Page 220 - Modern Spatiotemporal Geostatistics
P. 220
Single-Point Analytical Formulations 201
The following proposition deals with the situation in which the space/time
variability is expressed in terms of non-centered covariances.
PROPOSITION 10.2: The specificatory knowledge is as described in
Proposition 10.1. The general knowledge consists of the non-centered
ordinary covariance in space/time. The BMEmode estimate x.k is the
solution of the equation
are the ik-ib and fc-th elements, respectively, of the inverse
matrix C^ , where
is the matrix of the spatiotemporal non-centered covariances (7^ be-
and PJ (i, j = 1, ..., m, k); and the parameter
tween all points p t
is of the form of Equation 10.15 with
Proof: Working along the lines of the proof of Corollary 10.1 above, we find
that Equation 7.10 (p. 137) can be written as
where is the ij-th element of the inverse matrix
9ij(Xi,Xj) = XiXj (i,j = l,...,m and k), a
Equation 10.19 can be simplified as follows:
which—taking into account Equations 10.15 and 10.18—can be written in the
form of Equation 10.16.
In certain practical applications the spatiotemporal variogram can be cal-
culated more efficiently than certain other second-order statistical moments.
In such cases the following proposition is useful, for it provides the appropriate
BME formulation in terms of variogram functions.