Page 273 - Modern Spatiotemporal Geostatistics

254 Modern Spatiotemporal Geostatistics — Chapter 12

(Hi.) signal processing in which the test function q (also called a mother
wavelet) is chosen so that it characterizes the local smoothness proper-
ties of a signal. This gives rise to the wavelet random fields (WRF).

The class of coarse-grained RF

Generalized S/TRF provides the mathematical framework for a rigorous formu-
lation of physical coarse graining. Coarse-grained fields represent averaging ef-
fects in physical measurements due to finite instrument bandwidth or effects of
numerical averaging. Generalized S/TRF leads to well-defined representations
of discontinuous random fields that lack well-defined values at all space/time
points (e.g., for intermittent processes such as rainfall, coarse-grained values
can be obtained by averaging over specific space/time windows). The resulting
physical processes generally depend on the scale of the coarse-graining window.
Expressions such as Equations 12.33 and 12.34 provide mathematical represen-
tations of the coarse graining involved in measurements of random fields.
Test functions q with finite support provide appropriate models for the
observation effect (this is the case when the coarse-graining process can sig-
nificantly modify the properties of random fields; see, e.g., Cushman, 1984
and Baveye and Sposito, 1984). If the apparatus function is known, the point
values of the S/TRF are determined by the deconvolution of Equations 12.33
or 12.34, which can be realized by means of maximum entropy techniques.
Such methods have been used successfully for enhanced information recovery
by spectrum deconvolution in atomic spectroscopy (e.g., Davies et al, 1991;
Fisher et al, 1997).

The class of S/TRF-v/ji models in heterogeneity
analysis
The spatiotemporal variations of many natural processes exhibit considerable
heterogeneities (complicated space/time patterns, local trends, etc.}. In such
cases, the homogeneous/stationary S/TRF model is not the best option. In-
stead, a more general and powerful random field model that is capable of
handling complicated space/time heterogeneities is sought. It turns out that
by properly selecting the test function q, such a model can be constructed on
the basis of the generalized random field theory above. In particular, if the test
function q is chosen so that (a) it eliminates polynomial functions of degree v in
space and \i, in time representing space/time trends within each fl (q), and (b)
the expression (Eq. 12.34) is a zero-mean, spatially homogeneous/temporally
stationary field, then X(p) belongs to the useful class of S/TRF-z///^ models
(Christakos, 1991b). The variability of X(p) is in general characterized by the
two integer indices, v for space and /j, for time, which are called continuity
orders. Their values determine the degree of departure from homogeneity and
stationarity. The case v = /i = 0, e.g., denotes an S/TRF with homoge-
neous/stationary increments (RF of this type may have linear trends in space