3-D resistivity forward modeling and inversion using conjugate gradients

Abstract

We have developed rapid 3-D do resistivity forward modeling and inversion algorithms that use conjugate gradient relaxation techniques. In the forward network modeling calculation, an incomplete Cholesky decomposition for preconditioning and sparse matrix routines combine to produce a fast and efficient algorithm (approximately 2 minutes CPU time on a Sun SPARCstation 2 for 50 x 50 x 20 blocks). The side and bottom boundary conditions are scaled impedance conditions that take into account the local current flow at the boundaries as a result of any configuration of current sources.For the inversion, conjugate gradient relaxation is used to solve the maximum likelihood inverse equations. Since conjugate gradient techniques only require the results of the sensitivity matrix A or its transpose A T multiplying a vector, we are able to bypass the actual computation of the sensitivity matrix and the inversion of ATA, thus greatly decreasing the time needed to do 3-D inversions.We demonstrate 3-D resistivity tomographic imaging using pole-pole resistivity data collected during an experiment for a leakage monitoring system near evaporation ponds at the Mojave Generating Station in Laughlin, Nevada.

Multiscale electrical impedance tomography

“…Electrical impedance tomography is generally formulated as an inverse problem which aims at reconstructing the (possibly complex) electrical conductivity (or resistivity) distribution underground from electrical potential measurements made at the boundaries of the region to be imaged (see, for instance, Ward [1990] for a review). Both the inverse problem and the dual direct problem, which gives the electrical response for a known conductivity distribution, benefitted from recent numerous breakthroughs [e.g., Berryman and Kohn, 1990;Oldenburg, 1994a, 1994b;Li and Oldenburg, 1994;Zhang et al, 1995;Borcea et al, 1999;Li and Oldenburg, 2000;Torres-Verdin et al, 2000]. However, the inverse problem remains a notoriously difficult one because of both its highly nonlinear nature and its ill-posedness [Allers and Santosa, 1991;Molyneux and Witten, 1994;Cherkaeva and Tripp, 1996].…”

2.5D resistivity modeling of embankment dams to assess influence from geometry and material properties

“…This means that application of standard 2D techniques on embankment dams with measurement layouts along the crest of the dam cannot be used without cau-tion because of the obvious 3D effects from the dam geometry. It is possible to use 3D inversion techniques ͑Park and Van, 1991;Sasaki, 1994;Zhang et al, 1995;Loke and Barker, 1996͒. However, they still may not be convenient for repeated measurements, mainly because of limitations in computational resources and because data sets are 2D if only measured along the crest.…”

EIT reconstruction algorithms: pitfalls, challenges and recent developments

“…Until these problems are overcome it is unlikely, in the author's opinion, to be worth using iterative non-linear methods in vivo using individual surface electrodes. Note however that such methods are in routine use in geophysical problems [82,44,45]. Computational complexity of both forward solution and inversion of the linearized system meant that, although iterative nonlinear algorithms had been implemented for simulated data on modest meshes earlier [43] it was only in the mid 1990s that affordable computers had sufficient floating point speed and memory to handle sufficiently dense three-dimensional meshes to fit tank data adequately [71,74] The essence of non-linear solution methods is to repeat the process of calculating the Jacobian and solving a regularised linear approximation.…”