Page 142 - Modern Spatiotemporal Geostatistics

Mathematical Formulation of the BME Method 123

The subsidiary theories and parameters are combined with empirical evidence in
part (c), and then interpreted, tested, and evaluated in detail in parts (d) and
(e). The testing and evaluation may involve (but are not limited by) the Pop-
perian falsification principle, which attempts to define the applicability limits
of the models (e.g., the areas in which the models are falsified, or they make
incorrect predictions).
While parts (a), (b), and (e) belong to the theoretical world, parts (c)
and (d) belong to the empirical world. These two worlds are linked through
a feedback process, which allows modern geostatisticians to learn from their
own mistakes. Indeed, if the predictions are not verified or an appropriate level
of understanding and explanation is not achieved, the implication is that there
must be unknown influences which are relevant and which should be sought at
the ontological level of part (b) or there is a lack of sufficient data in part (c).
In rare cases one may need to modify the core epistemology of part (a).
The core program aims at drawing attention to salient features of the
observational world and its relation to the understanding process of the observer
(the geostatistician). The program involves a strong formal part (mathematical
concepts and tools, etc.). Depending on the results obtained (accuracy of the
predictive pattern, explanatory content of the maps, etc.), a subsidiary theory
may be modified or replaced by another, if necessary. This was the meaning,
e.g., of Comment 1.4 (p. 21; e.g., information measures other than entropy
may be considered).

The Two Legs on Which the BME
Equations Stand

To summarize, what we have demonstrated is that the BME approach of mod-
ern spatiotemporal geostatistics forces the researcher to determine explicitly
the physical knowledge bases that are objectively available, and to develop
logically plausible rules for knowledge integration and processing. All of this
is epistemically incorporated in the mapping process; nothing is swept under
the carpet. As a consequence, the conclusions that can be reached by the
BME method have a security that is denied to conclusions reached merely by
empirical procedures.
We have also established that there are two fundamental operators in-
volved in the mathematical formulation of the BME approach:
(i.) the 9£-operator that incorporates general knowledge §, and
(ii.) the 9£-operator that incorporates specificatory knowledge S-
These two operators are, indeed, the two legs on which the BME equations
stand. Much of the following chapters are essentially mathematical results in
terms of these two operators. The mathematics that we will use, however, are