Spatio-Temporal Model of a Product–Sum Simulation on Stream Network Based on Hydrologic Distance

Abstract

The modeling of spatio-temporal processes is crucial in many fields such as environmental science, hydrology, and water storage engineering. A basic concept of the spatio-temporal model is an auto-covariance function. The stream network model based on the Euclidean distance as the auto-covariance function for investigating river pollution will potentially face a serious accuracy problem, due to the material or water pollutant movement on the water only occurring within the river network and the distance travelled not following the path of the river network. The use of the hydrologic distance may overcome the problem because, statistically, the Euclidean distance-based auto-covariance function is not valid if it is a negative-definite. Therefore, this study develops a spatio-temporal model of the generalized product–sum based on the hydrologic distance with a positive-definite property. The proposed model is built and, subsequently, it is then verified mathematically to be positive-definite. Furthermore, the relationships between the hydrologic distance, the Euclidean distance, and spatial configurations are also investigated. The simulation study result shows the proposed spatio-temporal model is positive-definite for some semivariogram models. The relationships between spatial configurations in the stream network follow an exponential model.