Appendix A: Calculating covariance matrices using the tail-up and tail-down models

A mixed-model moving-average approach to geostatistical modeling in stream networks. Ecology 91:644–651. A distance matrix that contains the hydrologic distance between any two sites in a study area is needed to fit a geostatistical model using the tail-up and tail-down autocovariance functions. However, the hydrologic distance information needed to model the covariance between flow-connected and flow-unconnected locations differs in most cases. The total hydrologic distance is a directionless measure; it represents the hydrologic distance between two sites, ignoring flow direction. The hydrologic distance from each site to a common confluence is generally used when creating models for flow-unconnected pairs (Fig. A1.A), which we term " downstream hydrologic distance ". In contrast, the total hydrologic distance is used for modeling flow-connected pairs (Fig. A1.B), which we term " total hydrologic distance ". A downstream hydrologic distance matrix provides enough information to meet the data requirements for both the tail-up and tail-down models. When two locations are flow-connected, the downstream hydrologic distance from the upstream location to the downstream location is greater than zero, but it is zero in the other direction. Using the data contained in Fig. A1 as an example, the