Page 73 - Modern Spatiotemporal Geostatistics

54 Modern Spatiotemporal Geostatistics — Chapter 2
investigation of space/time as a whole. The term "empirical" includes
all available physical knowledge bases.
According to Postulate 2.7, the basis of metrical determination should be
sought outside the abstract geometric objects in the natural processes that
act upon them. Postulate 2.7 implies that one's physical knowledge f^ (ob-
servational data, correlation functions, physical laws, empirical relationships,
etc.; see Chapter 3) must make sense within one's view of the structure of
space/time. In this context, Postulate 2.7 is closely related to Postulates 2.3,
2.4, and 2.6 considered earlier. Before proceeding with a more detailed analysis
of the metrical structures, it is necessary to introduce the concept of "field,"
which is essential in the geostatistical modeling of physical data and processes.

The Field Idea

Another fundamental idea of geostatistics is the idea of a field that associates
mathematical entities—such as the scalar, vector, or tensor—with spatiotem-
poral points. A field may be viewed as a model, i.e., as a mathematical con-
struction for representing the distribution of natural variables in space/time,
which leads us to the following postulate.

POSTULATE 2.8: A field presupposes a continuum £ of spatiotemporal
points and then attributes values of a natural variable to these points.
Specification of the values at all points in 'L specifies a realization of
the field.
From the viewpoint of physical modeling, fields have a number of inter-
esting features. There are many kinds of fields—some represent materialistic
variables (e.g., the spatiotemporal distribution of a material in soil or the con-
centration of a contaminant in water), others express non-materialistic regions
of influence through which values of natural variables can affect each other
(e.g., the Earth's gravitational field or the electromagnetic field). It is also
possible that field-based representations of reality may involve a hierarchy of
fields. There is no unique field representing every aspect of reality. Instead, one
field describes one characteristic of reality and another field some other char-
acteristic (a detailed discussion of the field concept and its various applications
can be found in Christakos and Hristopulos, 1998).

Restrictions on spatiotemporal geometry imposed by
field measurements and natural media

The crux of Postulates 2.7 and 2.8 is that there is no space/time without
natural fields. When we seek to develop a self-consistent geostatistical analysis
we must take into consideration the physical characteristics of the domain
and the diverse systems of measurement. The examples below show that the
spatiotemporal distance may depend on the properties of the physical medium