Page 238 - Modern Spatiotemporal Geostatistics

Multipoint Analytical Formulations 219

vector estimates x/t are solutions of the equations

The following proposition is a straightforward extension in the multipoint
case of the single-point situation treated in Proposition 10.2 (p. 201).

PROPOSITION 11.2: Assume that hard data are given at points p i
(2 = 1,2,..., rn h) and soft data of various possible forms (Table 6.1)
at points p t (i = rrih + 1, ..., m). General knowledge is the (non
centered) ordinary covariance. The multipoint BMEmode estimate
(Xki, • • • Xk ) is a solution of the set of equations

C ik* is the ikf-th element of the inverse matrix C m* pt where C map is the
matrix of the non-centered covariances C 13 between all the points p i
and PJ (i, j = 1, ..., m, k\ ..., k p); and the parameter xl (\ k) is of the
form of Equation 11.6 above.

Proof: Equation 10.18 (p. 201) can be extended to the multipoint case where
now X map = (Xi, • • » Xm, Xfen • • •, Xk p) and
•

is the non-centered covariance matrix between all points p t(i = 1,..., m, fci,
... ,k p). Equation 9.32 (p. 176) is valid with

where is the ij-th element of the inverse matrix
•
9ij (Xi,Xj) = XiXj (*, j = 1, • • •, "». fei, • • , k p). After some manipulations

and