Approximately optimal spatial design approaches for environmental health data

Environmental health research considers the relationship between exposures to environmental contaminants and particular health endpoints. Due to spatial structure associated with either exposures or outcomes, spatial modeling is making rapid inroads in environmental health. Our focus in this paper is on approximately optimal spatial design in the case of one-time sampling at a large number of spatial locations. If we plan to use spatial processes in building models to analyze the data, it seems equally appropriate to use such models in developing the sampling design. For a given study region, our contribution is to develop an approximately optimal sampling strategy to learn about the spatial distribution of a contaminant across the region. Optimal design, working with a continuum of locations, is intractable so, as is customary, we presume that the region has been gridded to high resolution. The criteria we focus on, are developed from the Fisher information matrix with the goal of learning not only about the regression structure in the model but also about the dependence structure. Under a criterion that attempts to maximize information gain, we consider three strategies to develop an approximately optimal design-sequential sampling, block sampling, and stochastic search. We also discuss utility-based modification of these strategies to achieve oversampling with regard to specified objectives. We present some theoretical and empirical properties and relationships among these strategies and provide an illustrative implementation for a simulated dataset. We also describe a real application in the context of the toxics release inventory (TRI). Copyright (c) 2005 John Wiley & Sons, Ltd.