A Multi-Scale-Oriented Blocking Markov Chain Monte Carlo Method for Inverse Stochastic Simulation

A Monte Carlo method for high-resolution reservoir characterization typically invokes a stochastic simulation to generate i.i.d realizations that honor both hard data and dependent state data. The blocking Markov chain Monte Carlo (BMcMC) method has been proved to be an effective scheme to carry out such conditional and inverse-conditional simulation by sampling directly from a posterior distribution that incorporates the prior information and the posterior observations. However, the usefulness of the previous BMcMC method suffers from the limited capability of the LU-decomposition of the covariance matrix. In this study, a multi-scale blocking McMC scheme is presented to generate high-resolution, conditional and inverse-conditional realizations. What make this method quite efficient in exploring the parameter space are that the proposal kernel is an appropriate approximation to the target posterior distribution, that the fast generation of candidate realizations is based on the spectral decomposition of the covariance matrix with aid of fast Fourier transform, and that a multi-scale procedure is used to calculate the likelihood quickly. The independent realizations generated in this way are not only conditioned to the conductivity, dependent state data, and other measurements available, but also have the expected spatial statistics and structure.