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Mathematical Formulation of the BME Method 105

The epistemic rules mentioned above have, thus, a quantitative side that
can be expressed by means of the information measure (Eq. 5.1). Postulate
5.1 also represents the uncertainty regarding the random vector a3 mop, which is
characterized by the pdf and provides the formal ground for the epistemic rule:
The higher the probability, the lower the uncertainty about x map and the lesser
the amount of information provided by the pdf about x map. The expected
information is then given naturally by

where all the components of the vector x map are assumed to be integrated
in the above. In other words, on the basis of the available general knowledge
Q, we construct a probability model of the mapping situation (how we do this
is discussed in the following section). The amount of information about the
natural map provided by Q and carried in the model is expressed by Equation
5.2, which is called the entropy function. In fact, the possible interpretation
of Equation 5.2 in terms of entropy was responsible for the third letter in the
acronym "BME," used originally to name the spatiotemporal mapping approach
(Christakos, 1990). Intuitively, the entropy (Eq. 5.2) of a pdf is its degree
of diffuseness, so that the more concentrated it is, the smaller its entropy.
Various bases can be used for the logarithm in Equations 5.1 and 5.2; in most
applications the natural basis e, is assumed. Equation 5.2 is the continuum
entropy. In the discrete-domain formulation, the integral should be replaced
simply by a summation (the latter is used in the computer implementation of
BME). In a different setting than the space/time mapping problem considered
above, entropy-based analyses have been used to study a variety of problems,
including the inverse problem of groundwater hydrology (Woodbury and Ulrych,
1998).

COMMENT 5.1 : Note that while th e continuum entropic definition (Eq.
5.2) as a measure of uncertainty is the subject of some theoretical debate,

nevertheless, it is fully justified from a practical point of view (i.e., by the

physical arguments which support i t and the results it yields; e.g., Jumarie,
1990). In certain cases group-invariance requirements may make it more

appropriate to replace fogf with
where fo is a noninformative prior pdf (i.e., one that is form-
invariant under the transform groups that leave the physical laws invariant;

Jaynes, 1983). Usually, an x map parameterization is sought for which the
corresponding / o i s a natural constant. Moreover, Eddington (1959, p. 133)

notices that in real-world applications involving natural phenomena: "Prob-

ability is always relative to knowledge (actual or presumed) and there is no

a prior i probability of things in a metaphysical sense, i.e., a probability rel-

ative to complete ignorance." In any case, the theoretical issues associated

with the representation of complete ignorance (noninformativeness) are not

of practical concern in BME mapping applications. Even if it is the case