Page 249 - Modern Spatiotemporal Geostatistics

230 Modern Spatiotemporal Geostatistics — Chapter 12

estimates (map x map) that are highly probable in light of the specificatory
knowledge base S being considered. A symbolic representation of the BME
process may be written:

The integration of the S base into the prior probability model fg yields the new
(posterior) model f K. Hence, the mathematical form of f K depends on that
of S and jg. The space/time map x map may also depend on the knowledge-
based conditionals considered in the probability model (i.e., Bayesian, truth-
functional, etc. conditionals). A central feature of the BME approach is it
considerable generality. In sciences, the desire for generality constantly calls
for relentless and coordinated exploration of their foundations, and for greater
freedom as well. Modern geostatistics is not merely a collection of hard-data
processing techniques. Indeed, as discussed in the previous chapters, many
applications involve plenty of physical knowledge that must be taken into con-
sideration by the mapping method. Such a consideration requires sounder
foundations and new additions to the mathematical structure of geostatistics.
Certainly, there are situations (e.g., in the early stages of development
of a scientific field) in which only limited sets of observations are available,
and in such instances, the use of popular data-processing techniques may be
justified. The present chapter examines certain of these data-processing tech-
niques in the light of modern spatiotemporal geostatistics. As happens with
any novel theory that seeks to successfully replace old ones, the techniques of
classical geostatistics are special cases of the considerably more general theory
of modern geostatistics. This is a process that shows up many times in the
development of sciences: obtaining certain already-known results as logical or
mathematical consequences of newly stated principles. Comparative studies o
BME approaches and those of classical geostatistics are also discussed in this
chapter, and the powerful features of BME are demonstrated through synthetic
examples as well as real-world applications. Modern spatiotemporal geostatis-
tics includes a variety of mathematical models and stochastic techniques. As
will be shown in this chapter, a unified framework is provided by the general-
ized spatiotemporal random field theory. A large number of popular models,
including coarse-grained random fields, wavelet random fields, and fractal ran-
dom fields, can be derived as special cases of the generalized spatiotemporal
random field theory.

Minimum Mean Squared Error Estimators

Assume that the specificatory knowledge consists of only a set of hard data
Xhard about the natural variable X(p) at the space/time points p i (i —
1, ..., m). It is a well-known result (e.g., Cramer and Leadbetter, 1967)
that the best of all spatiotemporal minimum mean squared error (MMSE)