Interpolation: Kriging, Cokriging, Factorial Kriging, and Splines
In the previous two chapters, we discussed the meaning of the geostatistical model and of its parameters. We will now discuss how this model can be applied. We will start with deterministic techniques, known under the generic name of kriging. Here, “deterministic” should be understood in the sense of “providing only one solution.” We will see that, although the model is probabilistic, kriging produces only one solution. Kriging covers a wide range of applications. The first one consists of interpolating one single variable in one, two, or three dimensions, and the second one consists of interpolating one variable but using the extra information provided by another variable that is related, of course, to the first one.
Recall the probabilistic model we defined in the previous chapter. The variable Z(x) is interpreted as the sum of a polynomial trend, m(x), plus a residual, R(x), of mean zero. Under this model, universal kriging (Matheron, 1970) addresses the problem of interpolating a variable on the basis of a number of scattered data. This can be the interpolation of layer-averaged porosity from well data or the interpolation of seismic times from a 2D seismic campaign.
To understand kriging, let us consider the variogram from another perspective (Fig. 3-1). Suppose that layer-averaged porosity has been calculated at a well and that we want to estimate porosity 1 km away from that well by using the value at the well. Obviously, we will make an error, which can be directly read from