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.

Review: Some low-frequency electrical methods for subsurface characterization and monitoring in hydrogeology

“…These approaches have been adopted to solve the resistivity inverse problem in both 2D (e.g., Sasaki 1992;LaBrecque et al 1996) and 3D (e.g., Park and Van 1991;Sasaki 1994;Zhang et al 1995) A good review of these type of approaches for resistivity imaging, including options for image appraisal, can be found in Binley and Kemna (2005). Recent inversion methods involve also the use of stochastic techniques (e.g., Malinverno and TorresVerdin 2000; Andersen et al 2001;Yeh et al 2002;Ramirez et al 2005).…”

A hybrid method for inversion of 3D AC resistivity logging measurements

“…There exist several numerical methods for the simulation of forward resistivity measurements, including the transmission line matrix method [40], integral equation based methods [28], Lanczos decomposition based methods [23], Fourier based methods [12,13,41], finite difference methods [22,[24][25][26][27], and finite element methods [29,42,43].…”

Fusion of hydrologic and geophysical tomographic surveys

“…The resistivity field is then estimated by inversion of the data sets using a model without the assumption of a homogeneous earth, and using a regularized optimization approach (e.g., Daily et al, 1992;Ellis and Oldenburg, 1994;Li and Oldenburg, 1994;and Zhang et al, 1995).…”