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Popular Methods in the Light of Modern Geostatistics 259

respectively. Then, Equation 12.52 leads to the definition of the continuou
1
WRF-V> on R as follows

The WRF X[il>](t) is sometimes denoted X(a, t) to emphasize the scale param-
eter a of the wavelet. In certain applications, these WRF constitute efficient
tools for local analysis of nonstationary and fast transient signals. The choice of
the mother wavelet is of utmost importance in applications. Localized mother
wavelets -fi(t, u, a), which are such that they decay rapidly to 0 as t —> ±00, are
of special interest. Test functions used as wavelets are known in the literature
under several names: Morlet wavelet, Meyer wavelet, Daubechies wavelet,
and the "Mexican hat" wavelet (see, e.g., Poularikas, 1996). One-dimensional
WRF of the form of Equation 12.55 with A(a) = 1/v/a have been studied
extensively in the literature. The wavelet mean and (non-centered) covariance
for the WRF (Eq. 12.56) are given by

and

where mother wavelets of a suitable form may be used.

In addition to its physical features, the class of mother wavelets determined
by Equation 12.55 has certain interesting analytical properties, some of which
we may discuss very briefly.

EXAMPLE 12.19: Let X(t) be an FRF-tf in the sense that C x(ct, ct') =
c 2H C x(t, t'), where c, H > 0; the C x(t, t') and if>(t) satisfy some well-
defined conditions (Cambanis and Houdre, 1995). Then, it is easily shown
1
that C x(ca, cb; ct, ct') = c' 2H+l C x(a, b; t, t ), and vice versa. This im-
plies that X(t) is an FRF-H if the corresponding WRF, X[V>](t) = X(a, t),
is an FRF-H + 5. A similar result is valid in terms of RF with homogeneous
increments of order v.

A symbolic summary of some of the results above is presented in Figure
12.17. This figure illustrates how certain classes of S/TRF-z///u, FRF-.ff, and
WRF-V> models are derived from the generalized random field theory of modern
spatiotemporal geostatistics.