Introduction

The inversion of an electromagnetic data set to determine the electrical conductivity structure of the subsurface requires numerical methods. In three dimensions and for practically relevant models, several tens of thousands of data and model parameters are involved. Presently, such computationally intensive problems require high performance or supercomputers. Börner (2010) and Avdeev (2005) review state of the art three-dimensional electromagnetic numerical simulation and inversion strategies.

FD methods simulating the diffusion of transient fields in three-dimensional structures were first developed in the late 1980s (Newman et al., 1986). Wang und Hohmann (1993) introduced a time-stepping approach to compute the electromagnetic fields on a staggered grid (Finite Differences in the Time Domain, FDTD). The FDTD method is attractive because explicit time-stepping methods avoid solving linear equation systems. Commer und Newman (2004) demonstrated a numerical solution of the FDTD method on parallel computers.

A different approach to solving the TEM forward problem was presented by Druskin and Knizhnerman (1994), who introduced the spectral Lanczos decomposition method (SLDM). Here, Maxwell's equations are approximated on an FD grid, which leads to a system of ordinary differential equations. The solution of this system is reduced to the products of functionals of the stiffness matrix and a vector containing the initial conditions. Börner et al. (2008) developed the method of model reductions in the frequency domain (MRFD), which extends the SLDM method by combining it with the advantages of the FE methods on unstructured grids.

Adaptive grids with local refinements have predominantly been applied for FE methods (e.g. Franke et al. 2007; Rücker et al, 2006, Li and Key, 2007, Schwarzbach, 2009). For all of these methods the subsurface is subdivided into a large number of homogeneous cells organized in ordinary or staggered grids (Yee, 1966) for which Maxwell's equations are solved.

Over the last few years, approaches to solve the 3D inverse problem have been suggested based on Newton-, Quasi-Newton-, Gauss-Newton and non-linear conjugate gradient methods (Rodi and Mackie, 2001; Newman and Boggs, 2004; Commer and Newman, 2008, Avdeev and Avdeeva, 2009). All-at-once approaches attempt to solve the forward and the inverse problem simultaneously, applying an approximation to the forward solution (Haber et al., 2004). If the amount of data is less than the amount of model parameters, transformation to the data space can be advantageous (Siripunvaraporn and Egbert, 2005). Most inversion procedures are based on Gauss-Newton approaches using various forms of regularization (Smith and Booker, 1991; Oldenburg and Ellis, 1991; Sasaki, 2001; Günther et al., 2006). Often, the objective function represents a trade-off between optimum data fit and minimum structure of the model based on some smoothness constraint (de Groot-Hedlin and Constable, 1990). Newton techniques have favourable convergence characteristics, but require computationally costly second order derivatives of the objective function or derivatives of the sensitivities with respect to the model parameters.

Previous attempts to fit electromagnetic observations by 3D inversion include Patro and Egbert (2008) for magnetotelluric, Haber et al. (2007) for TEM, and Newman et al. (2010) for CSEM data. Commer and Newman (2009) consider the joint inversion of MT and CSEM data.