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Institut für Wasser- und Umweltsystemmodellierung - IWS



"Bayesian Geostatistical Design: Optimal Site Investigation When the Geostatistical Model is Uncertain"

Abstract Geostatistical optimal design seeks to find sampling patterns for subsurface exploration with a maximum expected gain towards a task-specific objective or prediction goal. Past studies of geostatistical optimal design have assumed that the structural parameters (mean values, trend coefficients and covariance parameters) of the involved geostatistical models are given a priori. We believe that this is not justifiable, especially when only few or no data are available yet. Instead, we formulate four postulations: geostatistical optimal design should (1) treat structural parameters as uncertain to accurately represent prediction uncertainty, (2) be robust towards mis-specified structural parameters, (3) maximize the potential of planned investigations to identify the structural parameters, and (4) lead to optimal resource allocation between collecting spatial information and information on the structural parameters. Towards this end, we transfer the concept of Bayesian Geostatistical Design, introduced by Diggle und Lophaven (2006) for kriging-like problems, to geostatistical inverse problems. Bayesian Geostatistical Design differs from conventional geostatistical optimal design by accounting for uncertainties in the geostatistical model. We also deem it inappropriate to assume a fixed parametric covariance model. The Matérn family of covariance functions has an additional shape parameter, including the exponential, Whittle and Gaussian models as special cases. By treating the shape parameter as uncertain, we convert the model selection problem to parameter identification, which resembles Bayesian model averaging over a continuous spectrum of covariance models. For illustrative purposes, we derive a sequential linearized approach to Bayesian Geostatistical Design, and apply it to a series of synthetic test cases, where we minimize the prediction variance of solute concentration at an ecologically sensitive location by optimal placement of hydraulic head and log-conductivity measurements. The individual cases feature a different number of structural parameters. On this basis, we demonstrate the principle and show that Bayesian Geostatistical Design fulfills our four postulations.