Page 246 - Modern Spatiotemporal Geostatistics

Multipoint Analytical Formulations 227

c x(r) models, a variety of space/time covariances can be derived, such as the
following two examples: the covariance

associated with
where k = \ k \ and F is the gamma function; and the covariance

associated with the density
EXAMPLE 11.8: A certain class of random fields includes models of growth
and pattern formation in which the spatiotemporal evolution is governed by
a set of dynamic rules. In such a physical context, it is worth mentioning a
nonseparable covariance model that satisfies the dynamic scaling form

where z = 1.82, and g(x) ~ xa (x « 1), ~ x b (x » 1) with exponents
a = 1.4 and b = 0.6 (for details, see Christakos and Hristopulos, 1998
EXAMPLE 11.9: A class of /rocte/-related nonseparable covariance models in
space/time are available, such as

where:

with v = z, a; x = r/r^, r, and y — u c, w c. The function has
an unusual dependence on the space and time lags through r/r@. For large r,
the ratio r/r 13 is close to zero (for r sufficiently large), and the value of the

function f z is close to 1. With regard to / z, two pairs of space/time points are
equidistant if TI/T-J = rg/rf • Hence, the equation for equidistant space/time
contours is r/r@ = c. This dependence is physically quite different than that
implied by, e.g., a Gaussian space/time covariance function (in the latter case,
equidistant lags satisfy the equation
EXAMPLE 11.10: A variety of nonseparable generalized spatiotemporal covari
ance models can be defined on the basis of the S/TRF-i///i theory. One such
model is as follows:

where a p/£ are coefficients such that the permissibility conditions are satisfied.
The s(p), s(C) are sign functions given by where
x = p, C (Christakos and Hristopulos, 1998).