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Popular Methods in the Light of Modern Geostatistics 253

1926-27; Schwartz, 1950-51). A practical space <D contains functions that
are continuous, integrable, and infinitely differentiable. It is also customary to
require that the test functions and all their derivatives vanish outside a certain
interval in n+1-dimensional support fi (q) (Gel'fand and Vilenkin, 1964). Other
ID-spaces are chosen on the basis of the physics of the situation (see below).
The stochastic moments of a generalized S/TRF are defined in a straight-
forward manner: the point values x of the random field are replaced by the
functionals X(q}. Several properties and classifications that are valid for ordi-
nary fields can be extended to generalized fields, as well (see, e.g., Christakos
and Hristopulos, 1998). Generalized random fields share the stochastic sym-
metries of ordinary fields. Thus, homogeneous/stationary generalized random
fields are defined by means of the second order moment functionals (in the
weak sense) or the pdf (strict sense), etc.

EXAMPLE 12.14: Second-order moments of the generalized fields (Eq. 12.34)
include the space/time mean functional

and the (non-centered) covariance functional

Higher order moment functionals may be defined in a similar manner.
Although the derivative of an ordinary random field may not exist in the
usual sense, it may still be defined in terms of generalized random fields. The
derivatives of generalized random fields in 'D always exist as generalized fields.
The partial derivatives of any order of the generalized field are obtained by

Therefore, the derivatives of the generalized field can be evaluated by means
of the derivatives of the test function.
Generally, the 2?-spaces of test functions q are selected on the basis of
the physics of the situation and the goals of the analysis. Below we will briefly
discuss three special cases of generalized S/TRF that are derived by properly
specifying the desirable properties of the test function q (again, for a detailed
discussion, the interested reader is referred to the relevant literature). These
special cases include:
(i.) coarse-graining applications in which q is chosen so that it represents
averaging effects in physical measurements,
(ii.) heterogeneity analysis in which q is chosen so that the resulting random
field is spatially homogeneous/temporally stationary [this class of random
fields includes fractal random fields (FRF) as a special case], and