Page 285 - Modern Spatiotemporal Geostatistics

266 Modern Spatiotemporal Geostatistics — Chapter 13

The formal part involves mathematical tools and logical rules on how to use
them. This part is independent of empirical data and natural language. The
interpretive part, on the other hand, provides meaningful justifications for the
mathematical assumptions made in the formal part, connects the formal part
to experience, and relies on a natural language which guides experimentation
and involves physically testable statements. Each of these two parts of scien-
tific inquiry has its own challenging problems and intriguing research questions
to deal with. Some geostatisticians may chose to concentrate their efforts on
formal geostatistics, whereas some others may find it more profitable to focus
on the interpretive approach. Both groups should serve the field of geostatistics
well. Below we continue our discussion of the two fundamental components of
geostatistical research and development.

The Formal Part

From the formal point of view, the concentration is on the purely mathematical
structure of the space/time mapping approach. The main steps of this view-
point were discussed in Comment 6.3 (p. 134) and elsewhere in this volume.
Certain assumptions are made regarding the form of the general knowledge-
based probability distribution, the space/time estimation equations are formu-
lated and solved, and specificatory knowledge-based maps are derived from
a set of logical rules involving mathematical definitions, theorems, and proofs
(Chapters 9-11). The formal approach is a rigorous generalization that is based
on an internally consistent structure. Although the formal part of BME differs
profoundly from the formal part of MMSE mapping, it nevertheless contains
MMSE mapping as a limiting case, valid under specific conditions (Chapter
12). BME avoids several limitations of spatial statistics. As Stein (1999, p.
2) notes, "In practice, specifying the law of a random field can be a daunting
task.. .calculating this conditional distribution may be extremely difficult. For
these reasons, it is common to restrict attention to linear predictors." The
formal structure of BME, however, makes it possible to derive such probabil-
ity distributions in a way that guarantees consistency with physical knowledge
(Chapters 5 and 6). Moreover, there is no need to restrict spatiotemporal mod-
eling to linear predictors—nonlinear BME predictors that are more informative
than the linear ones can be considered (Chapter 7). In certain cases, standard
practice in spatial statistics based on the empirical covariance or variogram can
be seriously flawed (see Stein, 1999, p. 221). BME mapping can overcome
these kinds of problems by incorporating the covariance or variogram in terms
of physical laws (see Comment 3.3 on p. 81 and Comment 5.4 on p. 110 in
this volume).
The formal part presents the ambitious geostatistician with a series of
challenging research problems (existence and uniqueness of the solutions of
integrodifferential equations, formulation of logical conditionalization rules,
permissibility of space/time correlation functions, calculation of multiple pos-
terior integrals, etc.). The solution of these problems many times leads to the