Page 72 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 53

examine whether the geometry assumed is consistent with the physics of the
domain. As a result, the geostatistical analysis could be based on unjusti-
fied assumptions about space/time which may lead to unrealistic maps of the
phenomena under consideration (see section on the spatiotemporal random
field concept beginning on p. 59). Physically more meaningful geometries are
those that clearly distinguish the various space/time concepts so that scientific
theories can employ only the necessary ones. In modern spatiotemporal geo-
statistics, e.g., depending on the physical features of the domain, the metric
and the coordinate system used to describe that metric may be independent of
each other and not necessarily of the Euclidean form. This allows considerable
flexibility in the choice of the geometry that best represents physical reality.
The separate metrical structure of Equation 2.11, in particular, would be
suitable to represent our commonsense view of space as a container (within
which all events take place) and time as an absolute entity (that registers the
successive or simultaneous occurrences of these events); space and time exist
independently of natural processes and laws, as a kind of theater in which the
natural processes and laws enact their drama. On the other hand, the basic
idea underlying Equation 2.26 is that the theater (the space/time continuum)
is intimately linked to its actors (natural processes and laws) and cannot exist
independently of them.
In several natural applications the separate metrical structure (Eq. 2.11)
will suffice. In other applications, however, the more involved composite struc-
ture (Eq. 2.26) may be necessary. In the latter case, considering the several ex-
isting spatiotemporal geometries that are mathematically distinct but a priori
and generically equivalent, we must ask ourselves: Which mathematical spa-
tiotemporal metrical structure (i.e., what sort of function g) describes reality,
inasmuch as there is no need to single out the Euclidean metrical structure?
The answer to this question should take into account the fact that mathemat-
ics describes the possible geometric spaces, and physical knowledge determines
which one of them corresponds to real space. According to this approach the
question is subject to empirical investigation. It is the space of physical expe-
rience and intuition that gives rise to the concepts of metric and measurement
and presents the notion and possibility of spatiotemporal relationships, such as
Equation 2.26. In other words, a system of pure axiomatic geometry may not
suffice, if geometry is to be applied to physical space/time. What is required
in many cases is to establish a relationship between the geometric concepts of
the abstract system with the natural processes of the physical system.
How can such a relationship between the abstract geometry with the phys-
ical system be established? Again, any answer to this question must take into
consideration the fact that mathematical geometry is purely logical, whereas
physical geometry is empirical. In other words, the following postulate makes
sense.

POSTULATE 2.7: The nature of the mathematical geometry that best
describes space/time should be disclosed in terms of the empirical