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Modifications of BME Analysis 177

which are important in the explanatory description of the natural phenomenon
the map represents.
EXAMPLE 9.8: If we say that a porosity map of the West Lyons field (Kansas)
is given by the vector

with a probability of 0.7, we are making a generalization that goes beyond the
small data set available (76 data points, see p. 143-147). In assigning this
probability, we understand that the little we actually observe is part of a much
larger natural process.

Multipoint BME uncertainty assessment

A complete characterization of mapping uncertainty is provided by the multi-
point BME pdf. In many applications, a realistic assessment of the multipoint
mapping error is achieved using the concept of the multipoint BME confidence
set, which is an extension of the confidence width of single-point mapping
(Chapters, "Asymmetric Posteriors," p. 153). As we did in previous situations,
we choose an appropriate confidence level r) (with 0 < r? < 1). The choice of
ry depends on the mapping situation at hand. Then, a multipoint confidence
set $„ is determined such that

Clearly, for a given level of probability r\ there are several sets $,, satisfying
Equation 9.36 (intuitively, as the 77 increases, so should the corresponding $,,)
The multipoint BME confidence set is defined as the set with the smallest size

for a given 77. These concepts are best illustrated by means of an example.
EXAMPLE 9.9: If, e.g., p = 1 (single-point analysis), the BME confidence set
is simply the confidence interval $,, = [xk,i, Xk,u] with the smallest length
II ^»j II — Xk,u ~ Xk,l suc h that Equation 9.36 is satisfied.
A multipoint confidence set requires a multidimensional (p-dimensional)
plot; e.g., a contour map in the case of two points (p = 2). A confidence set
takes into consideration dependencies between all variables, as expressed by the
posterior pdf. The following result is useful in determining BME confidence
sets (Serre and Christakos, 1999a).

PROPOSITION 9.1: The A^-probability density set defined as