Sequential multiple-point statistics simulations conditioned on arithmetic averages

We seek to develop a methodology enabling fast geostatistical simulations honoring both geophysical data and a complex prior model. Particularly, we consider a multiple-point statistics (MPS) framework in which a training image (TI) describes the available prior knowledge. Accurate posterior sampling is then possible by using a so-called extended Metropolis algorithm in which proposals are drawn from the prior using sequential geostatistical resampling. Such a Markov chain Monte Carlo (MCMC) algorithm will eventually locate and sample proportionally to the posterior distribution, however, it is often exceedingly slow and typically demands millions of MCMC iterations before the posterior is sampled sufficiently. We are developing a methodology in which the MPS simulation is built up iteratively pixel-by-pixel starting from an empty grid. At each pixel, multiple proposals are generated using an MPS algorithm and the proposals are accepted proportionally to the likelihood considering conditioning data in terms of linear averages (for instance geophysical data). The likelihood function is generally intractable as it depends on the pixels that have not yet been sampled. We approximate the likelihood function using a Gaussian model in which the posterior mean and covariance are updated sequentially as the simulation builds up. The posterior statistics are approximated by running the algorithm multiple times (sequentially or in parallel). Considering crosshole first-arrival ground-penetrating radar data, we assess the accuracy of our methodology both for multi-Gaussian priors for which analytical posteriors are available and for more complex training images against the extended Metropolis method. Our approach is inherently approximate due to the use of a finite training image, a finite number of candidates for each pixel and the need to approximate intractable likelihood functions. Nevertheless, preliminary results are promising as this method allows directly obtaining a reasonable estimation at a reduced computational cost compared to MCMC.