Page 57 - Modern Spatiotemporal Geostatistics

38 Modern Spatiotemporal Geostatistics — Chapter 2

A summary of the main features of the orthogonal curvilinear coordinate sys-
tems is provided later in this chapter in Table 2.3 (p. 48)

COMMENT 2.3: Th e spherical coordinates (Eq. 2.8) are similar to the coor-

dinates used in geographical studies (e.g. , Langran, 1992; Bonham-Carter,

1994)- In geographical coordinates, \9\ denotes the longitude (meridian an-
gle) and is called east or west longitude according to whether 6 is positive

or negative. \ ?r/2 — (f>\ denotes th e latitude (equatorial angle) and i s called

north or south latitude according to whether n/2 — (p is positive or negative.

The coordinate system used in a physical application can have important
consequences. As shown later in Example 2.32 (p. 67), geostatistical analy-
sis in terms of spherical vs. Cartesian coordinates can lead to very different
variographies and spatial maps.
Non-Euclidean coordinate systems
Euclidean geometry and the associated coordinate systems are not appropri-
ate for several types of physical space. As emphasized in Postulate 2.4 and
Example 2.4, when we seek an internal visualization of arbitrary curved sur-
faces, the geometry is generally non-Euclidean and local descriptions of the
domain may be considered (this is the case, e.g., with spherical and saddle
surfaces). The two-fold requirement (i.e., local and internal characterization
of the space/time domain) has led to the development of non-Euclidean coor-
dinate systems which generally are not tied to rectangular coordinates. Among
the most well-known non-Euclidean coordinate systems are the Gaussian sys-
tem and its generalization, the Riemannian coordinate system. We will study
these two systems next.
The Gaussian coordinate system was developed for the internal visualiza-
tion of two-dimensional surfaces (i.e., so that we can derive the coordinates
of any point on the surface by means of measurements carried out without
leaving the surface and moving into a third dimension into which the surface is
embedded). In the Gaussian coordinate system, the network of parallel lines of
the Euclidean plane is replaced by an arbitrary dense network of ordered curves
(Fig. 2.8).
Fixing the value of one of the variables, u\ or u^, produces a curve on
the surface in terms of the other variable, which remains free. In this way,
a parametric network of two one-parameter families of curves on the surface
is created, so that just one curve of each family passes through each point