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Modern Spatiotemporal Geostatistics
By GEORGE CHRISTAKOS
Dover Publications, Inc.Copyright © 2000 Oxford University Press, Inc.
All rights reserved.
SPATIOTEMPORAL MAPPING IN NATURAL SCIENCES
"Science is built of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house." H. Poincaré
The urge to map a natural pattern, an evolutionary process, a biological landscape, a set of objects, a series of events, etc., is basic in every scientific domain. Indeed, the mapping concept is deeply rooted in the human desire for spatiotemporal understanding: What are the specific distributions of proteins in cells? What are the locations of atoms in biological molecules? What is the distribution of potentially harmful contaminant concentrations in the sub-surface? What are the genetic distances of human populations throughout a continent? What are the prevailing weather patterns over a region? How large is the ozone hole? How many light-years do galaxies cover? Answers to all these questions—extending from the atomic to the cosmic—are ultimately provided by means of good, science-based spatiotemporal maps.
Furthermore, studies in the cognitive sciences have shown that maps are particularly suitable for the human faculty of perception, both psychological and neurological (Anderson, 1985; Gregory, 1990). These faculties can most efficiently recognize characteristic elements of information when it is contained in a map that helps us build visual pictures of the world. Every scientific discipline depends fundamentally on the faculty of perception in order to interpret a process, derive new insights, conceptualize and integrate the unknown.
What exactly is a spatiotemporal map? The answer to this question depends upon one's point of view, which is, in turn, based on one's scientific background and practical needs. From a geographer's point of view, a map is the visual representation of information regarding the distribution of a topographic variable in the spatiotemporal domain (e.g., ozone distribution, radon concentration, sulfate deposition, disease rate). From an image analyst's perspective, a map is the reconstruction of some field configuration within a confined 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 detailed spatiotemporal maps, such as those shown in Figure 1.1. Interpreted with judgment (i.e., keeping in mind the underlying physical mechanisms, assumptions, 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 parameters, 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).
(iii.) Measuring the travel times of earthquake waves and using a seismic tomography technique, the Earth's core and mantle are mapped in Figure 1.2 (Hall, 1992). As is shown in this figure, the cutaway map that covers the area of the mid-Atlantic ridge is completely different than the cutaway map that covers the area around the East Pacific Rise.
(iv.) Finally, the two-dimensional porous medium map plotted in Figure 1.3 consists of oil-phase isopressure contours for an anisotropic intrinsic permeability field. This map represents the solution of a set of partial differential equations and constitutive relations modeling two-phase (water–oil) flow in the porous medium (Christakos et al., 2000b).
The salient point of our discussion so far is properly expressed by the following postulate. (Postulates presented throughout the book should not be considered as self-evident truths, but rather as possible truths, worth exploring for their profusion of logical consequences. Indeed, a proposed postulate will be adopted only if its consequences are rich in new results and solutions to open questions.)
POSTULATE 1.1: In the natural sciences, a map is not merely a data-loaded artifact, but rather a visual representation of a scientific theory regarding the spatiotemporal distribution of a natural variable.
According to Postulate 1.1, a map is a representation of what we know (a theory) about reality, rather than a representation of reality itself. In view of this representation, scientific explanation and prediction are to some extent parallel processes: a cogent explanation of a specific map should involve demonstrating that it was predictable on the basis of the knowledge and evidential support available. Maps represent one of the most powerful tools by which we make sense of the world around us. In fact, once our minds are tuned to the concept of maps, our eyes find them everywhere.
Why is mapping indispensable to the natural sciences? If a convincing answer to this question is not offered by the discussion so far, the following examples can provide further assistance in answering the question by describing a wide range of important applications in which spatiotemporal mapping techniques play a vital role. The reality is that significant advances in various branches of science have made it possible to measure, model, and thus map a breathtaking range of spatiotemporal domains. Examples 1.2–1.5 below refer to the various uses of maps in agricultural, forestry, and environmental studies.
EXAMPLE 1.2: Thermometric maps (see Fig. 1.4) provide valuable information for a variety of atmospheric studies, agricultural activities, pollution control investigations, etc. (Bogaert and Christakos, 1997).
EXAMPLE 1.3: In forestry, ground inventory provides important information on biodiversity that cannot be obtained by remote sensing (Riemann Hershey, 1997). A ground inventory, however, is expensive and labor intensive, especially when it covers large areas. Mapping techniques provide the means for estimating unsurveyed areas using a limited number of sample points in space/time.
EXAMPLE 1.4: Assessment of environmental risk due to some pollutant often requires information regarding the pollutant's distribution on grids covering large spatial domains and multiple time instances (e.g., Bilonick, 1985). This information can be provided most adequately by means of mapping techniques, which on the basis of a limited number of existing measurements and mathematical modeling lead to estimates of the pollutant at other locations and time periods. Also, in studies relating health status to pollutant distribution, an air-quality sampling network usually consists of fewer points in space than are available for health data sets (Briggs and Elliott, 1995). Mapping techniques must then be employed in order to derive pollutant estimates in wider area units.
