Modeling space and time dependence in environmental studies

This paper analyzes the use of stratification in the modeling of dependence for regionalized variables in space and time. Stratification enables optimal use of available information, and it is used when deterministic, small-scale information is combined with large-scale random variation: on the one hand, the problem of having too few data to reliably estimate statistical parameters may be solved by stratification; on the other hand some variation may be so obvious that it serves as a prerequisite for obtaining any form of stationarity. The paper is illustrated with four case-studies. The first study analyzes optimization of an extant groundwater monitoring network within a catchment of the river Rhine. Stratification in space enabled 5 cm precision of predicted levels at critical locations, something that could not be reached without stratification. The second study analyzed variability in space and time of wind erosion in a field. Erosion by four storms was measured with 21 devices. Use of temporal stratification to overcome lack of spatial data resulted, in one instance, in a decrease of 17 percent in mean absolute error. The third study addressed changes in the soil solution influenced by environmental deposition over a period of six years. Stratification was made in space to enable analysis in the time domain and to obtain spatio-temporal maps. The fourth study used a general space-time model to evaluate the development of root-rot in soybean, caused by naturally occurring pathogens. Predictions were made at unvisited times to estimate the probability of occurrence of root rot. We conclude that use of stratification enhances the utility of space-time statistics for the solution of various environmental and agricultural problems.