A mixed-model moving-average approach to geostatistical modeling in stream networks

Spatial autocorrelation is an intrinsic characteristic in freshwater stream environments where nested watersheds and flow connectivity may produce patterns that are not captured by Euclidean distance. Yet, many common autocovariance functions used in geostatistical models are statistically invalid when Euclidean distance is replaced with hydrologic distance. We use simple worked examples to illustrate a recently developed moving-average approach used to construct two types of valid autocovariance models that are based on hydrologic distances. These models were designed to represent the spatial configuration, longitudinal connectivity, discharge, and flow direction in a stream network. They also exhibit a different covariance structure than Euclidean models and represent a true difference in the way that spatial relationships are represented. Nevertheless, the multi-scale complexities of stream environments may not be fully captured using a model based on one covariance structure. We advocate using a variance component approach, which allows a mixture of autocovariance models (Euclidean and stream models) to be incorporated into a single geostatistical model. As an example, we fit and compare "mixed models," based on multiple covariance structures, for a biological indicator. The mixed model proves to be a flexible approach because many sources of information can be incorporated into a single model.