Stochastic inversion of 3-D ERT data

Abstract

A new stochastic inverse algorithm for the inversion of three-dimensional (3-D) electrical resistivity tomography (ERT) data has been developed and tested using both synthetic and field data. My stochastic inverse algorithm produced satisfactory inverse solutions that were very similar to those of the commonly used Occam's inversion. The ill-posed 3-D stochastic inverse problems were stabilized by incorporating a-priori information in the algorithm in the form of an a-priori model, and data and model covariance matrices. There were several novel features in my algorithm. First, a very fast successive over-relaxation (SSOR) preconditioner was implemented in the conjugate-gradient forward solver. Second, a trade-off factor adapted from the Occam's inversion was employed in the algorithm for better control of convergence. Third, the QMRCGSTAB inverse solver, a quasi-minimal residual (QMR) variant of the stabilized bi-conjugate gradient method (Bi-CGSTAB), was used to solve an asymmetric linearized iterative system, so the inversion of a large model covariance matrix was sidestepped. Therefore my algorithm can handle a variable scale of model correlation. Fourth, much better convergence was achieved by using the data standard deviation instead of the data variance as the entries of data covariance matrix. Fifth, a model covariance weighting scheme using the diagonal of the transposed sensitivity matrix times the sensitivity matrix improved the model resolution greatly in the region where is usually poorly resolved. The run-time of stochastic inversion with a biconjugate-gradient inverse solver doubled that of the Occam's inversion with a conjugate-gradient solver. To speed up the stochastic inversion in in-situ monitoring applications, I developed an efficient difference inversion algorithm to invert the difference between the background and subsequent data sets. The new difference inversion can be two to three times faster than the stochastic inversion. I also developed a unique cokriging approach to estimate a 3-D moisture content distribution quantitatively from 3-D ERT data and a limited number of neutron-derived moisture content data for an infiltration experiment in Socorro, New Mexico. I found that one neutron well in the center of ERT mesh, where model parameters are usually poorly resolved in ERT inversion, played an indispensable role in cokriging.