Page 101 - Modern Spatiotemporal Geostatistics

82 Modern Spatiotemporal Geostatistics — Chapter 3

There are several ways to incorporate such knowledge into BME analysis. As-
suming that the univariate pdf is the same for all points in space/time, a sim-
ple approach is to consider a transformation of the original random field into
a Gaussian field with known mean and covariance functions, which brings us
back to the earlier section on general knowledge in terms of statistical moments
(p. 75).
In some other circumstances, while the multivariate pdf } ? is unknown, an
equation that describes the evolution of a lower level pdf may be derived from
physical or mathematical considerations (e.g., a diffusion-type equation). In
order to incorporate this knowledge into BME analysis, one may select a set
of g a functions so that the stochastic expectations <^ with respect to /§ are
calculable (e.g., with the help of the evolution equation of/ §). The statistical
functions Tfe are now tne general knowledge, i.e., the original situation again
has been converted to that of the earlier section (p. 75). It may be instructive
to demonstrate the approach in terms of a very simple example.

EXAMPLE 3.11: Suppose that the prior pdf / ? satisfies the equation

X € [a, 6], with initial condition f s(x', 0) = <Kx)- We cnoose 9o = 1 (as
n
usual) and let gi(x) De a known function of x ° \y, with <7i(0) = CQ. We can
then write

In light of the initial condition, Equation 3.28 gives g\ (t) = CQ exp[/3t].

Finally, in some applications, useful physical knowledge may be expressed
in the form of integral equations. This is the case, e.g., of perturbation ap-
proximations (ordinary or diagrammatic) to groundwater flow and solute trans-
port problems (for a detailed study of such perturbations see Christakos and
Hristopulos, 1998).

The Specificatory Knowledge Base
The specificatory or case-specific knowledge base S is knowledge about the
specific situation. For example, the singular statements, "the litmus paper that
Mr. Nicolaou gave me turned red when immersed in the liquid" and "the distri-
bution of the West Lyons field porosity data is shown in Figure 7.4," constitute
specificatory knowledge. Unlike general knowledge, the specificatory knowl-
edge base S refers to a particular occurrence or state of affairs at a particular
location and at a particular time. The S includes both external or demon-
strative evidence (actual measurements, perceptual or data of sense, etc.) and