Page 147 - Modern Spatiotemporal Geostatistics

128 Modern Spatiotemporal Geostatistics — Chapter 6

combinations of sub-intervals so that where
q = l,...,p. We can thus write

since R g n R g> (S) = 0 for q ^ q'. Furthermore, the events
Xhardt x aoft € Rg] and x soft € R q (S) are independent (they are associated
with independently derived probability laws). Therefore, by letting p —> oo

(6.5)
and in view of Equations 6.4, 5.6, and 5.35, Equation 6.5 yields Equation 6.3.
Similarly, it is found that A = f f dF s (Xsojt) fg(Xdata)- This equation together
with Equations 6.4 and 6.5 gives the corresponding posterior pdf.

The following example serves a specific instructional purpose: it demon-
strates the role of combinations R q in Proposition 6.2.

EXAMPLE 6.2: Assume that soft data \soft — (xi> X2, Xz) are available of
the probabilistic form (Eq. 3.35, p. 86). The intervals 7j of these soft data are
partitioned as follows (see Fig. 6.1),

The possible combinations Rq, q = 1, . . . , 4 of subsets for xsoft are

i.e., .R g is a vector of sets which belong to the domain of the soft data.
Note that the R q's are mutually exclusive and exhaustive sets. Each R q is
a vector with elements that are subsets (intervals) of the Xsojt domain; so,
even though the RI and R±, e.g., have one interval in common, they do not