Page 231 - Modern Spatiotemporal Geostatistics

212 Modern Spatiotemporal Geostatistics — Chapter 10

where 0 is the number of the spatiotemporal increments considered, Q
is the vector operator associated with the S/TRF-Z///4 , and K, nap is the
generalized covariance matrix between the space/time points p l such
that

Proof: The 9^-operator is given by

where Vi = Qi(Xmap)' * = li 2, ... , 0, are spatiotemporal increments de-
termined on the basis of the theory of S/TRF-i///^ (Christakos, 1992), and
- ln view °f Equation 10.31, and letting juy =
Equations 5.6 and 5.7 (p. 106-107) lead to

where A = {\ij}, i, j = 1, ..., 0. The partition function is now written as

Since Equation 10.32 is of a Gaussian form and c^y = CQ^j(n map), we imme-
diately get

In light of Equations 10.33 and 10.34 and taking into account the relation
Z~ l — exp ^Oi the 9£ and p.o can be expressed by Equations 10.28 and 10.29,
respectively.
In order to illustrate the application of Proposition 10.4, let us consider
the following example.

EXAMPLE 10.3: A hard datum xi is available at point p l and a soft datum
% 2 at point p 2 in space/time. We seek the BME estimate at the point p k.
Assume that the operator associated with the underlying S/TRF-1/1 model is
given by

in which case the prior statistic is written as