We have developed a novel goal-oriented adaptive mesh refinement approach for finite-element methods to model plane wave electromagnetic (EM) fields in 3-D earth models based on the electric field differential equation. To handle complicated models of arbitrary conductivity, magnetic permeability and dielectric permittivity involving curved boundaries and surface topography, we employ an unstructured grid approach. The electric field is approximated by linear curl-conforming shape functions which guarantee the divergence-free condition of the electric field within each tetrahedron and continuity of the tangential component of the electric field across the interior boundaries. Based on the non-zero residuals of the approximated electric field and the yet to be satisfied boundary conditions of continuity of both the normal component of the total current density and the tangential component of the magnetic field strength across the interior interfaces, three a-posterior error estimators are proposed as a means to drive the goal-oriented adaptive refinement procedure. The first a-posterior error estimator relies on a combination of the residual of the electric field, the discontinuity of the normal component of the total current density and the discontinuity of the tangential component of the magnetic field strength across the interior faces shared by tetrahedra. The second a-posterior error estimator is expressed in terms of the discontinuity of the normal component of the total current density (conduction plus displacement current). The discontinuity of the tangential component of the magnetic field forms the third a-posterior error estimator. Analytical solutions for magnetotelluric (MT) and radiomagnetotelluric (RMT) fields impinging on a homogeneous half-space model are used to test the performances of the newly developed goal-oriented algorithms using the above three a-posterior error estimators. A trapezoidal topographical model, using normally incident EM waves at both MT and RMT frequencies, is adopted to further test the convergence of the newly developed algorithms against a surface integral approach. Next, the 3D-1 benchmark model from the COMMEMI project is used to show the efficiency of the goal-oriented adaptive algorithm and to compare our solutions against volume integral solutions and other finite-element solutions. For all three test cases, we found that the error estimator using face jumps of normal components of current density embedded in the goal-oriented adaptive refinement procedure shows the most robust performance.

A new adaptive finite-element method for solving 3D direct-current resistivity modeling problems is presented. The method begins with an initial coarse mesh, which is then adaptively refined wherever a gradient-recovery-based a posteriori error estimator indicates that refinement is necessary. Then the problem is solved again on the new grid. The alternating solution and refinement steps continue until a given error criterion is satisfied. The method is demonstrated on two synthetic resistivity models with known analytical solutions, so the errors can be quantified. The applicability of the numerical method is illustrated on a 2D homogeneous model with a topographic valley. Numerical results show that this method is efficient and accurate for geometrically complex situations.

A meaningful solution to an inversion problem should be composed of the preferred inversion model and its uncertainty and resolution estimates. The model uncertainty estimate describes an equivalent model domain in which each model generates responses which fit the observed data to within a threshold value. The model resolution matrix measures to what extent the unknown true solution maps into the preferred solution. However, most current geophysical electromagnetic (also gravity, magnetic and seismic) inversion studies only offer the preferred inversion model and ignore model uncertainty and resolution estimates, which makes the reliability of the preferred inversion model questionable. This may be caused by the fact that the computation and analysis of an inversion model depend on multiple factors, such as the misfit or objective function, the accuracy of the forward solvers, data coverage and noise, values of trade-off parameters, the initial model, the reference model and the model constraints. Depending on the particular method selected, large computational costs ensue. In this review, we first try to cover linearised model analysis tools such as the sensitivity matrix, the model resolution matrix and the model covariance matrix also providing a partially nonlinear description of the equivalent model domain based on pseudo-hyperellipsoids. Linearised model analysis tools can offer quantitative measures. In particular, the model resolution and covariance matrices measure how far the preferred inversion model is from the true model and how uncertainty in the measurements maps into model uncertainty. We also cover nonlinear model analysis tools including changes to the preferred inversion model (nonlinear sensitivity tests), modifications of the data set (using bootstrap re-sampling and generalised cross-validation), modifications of data uncertainty, variations of model constraints (including changes to the trade-off parameter, reference model and matrix regularisation operator), the edgehog method, most-squares inversion and global searching algorithms. These nonlinear model analysis tools try to explore larger parts of the model domain than linearised model analysis and, hence, may assemble a more comprehensive equivalent model domain. Then, to overcome the bottleneck of computational cost in model analysis, we present several practical algorithms to accelerate the computation. Here, we emphasise linearised model analysis, as efficient computation of nonlinear model uncertainty and resolution estimates is mainly determined by fast forward and inversion solvers. In the last part of our review, we present applications of model analysis to models computed from individual and joint inversions of electromagnetic data; we also describe optimal survey design and inversion grid design as important applications of model analysis. The currently available model uncertainty and resolution analyses are mainly for 1D and 2D problems due to the limitations in computational cost. With significant enhancements of computing power, 3D model analyses are expected to be increasingly used and to help analyse and establish confidence in 3D inversion models.

We developed a novel parallel domain-decomposition approach for 3D large-scale electromagnetic induction modeling in the earth. We used the edge-based finite-element method and unstructured meshes. Unstructured meshes were divided into sets of nonoverlapping subdomains. We used the curl-curl electric field equation to carry out the analysis. In each subdomain, the electric field was discretized by first-order vector shape functions along the edges of tetrahedral elements. The tangential components of the magnetic field on the interfaces of the subdomains were defined as a set of Lagrange multipliers. The unknown Lagrange multipliers were solved from an interface problem defined on the interfaces of the subdomains. With the availability of the Lagrange multipliers, the electric field values in each subdomain were solved independently. Three synthetic examples were evaluated to verify our code. Excellent agreement with previously published solutions was obtained. Synthetic examples revealed that our domain decomposition technique is scalable with respect to the number of subdomains and robust with regard to frequency and the heterogeneous distribution of material parameters, i.e., electric conductivity, electric permittivity, and magnetic permeability.