Page 151 - Modern Spatiotemporal Geostatistics

132 Modern Spatiotemporal Geostatistics — Chapter 6

BME analysis is easily modified to handle various other cases of practical
importance arising in physical applications in which soft data are available not
only at the data points p aoft, but also at the estimation point p k itself (see
Example 3.17, p. 86). Such a situation is presented in the following proposition.
PROPOSITION 6.5: Assume that the specificatory knowledge includes
the hard data (Eq. 3.30) and the probability law fs(x sojt^ Xk)> where
So : X SOft € I and Si ' Xk € 4 so that S= So U Si- Then, the posterior
operator has the form

Similar expressions for the f)f-operator can be derived in cases where other
forms of soft data are available at the estimation points.
EXAMPLE 6.6: From Equation 6.16 one obtains the posterior pdf f K(xk) =
^"^[Xtnap! PfeL where

is the normalization parameter. In the special case that fs(x softi Xk) =
fso (X i0ft)fsi (Xk), Equation 6.16 yields the recursive relationship f K(xk) =
1
^~ /5i(Xfe)/3Co(Xfe)- where f Ko(xk) is the posterior pdf associated with
and
Depending on the practical application, various combinations of the pre-
ceding results are possible. Furthermore, additional classes of ^-operators can
be created that incorporate many other kinds of soft data, including the intu-
ition, beliefs, and subjective assessments of laymen. However, these kinds of
soft data must be fitted into an organized and coherent system of knowledge
before they can attain significance or applicability. Indeed, it is the rich network
of physical knowledge bases created by generations of scientists and engineers
that gives meaning to such soft data as is occasionally provided by laymen.

Discussion
Certain of the previous results can be summarized in terms of the following
useful expressions of the BME posterior operator and pdf

with

where A is the normalization parameter; and the B, D and E s determine the
form of the ^-posterior operator (Table 6.1). This form depends, of course,
on the specificatory knowledge available.