Page 74 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 55

and the particular experimental procedure that we decide to use in order to
obtain measurements of the natural field.
EXAMPLE 2.21: One cannot decide by means of purely geometrical notions
how the distance di 2 between the points p l and p 2 at a 3-km spatial lag
and a 2-day time lag in a porous medium compares with the distance d\y
between the points pj and p 2 at a 2-km spatial lag and a 3-day time lag.
Which spatiotemporal distance is larger can be decided only by means of a
physical process as follows: An experiment is performed during which a tracer
is released at p 1. If the tracer is detected at p 2 but not at p 2, then di 2 < A\y
with respect to the particular medium and experimental setup. Also, measuring
distance by means of fluid-tracer dispersion can lead to very different results
than measuring it by means of electromagnetic propagation.
EXAMPLE 2.22: Consider two geostatisticians A and B who both use the
same Euclidean system, say, rectangular coordinates. As a result of physical
considerations, the two geostatisticians use different metrical systems. Geo-
statistician A employs the usual linear Euclidean metric system s itA = r)i,AU
(with a unit distance u), and geostatistician B uses the nonlinear metric system
s itB = exp^s] w (with a unit distance v); rjj^ and 77^5 are the number of
distance units for A and B, respectively. Since both systems are Euclidean,
the spatial coordinates of a specified point P in R 3 provided by the two geo-
statisticians are related as follows

where EJ are real-valued coefficients. By adjusting the unit distances of the
two geostatisticians along the si axis so that SI,B = Si tA, the remaining
coordinates are altered in the same ratio; then, Equation 2.30 yields

In other words, since the two geostatisticians use different metric systems,
there is no way to adjust their measurements for all three coordinates. For this
reason, the two geostatisticians obtain very different representations of objects
=
s
and processes in space/time. The sphere J^ILi f A P* °f the first geostatis-
tician, e.g., is for the second geostatitician an ellipsoid
the angles measured by the two geostatisticians are different; etc.
The situation presented in the following example has interesting applica-
tions in the domain of relativistic geostatistics and has already been mentioned
in Example 2.20.
EXAMPLE 2.23: In space/time, distances are physically measured most effi-
ciently using light pulses. Let a flash of light be emitted from point PI at time
t and reach point P 2 at time t + dt. The spatial distance \ds\ between the two
points is given by

where c is the speed of light are the orthogonal projections