Bayesian inverse problem and optimization with iterative spatial resampling

Abstract

[1] Measurements are often unable to uniquely characterize the subsurface at a desired modeling resolution. In particular, inverse problems involving the characterization of hydraulic properties are typically ill-posed since they generally present more unknowns than data. In a Bayesian context, solutions to such problems consist of a posterior ensemble of models that fit the data (up to a certain precision specified by a likelihood function) and that are a subset of a prior distribution. Two possible approaches for this problem are Markov chain Monte Carlo (McMC) techniques and optimization (calibration) methods. Both frameworks rely on a perturbation mechanism to steer the search for solutions. When the model parameters are spatially dependent variable fields obtained using geostatistical realizations, such as hydraulic conductivity or porosity, it is not trivial to incur perturbations that respect the prior spatial model. To overcome this problem, we propose a general transition kernel (iterative spatial resampling, ISR) that preserves any spatial model produced by conditional simulation. We also present a stochastic stopping criterion for the optimizations inspired from importance sampling. In the studied cases, this yields posterior distributions reasonably close to the ones obtained by a rejection sampler, but with a greatly reduced number of forward model runs. The technique is general in the sense that it can be used with any conditional geostatistical simulation method, whether it generates continuous or discrete variables. Therefore it allows sampling of different priors and conditioning to a variety of data types. Several examples are provided based on either multi-Gaussian or multiple-point statistics.

Inverse problems with non-trivial priors: efficient solution through sequential Gibbs sampling

“…As mentioned earlier the 'resampling' algorithm presented above was originally proposed by Hansen et al [19], and subsequently an almost identical method was proposed by Mariethoz et al [30]. Neither of these works present a theoretical background for using the method, and provide no proof that the resulting algorithm samples an equilibrium distribution, nor that such an equilibrium distribution would in fact be the requested prior model.…”

“…Note that the same conclusions can be made using optimization based on the gradual deformation method [29] and the probability perturbation method [6], which are two methods that can be used to gradually change a realization of a random function based on 2-point and multiple-point geostatistics respectively. Likewise the optimization method proposed by Mariethoz et al [30], based on a prior sampler resembling the sequential Gibbs sampler, will not locate the model with maximum posterior probability.…”

“…Mariethoz et al [30] essentially describes an optimization application of the resampling technique used together with a version of the Metropolis sampler, where only models that increase the likelihood are accepted.…”

Probabilistic inference of multi‐Gaussian fields from indirect hydrological data using circulant embedding and dimensionality reduction

“…Before proceeding with our results, we would like to emphasize that SGS (or the very similar but independently developed iterative spatial resampling (ISR) scheme by Mariethoz et al [2010]) is a powerful MCMC algorithm for sampling from complex geologic prior models [e.g., Mariethoz et al, 2010;Hansen et al, 2012]. This method creates candidate points by conditioning a field realization drawn from the prior to a randomly chosen set of points from the current state (and hence model/field) of the Markov chain.…”

Stochastic Inverse Methods to Identify non-Gaussian Model Parameters in Heterogeneous Aquifers

“…Two such examples that are capable of dealing with curvilinear structures are the blocking moving window algorithm (Alcolea and Renard, 2010) and the iterative spatial resampling (Mariethoz et al, 2010a). Another avenue is treating the inverse problem as a search problem, e.g., a distance-based inverse method (Suzuki and Caers, 2008).…”