Page 49 - Modern Spatiotemporal Geostatistics

30 Modern Spatiotemporal Geostatistics — Chapter 2

Table 2.2. Some Euclidean and non-Euclidean geometries
in two-dimensional space.
Geometry Euclidean Riemannian Bolyai-Lobachevskian

Surface Plane Sphere Saddle

Parallels 1 0 Many

Curvature 0 >0 <0

Angular sum
for triangles 180° >180° <180°

Ratio of
circumference/diameter 7T <7T >7T
of circle

1983). The geometry of Riemann is exemplified by a spherical surface and that
of Bolyai-Lobachevski by a saddle surface.
Implicit in a spatiotemporal geometry are certain hypotheses concerning
the way space and time operate. The goal of modern geostatistics is to find
what kind of spatiotemporal structure we can choose on £. Now, in other
words, given that the set of events £ forms the basic set in space/time, the
problem is to equip modern spatiotemporal geostatistics with a mathematical
structure that captures the significant physical relations. This important issue
is addressed by the following postulate.

POSTULATE 2.3: Since a set of physical relationships between events is
associated with a set of geometrical relationships between points in 'E,
a spatiotemporal structure is imposed on £ by means of these physical
relationships.
Starting from the assumption that relationships between events express
physical knowledge 9£ (laws of nature, scientific theories, empirical correlations,
etc.; see Chapter 3) and that relationships between points are geometrical, the
implication of Postulate 2.3 is that the geometrical structure we decide to use
on space/time £ has a strong influence on which kinds of knowledge ^C we
can consider. Therefore, while all of the geometries are on an equal footing
from a logical point of view, they are not on an equal footing epistemically.
Another important factor in the choice of a geometry on the space/time
continuum £ is the concept of £ visualized intrinsically (or internally) vs. £
visualized extrinsically (or externally). To visualize £ internally is to imagine
the kinds of experiences we would have if we were living in such a space/time.