Page 76 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 57

2.34 into geostatistical analysis, one cannot specify the spatiotemporal metric
and the natural field values independently. Rather these two entities must be
connected via Equation 2.34.
In some other applications, the form of the metric is obtained indirectly
from the field equations. This may happen in the case in which the solution of
the physical law can be written as

The solution (Eq. 2.35) puts restrictions on the geometrical features of space/
time and suggests a metric of the form |p| = g(si, ...,s n,t), where |p| defines
the space/time distance from the origin. In fact, in the section, "Separate
metrical structures" (p. 43), we discussed the formulation of physical equations
in terms of general curvilinear coordinates involving the metric coefficients
Qij. It is possible that the physical law could lead to a solution that offers
information about the coefficients gjj of the metric. These possibilities are
demonstrated with the help of the following examples.

EXAMPLE 2.24: Consider the natural field X(si, s 2) whose spatial distribution
is governed by the physical law

The solution of Equation 2.36 is expressed as
Hence, a spatial metric suggested by the physical law above
is the Euclidean \s\ = ^s\ + s%. On the other hand, the physical equation
sidX/dsi—s^dX/dsz = 0 has a solution of the form X(si, s^) = X(si 82),
which means that the above physical equation may be associated with the
metric \a\ — s\ s?.

EXAMPLE 2.25: The governing flow equations for phases a (= water and oil)
in a porous domain are (Christakos, et al., 2000b)

where e a is the direction vector of the a-flowpath trajectory, Ca is the mag-
nitude of the pressure gradient along e a, K a are intrinsic permeabilities, and
0 is a function of e a and K a. The solution of the flow equation above is of
the form (, a = C, a(\s\), where the metric \s\ = £ a is the distance along the
a-flowpath.
EXAMPLE 2.26: Consider the R 2 x T space/time continuum considered in
Example 2.19. The question is: How can we determine a suitable spatiotem-
poral metric? According to the preceding discussion, the physical knowledge
available can be very helpful in this respect. Let us suppose that the natural
2
field X(si, s 2, t) is governed in R x T by the flux-conservative law

where v = (vi, v%) is an empirical velocity to be determined from the data.
The physical law (Eq. 2.38) puts certain restrictions on the geometrical features