3-D resistivity forward modeling and inversion using conjugate gradients

Abstract

We have developed rapid 3-D dc 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 SPARC‐station 2 for 50 × 50 × 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 [tilde under A] or its transpose [tilde under A][superscript T] multiplying a vector, we are able to bypass the actual computation of the sensitivity matrix and the inversion of [tilde under A][superscript T] [tilde under A], 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.