Page 275 - Modern Spatiotemporal Geostatistics

256 Modern Spatiotemporal Geostatistics — Chapter 12

exist. Two such examples were given in this volume in the previous chapter
(p. 227-228); other examples can be found in the relevant literature.

The class of space/time fractal RF models

It is noteworthy that members of the class of fractal RF (FRF) models can be
derived from the generalized S/TRF theory. Let a spatiotemporal FRF X(s, t)
satisfy the relationship

in a stochastic sense (c > 0; rj, £ and H are suitable scaling exponents), i.e.,
the X(s, t) and X^s, c*>t) have the same probability law. Sometimes we
write FRF-7? to denote the associated fractal exponent. In many cases, the
FRF-/? can be characterized equivalently by the weaker condition

(e.g., when the underlying distribution is Gaussian). To show that a S/TRF-
v/H X(p) can admit FRF-H representations, one can use (i.) generalized
X(p) representations and Equation 12.43 or (ii.) Equations 12.41 and 12.44.
These two approaches are illustrated by means of the following two examples.

EXAMPLE 12.15: Let X(p) be a generalized S/TRF-Z//M with the representa-
tion (Eq. 12.39) in the R l x T domain, i.e., p = (s, t) and

where Y(p) is a Gaussian white-noise field with zero mean and C y(r,r] oc
6(r)6(T). A corresponding generalized covariance is

+
where A(v, /x) = (-1)" 'V [(2i> + 1)! (2/x + 1)!]. If Equation 12.43 holds, the
generalized S/TRF-f/^ X(p) with the representation of Equation 12.45 can
be considered as an FRF-H with H = (y + 5)77 + (/x + ^K, where the range
of values for v, n, H, etc., satisfy the appropriate conditions (integrability,
continuity, etc.). Note that one could study an FRF using S/TRF-i//// repre-
sentations other than Equation 12.45.
EXAMPLE 12.16: Let X(t) be a generalized temporal RF of order v, in which
case Equation 12.41 reduces to

where