Page 77 - Modern Spatiotemporal Geostatistics

58 Modern Spatiotemporal Geostatistics — Chapter 2

of the space/time domain. Indeed, by considering a coordinate transformation
from the rectangular Euclidean system (s^) to the system of coordinates defined
by ~Si = Si — Vit, the solution of Equation 2.38 has the following form

i.e., it depends on the space/time vector p = s-vt. This dependence implies
that in the rectangular coordinate system, a natural field with the physics
described by Equation 2.39 may be assigned a metric of the form of Equation
2.27 above, where n = 2, 500 = (v\ + v$), g n = £22 = 1, 9w = 9oi = -2vi,
520 = 902 = -2«2, and 512 = 521 = 5oi = 9w = 902 = 9w = 0 ; i.e., we have

Equation 2.40 demonstrates how the physical law determines the geometrical
T
metric through the empirical vector v = [vi, V2] . Also, a relation between
the mathematical geometry and the physical parameters can be established by
solving the law with respect to v, which gives

Equation 2.41 expresses the empirical vector v in terms of the natural field
X(si, 82, £)-values in space/time. Also, later in this chapter (see "Corre-
lation analysis and spatiotemporal geometry," p. 61) we will discuss how the
covariance functions can be instructive in determining the appropriate geometry
in a spatiotemporal continuum.

COMMENT 2.8 : Th e following notational remark i s important fo r future

reference, as well. In vector calculus, the vector x = (xi,..., x n) is consid-

ered as a column vector

when matrix or vector multiplications ar e involved (see, e.g., Marsden an d

Tromba, 1988).

The Complementarity Idea
In matters of scientific investigation, we need to consider alternatives where
facts are unknown. In view of uncertainty and imperfect knowledge, actuality
is, indeed, surrounded by an infinite realm of possibilities. The spatiotemporal
distributions of most natural variables are not sharply defined but, instead,
they have an uncertain or indeterminate structure. A theoretical as well as a
practical need to account for this uncertainty gives rise to the complementarity
idea in the following postulate.