Spatial sampling design based on spectral approximations to the random field

Abstract

During the last 20 years several software packages have become available for spatial statistics. Spatial statistics deals with geo-referenced data and loosely speaking may be subdivided into the areas point processes analysis, areal and lattice data analysis and geostatistics. The topic of this article is geostatistics, the science of continuous stochastic processes that are defined either over some region in 2- or 3-dimensional geographic space or in space-time. Geostatistics is best known under the heading of kriging and covariance function estimation. A lot of free and commercial software packages are nowadays available for these tasks of optimal spatial interpolation and determination of the roughness of spatial random fields. When interpolating a spatial random field by means of kriging the uncertainty and accuracy of the kriging predictions are communicated by means of the so-called kriging variances. The kriging variances are dependent on the number and the density of the available gauged data locations. The denser the grid of available gauged data locations the smaller become the kriging variances and the better become the kriging predictions. Unfortunately, for the task of optimal planning prior to data gathering where to locate the monitoring stations or samples almost no software is freely available up to date. This article reports on a MATLAB and Octave toolbox whose main task is the optimal planning of monitoring networks. Both, addition of sampling locations to available networks and the reduction of monitoring networks are considered in an optimal way by means of borrowing ideas from convex experimental design theory and regression models with random coefficients. Both, design criteria for optimal interpolation with the covariance function assumed to be fixed and certain as well as a criterion where the uncertainty of the covariance function estimation is taken into account are developed and optimal designs are calculated by means of deterministic algorithms that fully make use of the mathematical structure of the considered design criteria.