Page 61 - Modern Spatiotemporal Geostatistics

42 Modern Spatiotemporal Geostatistics — Chapter 2

Figure 2.10. An illustration of the geodesies coordinate system.

(e.g., toroidal coordinates are such that the equation of a magnetic-field line
is that of a straight line in these coordinates). This section concludes with a
postulate that summarizes an important conclusion from our examination of
spatiotemporal analysis thus far and foreshadows results to be obtained in the
following sections.
POSTULATE 2.6: The geostatistician should choose a coordinate sys-
tem (Euclidean or non-Euclidean) that provides a physically meaningful
representation of the situation and works as simply as possible.
In other words, in many geostatistical applications, the physical interpre-
tation of the subject matter (i.e., "In what physical situations do we use which
system of coordinates?") may be equally as important as the computational
convenience resulting from a particular choice of the coordinate system.

Metrical Structure

Central among the quantitative features of a spatiotemporal geometry is its
metrical structure, that is, a set of mathematical expressions that define dis-
tances. Metrical properties (or distance relations) must be "added" once we
have a set of points in space/time; the same set of points may be compatible
with many metrical properties.
DEFINITION 2.5: A spatiotemporal metric is a function defined for a
coordinate system such that the spatiotemporal distance between any
two points in that system is determined from their coordinates.

Distance expressions cannot always be defined unambiguously. The ex-
pression for the metric of any continuum is dependent on two entirely different
factors: (i.) a "relative" factor—the particular coordinate system; and (ii.) an
"absolute" factor—the nature of the continuum itself (whether it forms a
plane, a sphere, or an ellipsoid; the physical laws governing the natural vari-
ables occurring within the continuum, etc.).
In modern spatiotemporal geostatistics, it is usually convenient to con-
sider two prime metrical structures: one is the separate metrical structure and
the other is the composite metrical structure. Both metrical structures are
discussed below in considerable detail.