Expanded uncertainty quantification in inverse problems: Hierarchical Bayes and empirical Bayes

A common way to account for uncertainty in inverse problems is to apply Bayes' rule and obtain a posterior distribution of the quantities of interest given a set of measurements. A conventional Bayesian treatment, however, requires assuming specific values for parameters of the prior distribution and of the distribution of the measurement errors (e.g., the standard deviation of the errors). In practice, these parameters are often poorly known a priori, and choosing a particular value is often problematic. Moreover, the posterior uncertainty is computed assuming that these parameters are fixed; if they are not well known a priori, the posterior uncertainties have dubious value. This paper describes extensions to the conventional Bayesian treatment that assign uncertainty to the parameters defining the prior distribution and the distribution of the measurement errors. These extensions are known in the statistical literature as “empirical Bayes” and “hierarchical Bayes.” We demonstrate the practical application of these approaches to a simple linear inverse problem: using seismic traveltimes measured by a receiver in a well to infer compressional wave slowness in a 1D earth model. These procedures do not require choosing fixed values for poorly known parameters and, at most, need a realistic range (e.g., a minimum and maximum value for the standard deviation of the measurement errors). Inversion is thus made easier for general users, who are not required to set parameters they know little about.