Page 89 - Modern Spatiotemporal Geostatistics

70 Modern Spatiotemporal Geostatistics — Chapter 2

our physical concepts and measurements have a particular meaning. Before
attempting to apply any sort of geostatistical approach to the spatiotemporal
analysis and mapping of natural variables, a system of coordinates and metric
properties needs to be established whereby these variables can be quantified.
The selection of the appropriate coordinates and metric properties depends on
the physical knowledge relevant to the specific situation, and is not a purely
geometric affair. Nature does not allow the natural processes to vary arbitrar-
ily, but imposes constraints in the form of physical laws which, in turn, imply
restrictions on the background space/time geometry. Geometrical approaches
based on the intrinsic (internal) as well as the extrinsic (external) visualization
of the spatiotemporal domain were considered in this chapter.
We have examined several groups of coordinate systems and metrical
structures. The coordinate systems allow representations of space/time ge-
ometry on the basis of the underlying symmetry of the natural process under
consideration. The use of a specific system also depends on the mathematical
convenience afforded by the system (e.g., in the case of a natural process that
has cylindrical symmetry, the cylindrical coordinate system may be more con-
venient for mathematical manipulations than a rectangular coordinate system).
In addition to coordinate systems, an important issue for geostatistical appli-
cations is the introduction of a metric that measures distances in space/time.
The definition of an appropriate metric depends on the local properties of space
and time (e.g., the space/time curvature), as well as on rules imposed by the
specific natural process (e.g., diffusion). The space/time metric is used in for-
mulating parametric models for the covariance function, which are then used in
geostatistical mapping studies. In some cases it is possible, based on invariance
principles and other physical considerations, to obtain explicit expressions for
the metric. If explicit expressions for the metric are not available, it is still
possible to obtain the space/time correlation functions for specific processes
based on numerical simulations or experimental observations (e.g., this is the
case with processes that occur in fractal spaces).
It is possible that future experience will show that our tried and trusted
systems and metric structures are inadequate, and that a new, conceptually
different spatiotemporal geometry will arise. To properly describe observational
evidence one may need to introduce, e.g., a fourth spatial dimension. It must
be emphasized, however, that four-dimensional space should not be confused
with four-dimensional space/time. Time has some physical properties that
space does not have, and vice versa. Certain activities and processes that
are possible in pure four-dimensional space are not possible in four-dimensional
space/time. To give another example, the Riemannian geometry we discussed
in this chapter needs to be modified in order to describe consistently the new
short-distance physics of superstring theory. Future developments may also
show that all physical knowledge can be unified into a single science expressed
in terms of space/time geometrical conceptions.