Page 241 - Modern Spatiotemporal Geostatistics

222 Modern Spatiotemporal Geostatistics — Chapter 11

Ordinary Variogram — Hard and Soft Data

Next we consider the common case in which the ordinary variogram function
is available together with a set of hard and soft data.

PROPOSITION 11.5: Assume that hard data exist at points p i (i =
.
1, 2, . . , mh) and soft data of various possible forms (Chapter 3) at
points p i (i — rn h + 1, . . . , m). General knowledge is expressed in terms
of the variogram. The multipoint BMEmode estimate \ k is the solution
of the set of equations ((. = 1, . . . , p)

1
where 7 ij- is the ij-th element of the inverse matrix 7 m^ p such that

is a variogram matrix; 7j is an (m x p) x (m x p) diagonal matrix of
variograms between the points p i (i = 1, .., m) and p ke (i — 1, . . , p);
.
.
7 2 is a p x p diagonal matrix of variograms between the points p kf
themselves; and

with

Proof: The prior pdf is of the form

where

Since the is given ty Equation 10.25 (p. 202), it
follows that

where

In light of the aforementioned analysis, the BME equation is given by Equa-
tion 9.29 (p. 175), where the operator £K is of the form of Equation 11.21 and