Abstract
We apply a stochastic Newton (SN) approach to solve a high-dimensional hydraulic inverse problem in highly heterogeneous geological media. By recognizing the connection between the cost function of deterministic optimizations and the posterior probability density of stochastic inversions, the Markov chain Monte Carlo (MCMC) sampler of SN is constructed by two parts: a deterministic part, which corresponds to a Newton step of deterministic optimization, and a stochastic part, which is a Gaussian distribution with the inverse of the local Hessian as the covariance matrix. The hybrid inverse method exploits the efficient tools for fast solution of deterministic inversions to improve the efficiency of the MCMC sampler. To address the ill-posedness of the inverse problem, a priori models, generated by a transition-probability geostatistical method, and conditioned to inter-well connection data, are used as regularization constraints.The effectiveness of the stochastic Newton method is first demonstrated by a synthetic test. The transmissivity field of the synthetic model is highly heterogeneous, and includes sharp variations. The inverse approach was then applied to a field hydraulic tomography investigation in a fractured and karstified aquifer to reconstruct its transmissivity field from a collection of real hydraulic head measurements. From the inversions, a series of transmissivity fields that produce good correlations between the inverted and the measured hydraulic heads were obtained. The inverse approach produced slightly different a posteriori transmissivity patterns for different a priori structure models of transmissivity; however, the trend and location of the high-transmissivity channels are consistent among various realizations. In addition, the uncertainty associated with each realization of the inverted transmissivity fields was quantified.