Geostatistical modelling on stream networks: developing valid covariance matrices based on hydrologic distance and stream flow

1. Geostatistical models based on Euclidean distance fail to represent the spatial configuration, connectivity, and directionality of sites in a stream network and may not be ecologically relevant for many chemical, physical and biological studies of freshwater streams. Functional distance measures, such as symmetric and asymmetric hydrologic distance, more accurately represent the transfer of organisms, material and energy through stream networks. However, calculating the hydrologic distances for a large study area remains challenging and substituting hydrologic distance for Euclidean distance may violate geostatistical modelling assumptions. 2. We provide a review of geostatistical modelling assumptions and discuss the statistical and ecological consequences of substituting hydrologic distance measures for Euclidean distance. We also describe a new family of autocovariance models that we developed for stream networks, which are based on hydrologic distance measures. 3. We describe the geographical information system (GIS) methodology used to generate spatial data necessary for geostatistical modelling in stream networks. We also provide an example that illustrates the methodology used to create a valid covariance matrix based on asymmetric hydrologic distance and weighted by discharge volume, which can be incorporated into common geostatistical models. 4. The methodology and tools described supply ecologically meaningful and statistically valid geostatistical models for stream networks. They also provide stream ecologists with the opportunity to develop their own functional measures of distance and connectivity, which will improve geostatistical models developed for stream networks in the future. 5. The GIS tools presented here are being made available in order to facilitate the application of valid geostatistical modelling in freshwater ecology.