The structure of a variable may depend on the direction. This chapter provides an overview of selected techniques for modeling such behavior.

Examples of anisotropy

The variograms studied in Chapters 3 and 4 are examples of isotropic models, in which the correlation structure (in particular, the variogram) does not differ with the orientation. The variogram depends only on the separation distance h and not on the orientation of the linear segment connecting the two points; that is, the average square difference between two samples at distance h is the same whether this distance is in the horizontal or in the vertical direction.

There are many cases, however, where the structure of a certain variable does depend on the direction (anisotropy). The best examples of anisotropic structures can be found in stratified formations. Consider, for example, an alluvial unit formed by the deposition of layers of gravel, sand, and silt. The layers and lenses of the various materials are oriented in the horizontal direction; as a result, the mean square difference of hydraulic conductivity measured at two points at a short distance h in the horizontal direction is smaller than the mean square difference at the same distance in the vertical direction. The same holds true for other hydrogeologic parameters, such as storativity.

Figure 5.1 is a vertical cross section showing a sketch of the cross section of an alluvial formation. Assume that we compute the experimental variogram γ1 as a function only of the separation distance h1 in the vertical direction and then we compute the experimental variogram γ2 as a function of the horizontal distance h2.