Hydraulic Inverse Modeling Using Total-Variation Regularization with Relaxed Variable-Splitting

Inverse modeling seeks model parameters given a set of observed-state variables. For many practical hydrogeological problems, because the data coverage is limited, the inversion can be ill-posed and unstable. Typically, inverse analyses are applied to characterize aquifer heterogeneity where the hydraulic permeability is estimated throughout the model domain. To stabilize the inversion, regularization techniques can be employed to eliminate the ill-posedness. The most commonly used type of regularization include Tikhonov and Total-Variation (TV). The hydraulic tomographic analyses of aquifer heterogeneity with Tikhonov regularization tends to yield smoothed inversion results, while the ones with TV regularization can preserve the sharp contrast between low and high permeability regions. However, hydraulic inverse modeling with the conventional TV regularization can be computationally unstable and yield unwanted artifacts because of the non-differentiability of the TV norm. We develop a novel hydraulic inverse modeling method using a TV regularization with relaxed variable-splitting scheme to preserve sharp interfaces in piecewise-constant structures and improve the accuracy of inversion. We use an alternating-minimization algorithm to solve the minimization problem. Specifically, we decouple the original problem into two simple sub-problems: a standard inverse modeling sub-problem with the Tikhonov regularization, and a standard L2-TV sub-problem. We solve these two sub-problems using the standard linear solver and Alternating Direction Method of Multipliers (ADMM) iterative methods, respectively. Our new inversion algorithm is implemented in the MADS computational framework (http://mads.lanl.gov). The computational cost of our new inversion methods is comparable to those of conventional inversion approaches. Our numerical examples using synthetic data show that our new methods not only preserve sharp interfaces of subsurface permeability distribution, but also significantly improve the accuracy of the inversion.