Page 244 - Modern Spatiotemporal Geostatistics

Multipoint Analytical Formulations 225

Several other examples of separable space/time covariances and vari-
ograms can be found in Whittle (1954), Christakos (1992), Johnson and
Dudgeon (1993), Christakos and Hristopulos (1998), and references therein.

Nonseparable models

Nonseparable spatiotemporal covariances for homogeneous/stationary ran-
dom fields have been proposed by a number of researchers in various scientific
disciplines. Let us discuss a few examples.

EXAMPLE 11.3: Heine (1955) developed the following nonseparable covariance
1
model in R x T,

where a, c, and cr 2 are suitable coefficients associated with a parabolic-
type partial differential equation; and Erfc is the complementary error
function.
EXAMPLE 11.4: Jones and Zhang (1997) suggested a set of covariances in
R n x T having the spectral density

2
Equation 11.36 represents homogeneous/stationary random fields. In R x T,
for p = 2, Equation 11.36 leads to a covariance that can be calculated as
fr»llri\A/c

where JQ is the Bessel function of the 1st kind of order 0.
While the above covariance models represent homogeneous/stationary
random fields only, Christakos (1992) and Christakos and Hristopulos (1998)
presented several classes of nonseparable models which are valid for homoge-
neous/stationary fields as well as models for nonhomogeneous/nonstationary
fields. The examples below include nonseparable covariance models derived on
the basis of stochastic partial differential equations, nonlinear systems, suitably
chosen spectral densities, dynamic rules, fractal processes, etc.

EXAMPLE 11.5: Nonseparable spatiotemporal covariance models can be asso-
ciated with the stochastic partial differential equation