We develop a forward modelling code to simulate 3D land-based controlled-source electromagnetic (CSEM) problems in frequency domain with unstructured tetrahedral meshes. The algorithm accounts for isotropic electrical resistivity and magnetic permeability variations in the subsurface. The latest addition to the software is a goal-oriented adaptive mesh refinement strategy driven by error estimators based on “face-jumps” of current density and magnetic flux density. In this study, we demonstrate that the goal-oriented adaptive refinement approaches are suitable to design a problem-specific mesh, which helps to solve 3D CSEM forward problems efficiently and accurately. In mineral exploration, ore bodies often exhibit a strong resistivity contrast and sometimes a non-negligible contrast in magnetic permeability to their host rock. Accurate 3D modelling of electromagnetic measurement setups is therefore needed for feasibility studies and incorporation of the forward modelling in inversion approaches. To obtain sufficiently accurate solutions in time- and memory efficient computations, one option is to employ guided mesh refinement strategies. The so called goal-oriented adaptive mesh refinement method aims at designing a mesh, which is fine where necessary and coarse where discretisation errors do not influence the accuracy of the solution at the points of interest, typically the receiver sites. We apply the total electric field approach and first order Nédélec basis functions as interpolation functions defined on the edges of the finite elements to solve the electromagnetic diffusion equations. Thus, we achieve continuity of the electric and magnetic fields inside the elements and tangential to the edges and faces. However, the continuity of the normal components of current density and magnetic flux density across element interfaces cannot be ensured, resulting in small errors in the solution. We calculate these so called “face-jumps” and use them in combination with the elemental residuals and the dual solution of the problem to obtain error estimators that guide our adaptive refinement approach. The dual problem simulates influence sources at the receiver sites to weight the elemental error estimators with their influence to the solution accuracy at the receivers. We utilise a model of an iron ore body in central Sweden with a known magnetic permeability contrast and unknown electrical resistivity to study the behaviour of our implemented adaptive mesh refinement approaches. This is combined with a feasibility study to investigate the detectability of the ore body with CSEM. From literature examples on magnetotelluric forward modelling we know, that the error estimator based on the continuity of the normal current density shows robust performance, when modelling for electrical resistivity. We observe the same behaviour after adapting it to the controlled-source problem. The error estimator using the continuity of the magnetic flux density seems mathematically most promising to improve the mesh, when variations in magnetic permeability are significant. Numerical experiments with the ore body model indicate, that best results can be achieved, when mesh refinement guided by both error estimators is applied.@en

We developed a 3-D forward modelling code, which simulates controlled source electromagnetic problems in frequency domain using edge-based finite elements and a total electric field approach. To evaluate electromagnetic data acquired across complex subsurface structures, software performing accurate 3-D modelling is required, especially for incorporation in inversion approaches. Our modelling code aims at finding a good compromise between the necessary solution accuracy at the points of interest and the general problem size by using a goal-oriented mesh refinement strategy designed for models of variable electric conductivity and magnetic permeability. To formulate an improved error estimator suitable for controlled source electromagnetic problems, we developed literature approaches of mesh refinement further targeting three aspects. First, to generate a roughly homogeneously fine mesh discretization around all receiver sites, our new error estimator weights the adjoint source term by the approximate decay of the electric field with increasing distance from the primal source using the expression for a homogeneous half-space. This causes almost no additional computational cost. Second, the error estimator employed in the refinement approach can be optimized for models with pronounced conductivity and magnetic permeability contrasts as often encountered in, for example, mineral prospecting scenarios by optionally including terms that measure the continuity of the normal component of current flow and the tangential component of the magnetic field across interfaces of abutting elements. Third, to avoid amplitude-dependent over-refining of the mesh, we formulate our element-wise error estimators relative to the local amplitude of the electromagnetic field. In this work, we evaluate the implemented adaptive mesh refinement approach and its solution accuracy comparing our solutions for simple 1-D models and a model with 3-D anomalies to semi-analytic 1-D solutions and a second-order finite-element code, respectively. Furthermore, a feasibility study for controlled-source electromagnetic measurements across ferrous mineral deposits is conducted. The numerical experiments demonstrate that our new refinement procedure generates problem-specific finite-element meshes and yields accurate solutions for both simple synthetic models and realistic survey scenarios. Especially for the latter, characteristics of our code, such as the possibility of modelling extended sources as well as including arbitrary receiver distributions and detailed subsurface anomalies, are beneficial.@en

