We preview the general methodology underlying geostatistical modeling and apply it to the most common model, which is known as the intrinsic isotropic model and is characterized by the variogram. This chapter introduces kriging, which is a method for evaluating estimates and mean square estimation errors from the data, for a given variogram. The discussion in this chapter is limited to isotropic correlation structures (same correlation in all directions) and focuses on the methodology and the basic mathematical tools. Variogram selection and fitting will be discussed in the next chapter.

Methodology overview

Consider that we have measured porosity along a borehole at several locations (see Figure 3.1). To estimate the value of the porosity at any location from the measured porosity values, we need a mathematical expression (or “equation” or “model”) that describes how the porosity varies with depth in the borehole. In other words, we need a model of spatial variability.

However, hydrologic and environmental variables change from location to location in complex and inadequately understood ways. In most applications, we have to rely on the data to guide us in developing an empirical model. The model involves the concept of probability in the sense that spatial variability is described coarsely by using averages. For example, the best we can do might be to specify that the porosity fluctuates about some mean value and to come up with a formula to correlate the fluctuations at two locations depending on their separation distance.