Advances in three-dimensional geoelectric forward solver techniques

Abstract

S U M M A R YModern geoelectrical data acquisition systems allow large amounts of data to be collected in a short time. Inversions of such data sets require powerful forward solvers for predicting the electrical potentials. State-of-the-art solvers are typically based on finite elements. Recent developments in numerical mathematics led to direct matrix solvers that allow the equation systems arising from such finite element problems to be solved very efficiently. They are particularly useful for 3-D geoelectrical problems, where many electrodes are involved. Although modern direct matrix solvers include optimized memory saving strategies, their application to realistic, large-scale 3-D problems is still somewhat limited. Therefore, we present two novel techniques that allow the number of gridpoints to be reduced considerably, while maintaining a high solution accuracy. In the areas surrounding an electrode array we attach infinite elements that continue the electrical potentials to infinity. This does not only reduce the number of gridpoints, but also avoids the artificial Dirichlet or mixed boundary conditions that are well known to be the cause of numerical inaccuracies. Our second development concerns the singularity removal in the presence of significant surface topography. We employ a fast multipole boundary element method for computing the singular potentials. This renders unnecessary mesh refinements near the electrodes, which results in substantial savings of gridpoints of up to more than 50 per cent. By means of extensive numerical tests we demonstrate that combined application of infinite elements and singularity removal allows the number of gridpoints to be reduced by a factor of ≈6-10 compared with traditional finite element methods. This will be key for applying finite elements and direct matrix solver techniques to realistic 3-D inversion problems.

3-D electrical resistivity tomography using adaptive wavelet parameter grids

“…Dey & Morrison 1979b;Rücker et al 2006;Blome et al 2009). There exists a plethora of numerical methods for solving eq.…”

A finite-element-based domain-decomposition approach for plane wave 3D electromagnetic modeling

“…Numerical modeling methods for geoelectromagnetic induction problems can be principally categorized into the following classes: volume integral methods (Hohmann, 1975;Avdeev et al, 2002;Kuvshinov et al, 2002;Zhdanov et al, 2006), surface integral methods (Xu et al, 1997;Liu and Lamontagne, 1998;Ren et al, 2013b), finite-difference methods (Mackie et al, 1994;Haber et al, 2000;Weiss and Newman, 2003), finite-volume methods (Haber et al, 2000;Jahandari and Farquharson, 2013), finite-element methods (FEMs) (Mitsuhata and Uchida, 2004;Key and Weiss, 2006;Rücker et al, 2006;Franke et al, 2007;Li and Key, 2007;Nam et al, 2007;Blome et al, 2009;Ren and Tang, 2010;Farquharson and Miensopust, 2011;Mukherjee and Everett, 2011;Schwarzbach et al, 2011;Ren et al, 2013a;Schankee et al, 2013;Wang et al, 2013), and hybrid methods (Erdoğan et al, 2008;Vachiratienchai et al, 2010;Vachiratienchai and Siripunvaraporn, 2013;Ren et al, 2014). The FEMs are generally recognized as the most suitable approach for 3D complicated electromagnetic induction problems in the earth (Avdeev, 2005;Börner, 2010;Everett, 2012).…”

Fast 3‐D large‐scale gravity and magnetic modeling using unstructured grids and an adaptive multilevel fast multipole method

“…Bapat et al ., ]. Its concept has only been sparsely used by the geophysical community with applications reported for the 3‐D direct current resistivity problem [ Blome et al ., ], the seismic wave propagation problem [ Çakir , ; Çakır , ], and the 3‐D electromagnetic induction problem [ Ren et al ., , ]. As for gravity modeling problems, only geometrically simple Earth models with 3‐D structured grids were considered [ May and Knepley , ].…”