Page 243 - Modern Spatiotemporal Geostatistics

224 Modern Spatiotemporal Geostatistics — Chapter 11

elaborate as the number of knowledge sources that BME takes into consid-
eration increases. The reader, however, should remember: "There's no free
lunch!" It is, after all, a matter of choice. If we decide to limit our analysis to
a few low-order statistical moments and a set of hard data, BME will have no
difficulty generating simple, linear estimators. If, however, we believe that the
available knowledge bases cannot be ignored, we have no choice but to use the
more elaborate BME formulations. Modern spatiotemporal geostatistics allows
such a choice, something that is not possible with most classical techniques.

Spatiotemporal Covariance and
Variogram Models

The BME framework is very general, and one has considerable freedom in the
choice of the covariance and variogram models (ordinary and generalized). In-
deed, the covariance and variogram models used in the BME equations can
be separable or nonseparable functions, they may be associated to homoge-
neous/stationary or nonhomogeneous/nonstationary random fields, etc.

Separable models

Separable covariances and variograms (ordinary or generalized), which are ob-
tained by combining permissible spatial and temporal models, offer useful so-
lutions in a variety of applications. In particular, a wide variety of space/time
separable covariance models are obtained by means of the product

where c s(h) and c t(r) are valid spatial and temporal models.

EXAMPLE 11.2: The Gaussian model

and the exponential model

are among the most popular separable models (see also p. 64-65 in this vol-
ume). Another interesting separable model is given by

where r = | h \ and r = t—t 1 (the spectral density of the above model decreases
as either the spatial frequency k or the temporal frequency u> increases). In
2
an effort to study rainfall fields in R x T, Rodriguez-lturbe and Mejia (1974)
used the model

where K\ is the modified Bessel function of the 2nd kind.