Subproject II: Joint inversion of transient electromagnetic and DC resistivity methods

WP 2.01 – Further development and parallelization of forward modelling operators for DC resistivity methods: (see WP 2.02).

WP 2.02 – Further development and parallelization of forward modelling operators for transient electromagnetic (TEM): The working group in Freiberg have developed their own, fast forward operators for DC electrics (Rücker et al., 2006) and TEM (Börner et al., 2008). Both codes use an unstructured grid finite element approach (Lagrange type for DC resistivity and Nédélec type for TEM), such that given structures can be transferred to synthetic models in great detail. These codes are powerful tools for practical use. Within this work package we want to assess if the originally sequentially structured codes can be parallelized or have to be reengineered to facilitate efficient parallelization.

WP 2.03 – Further development of the spectral Lanczos method: The interpretation software for TEM data, which has been developed during the last three years within the DFG project ‘Numerical simulation of the propagation of transient electromagnetic fields for the exploration of the subsurface', is based on the finite element discretization of the three-dimensional quasi-static Maxwell's equations in space (Nédélec elements) and the solution of the semi-discrete problem by advancing the spectral Lanczos decomposition method (SLDM) using restarts (Eiermann and Ernst, 2006). Restarted algorithms reduce the memory requirement of SLDM significantly (or supersede a second run of the Lanczos algorithm, respectively). However, they converge generally slower than the variant without restarts. Another newly developed approach called ‘thick restarts' may compensate the loss of speed without waiving the advantages of restarts. This technique will further accelerate our forward solver. In recent years, so called rational Lanczos methods were investigated at the Institute of Numerical Analysis and Optimization as an alternative to the classical Lanczos method. The crucial advantage of this method is the fact that its convergence rate does not depend on the size of the problem (at least for the problems at hand). Whereas classical SLDM needs additional iteration steps with increasing resolution to reach a given accuracy level, the number of iterations remains constant for the rational Lanczos method – an important advantage for the solution of very large problems. Since for the rational Lanczos method one linear system of equations of the type (A − λI )x = b has to be solved in each iteration step with A being the discrete version of Maxwell's equations and possibly including a complex shift λ, fast solvers are an essential prerequisite for the application of these methods. Therefore, we need to apply multi-grid methods that are specifically developed for the curl-curl operator and have been shown to exhibit robust convergence rates with respect to the choice of the shifts.

WP 2.04 – DC resistivity method: resolution analyses, optimization of the experimental design (see WP 2.05).

WP 2.05 – TEM: resolution analyses, optimization of the experimental design: Apart from the actual simulation, the computation of sensitivities is complicated and numerically very costly. Günther et al. (2006) und Baranwal et al. (2007) have developed working inversion codes which have increased our knowledge about resolution characteristics of the DC and TEM methods (see also Spitzer, 1998). For both applications, we employ adjoint field techniques, which have partly been implemented (Ullmann, 2008), but require further development and adaptation to large and powerful computers. The envisaged modular, flawless inversion code would simplify an efficient parallelization. By means of a joint DC/TEM inversion we can investigate the resolution characteristics of different experimental designs in order to optimize the resolution power of field experiments. In other words, we intend to develop a tool telling us where to place receivers and sources to obtain an optimal (with the highest resolution) image of the target structures (see Stummer et al., 2004).

WP 2.06 – Combination of both methods in a joint inversion: After developing the individual inversion algorithms for TEM and DC, we will start implementing a joint inversion scheme. In this context, problems with the combined parameterizations have to be solved. By means of sensitivities we can compute resolution matrices to assess the resolution characteristics of the individual methods and subsequently of their combination.