Page 71 - Modern Spatiotemporal Geostatistics

52 Modern Spatiotemporal Geostatistics — Chapter 2

EXAMPLE 2.20: In the relativistic representation of space/time a composite
(Riemannian) metric (Eq. 2.27) is defined as follows (Chavel, 1995)

The distance (Eq. 2.28) is obtained directly from Equation 2.27 by letting
2
9oo = —c , gn = 1 (i = 1, 2, 3; c is the speed of light) and 5^ = 0 (i ^ j).
This space/time distance is known as the Minkowski metric (for a physical
interpretation, see Example 2.23 below).
Riemann has shown that Equation 2.27 provides a sufficient but not nec-
essary specification that satisfies the fundamental requirements of a metric
(Weber, 1953). The metric (Eq. 2.27) can be used for an extrinsic (external)
as well as an intrinsic (internal) characterization of space/time, where the <?».,•
coefficients are obtained from the available physical knowledge (measurements,
laws, etc.) and are, in general, functions of space and time. In the case of
the external geometry model, a point is defined in space/time using Euclidean
coordinates. In the case of the intrinsic geometry model, the Riemannian coor-
dinates of a point can be defined without knowledge of distances. Then, spatial
distances and time lags are calculated within small local meshes using an ex-
pression of the form of Equation 2.27, where the <?y coefficients may vary from
one mesh to the other. These coefficients essentially convert the increments
of Riemannian coordinates in each small local mesh into spatial distances and
time lags.

Some comments on physical spatiotemporal
geometry

By way of a summary, in modern geostatistics the spatiotemporal geometry is
generally represented by the ordered pair

where £ is a space/time continuum, each point of which is associated with a
set of coordinates (Euclidean or non-Euclidean, Cartesian or non-rectangular,
etc.), and \dp\ denotes a suitable space/time metric (separate or composite,
Euclidean or Riemannian, etc.). As we saw above, Newtonian space/time
requires two distinct structures to be specified: \ds\ and dt—see Equation 2.11.
Minkowskian space/time, on the other hand, involves one structure: the metric
in Equation 2.28.
As far as the representation of physical knowledge is concerned, the exist-
ing spatiotemporal geometries display some important differences. Traditional
geometries roll together metrical and other spatiotemporal concepts, which
may hamper the development of physics. In classical geostatistics, e.g., the
spatiotemporal continuum is routinely covered by a single coordinate system
which implies a specified metrical structure (i.e., rectangular coordinates with
a Euclidean metric defined by the inner product) and no attempt is made to