Bayesian inversion and visualization of hierarchical geostatistical models

<p>Geostatistical inversion methods estimate the spatial distribution of heterogeneous soil properties (here: hydraulic conductivity) from indirect information (here: piezometric heads). Bayesian inversion is a specific approach, where prior assumptions (or prior models) are combined with indirect measurements to predict soil parameters and their uncertainty in form of a posterior parameter distribution. Posterior distributions depend heavily on prior models, as prior models describe the spatial structure of heterogeneity. The most common prior is the stationary multi-Gaussian model, which expresses that close-by points are more correlated than distant points. This is a good assumption for single-facies systems. For multi-facies systems, multiple-point geostatistical (MPS) methods are widely used. However, these typically only distinguish between several facies and do not represent the internal heterogeneity inside each facies.</p><p>We combine these two approaches to a joint hierarchical model, which results in a multi-facies system with internal heterogeneity in each facies. Using this model, we propose a tailored Gibbs sampler, a kind of Markov Chain Monte Carlo (MCMC) method, to perform Bayesian inversion and sample from the resulting posterior parameter distribution. We test our method on a synthetic channelized flow scenario for different levels of data available: A highly informative setting (with many measurements) where we recover the synthetic truth with relatively small uncertainty invervals, and a weakly informative setting (with only a few measurements) where the synthetic truth cannot be recovered that clearly. Instead, we obtain a multi-modal posterior. We investigate the multi-modal posterior using a clustering algorithm. Clustering algorithms are a common machine learning approach to find structures in large data sets. Using this approach, we can split the multi-modal posterior into its modes and can assign probabilities to each mode. A visualization of this clustering and the according probabilities enables researchers and engineers to intuitively understand complex parameter distributions and their uncertainties.</p>