Combining information across spatial scales

Spatial and spatiotemporal processes in the physical. environmental. and biological sciences often exhibit complicated and diverse patterns across different space-time scales. Both scientific understanding and observational data vary in form and content across scales. We develop and examine a Bayesian hierarchical framework by which the combination of such information sources can be accomplished. Our approach is targeted to settings in which various special spatial scales arise. These scales may be dictated by the data collection methods, availability of prior information, and/or goals of the analysis. The approach restricts to a few essential scales. Hence we avoid the challenging problem of constructing a model that can be used at all scales. This means that we can provide inferences only at the preselected special scales. However, problems involving special scales are sufficiently common to justify the trade-off between our comparatively simple modeling and analysis strategy with the formidable task of forming models valid at all scales. Specifically. our approach is based on a simple idea of conditioning the spatially continuous process on an areal average of the process at some resolution of interest. In addition, the data at prescribed resolutions are then conditioned on this areal-averaged true process. These conditioning arguments fit nicely into the hierarchical Bayesian framework. The methodology is demonstrated for the spatial prediction of an important quantity known as streamfunction based on wind information from satellite observations and weather center, computer model output.