Page 93 - Modern Spatiotemporal Geostatistics

74 Modern Spatiotemporal Geostatistics — Chapter 3

other which varies inversely as the square of the distance between them, and
varies directly as the product of their masses."
A theory is an intellectual construct that enquires deeply into the fun-
damental significance of a phenomenon and has explanatory power. A theory
may also include nonobservable quantities. A law is a representation of a rela-
tion that has been observed consistently in the past and is expected to occur
consistently in the future (e.g., Pick's law of mass diffusion, or Fourier's law
for heat flow). In this sense, a law is a summary description of numerous facts
and is neither fundamental nor explanatory. A law may be derived from a
theory (e.g., under certain conditions Darcy's law may be deduced from first
principles). Local (or phenomenological) laws could be obtained by means
of well justified approximations or intuitive interpretations of a larger theory.
Phenomenological laws may also include local experimental results.
A scientific theory, a physical law, etc. may be incorporated directly or
indirectly into the BME framework. In the second case, a law may be used
to derive a logical construct that expresses general knowledge about certain
feature of the natural phenomenon. Such a construct may be, e.g., a statisti-
cal moment that represents knowledge about spatiotemporal correlations and
interactions.
Other forms of general knowledge may be obtained from sources such
as literature searches (reports from state and federal agencies, printed material
from professional organizations, computerized data bases, etc.), and experience
with similar physical situations.

A mathematical formulation of the general
knowledge base

In natural sciences our data sets, concepts, and theories strongly depend on spa-
tiotemporal correlations and probabilistic dependencies. Therefore, it is useful
to express the general knowledge § mathematically in terms of stochastic func-
tions or operations involving the natural variables of interest. Assume that the
general knowledge base Q involves a series of funtionals Q a (a = 0,1,..., N c)
such that

The right-hand side of Equation 3.1 depends on the \ map values, the j> map
coordinates, and the pdf f s associated with the general knowledge Q. The left-
hand side represents a set of stochastic expectations of the natural variables
involved. As we shall see later, the h a can be formulated in a number of ways,
depending on the general knowledge base available. Equation 3.1 accounts for
a variety of sources of general knowledge, including: (i.) statistical correlation
(or moment) models, (ii.) empirical relationships, and (Hi.) physical laws and
scientific theories. Equation 3.1 is, thus, the kind of quantity that makes
scientists feel at home and enables them to pull out the tricks of their trades.
An important function of modern spatiotemporal geostatistics is to endeavor
constantly to refine and improve formulation (Eq. 3.1) so that it may continually