Page 65 - Modern Spatiotemporal Geostatistics

46 Modern Spatiotemporal Geostatistics — Chapter 2

geometry is the existence of coordinates satisfying the Pythagorean formula.
The curvilinear coordinate systems presented above (polar, cylindrical, and
spherical) are associated with the Euclidean metric in Equation 2.12—since
they are connected via Equation 2.4 with rectangular coordinates and Eu-
clidean space. However, these same systems can be formally adopted in a
non-Euclidean space if there are physical and/or mathematical reasons for do-
ing so. This implies that the spatiotemporal metric and the coordinate system
used to describe that metric are independent of each other (e.g., we may have
a Euclidean coordinate system with a non-Euclidean metric).

EXAMPLE 2.12: On the Euclidean plane one can always transform any coor-
dinate system into coordinates that satisfy the Pythagorean metric (Eq. 2.12),
e.g., by transforming polar coordinates into Cartesian coordinates. By con-
trast, on a curved non-Euclidean surface it is not possible to perform such a
transformation, since Cartesian coordinates do not exist.
Several of the spatial distances—Euclidean and non-Euclidean—can be
summarized in terms of the following definition.

DEFINITION 2.7: The Riemannian distance is defined as

where g iy are coefficients that generally are dependent on the spatial
location.

Equation 2.16 may be viewed as Riemann's generalization of Pythagoras'
metric (Eq. 2.12)—although one may wonder if Pythagoras would have rec-
ognized this. A Riemannian coordinate system (Definition 2.4) together with
a metric (Definition 2.7) determine a Riemannian space. The following two
examples present some Euclidean and non-Euclidean metrics as special cases
of Equation 2.16.

EXAMPLE 2.13: The Euclidean metric in a rectangular coordinate system—
Equation 2.12—is a special case of Equation 2.16 for gu = 1 and g tj = 0
(i ^ j}. In a polar coordinate system, the Euclidean metric is obtained from
Equation 2.16 for n = 2 and g\\ = 1, e/ 22 = sf, <?„ = 0 (i ^ j). For n = 3
and gu = #33 = 1, 522 = s\, gy = 0 (i ^ j), Equation 2.16 provides th
metric in a cylindrical system. In a spherical coordinate system, the Euclidean
metric is obtained from Equation 2.16 for n — 3 and gu — 1, 522 = s\,
2
933 = [si sin(s 2)] , 9a = 0 (i ± j).
As is demonstrated in the following example, as far as the Gaussian or
Riemannian geometries are concerned, the physical surface leaves its stamp on
the metric.