Page 29 - Modern Spatiotemporal Geostatistics

10 Modern Spatiotemporal Geostatistics — Chapter 1

the more dissimilar the genetic composition of the populations involved. The
contours of the map closely match those of a map developed independently by
archeologists studying the spread of agriculture from the Near East to hunt-
ing/gathering tribes of Europe. Thus, the conclusion that early farmers spread
their genes (by intermarriage with local inhabitants), as well as their grains and
agricultural know-how, is justified (Menozzi et al., 1978; Wallace, 1992).
Correlations of gene-frequency maps with health parameters at the geographic
level have been instrumental in the discovery of specific genetic adaptations.
A map of the sickle-cell anemia gene, e.g., showed a correlation with that
of malaria, leading to the hypothesis that this gene may confer resistance to
malaria. This hypothesis was later confirmed by more direct tests (e.g., Cavalli-
Sforza et al., 1994).

The maps discussed above, even when they do not offer the ultimate
answer to fundamental questions related to the natural phenomena they rep-
resent, certainly suggest where answers should be sought. In this section we
have selected a representative sample of maps that cover some of science's
most fascinating frontiers. This selection is, though, by no means complete.
Many other important mapping applications have been omitted (for more de-
tailed accounts of various mapping projects—past, present and future—the
interested reader is referred to, e.g., Bagrow, 1985, and Hall, 1992).
Having demonstrated the importance of spatiotemporal maps in every
branch of science, we can now address the next question: What constitutes a
spatiotemporal mapping approach? Formally, a mapping approach consists of
three main components:
1. The physical knowledge ^available, including data sets, physical mod-
els, scientific theories, empirical functions, uncertain observations, justified
beliefs, and expertise with the specified natural phenomenon.
2. The estimator X which denotes the mathematical formulation used to
approximate the actual (but unknown) natural variable X.
3. The estimates x of the actual values x generated from the estimator
X, usually on a regular grid in space/time. These grid values constitute a
spatiotemporal map.
There are various methods that can be used to construct accurate maps
in space/time. Among other things, a useful mapping approach should ex-
plain when and how one can cope rationally with the uncertainty of natural
variables. For many years, because of their versatility, classical geostatistics
methods emerged as the methods of choice for many spatial estimation appli-
cations (Matheron, 1965; Journel, 1989; Cressie, 1991; Kitanidis, 1997; Olea,
1999). However, when it comes to scientific mapping (i.e., mapping that
proceeds on the basis of scientific principles and laws), these methods suffer
from certain well-documented limitations (restrictive assumptions and approx-
imations are often used to compensate for the absence of a sound theoretical
basis, a rigorous approach is lacking for incorporating important knowledge
sources, physically inadequate space/time geometries are sometimes assumed,