Controlled-source electromagnetic methods are applied to survey the electrical resistivity distribution of the subsurface. This work compares normalized electromagnetic field components and transfer functions such as impedance tensors and vertical magnetic transfer functions generated by two independent source polarizations as input data for three-dimensional inversion. As most other available inversion codes allow for inverting only one of the mentioned input data types, it is unclear which data type is preferable for controlled-source electromagnetic inversion. Our three-dimensional non-linear conjugate gradient inversion code can handle both input data types, facilitating a comparison of normalized electromagnetic field components and transfer functions inversion. Examining inversion results for a three-dimensional synthetic model with two anomalies, we infer that the transfer functions inversion is favourable for recovering the overall resistivity distribution below the receiver sites in fewer iterations. The inversion of normalized electromagnetic field components produces a sharper image of the anomalies and may be capable of detecting the resistivity distribution below the extended sources, which comes at the price of introducing a more heterogeneous background resistivity in the model.@en

We developed two forward modelling approaches to simulate 3-dimensional land-based controlled-source electromagnetic (CSEM) problems in frequency domain with hexahedral spectral-element meshes and tetrahedral finite-element meshes. In recent years, the geo-electromagnetic community made a lot of progress in modelling and inversion of EM data in three dimensions using a variety of approaches. The available software is used to verify the accuracy of newly developed codes, which apply e.g. different element shapes or interpolation schemes. However, a direct comparison in terms of advantages and disadvantages of different modelling strategies, especially discretisation methods in 3D, is often not focused on in publications. Having two modelling codes and their developers available at the same place, gives us the unique opportunity to compare the approaches in a very detailed way. Our spectral-element as well as our finite-element solution is based on Galerkin’s weighted residual method and we solve the electromagnetic diffusion equations for the total electric field on the element edges. The main differences between both codes are the choice and order of the interpolation functions and the discretisation of the modelling domain employing hexahedral and tetrahedral elements. While the tetrahedral meshes used in our finite-element approach are known for being able to properly resolve complex structures in the subsurface, this issue is addressed in the spectral-element method by utilising curvilinear instead of orthogonal hexahedral elements. In this contribution, we focus on the comparison of both approaches for a simple 1D model and a complex 3D model in terms of accuracy, effort in mesh generation and computational resources such as simulation time and memory requirement. Moreover, we contrast the influence of mesh discretisation on the solution for the two methods as well as the order of approximation. A preliminary test simulation of a model consisting of a conductive body buried within a resistive background covered by a thin conductive layer yielded comparable results in terms of accuracy. It also revealed significant differences concerning the mesh discretisation meaning the solution's dependency on the meshing of the model domain.@en

We present a synthetic inversion study illustrating two approaches which help to deal with heterogeneous sensitivities in 3D frequency-domain controlled-source electromagnetic inverse problems. Using edge-based vector finite-element approximations on tetrahedral meshes and a preconditioned non-linear conjugate gradient algorithm, we invert for impedance tensor elements generated by a set of two coincident perpendicularly oriented horizontal electric or horizontal magnetic dipole sources. Depending on the number and locations of sources and the choice of impedance tensor components used for inversion, the sensitivity patterns can differ significantly. Measurement setups with a small number of sources, but many receiver stations at the surface covering near-field, transition zone and far-field, are often deployed for land-based controlled-source electromagnetic measurements. Such a setup can result in accumulated sensitivities close to the sources and receivers, which implies strongest model updates in these regions and can mislead the inverse algorithm to a search direction, where no physically meaningful model can be produced nor the data are fitted. In order to mitigate the influence of strong sensitivities near sources and receivers on the inversion process, we apply an efficient preconditioner and customisable weights in the model regularisation matrix. The preconditioner is updated with the Broyden-Fletcher-Goldfarb-Shanno algorithm using the diagonal of the approximate Hessian matrix as start preconditioner. The latter is computationally expensive to obtain, but aims at finding a favourable search direction for the inverse algorithm already in early iterations and distributing the model update more evenly in the domain. To account for the sensitivity loss with depth, we implemented a depth weighting functional in the model regularisation term. The approach is based on counteracting the exponential and geometrical decay of the electromagnetic fields with depth and distance from the sources. In practical, we increase the smoothing in the shallow part of the model close to the source locations, where no structure is expected. We present synthetic examples indicating that this approach is an efficient way of helping the inversion to converge, obtaining a reliable model and resolving structure at depth.@en