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Spatiotemporal Mapping in Natural Sciences 19

COMMENT 1.3: Most studies of the scientific method (e.g. , Lastrucci, 1967;
Kosso, 1998) show that the most ardent "believer in data" eventually real-

izes that these data are valid and meaningful because they rest upon some

theory of significance and credibility, even though he/she may not have rea-

soned through the surface of the data to their origin in the theoretical notion

which supports them. I n th e words o f Wang (1993, p. 168): ".. . a safe be t

is that, without a good theory, even if the data are thrown in the face of the

'data analyst,' the chance that he will find anything significant is next to
zero."
Spatiotemporal geometry
Spatiotemporal structures differ, as do the geometries which describe them.
Euclidean formulas, e.g., are appropriate for geometric figures on flat surfaces,
but generally are inappropriate for describing intrinsically geometric relations
on curved surfaces. Several other examples will be discussed in Chapter 2.
Classical geostatistics routinely employs the Cartesian coordinate system
equipped with a Euclidean metric (distance), and considers time merely as an
additional dimension. Within such a framework, space and time have an exis-
tence all on their own, independent of the existence of physical processes and
objects. In many applications, however, Cartesian geometry is not a realistic
model of the physical situation, which requires a higher level of understanding
of the Spatiotemporal domain. In many cases, this understanding involves non-
Cartesian coordinate systems and non-Euclidean metrics. Also, the concepts of
absolute space and time—independent of the underlying physical processes—
usually do not exist in the real world. Applications discussed in Schafer-Neth
and Stattegger (1998) as well as in Christakos et al. (2000a) have shown that
the mapping techniques of classical geostatistics—which rely on Cartesian co-
ordinate systems or use a physically inappropriate Euclidean metric—can lead
to inaccurate maps which imply false conclusions (see Chapter 2, Examples
2.31 and 2.32, p. 67). The inadequacy of classical geostatistics to handle these
situations requires the development of new methods, as is suggested by the
following postulate.

POSTULATE 1.4: Modern Spatiotemporal geostatistics recognizes that
Spatiotemporal geometry is not a purely mathematical affair and relies
on physical knowledge in order to decide which mathematical geometry
best describes reality.

Postulate 1.4 essentially expresses the view that the Spatiotemporal ge-
ometry of modern geostatistical applications is intimately connected with the
laws of the physical domain and cannot exist independent of them. As a result,
in practical applications it is necessary to investigate whether the traditional
geometrical structures need to be replaced by a physically more meaningful spa-
tiotemporal geometry. The appropriate coordinate system should allow, e.g.,
representations of the Spatiotemporal geometry on the basis of the underlying