Page 123 - Modern Spatiotemporal Geostatistics

104 Modern Spatiotemporal Geostatistics — Chapter 5

This paradigm suggested certain essential directions for studying space/time
variability and producing a map that can be formalized in both the continuum
and the discrete domains. In the following theoretical analysis, the continuum
mapping formalization is presented for reasons of mathematical convenience
and generality, but its discrete version may be used in the implementation of
BME by means of computer algorithms.

The Prior Stage

Each stage of the BME analysis processes physical knowledge. We do not
do scientific reasoning in a void. Before we reason from a specificatory data
set to a particular map, we already have some general knowledge about the
distribution of the natural variable or the phenomenon been mapped. This
general knowledge Q is the result of earlier instances of scientific reasoning, as
well as background beliefs relative to the situation overall (Chapter 3, p. 73,
"The General Knowledge Base").

Map information measures in light of general
knowledge

Let fg(Xmap) ^e tne multivariate pdf model associated with the general knowl-
edge (3, before any specificatory knowledge S (e.g., hard and/or soft data) has
been taken into consideration. As we saw in Chapter 3, the general knowledge
(j considered at the prior stage may be expressed mathematically in terms of
a series of functions g a which represent known statistics of the S/TRF X(p).
The fir a's can be associated with various types of knowledge about X(p). Ex-
amples were given in Chapter 3.
At the prior stage, the knowledge contained in the pdf about the random
vector x map can be expressed mathematically in terms of information mea-
sures. Generally, the more probable a model of x map is, the more alternatives
it allows; but, it is also less informative. Conversely, the more informative the
model is, the more alternatives it excludes. These standard epistemic rules im-
ply the inverse relationship between prior information and probability, and have
already been discussed in the previous chapter. There are various information
measures satisfying this inverse relation. One such measure is suggested by the
following postulate.

POSTULATE 5.1: Given the general knowledge base §, the information
contained in the map x map will be expressed as follows

which is sometimes referred to as the Shannon information measure (see
also Eq. 4.2, p. 93).