Block Kriging

Abstract

In the Introduction I mentioned that kriging was formulated from the outset to generalize linear regression, partly in order to consider different supports for the estimate and the sampling. In Chapter 5 we defined support as the shape, size, and orientation of the volume associated with any observation. So far we have not used that potential of kriging. For practical and pedagogic reasons we have been dealing with point estimates or estimates whose supports are small enough at the scale of the sampling domain that one can regard such small supports-cores, plugs, jugs-as points. The practical reasons are obvious. We have been able to cover ten chapters of applications devoted to point formulations. The pedagogic reasons will be clear at the end of this chapter, but in a nutshell, the generalization to estimates with non-point supports does not come without a price. The models are more complex than those for point kriging. Hopefully, at this stage of the exposition, the reader has understood the fundamentals of point kriging and the generalization will be easier to grasp than if we had followed the historical sequence. Block kriging is the generic name given to any form of kriging in which the interest lies in estimation of a linear average for an attribute inside supports that are intermediate in size between the support of the sampling and the sampling domain. Definition 12.1 Let Z (x) be a point support random function in D and let S (xo) c D be a support of finite size S centered around xo. The true block average Zs (.) at location Xo is Zs(xo) = ~ 1 Z(x)dx. 0 s(X o) When Matheron formulated kriging, he presented it in the form of block kriging (Matheron, 1963) partly for mathematical generality but mostly because his immediate concern was estimation of in situ resources, that is, estimates