Page 51 - Modern Spatiotemporal Geostatistics

32 Modern Spatiotemporal Geostatistics — Chapter 2

Viewed separately, both space and time are continua. Thus, in general,
they share all the properties possessed by the abstract notion of a continuum.
But there are important differences, as well. Time has certain extra-continua
physical properties not shared with any other continuum, by virtue of which
time is specifically time and not just a continuum (e.g., recursivity is not a
property that continua have in general; recursivity is, indeed, an extra-continua
property of time, but not of space). The same is true with certain properties of
space. These extra-continua physical properties—some of them known, some
of them not—may contribute considerably to the behavior of the space/time
system as a whole. Indeed, the difference between the degree of importance
that space and time play in natural processes may depend on their extra-
continua properties.
When we bring space and time together, the extra-continua physical prop-
erties of space integrate with those of time, producing a holistic space/time in
which the whole is greater than the sum of its parts. In such a holistic environ-
ment, the spatiotemporal connections and cross-effects could control natural
variations. We are not, e.g., concerned merely about the distance between two
geographical locations l\ and 1%, but rather about the distance between the
location l\ at a specified time t and the location /2 at another time t''. In the
case of many natural variables, some aspects of time have repetitive or cyclical
features which result because of holistic relationships between the spatial and
temporal domains (not because time actually repeats). The cyclic behavior of
precipitation profiles, e.g., is due to the occurrence of certain spatiotemporal
climatic processes. Also, the intimate connection between space and time is
embodied in the astronomical unit of distance: the light year (i.e., the distance
traveled by light in one year).
Postulates 2.1-2.4 express rather broad, qualitative features of the space/
time continuum £. In order for £ to be useful in real-world applications it must
be equipped with numerical information and operational concepts. With this
in mind we will examine these postulates in more detail in the following sections.

The Coordinate System

Postulate 2.3 involves a combination of three components: (i.) an axiomatic
component (i.e., a set of geometric objects, axioms, relations, and their logical
consequences); (ii.) an analytical component (points represented by coordi-
nate systems, relations expressed in terms of algebraic equations, etc.); and
(Hi.) an empirical component (a means of investigating which combination of
axiomatic and analytical geometrical constructions best describes the observed
facts).
The introduction of a coordinate system is essential in making a decision
about how to assign "addresses" to different points in the space/time contin-
uum £. Generally, a point p in £ can be identified by means of two separate