Page 83 - Modern Spatiotemporal Geostatistics

64 Modern Spatiotemporal Geostatistics — Chapter 2

C O M M E N T 2.12: Th e empirical parameter vi n Example 2.28 relates space

and time intrinsically and prevents the dissolution of the spatiotemporal

structure into independent spatial and temporal components. In this case,

the spatial distance and the time interval are replaced b y the so-called proper
2
2
2
2
time interval between two space/time points defined a s drj = r + h v~ (a

proper spatial interval may be defined in a similar fashion).

In light of the preceding analysis, the choice of a spatiotemporal geometry
in geostatistical applications must avoid discrepancies between the "natural"
geometry—as revealed by the physical equations and data—and the appropriate
mathematical geometry. Spatial arrangements in the domain under consider-
ation should be combined with the temporal order of events in a way that
reflects relationships determined by physical knowledge.
Permissibility criteria and spatiotemporal geometry
The choice of a geometry may have significant consequences in geostatisti-
cal analysis. One such important consequence is related to the covariance
n
permissibility criteria in R x T (i.e., the criteria for a function to be a covari-
ance model); similar consequences are valid for other spatiotemporal correlation
functions, including the variogram and the generalized covariance models. In
fact, the validity of the following postulate will be demonstrated in this section.
POSTULATE 2.10: The permissibility criteria for spatiotemporal corre-
lation models depend on the geometry assumed.
Mathematically, a necessary and sufficient condition for a function to be a
permissible covariance model is that it be nonnegative definite. Furthermore,
Bochner's theorem shows that for a function c x(h,r) to be nonnegative def-
inite it is necessary and sufficient that its spectral density C X(K,UJ) be a real-
valued, integrable, and nonnegative function of the spatial frequency K and
the temporal frequency u (for details, see Christakos and Hristopulos, 1998;
Gneiting, 1999). Postulate 2.10 implies that the formulation of Bochner's the-
orem depends on the coordinate system and the geometric metric used and,
therefore, a model that is a permissible covariance for a specific spatiotemporal
geometry may not be so for another geometry. As we will see below, this is
indeed the case with certain commonly used models.
The Gaussian function is a permissible covariance model for the Euclidean
distance (Eq. 2.12). However, as is demonstrated in the following proposition,
the same model is not permissible when a non-Euclidean distance is assumed.

PROPOSITION 2.1: The Gaussian function in R 2 x T

where the spatial distance is defined as in Equation 2.13, i.e.

is not a permissible covariance.