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Modifications of BME Analysis 175

Multipoint BME Analysis

In Chapters 6-8, we mainly considered single-point mapping, i.e., we were
seeking an estimate Xk of Xk at a single point p k, given the physical knowledge
bases (3 (general) and 5 (specificatory). The multipoint mapping approach,
on the other hand, is seeking the estimates Xfej °f Xfc 3 a * several points p k.
•
(.7 = 1, • • , p) simultaneously, given the physical knowledge available (Fig. 5.4,
p. 121). When practically possible, multipoint mapping could be a considerable
improvement over single-point mapping. Indeed, while multipoint mapping
offers several interrelated estimates at a time, single-point mapping gives one
estimate at a time, independently of its neighboring estimates. The former
involves a multivariate pdf that is more informative than the univariate pdf
used in the latter.

Multipoint BME estimation

In principle, the BME approach can provide multipoint estimates by replacing
the single Xk with the vector

and the vector with the vector

Then, the basic BME equations of multipoint space/time mapping are
derived as follows. The Lagrange multipliers /j. a are calculated from the solution
of the following system of (p-point) equations

The multipoint (p-point) posterior pdf is given by

where and, as usual, the forms of A and y s depend on the
specificatory knowledge considered (see also Example 9.7). We now have a
system of p BMEmode equations

•
Solution of the system (Eq. 9.29) provides the estimates x fe = ( X k t , • • , Xk p).
The number p of points to be estimated simultaneously may depend on theoret-
ical, computational, and physical, as well as on decision-making considerations.