Page 271 - Modern Spatiotemporal Geostatistics

252 Modern Spatiotemporal Geostatistics — Chapter 12

Figure 12.16. Classification of spatiotemporal random field (RF) models.

The main classes of random field (RF) models used in modern spatio-
temporal geostatistics are outlined diagrammatically in Figure 12.16. This
classification distinguishes between ordinary and generalized random fields.
Ordinary models include homogeneous/stationary fields (H/S) and nonhomoge-
neous/nonstationary fields (n-H/n-S). Generalized models, on the other hand,
include S/TRF-Z//A* , wavelet RF (WRF), and fractal RF (FRF). In the re-
mainder of this section we will discuss very briefly the various parts of the
classification in Figure 12.16. The interested reader is referred to the relevant
literature for a more detailed presentation of the various RF models shown in
the illustration.
Ordinary S/TRF models were defined in Chapter 2 (Definition 2.10, p. 60)
and in references cited in that section. A generalized S/TRF X[q] is defined
by means of the following functional expression

where X(p) is an ordinary S/TRF with point values and q(p) is the so-called
test function that belongs to a properly chosen space 'D. The choice of 'D
reflects certain characteristics of the phenomenon under study and the goals
of the analysis. The set of linear functionals on 'D is called its dual space
1
and is denoted by 'D . The generalized field (Eq. 12.33) corresponds to an
average of X(p) over a support J7 (q) and satisfies the linearity conditions
X(q\ +92] = -^[<7i] +-^fe] and Xfag] = ocX[q\, where a is an arbitrary (real
or complex) number. Useful generalized fields are also defined by means of the
convolution integral

The X[q](p) is a function of the space/time point which represents a non-
local average of the S/TRF X(p) over a window determined by the test function
q. A standard example of a space (D is that of delta functions 6(p) (Dirac,