Kriging and Conditional Geostatistical Simulation Based on Scale-Invariant Covariance Models

Kriging is a powerful spatial interpolation technique, especially for irregularly spaced data points, and is widely used throughout the earth and environmental sciences. The estimation at an unsampled location is given as the weighted sum of the circumjacent observed points. The weighting factors depend on a model of spatial correlation. Calculation of the weighting factors is done by minimizing the error variance of a given or assumed model of the auto-covariance for the data with regard to the spatial distribution of the observed data points. As part of this work, I have developed a flexible and user-friendly matlab-program called vebyk (value estimation by kriging), which performs ordinary kriging and can be easily adapted to other kriging methods. Extensive tests demonstrate that (i) heavily clustered data require an adaption of the search neighborhood, (ii) kriging may cause artefacts in anti-persistent media when using the “correct” auto-covariance model and (iii) best performance for kriging scale-invariant media is obtained when using smoother auto-covariance models than those indicated by the observed dataset. Conversely, my results indicate that kriging is relatively insensitive to the absolute value of the correlation lengths used in the auto-covariance model as long as the structural aspect ratio is approximately correct. Finally, this kriging algorithm has been used as the basis for conditional geostatistical simulations of the porosity distribution in two heterogeneous sedimentary aquifers. The stochastic simulations were conditioned by porosity values derived from neutron porosity logs and georadar and seismic crosshole tomography. The results indicate that conditional simulations in “hydrogeophysics” will prove to be similarly useful for quantitatively integrating numerous datasets of widely differing, resolution, coverage and “hardness” as it has found to be in more established fields of reservoir geophysics. Chapter