Page 270 - Modern Spatiotemporal Geostatistics

Popular Methods in the Light of Modern Geostatistics 251

Figure 12.15. The application domains of the methods of modern spatio-
temporal geostatistics.

where KB-cond. denotes the knowledge-base conditionalization process. Thus,
BME methods are special cases of the epistemic approach discussed in Chapter
4, beginning on p. 90, if we assume that the map information measure is of
the form of Equation 4.2 (p. 93) and the physical knowledge conditionalization
process uses the Bayesian principle (Eq. 4.9, p. 96). Epistemic analysis may not
be restricted to Bayesian conditionalization. Other forms of KB-cond. include
material conditionals, etc. (p. 98). The traditional kriging techniques are, in
turn, special cases of the BME concept, if the general knowledge is limited to
statistical moments up to order 2, a restricted (hard) data set is considered,
and a single-point estimator is sought.

Random Field Models of Modern Spatiotemporal
Geostatistics

A unified framework
The mathematical apparatus of modern spatiotemporal geostatistics includes
a variety of theories and models. Among them, a powerful unified framework
is provided by the generalized S/TRF theory. Indeed, a large number of popu-
lar stochastic models, including coarse-grained random fields, wavelet random
fields, and fractal random fields, can be derived from the theory of general-
ized S/TRF. Mathematically, generalized fields were introduced in the works
of Dirac (1926-27), Sobolev (1938), and Schwartz (1950-51) on distribution
theory. Studies on generalized random fields in a purely spatial domain were
spearheaded by Ito (1954) and Gel'fand (1955) with important contributions
by Yaglom (1957, 1987) and Matheron (1973). An extension of the generalized
random field theory in the spatiotemporal domain can be found in Christakos
(1991a, 1991b, 1992) and in Christakos and Hristopulos (1998).