Page 59 - Modern Spatiotemporal Geostatistics

40 Modern Spatiotemporal Geostatistics — Chapter 2

where MI, w, 2 is a pair of real-valued parameters that serve as registration
numbers or identification marks for the curves of the local mesh under
consideration, and dui, du- 2 vary between 0 and 1. Within this mesh the
corresponding distances are

where Cn and {22 are parameters obtained from physical measurements.

The parameters £u and £22 do not change as long as we stay within
the specific mesh, although they may change when we move from one mesh to
another (see also the following section, "Metrical Structure"). The coordinates
of any point on the surface covered by the Gaussian system (Fig. 2.8) are
then known precisely if we know the values of £n and £22 for every mesh.
These values are obtained from physical measurements carried out by always
remaining on the surface and never going outside of it.
EXAMPLE 2.9: As we saw above, Gaussian coordinates are useful when it is
desirable to establish a coordinate system without leaving the surface under
study, i.e., internally. Consider a sphere, say, the Earth. If we think of the
space inside and outside the Earth, we have a three-dimensional continuum with
straight-line geodesies (the shortest distances between points) and we can use
Euclidean geometry: a Cartesian coordinate system with origin at the Earth's
center and axes along three mutually perpendicular diameters; and we have to
refer to external points, lines, and planes. This would be very inconvenient,
since it would necessitate taking measurements below the Earth's surface near
its hot center, flying out into the atmosphere, etc. Things could be simplified
considerably by having our coordinate system right on the surface of the Earth.
In this case, straight lines will be replaced by arcs, for these are the geodesies.
A triangle will consist of three intersecting arcs, and the sum of its angles
will be greater than 180°. Longitude and latitude and all geographical facts
related to them are defined in terms of the Earth's surface, i.e., they are all
internal aspects of the surface. Therefore, on the surface of the Earth we have
a two-dimensional continuum with a non-Euclidean geometry of the Gaussian
type.
The preceding analysis reveals an important aspect regarding the choice of
a coordinate system to describe natural phenomena. This aspect concerns the
distinction between extrinsic vs. intrinsic coordinate systems (which is related
to a similar distinction between visualization viewpoints—see Postulate 2.4)
and is summarized in the following postulate.
POSTULATE 2.5: In certain situations understanding may be enhanced
if the conception of points extrinsic (external) to physical space can be
avoided and, instead, the analysis is performed in terms of intrinsic
(internal) properties of the physical space.