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Popular Methods in the Light of Modern Geostatistics 257
In light of the expression
2
and letting t = t', Equation 12.47 yields X (t) = K x(0) + const, x t 2H ',
where H = v + 1/2. In other words, for an appropriate range of /f-values the
generalized RF-i/ X(t} can be considered as an FRF with H = v + 1/2. The
determination of the H and v values depends on the situation. Assume, e.g.,
that X(t) is a Wiener (Brownian) RF. This process is a generalized temporal
RF with v = 0, which has a zero mean and covariance
where a is a constant and r = \t — t'\. In light of Equation 12.47, Equation
12.50 implies that K X(T) = —ar, bo(t) = at, and &o(*') = ott'; hence,
H = 1/2. Equation 12.49 is satisfied for H = 1/2 and, thus, the Wiener
(Brownian) process X(t) is an FRF-1/2, i.e., it is valid that X(ct) = ^/cX(t}.
Furthermore, a Gaussian RF with v = 0 and K X(T) = —ar 2H (0 < H < 1) is
an PRP-H, also called a fractional Brownian RF. D
The greater part of the above analysis can be extended to multi-FRF
models. The latter include random fields that can be characterized by Equation
12.44. In this case, the fractal exponent is a function of the space/time point
p, i.e., the exponent must be written as H(p). The class of S/TRF-i///Lt is
considerably wider than these of FRF. The former, e.g., can be characterize
by generalized covariances that have forms more complicated than Equations
12.46 and 12.48.
The class of wavelet RF
As we saw above, the £>-spaces of test functions q in generalized S/TRF are
selected on the basis of the physics of the situation and the goals of the analysis.
In several physical applications, the test functions that are selected possess
properties that can be used to study certain important characteristics of the
random field of interest (e.g., singularities caused by physical laws, multiscale
features, and sharp variations).
We start by using the fundamental generalized S/TRF concept to define
the wavelet random field (WRF) model in the space/time domain. For this
purpose, we select a test function such as
where ^(p, u, a) is called the mother space/time wavelet, and the coefficient
A(a) is chosen so that it magnifies or reduces the sensitivity of the field to
different scale parameters; Oj and s, (i = 1, ..., n) are the spatial scale and
translation parameters, respectively, whereas the at and Ut are the temporal