Page 62 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 43

Separate metrical structures

These metrics are convenient for many geostatistical applications because they
treat the concept of distance in space and time separately. The main idea is
introduced by the following definition.

DEFINITION 2.6: The separate metrical structure includes a spatial
distance \ds\ € -R+ 0 and an independent time interval dt = t-z —1\ e T,
en that

In Equation 2.11, the structures of space and time are posited indepen-
dently. Thus "distance" has meaning only at a fixed point in space (when it
means "time elapsed") or at a fixed time (when it means "distance between spa-
tial locations"). The formulation (Eq. 2.11) includes two celebrated space/time
structures: Newtonian and Galilean. Newtonian space/time is defined by the
following properties: between any two points in space/time (events) p and p'
there exist a Euclidean spatial distance \da\ and a temporal interval dt. An
alternative conception to Newtonian space/time is Galilean space/time, which
is defined so that between any two points p and p' there always exists a tem-
poral interval dt, but only if the two points are simultaneous is there any spatial
distance \ds\ between them.
The distance \ds\ can take on different meanings depending upon the
particular topographic space under consideration. The following examples ex-
plore several Euclidean and non-Euclidean spatial distances in two and three
dimensions.

EXAMPLE 2.10: In the Euclidean plane and in ordinary three-dimensional Eu-
clidean space, \ds\ is defined as the length of the line segment between the
spatial locations Si and 82 = Si +ds, i.e., the Euclidean distance in a rect-
angular coordinate system is defined by the Pythagorean formula

Frequently, the distance concept most useful in a particular application is non-
Euclidean. In some physical situations it makes sense to define the absolute
distance between the spatial locations PI and P% with coordinates «i and
«2 = si +ds, respectively, as follows (see, also, Fig. 2.11)

Distance as defined in Equation 2.13 could represent the length of the shortest
path traveled by a particle that moves from PI to P^, if the particle is con-
strained by the physics of the situation to move along the sides of the grid; or,
it could represent the shortest path that a car must travel to get from point