Geostatistical modeling for stream networks: Capturing multiple patterns of spatial autocorrelation in stream water temperature

Geostatistical models are typically based on Euclidean distance, which fails to represent the spatial configuration, connectivity, and directionality of sites in a stream network. Instead, hydrologic distances may represent the transfer of organisms, material, and energy through networks more accurately. In this study, we generated a geostatistical model and used it to quantify multiple patterns of spatial autocorrelation in stream water temperature data. We used a geographic information system (GIS) to calculate the spatial data necessary for geostatistical modeling. The autocovariance structure was modeled using three distance measures:  Euclidean distance, symmetric hydrologic distance, and flow-weighted asymmetric hydrologic distance. Many spatial autocovariance functions are not valid when hydrologic distance is substituted for Euclidean distance. To address this issue, we used moving average constructions to develop valid covariance matrices and fitted them using restricted maximum likelihood. We also generated a non-spatial general linear model. When we compared the r2 for the observed values and the cross-validation predictions, we found that the geostatistical model explained significantly more variability in the data than the same model without the spatial component (0.68 and 0.48, respectively). Temperature exhibited multiple patterns of spatial autocorrelation, which were best explained using a combination of distance measures. Weighted asymmetric hydrologic distance accounted for the largest proportional variance component (0.39), followed by symmetric hydrologic distance (0.34), the nugget effect (0.28), and Euclidean distance (0.0005). We believe that geostatistical models, which account for multiple patterns of spatial autocorrelation, are a valuable tool because they enable us to more accurately represent the ecological complexity in natural systems.