Page 70 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 51

Figure 2.14. A point in the space-time domain R 2 x T.

structure is used, the function g should be determined by means of the physical
knowledge available about the natural variable X(si, s%, t).

The wavy line in Figure 2.14 is used to denote that the spatiotemporal vector
OP = p does not, in general, have a Euclidean metrical structure, i.e., the
2
function g in Equation 2.26 is not necessarily of the form ^/s\ + s^ + t . In
fact, the latter form may have no physical meaning. We will continue the
discussion of Figure 2.14 in Example 2.26.

COMMENT 2.7: Th e situation with the physical meaning o f vector OP = p

in Figure 2.14 raises an important —although sometimes ignored —logical

connection between a vector and its components, in general. In many cases,

it is the vector itself that carries the fundamental meaning. In some other

cases, however, while its components are physically meaningful, the vector

itself has no physical meaning independent of its components. An exam-

ple of the former situation is the velocity vector, which has a meaningful

geometric representation in terms of its coordinates (directional velocities)

and i t also carries a fundamental meaning b y itself ('i.e. , even i f it s coor-

dinates are erased from the geometric representation). An example of the
reverse situation is a vector that characterizes an individual so that its three

components represent "age," "weight," and "height." If these components
are erased from the geometric representation, nothing is signified by the

remaining vector, which becomes totally meaningless.
A special case of Equation 2.26 is the space/time generalization of the
spatial distance in Equation 2.16 that leads to the spatiotemporal Riemannian
metric defined below.
DEFINITION 2.9: The spatiotemporal Riemannian metric is defined as
where the metric coefficients gij (i, j = l,...,n) are functions of the
spatial location and time.
A well-known spatiotemporal metric of the Riemannian type is discussed in the
following example.