Practical Scheffe‐type credibility intervals for variables of a groundwater model

Simultaneous Scheffé‐type credibility intervals (the Bayesian version of confidence intervals) for variables of a groundwater flow model calibrated using a Bayesian maximum a posteriori procedure were derived by Cooley [1993b]. It was assumed that variances reflecting the expected differences between observed and model‐computed quantities used to calibrate the model are known, whereas they would often be unknown for an actual model. In this study the variances are regarded as unknown, and variance variability from observation to observation is approximated by grouping the data so that each group is characterized by a uniform variance. The credibility intervals are calculated from the posterior distribution, which was developed by considering each group variance to be a random variable about which nothing is known a priori, then eliminating it by integration. Numerical experiments using two test problems illustrate some characteristics of the credibility intervals. Nonlinearity of the statistical model greatly affected some of the credibility intervals, indicating that credibility intervals computed using the standard linear model approximation may often be inadequate to characterize uncertainty for actual field problems. The parameter characterizing the probability level for the credibility intervals was, however, accurately computed using a linear model approximation, as compared with values calculated using second‐order and fully nonlinear formulations. This allows the credibility intervals to be computed very efficiently.