Page 66 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 47

EXAMPLE 2.14: In the case of a two-dimensional curved physical surface (e.g.,
the surface of a hill), the Riemannian distance (Eq. 2.16) reduces to the Gaus-
sian distance \OP\ of Figure 2.9; i.e., for n = 2, Equation 2.16 gives

where, for notational generality, the u\ and U2 are replaced by si and s-^.,
respectively; the values of gn, 5221 and 512 = 521 are found from actual
measurements (P\M is the projection of P\P on OQ). The internal geometry
of the surface is determined completely if the values of q\\, g^, and 512
are known for every local mesh (recall that the g's are generally functions
of the spatial location coordinates Si, i = 1, 2). Suppose that si = 9 and
= 7T/2 — (p represent the longitude and latitude of a point on a spherical
s 2
2
surface (Fig. 2.12) whose radius r is one unit of length; then, g\\ = cos 52,
512 = 521 = 0, and 522 = 1.

If the surface is a hyperbolic paraboloid, one can choose Gaussian coordinates
such that 5n = 1 + 4s?, gi 2 = # 2i = -4«i s 2< and g 22 = 1 + 4s|.
A surface of constant curvature a is determined by the following metric coeffi-
cients 5ii = 522 = [l+0.25a(sf + S2)]~ 2 an ^ 9i2 = <?2i = 0. The metric coef-
2
2
2
2
2
2
2
2
2
2
2
2
ficients 511 = fc (fc + s )(fc + s + s )- , 922 = fc (fc + 5 )(A; + s + si)-
2
and #12 = 521 — —siS2(^ 2 + s + s%)~ 2 have been used to characterize a
2
certain surface of curvature k~ . Note that as k tends to infinity, this metric
converges to the Pythagorean metric.
The tensor g = (gij) is called the metric tensor. Although from the
viewpoint of differential geometry the metric tensor gives infinitesimal length
elements, the mathematical form of Equation 2.16 may be used to define fi-
nite distances as well (see, e.g., the section entitled "Correlation analysis and
spatiotemporal geometry" on p. 61). The choice of g should satisfy certain
physical and mathematical conditions. A possible set of mathematical condi-
tions is described in the comment below.
C O M M E N T 2.4: Th e metric tensor g should be: (i.) of differentiability class

2
C (all second-order derivatives of g^ exist and are continuous); (ii.) sym-

metric (g^ = Qji); (in.) nonsingular (\Qij\ ^ Oj ; an d (iv.) such that the

Riemannian distance \ds\ is invariant with respect to a change of coordi-

nates.
EXAMPLE 2.15: Consider the transformation from a special {sj} coordinate
system to the rectangular {?»} coordinate system defined by ~si = si and
^2 = si + In s 2. The Jacobian is