Page 245 - Modern Spatiotemporal Geostatistics

226 Modern Spatiotemporal Geostatistics — Chapter 11

where L a is a linear spatial differential operator. These models include the
following (Christakos and Hristopulos, 1998)

where xin and X2n represent eigenfunctions (modes) of Equation 11.38, the
coefficients c nm represent correlations of the mode coefficients, i.e., c nm =
A nA m, the two-point function c x( nim)(s,s') denotes the correlation
Xin(s)xim(s'), and A n are random variables to be determined from the initial
and boundary conditions. Randomness in the three models of Equation 11.39
can be introduced, respectively, by: (i.) the initial or boundary conditions lead-
ing to random coefficients A n ; (ii.) the differential operator L s leading to
random eigenfunctions Xin(s); and (Hi.) by both of the above. Models (Eq.
11.39) may be homogeneous/nonstationary due to a number of reasons in-
cluding the boundary and initial conditions. Similarly, on the basis of the noisy
Burgers equation, an interesting nonseparable covariance model in R l x T is
given by

EXAMPLE 11.6: A diffusion-inspired covariance model is associated with the
2
2
2 2
spectral density c x(k,u) = 2a<7 /[u; + (afc ) ], which satisfies Bochner's
theorem. By calculating the inverse transform we find the nonseparable
space/time covariance model

in R n x T. The nonseparable variogram below that is used in several ap-
plications in the R 1 x T domain has been obtained from the application of
Bochner's theorem

The a and b are coefficients corresponding to the spatial and temporal scales
(see, e.g., Christakos, 1992); this model represents homogeneous/stationary
random fields.
n
EXAMPLE 11.7: A rich class of nonseparable covariance models in R x T is
generated from the spectral density

where v is a vector of known coefficients and c s(k) is the spectral density
of a purely spatial covariance c x(r). Thus, based on the already available