Page 63 - Modern Spatiotemporal Geostatistics

44 Modern Spatiotemporal Geostatistics — Chapter 2

Figure 2.11. Distance in Figure 2.12. Distance in
the sense of Equation 2.13. the sense of Equation 2.15.

PI to PI in a city. Therefore, this distance (Eq. 2.13) may be more appro-
priate for processes that actually occur on a discrete grid or network of some
sort (however, this may not be true for the simulation of continuous processes
by means of numerical grids, since in this case the grid is only a convenient
modeling device and does not change the spatiotemporal metric).
Yet another distance \ds\ is defined by

In the case of Figure 2.11, Equation 2.14 gives \ds\ = |PiM|. When dealing
with distances between two geographical points PI and P% on the surface of
the Earth (consider a sphere with center O and radius r, as shown in Fig. 2.12),
the arc distance \ds\ is defined as the length of the smaller arc of the great
circle joining these two points, i.e.,

where dip is the radian measure of the angle PiOP 2.

Finally, Euclidean metrics are not usually the most suitable measures of distance
in fractal spaces (e.g., processes taking place within porous media are more
adequately represented by fractal rather than Euclidean geometry). There is
not a general expression of the metric in fractal spaces, which rather depends
on the physics of the situation (Christakos et al., 2000a). In fact, formulating
explicit metric expressions (such as Eq. 2.12) is not always possible in fractal
spaces, since the physical laws may not be available in the form of differential
equations. Geometric patterns in fractal spaces are self-similar (or statistically
self-similar in the case of random fractals) over a range of scales (Mandelbrot,
1982; Feder, 1988); self-similarity implies that fractional (fractal) exponents
characterize the scale dependence of geometric properties. A common example
is the percolation fractal generated by random occupation of sites or bonds on
a discrete lattice. Length and distance measures on a percolation cluster,
denoted by f(r), scale as power laws with the Euclidean (linear) size of the