S U M M A R YWe present a new 3-D vector finite element code and demonstrate its strength by modelling a realistic marine CSEM scenario. Unstructured tetrahedral meshes easily allow for the inclusion of arbitrary seafloor bathymetry so that natural environments are mapped into the model in a close-to-reality way. A primary/secondary field approach, an adaptive mesh refinement strategy as well as a higher order polynomial finite element approximation improve the solution accuracy. A convergence study strongly indicates that the use of higher order finite elements is beneficial even if the solution is not globally smooth. The marine CSEM scenario also shows that seafloor topography gives an important response which needs to be reproduced by numerical modelling to avoid the misinterpretation of measurements.

During the last decade, tremendous advances have been observed in the broad field of numerical modelling for geo-electromagnetic applications. This trend received support due to increasing industrial needs, mainly caused by hydrocarbon and ore exploration industry. On the other hand, the increasing reliability and accuracy of data acquisition techniques further spurs this development. In this review, we will focus on advances and challenges in numerical modelling in geo-electromagnetics. We review recent developments in the discrete solution of the 3-D induction problem in the time and frequency domains. Particularly, advantages and disadvantages of the common numerical techniques for solving partial differential equations such as the Finite Difference and Finite Element methods will be considered.

SUMMARY We present an adaptive unstructured triangular grid finite element approach for effectively simulating plane‐wave diffusive electromagnetic fields in 2‐D conductivity structures. The most striking advantage of irregular grids is their potential to incorporate arbitrary geometries including surface and seafloor topography. Adaptive mesh refinement strategies using an a posteriori error estimator yield most efficient numerical solutions since meshes are only refined where required. We demonstrate the robustness of this approach by comparison with analytical solutions and previously published numerical simulations. Maximum errors may systematically be reduced to, for example, 0.8 per cent for the apparent resistivity and 0.2° in the phase. An additional accuracy study of the thickness of the air layer in E‐polarization suggests to keep a minimum thickness depending on lateral conductivity contrasts within the earth. Furthermore, we point out the new quality and flexibility of our simulation technique by addressing two marine magnetotelluric applications. In the first case, we discuss topographic effects associated with a synthetic sinusoidal sea bottom model and in the second case, we show a close‐to‐reality scenario using real bathymetry data from the East Pacific Rise at 17°S.

S U M M A R YWe introduce the concept of multi-objective optimization to cast the regularized inverse direct current resistivity problem into a general formulation. This formulation is suitable for the efficient application of a genetic algorithm, which is known as a global and non-linear optimization tool. The genetic inverse algorithm generates a set of solutions reflecting the trade-off between data misfit and some measure of model features. Examination of such an ensemble is highly preferable to classical approaches where just one 'optimal' solution is examined since a better overview over the range of possible inverse models is gained. However, the computational cost to obtain this ensemble is enormous. We demonstrate that at the current state of computer performance inversion of 2-D direct current resistivity data using genetic algorithms is possible if state-of-the-art computational techniques such as parallelization and efficient 2-D forward operators are applied.

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