Page 21 - Modern Spatiotemporal Geostatistics

2 Modern Spatiotemporal Geostatistics — Chapter 1

the visual representation of information regarding the distribution of a topo-
graphic variable in the spatiotemporal domain (e.g., ozone distribution, radon
concentration, sulfate deposition, disease rate). From an image analyst's per-
spective, a map is the reconstruction of some field configuration within a con-
fined region of space/time. From a physical modeler's standpoint, a map is the
output of a mathematical model which represents a natural phenomenon and
uses observations, boundary/initial conditions, and other kinds of knowledge
as input. While the viewpoints of the geographer and the image analyst are
more descriptive, that of the physical modeler is more explanatory. Therefore,
a variety of scenarios is possible regarding the way a physical map is produced
and the meaning that can be assigned to it:
(i.) The map could be the outcome of statistical data analysis based on a
set of observations in space/time.
(ii.) It could represent the solution of a mathematical equation modeling a
physical law, such as a partial differential equation (pde) given some
boundary/initial conditions.
(iii.) It could be the result of a technique converting physical measurements
into images.
(iv.) It could be a combination of the above possibilities.
(v.) Or, the map could be any other kind of visual representation documenting
a state of knowledge or a sense of aesthetics.
The following example illustrates some of the possible scenarios described
above.
EXAMPLE 1.1: (i.) Studies of ozone distribution over the eastern United
States that used data-analysis techniques include Lefohn et al. (1987), Casado
et al. (1994), and Christakos and Vyas (1998). These studies produced de-
tailed spatiotemporal maps, such as those shown in Figure 1.1. Interpreted
with judgment (i.e., keeping in mind the underlying physical mechanisms, as-
sumptions, and correlation models), these maps identify spatial variations and
temporal trends in ozone concentrations and can play an important role in the
planning and implementation of policies that aim to regulate the exceedances
of health and environmental standards. The use of data-analysis techniques is
made necessary by the complex environment characterizing certain space/time
processes at various scale levels (highly variable climatic and atmospheric pa-
rameters, multiple emission sources, large areas, etc.).
(ii.) While in these multilevel situations most conventional ozone distribution
models cannot be formulated and solved accurately and efficiently, in some
other, smaller scale applications, air-quality surfaces have been computed using
pde modeling techniques. In particular, the inputs to the relevant air-quality
models are data about emission levels or sources, and the outputs (ozone maps)
represent numerical solutions of these models (e.g., Yamartino et al., 1992;
Harley et al., 1992; Eerens et al, 1993).