Bayesian Inversion of Multi-Gaussian Log-Conductivity Fields With Uncertain Hyperparameters: An Extension of Preconditioned Crank-Nicolson Markov Chain Monte Carlo With Parallel Tempering

Abstract

In conventional Bayesian geostatistical inversion, specific values of hyperparameters characterizing the prior distribution of random fields are required. However, these hyperparameters are typically very uncertain in practice. Thus, it is more appropriate to consider the uncertainty of hyperparameters as well. The preconditioned Crank-Nicolson Markov chain Monte Carlo with parallel tempering (pCN-PT) has been used to efficiently solve the conventional Bayesian inversion of high-dimensional multi-Gaussian random fields. In this study, we extend pCN-PT to Bayesian inversion with uncertain hyperparameters of multi-Gaussian fields. To utilize the dimension robustness of the preconditioned Crank-Nicolson algorithm, we reconstruct the problem by decomposing the random field into hyperparameters and white noise. Then, we apply pCN-PT with a Gibbs split to this “new” problem to obtain the posterior samples of hyperparameters and white noise, and further recover the posterior samples of spatially distributed model parameters. Finally, we apply the extended pCN-PT method for estimating a finely resolved multi-Gaussian log-hydraulic conductivity field from direct data and from head data to show its effectiveness. Results indicate that the estimation of hyperparameters with hydraulic head data is very challenging and the posterior distributions of hyperparameters are only slightly narrower than the prior distributions. Direct measurements of hydraulic conductivity are needed to narrow more the posterior distribution of hyperparameters. To the best of our knowledge, this is a first accurate and fully linearization free solution to Bayesian multi-Gaussian geostatistical inversion with uncertain hyperparameters.