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Mathematical Formulation of the BME Method 121

Figure 5.4. Posterior pdf associated with: (a) multipoint mapping at all
space/time points simultaneously; and (b) single-point mapping at a
succession of points independently.

who have the right expertise and experience; see Chapter 3). Given 9£ and S,
a variety of ^-operators can be developed which express knowledge related to
actual measurements, uncertain evidence, probability distributions, functional
relationships, etc. Indeed, several examples of ^-operators are discussed in
Chapter 6.
On the basis of Equation 5.35, the effects of different prior ^-operators
can be assessed by comparing the corresponding posterior pdf related to single-
point vs. multipoint situations. Finally, in Chapter 7, we will discuss various
practical applications in which the general and specificatory knowledge opera-
tors are combined to provide informative spatiotemporal maps.

COMMENT 5.7: Th e importance o f notation in modern spatiotemporal geo-

statistics cannot be overemphasized. A well-chosen notation suggests the

right operations and liberates the mind from pointless distractions, while an
ill-chosen symbolism may be a hindrance to reasoning. For notational con-

venience, some o f the symbols 9§, S, \ map, o,ndp k that appear i n Equation

5.35 will be dropped occasionally in the following chapters. In some cases,
e.g., w e will write OslXmap'i Pk ] t° denote that th e operator involves th e

values o f a natural variable X, a s opposed t o OslXmapi VWo i Pk ] which

involves two natural variables X and Y (this is the multivariable or vector
case; see also Chapter 9). Furthermore, when its meaning is obvious from

the context, the operator will be denoted simply as y s.

In concluding this section, the following point is worth mentioning. In
light of the developments so far, the posterior BME information associated
with a spatiotemporal map can be defined naturally as
where the A parameter and the ^-operator have been defined above. There
is an interesting intuitive implication of Equation 5.36: The observation of the