Page 239 - Modern Spatiotemporal Geostatistics

220 Modern Spatiotemporal Geostatistics — Chapter 11

Hence, the BME equations can be written as

which leads to Equation 11.7.

In the following developments, it is convenient to define the partitioned
matrices

and

where the subscripts ft, s, and k denote hard points, soft points, and estimation
points, respectively. Computationally efficient formulations of the posterior (or
integration) pdf can be derived in some situations, as described by the following
two propositions (Serre and Christakos, 1999a).

PROPOSITION 11.3: Assume that hard data are given at points p,
.
(i = 1, 2, . . , m h) and soft data of the interval type (Eq. 3.32, p. 85
.
at points p i (i = m/, + 1, . . , m). General knowledge is the (centered)
ordinary covariance. The posterior pdf is as follows

B

where Xkh= (x k,Xha rd)> k\h =Ck,hC^ h,c k\ h = Ck,k-B k\ hCh,k,B s\ kh =

c Stkh,c^ kh,c s\ kh=c SiS-B 3\ khc kh, s,l = (lm h+i,---,l m)> an d u = (u TOh+1,

..., u m ); the < j ) ( x ' , x , c ) denotes a Gaussian distribution with mean vector

x and covariance matrix c; and A =
Note that the multiple integral in Equation 11.14 has the form of a multi-
variate Gaussian probability, which is very useful in numerical implementations
(see, e.g., Genz, 1992).