Geostatistics without Stationarity Assumptions within Geographical Information Systems

Abstract

The present work deals with two challenging problems of applied geostatistics: (i) Stationarity assumptions often do not hold under real-world conditions. (ii) Geostatistical methods have to be linked with spatial databases in order to be applicable in non-stationary situations. Solutions for both problems are proposed and implemented. (i) A central assumption in geostatistics is the stationarity of the process. However the spatial variability of many natural phenomena heavily depends on the local geology, which is nonstationary in most cases. To deal with this, the concept of process stationarity is replaced by a stationarity of the governing influence relating the local semivariogram and the local geology as stored in a Geographical Information System (GIS). A construction method is used, which can meaningfully incorporate additional spatial information from GIS, e.g. smoothly varying geology in the investigated area, spatially varying anisotropy induced by mountainous morphology, or geological faults interrupting continuity. Least-squares parameter estimation is used for fitting instationary semivariogram models in typical example situations, leading to non-linear optimization problems. Furthermore, a method for semivariogram parameter estimation in the present of linear trend is proposed. (ii) Geostatistical tools that make use of the local geology need direct access to the data stored in the GIS. A link between the presented geostatistical tools and the GIS software ArcView was established. Thus, spatial data such as measured contaminant concentrations, soil properties and morphology can be incorporated in geostatistical analyses. R code that fits instationary semivariogram models and performs kriging was implemented and can be obtained from the author. It is applied to simulated datasets.