Bayesian geostatistical design: Task-driven optimal site investigation when the geostatistical model is uncertain

Geostatistical optimal design optimizes subsurface exploration for maximum information toward task-specific prediction goals. Until recently, most geostatistical design studies have assumed that the geostatistical description (i.e., the mean, trends, covariance models and their parameters) is given a priori. This contradicts, as emphasized by Rubin and Dagan (1987a), the fact that only few or even no data at all offer support for such assumptions prior to the bulk of exploration effort. We believe that geostatistical design should (1) avoid unjustified a priori assumptions on the geostatistical description, (2) instead reduce geostatistical model uncertainty as secondary design objective, (3) rate this secondary objective optimal for the overall prediction goal, and (4) be robust even under inaccurate geostatistical assumptions. Bayesian Geostatistical Design follows these guidelines by considering uncertain covariance model parameters. We transfer this concept from kriging-like applications to geostatistical inverse problems. We also deem it inappropriate to consider parametric uncertainty only within a single covariance model. The Matern family of covariance functions has an additional shape parameter. Controlling model shape by a parameter converts covariance model selection to parameter identification and resembles Bayesian model averaging over a continuous spectrum of covariance models. This is appealing since it generalizes Bayesian model averaging from a finite number to an infinite number of models. We illustrate how our approach fulfills the above four guidelines in a series of synthetic test cases. The underlying scenarios are to minimize the prediction variance of (1) contaminant concentration or (2) arrival time at an ecologically sensitive location by optimal placement of hydraulic head and log conductivity measurements. Results highlight how both the impact of geostatistical model uncertainty and the sampling network design vary according to the choice of objective function.