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.

SUMMARY: Since electromagnetic (EM) fields diffuse more smoothly to greater depth, it physically makes sense to apply fine discretization to model structure at near surface with an increasingly coarser grid both in horizontal and vertical directions as the depth increases for the numerical solution of EM fields. For finite-difference magnetotelluric (MT) forward modelling on regular staggered grids, this can lead to difficulties in discretizing the curl–curl equation governing the EM diffusion in the earth at regions, where the grid resolution changes. In this paper, we propose an efficient algebraic multi-resolution sampling (MRS) method for MT forward modelling. In this method, we do not require the generation of physical subgrids and merely subsample the field components. The coefficient matrix for the subsampled components can be obtained by matrix multiplication based on the initial linear system of equations and field interpolation. To guarantee divergence-free current during the iterative solution process at low frequencies, which is considered crucial for the development of efficient iterative solvers, our forward modelling is based a regularization equation obtained by augmenting the curl–curl equation with an equivalent scaled grad–div operator for electrical fields (explicitly enforcing the divergence-free condition). The correctness of our algebraic MRS algorithm is examined based on a layered model. Its stability and efficiency is exploited by comparing our results with the forward modelling on unsampled staggered grids for the Dublin Test Model 1 (DTM1) and a complex model with realistic topography, indicating a time reduction of about 42–82 per cent.

Conventional 3D magnetotelluric (MT) inversion methods face substantial computational challenges when inverting large-scale data sets, demanding extensive processing time and computing resources. To address these limitations, we develop an advanced frequency-polarization joint parallel inversion framework for MT data that uses the nonlinear conjugate gradient (NLCG) method, integrating an adaptive algebraic multi-resolution sampling (AMRS) approach with a regularization technique. The AMRS approach adaptively discretizes modeling domains across different periods based on the normalized root-mean-square difference relative to the solution on the initial grid, which serves as a reference. The regularization technique ensures the conservation of current, which can significantly affect the efficiency of the forward-modeling problem and its adjoint when using iterative solvers, particularly at low frequencies. The reliability and efficiency of the regularization AMRS inversion are first examined by benchmarking it against the traditional NLCG inversion and the regularization inversion based on a two adjacent blocks model and a terrain model. Subsequently, the regularization AMRS inversion is applied to invert real MT data from the Cascadia region in the USA, demonstrating its capability to address real inverse problems.

Traditional Krylov subspace methods can be slow for electromagnetic modeling, particularly at lower frequencies. One of the commonly used remedies is to apply the divergence correction iteratively during the solution process, which requires solving an additional divergence correction equation. Alternatively, we have developed an efficient regularization technique to carry out the forward modeling of the anisotropic effect for 3D controlled-source electromagnetic data. Inside this scheme, we explicitly include the gradient of a scaled divergence correction term into the original curl-curl equation for general anisotropic conductivity structure. This inclusion leads to a significantly better-conditioned linear system of equations without changing the solution of the original system and avoids the solution of an additional equation. The correctness of the developed algorithm is examined based on a vertical transverse isotropic layered model with the semianalytical solution available. Then, we use a tilted transverse isotropic ocean canonical reservoir model and a general anisotropic 3D model to investigate the stability and efficiency of the algorithm. Numerical experiments demonstrate that the developed technique is 0.49–2.9 times faster than the standard algorithm at the considered frequencies.

Multigrid (MG) solvers, typically increasing the computational time linearly with the grid size, are suitable for large-scale forward modeling problems. However, for electromagnetic (EM) problems as frequency decreases and grid is increasingly stretched, MG solvers for EM modeling can converge slowly or even diverge. We propose an efficient four-color Line Gauss-Seidel (GS) MG for finite difference (FD) frequency-domain EM solution. In this algorithm, the edge components attached to nodes on each line in one particular direction are updated simultaneously, leading to that the solution in each local region satisfies divergence free condition. Due to the fact that each local linear system of equations is completely uncoupled with that formed for its disjoint lines, we can group all lines of grid nodes into four colors with the requirement that all local systems formed for lines with the same color are disjoint. This can be utilized to parallelize or vectorize our algorithm. The correctness is verified by comparing with the analytical solution based on a three-layered model. The numerical performance is examined by comparing with other commonly used state-of-art solvers based on three increasingly more complex models, indicating the efficiency dominance, good parallelization and excellent ability on handling grid stretching for our algorithm.