Page 52 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 33
n
numerical entities: the spatial coordinates s = (si,... ,s) £ S C R and the
n
l
temporal coordinate t along the temporal axis T C R , so that

In Equation 2.1 the union of space and time is defined in terms of their Carte-
sian product S x T. Every point in space R n is like every other point — a
property called homogeneity. The existing observational evidence requires the
introduction of up to three spatial dimensions (i.e., n — 1, 2, or 3). In two-
or three-dimensional space, the dimensions are equivalent to one another, a
property sometimes called isotropy. Isotropy is a simple but precious prop-
erty because it allows us to set up a suitable coordinate system. Time is also
homogeneous — origin indifferent. In many applications it is sufficient to inves-
tigate the temporal evolution after an initial time instant in the not-too-distant
past. Then, the initial time is set equal to zero and T C [0, oo). Exceptions are
processes with long-range correlations (e.g., the fractional Brownian motion).

COMMENT 2.2: It i s possible that unsolved physical problems in th e future

will oblige us to introduce a fourth spatial dimension. The fourth spatial

dimension would be of a different character than the others, one that pre-

sumably will be in contrast to ou r intuitive ideas about space (e.g. , i t will

question the idea of isotropy). Perhaps, while the existence of a fourth

dimension external to our world cannot be directly experienced by us, we

may nevertheless need to infer it from the geometry of our world. This

suggestion should not be surprising, since the human mind can arrive at

valid conclusions pertaining to things and events not directly within our

perceptual world or evolutionary history.
It can be seen from the above analysis that the "address" of a point in
space/time is characterized by n + 1 numbers: n (= 1, 2, or 3) for the spatial
coordinates plus 1 for the temporal coordinate. There are several methods for
specifying the n + 1 numbers, each of them associated with a different coor-
dinate system. It is mathematically important that any appropriate method
ensure continuity and unambiguity in the assignment of "addresses." In physi-
cal applications it is typical to choose a coordinate system that works as simply
as possible.
Space, though an obvious component of natural variation, requires care-
ful definition. Equation 2.1 suggests more than one way to define the spa-
tial location of a point, depending upon the choice of spatial coordinates
s = (si,...,s n). Essentially, the only constraint on the coordinate system
implied by Equation 2.1 is that it possess n independent quantities available
for denoting spatial position. We start with the so-called general curvilinear
system of coordinates for the spatial component of Equation 2.1. This system
is defined as follows.