Page 45 - Modern Spatiotemporal Geostatistics

26 Modern Spatiotemporal Geostatistics — Chapter 2

theories, in addition to empirical evidence: data do not always speak for them-
selves. Mathematics is an instrument of analysis, not a constituent of things. It
does not describe the behavior of natural phenomena, but only our knowledge
of that behavior. The real challenge in today's multidisciplinary scientific arena
is not merely to develop mathematical techniques to deal with numerical data,
but also to develop the theoretical means for interpreting and integrating these
data—as well as other important sources of physical knowledge—into the pro-
cess of understanding. The analysis in Chapter 4 suggests a powerful response
to this challenge, thus making possible the correction of the third fallacy that
estimation is an exercise of mathematical optimization.

The Spatiotemporal Continuum Idea

Physical theories parameterized by space and time variables are considered
more basic than those that are not (this is why, e.g., mechanics is considered
more basic than thermodynamics). Table 2.1 depicts the considerable ranges
of these variables (i.e., spatial distances and time intervals) encountered in
natural studies (Ridley, 1994). Spatiotemporal continuity implies an integration
of space with time and is a fundamental property of the mathematical formalism
of natural phenomena. In most applied natural sciences, the following postulate
is of vital importance for the purposes of Spatiotemporal analysis and mapping.
POSTULATE 2.1: Space/time £ is viewed as a set of points that are
associated with a continuous spatial arrangement of events combined
with their temporal order.
Events represent attributes related to a natural phenomenon or process
(e.g., contaminant concentration, hydraulic head, temperature, fluid velocity).
Within the space/time continuum £, space represents the order of coexistence
of events and time represents the order of their successive existence. This is,
in essence, the view of space/time held by the majority of scientists today.
While the concept of identifiable points in a continuum is most impor-
tant, it is often taken for granted because it seems so obvious. Each identifiable
point in a space/time continuum £ is associated with an event, but £ does
not merely serve as a repository of events. In addition, many other considerably
more complicated objects and processes can be described within £. A few ex-
amples follow, in an attempt to throw some light on these basic Spatiotemporal
continuum concepts.
EXAMPLE 2.1: The space/time path of an individual entity P (e.g., a particle)
in £ is represented by a set of points forming a space/time trajectory—also
called a world-line (see Fig. 2.1). In other words, a world-line defines the
spatial position of P at every instant of time with respect to a reference system
(a system of coordinates that will be defined mathematically in a following
section).
Just as points and curves constitute the basic elements of Euclidean geome-
try, events and world-lines are the basic elements of Spatiotemporal geometry.