EXAMPLE 1.5: Using data from satellites orbiting the Earth, spatiotemporal maps of radioactivity in the atmosphere (Fig. 1.5) revealed unusually high energy emissions which made the detection of the nuclear incident at Chernobyl possible, prior to its official Soviet acknowledgment (Sadowski and Covington, 1987; Arlinghaus, 1996).
A map can offer more information than merely the distribution of the spatiotemporal variable it represents. The distribution of an air pollutant, e.g., may be used in combination with an exposure-response model to predict the pollutant's impacts on human health and the ecosystem.
EXAMPLE 1.6: Figure 1.6 shows a health damage indicator map (expected number of representative receptors affected/km2) expressing damage due to ozone exposure in the New York City–Philadelphia area on July 20, 1995; a sublinear exposure-response model was assumed (Christakos and Kolovos, 1999). Interpreted with judgment (i.e., keeping in mind the assumptions made concerning exposure, biological and health response parameters, cohort characteristics of the representative receptor, etc.), such maps may offer valuable insight about the possible distributions of population health damage due to pollutant exposure.
Mapping applications also are abundant in the chemical, nuclear, and petroleum engineering fields.
EXAMPLE 1.7: Maps representing a type of material (e.g., chemical element) and the amount (e.g., concentration) of the material on a surface as a function of time are becoming increasingly important for determining inhomogeneities on and in solids (Schwedt, 1997). Nuclear waste facilities are interested in maps showing the migrations and activities of materials encapsulated in concrete barrels (Louvar and Louvar, 1998). The oil industry produces series of geological maps based on reflection seismic data for exploration and development purposes, etc. (Doveton, 1986).
Many applications in which mapping plays a vital role can be found in medical sciences and in genetic engineering.
EXAMPLE 1.8: Simulated spatiotemporal cell fields representing human organs damaged by exposure to chemical agents and other pollutants are increasingly important in environmental health studies (Christakos and Hristopulos, 1998).
Spatial maps simulating cell distribution at different times are shown in Figure 1.7. Note the change in number of normal cells (black) vs. affected cells (white) in space/time (some of the affected cells are repaired in time). These maps take into consideration the spatial and temporal correlations between cells.
EXAMPLE 1.9: Genetic distances between populations can be mapped based on gene frequencies in human beings. The map in Figure 1.8 shows a systematic pattern that slopes from the Near East and southeastern Europe to the northeastern portions of the European continent. The more dissimilar the shade, 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 hunting/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 represent, 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 detailed 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 K available, including data sets, physical models, scientific theories, empirical functions, uncertain observations, justified beliefs, and expertise with the specified natural phenomenon.
2. The estimator [??] which denotes the mathematical formulation used to approximate the actual (but unknown) natural variable X.
3. The estimates [??] of the actual values χ generated from the estimator [??], 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 explain 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 applications (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 approximations 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, constraints are often imposed on the form of the estimator, extrapolation is not reliable beyond the range of the data, computational problems emerge in the practical implementation of some methods, etc.; see, also, discussions in the following sections). In this book an attempt is made to develop a group of what might be called modern spatiotemporal geostatistics concepts and methods. The goal of modern spatiotemporal geostatistics is to remove some limitations of the older methods, thus providing a significant improvement in the field of scientific mapping. As is described in the following postulate, the book's approach to spatiotemporal modeling will be that of the physical scientist.
POSTULATE 1.2: Modern spatiotemporal geostatistics is concerned with stochastic analysis that functions at both the ontological level (i.e., building models for natural systems) and the epistemic level ([i.e., using what is known about the systems and how knowledge is integrated and processed from a variety of scientific disciplines), rather than with pure inductive procedures based on linear relationships between data and hypotheses using physical theory-free techniques.
This fundamental postulate—which will be discussed in more detail in the following section—is made necessary for the additional reason that, despite calls for closer interaction between scientific fields, some disciplines were never more narrow-minded and agoraphobic than today.
Some researchers may sound rather pessimistic when they argue that "we are slipping into a new form of darkness: one where it is popular, profitable, and politically expedient to suppress science" (Milloy and Gough, 1998, p. vi), but the phenomenon of people withdrawing in fear from novelty is not uncommon—especially when the new tools originate in disciplines other than their own. The current rigid system often discourages the vitally important cross-fertilization of concepts, models, and techniques between disciplines.
COMMENT 1.1: The term "epistemic" in Postulate 1.2 above signifies the scientific study of knowledge, as opposed to the philosophical theory of knowledge, which is known as epistemology (Bullock et al., 1977). A more extended definition of epistemic would include "constructing formal models of the processes by which knowledge and understanding are achieved, communicated, and integrated within the framework of scientific reasoning."
The Epistemic Status of Modern Spatiotemporal Geostatistics: It Pays to Theorize!
What is the main distinction between the modern geostatistics ideas advocated in this book and classical geostatistics? To answer this question, one must look into the different scientific reasoning frameworks underlying these two fields.
Classical geostatistics was designed to fit into a pure inductive framework. This framework basically involves the following stages: (i.) piling up experimental data, (ii.) fitting mathematical functions to these data, and (iii.) piling up more experimental data and tests.
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