A Sparse Bayesian Estimation Framework for Conditioning Prior Geologic Models to Nonlinear Flow Measurements

We present a Bayesian framework for reconstruction of subsurface hydraulic properties from nonlinear dynamic flow data by imposing sparsity on the distribution of the solution coefficients in a compression transform domain. Sparse representation of the subsurface flow properties in a compression transform basis lends itself to a natural regularization approach, i.e. sparsity regularization, which has recently been exploited in solving ill-posed nonlinear inverse problems that frequently encountered in subsurface flow and transport modeling. The Bayesian estimation approach allows for a probabilistic treatment of the sparse reconstruction problem by enforcing sparsity through Laplace priors on the distribution of the solution in the sparsifying transform basis. The methodology has its roots in machine learning and recently introduced relevance vector machine algorithm for linear inverse problems. We extend the application of this approach to nonlinear subsurface inverse problems where solution sparsity in a discrete cosine transform is assumed. The probabilistic fulfillment of solution sparsity, as opposed to deterministic regularization, avoids the nuisance of specifying a priori regularization parameter and allows for quantification of the estimation and prediction uncertainty. Several numerical experiments from subsurface multiphase flow and transport application are conducted to illustrate the performance of proposed method and compare it with the regular Bayesian estimation approaches that do not impose solution sparsity. While the examples are derived from a geophysical application, the proposed framework can be applied to nonlinear inverse problems in other fields such as medical imaging and electromagnetic inverse problem.