Bayes Optimal Design of Monitoring Networks

We are concerned with a particular problem of optimal design for random fields, namely the optimal placement of monitoring stations for monitoring regionalized environmental variables. The sparse literature on this problem is solely based on the variance of the well-known kriging predictor for spatial random fields, see e.g. Fedorov (1989) and Cressie et al. (1990). Here we consider a Bayesian version of the kriging predictor which allows the incorporation of prior knowledge about the trend behaviour of the regionalized variable under consideration. We derive a robust Bayes linear predictor which only requires approximate knowledge of the first and second order prior moments. Hereafter we consider the design problem for the (robust) Bayes linear predictor on the basis of the maximum and the integrated Bayes risk over the region of interest. Finally, we present some results for the case where there is only a finite number of potential sites from which the optimal choice is to be made. This particularly pertains to the problem of adding a few additional stations to an existing network.