Spatial Prediction of Stream Temperatures Using Top-Kriging with an External Drift

Top-kriging is a method for estimating stream flow and stream flow-related variables on a river network. Top-kriging treats these variables as emerging from a two-dimensional spatially continuous process in the landscape. The top-kriging weights are estimated by a family of variogram models (regularisations) for different catchment areas (kriging support), which accounts for the different scales and the nested nature of the catchments. This assures that kriging weights are distributed to both hydrologically connected and unconnected sites of the stream network according to the data situation: top-kriging gives most weight to close-by sites at the same river system, but when the next hydrologically connected site is far away, more weight is given to a close-by site at an adjacent river system. The distribution of weights is in contrast to ordinary kriging and stream distance-based kriging which does not account for both spatial proximity and network connectivity. We extend the top-kriging method by incorporating an external drift function to account for the deterministic patterns of the spatial variable. We test the method for a comprehensive Austrian stream temperature dataset. The drift is modelled by exponential regression with catchment altitude. Top-kriging is then applied to the regression residuals. The variogram used in top-kriging is fitted by a semiautomatic optimisation procedure. A leave-one-out cross-validation analysis shows that the model performs well for the study domain. The residual mean squared error (cross-validation) decreases by 20 % when using top-kriging in addition to the regression model. For regions where the observed stream temperatures deviate from the expected value of the drift model, top-kriging corrects these regional biases. By exploiting the topological information of the stream network, top-kriging is able to improve the local adjustment of the drift model for the main streams and the tributaries.