A Markov chain Monte Carlo method for the groundwater inverse problem

Abstract

In this study, we develop a Markov Chain Monte Carlo method (MCMC) to estimate the hydraulic conductivity field conditioned on the direct measurements of hydraulic conductivity and indirect measurements of dependent variables such as hydraulic head for saturated flow in randomly heterogeneous porous media. The log hydraulic conductivity field is represented (parameterized) by the combination of some basis kernels centered at fixed spatial locations. The prior distribution for the vector of coefficients θ are taken from a posterior distribution ρ(θ/d) that is proportional to the product of the likelihood function of measurements d given parameter vector θ and the prior distribution of θ. Starting from any initial setting, a partial realization of a Markov chain is generated by updating only one component of θ at a time according to Metropolis rules. This ensures that output from this chain has ρ(θ/d) as its stationary distribution. The posterior mean of the parameter θ (and thus the mean log hydraulic conductivity conditional to measurements on hydraulic conductivity, and hydraulic head) can be estimated from the Markov chain realizations (ignoring some early realizations). The uncertainty associated with the mean field can also be assessed from these realizations. In addition, the MCMC approach provides an alternative for estimating conditional predictions of hydraulic head and concentration and their associated uncertainties. Numerical examples for flow in a hypothetic random porous medium show that estimated log hydraulic conductivity field from the MCMC approach is closer to the original hypothetical random field than those obtained using kriging or cokriging methods. © 2004 Elsevier B.V.