Summary statistics from training images as model constraints in probabilistic geophysical inversion

A methodology is presented to incorporate complex a priori information on model morphology in probabilistic geophysical inversion by extracting statistical metrics from an ensemble of training image realizations. Training images are conceptual geological models that feature the expected lithological units and structural patterns. The inverse problem is tackled in a Bayesian formulation by Markov chain Monte Carlo sampling with the MT-DREAM(ZS) algorithm. To account for the extracted summary statistics, the classical likelihood function describing the data misfit is augmented by a term that quantifies the probability of each proposal state given the difference between the observed and simulated summary statistics. This procedure has the advantage of not requiring CPU-intensive geostatistical resimulation of the models during the inversion process. The model space is here parameterized by the discrete cosine transform of porosity fields where the dominant transform coefficients are defined by compressed sensing analysis of the training image realizations. This allows for a drastic reduction of the dimensionality of the inverse problem, while keeping the ability to recover realistic subsurface structures. In an application to crosshole ground-penetrating radar data, we additionally invert for petrophysical parameters that relate porosity to radar wavespeed. The summary metrics used are the frequency of occurrence of different geological facies and the total sum of discrete cosine transform coefficients as a global measure of model variability. In two synthetic case studies we show that the proposed method has the potential to (1) steer the inversion towards geologically more realistic a posteriori models, (2) prevent inversion artifacts and (3) improve the parameter estimates and reduce their uncertainty. Probabilistic inversion in high dimensional parameter spaces with large high quality data sets generally suffers the problem that the maximum likelihood estimate model is not necessarily a reasonable representation of the true subsurface structure. Incorporating summary statistics in the likelihood function substantially decreases this problem. The proposed approach is general and allows great flexibility in terms of the applied model constraints and the type of geophysical forward problem.