM in terms of the secondary electric fields inside the inhomogeneity. The linear equations from substituting the boundary values into the vector FEM linear equations then can be solved to obtain the secondary electric fields inside the inhomogeneity. The secondary electric fields at receivers are calculated by IEM based on the secondary electric field solutions inside the inhomogeneity. The hybrid algorithm is verified by comparison of simulated results with earlier works on canonical 3D disc models with a high accuracy. Numerical comparisons with two conventional IEMs demonstrate that the hybrid method is more accurate and efficient for high-conductivity contrast media.

The electromagnetic (EM) method is commonly used in mineral exploration due to the method’s sensitivity to conductive targets. Controlled source audio-frequency magnetotellurics (CSAMT) is developed from magnetotelluric (MT) method with an artificial EM source to improve the signal amplitude. It has been used for mineral exploration for many years. In this study, we performed a case study of the CSAMT application for the Eagles-Nest lead–zinc (Pb–Zn) ore deposits in Jianshui, China. The Eagles-Nest deposit is located in southwest in China in forest-covered complex terrain, making it difficult to acquire the geophysical data. Based on the previous dual-frequency induced polarization (IP) results, we designed four profiles for the CSAMT data acquisition. After data processing and inversion, we mapped the subsurface resistivity distribution. From the CSAMT results, we inferred the location of the ore body, which was verified by the drilling wells. The Pb–Zn ore body was found at a depth between 373.70 m to 407.35 m in the well.