Page 69 - Modern Spatiotemporal Geostatistics

50 Modern Spatiotemporal Geostatistics — Chapter 2

and

where g*i are elements of the inverse of the matrix g = (g^), and g = \g\.
EXAMPLE 2.18: Consider the special case of orthogonal curvilinear coordinates.
Then, fly = 0 (i ^ j), = I (i = j); 5« = 0 (i ^ j), = gf (i = j);
g = diag(ga) and g = Yl^gu- The form of Equation 2.21 remains the same,
but Equations 2.22 and 2.23 are now written as follows

and

where

Composite metrical structures
In the case of composite metrical structure, a higher level of physical under-
standing of space/time is assumed which may involve theoretical and empirical
facts of the natural sciences. According to this approach, the basis of metri-
cal determination should be sought outside the abstract geometric objects in
the physical processes that act on them. The composite metrical structure
approach is described by the following definition.
DEFINITION 2.8: In the composite metrical structure, space and time
parameters are connected by means of an analytical expression, i.e..

where g is a function determined from the knowledge available (topog-
raphy, physical laws, etc.).

EXAMPLE 2.19: Consider a point P in the space/time continuum R 2 x T with
coordinates p = (si, 82, t), as in Figure 2.14. A natural variable varying within
this continuum is written as X(p) = X(s\, s%, t}.
If the separate metrical structure were used, the distance | OP would be de-
fined indirectly in terms of two independent entities—space and time—forming
(
the pair \ s \ , t), where the distance s\ may have one of the spatial forms dis-
cussed in Examples 2.10 or 2.13 above. If, however, the composite metrical