Modified total variation methods for three-dimensional electrical resistance tomography inverse problems

Images from geophysical surveys and data inversion can be valuable information for hydrogeologic site characterization and monitoring. This paper focuses on solving nonlinear electrical resistance tomography (ERT) inverse problems; the methods are applicable to other inverse problems, such as groundwater flow. The total variation (TV) method is used to constrain ERT inverse solutions. The classic TV functions are modified to obtain differentiability with respect to parameters, which allows the use of gradient search optimization methods. The modified total variation (mTV) method constrains ERT inverse solutions effectively. Sometimes the mTV constraints cause important parameter features to be smoothed out. To eliminate excessive smoothing, two successive partial variation relaxation (SPVR) methods are devised for use with the mTV method. Computational experiments demonstrate that mTV works well in identifying very sharp discontinuities and smoothly distributed parameters and that SPVR improves mTV solutions for both problem types. The mTV and SPVR methods are applied to synthetic layered systems with and without measurement error, and a comparison of results for a large three‐dimensional ERT problem with block anomaly presented by Zhang et al. [1995] demonstrates the superiority of the mTV and SPVR methods. A final example examines a smoothly varying case and explores the impact of two computational parameters introduced in the development of mTV and SPVR. The example shows also that SPVR can be used directly to invert data and not necessarily as an adjunct to mTV.