Modern Spatiotemporal Geostatistics

By George Christakos

This scholarly introductory treatment explores the fundamentals of modern geostatistics, viewing them as the product of the advancement of the epistemic status of stochastic data analysis. The book's main focus is the Bayesian maximum entropy approach for studying spatiotemporal distributions of natural variables, an approach that offers readers a deeper understanding of the role of geostatistics in improved mathematical models of scientific mapping.
Starting with a overview of the uses of spatiotemporal mapping in the natural sciences, the text explores spatiotemporal geometry, the epistemic paradigm, the mathematical formulation of the Bayesian maximum entropy method, and analytical expressions of the posterior operator. Additional topics include uncertainty assessment, single- and multi-point analytical formulations, and popular methods. An innovative contribution to the field of space and time analysis, this volume offers many potential applications in epidemiology, geography, biology, and other fields.

• Language: English
• Publisher: Dover Publications
• Release date: Sep 26, 2013
• ISBN: 9780486310930

George Christakos

George Christakos is a Professor in the Department of Geography at San Diego State University (USA) and in the Institute of Island & Coastal Ecosystems, Ocean College at Zhejiang University (China). He is an expert in spatiotemporal random field modeling of natural systems, and his teaching and research focus on the integrative analysis of natural phenomena; spatiotemporal random field theory; uncertainty assessment; pollution monitoring and control; human exposure risk and environmental health; space-time statistics and geostatistics.

Book preview

Modern

Spatiotemporal

Geostatistics

Modern

Spatiotemporal

Geostatistics

George Christakos

Department of Geography

San Diego State University

San Diego, California

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright © 2000 by Oxford University Press, Inc.

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2012, is an unabridged republication of the work originally published under the auspices of the International Association for Mathematical Geology in 2000 by Oxford University Press, Inc., as Volume Six in the series, Studies in Mathematical Geology.

Library of Congress Cataloging-in-Publication Data

Christakos, George.

Modern spatiotemporal geostatistics / George Christakos.

              p. cm. — (Dover earth science)

Originally published: Oxford ; New York : Oxford University Press, 2000.

Includes bibliographical references and index.

eISBN-13: 978-0-486-31093-0

1. Earth sciences—Statistical methods. 2. Maximum entropy method. 3. Bayesian statistical decision theory. I. Title.

QE33.2.S82.C47 2012

550.72’7—dc23

2011053378

Manufactured in the United States by Courier Corporation

48818701

www.doverpublications.com

To my family and friends in space-time

PREFACE

If an old rule of thumb from the publishing industry is to be believed, every equation included in a book halves the number sold. Economist, Jan. 2–8, 1999

Stochastic characterization of spatial and temporal attributes began as a collection of mathematical concepts and methods developed originally (mostly in the 1930’s through 1950’s) by A.N. Kolmogorov, H. Wold, N. Wiener, A.M. Yaglom, K. Itô, I.M. Gel’fand, L.S. Gandin, B. Matern, P. Whittle, and others. G. Matheron coined the term geostatisticsto refer to these developments, brought them together, modified them in some cases, and then applied them systematically in the mining exploration context. Rapid commercial success allowed Matheron to establish the Fontainebleau Research Center in the late 1960’s outside Paris. Later, geostatistical techniques were used in other disciplines as well, including hydrogeology, petroleum engineering, and environmental sciences. Geostatistics was introduced in the 1970’s in Canada and the United States by geostatisticians including M. David, A.G. Journel, and R.A. Olea. References to selected publications of the above researchers may be found in the bibliography at the end of this book.

It is widely recognized that the techniques of classical geostatistics, which have been used for several decades, have reached their limit and the time has come for some alternative approaches to be given a chance. In fact, many researchers and practitioners feel that they may soon be faced with some kind of law of diminishing returns for geostatistics, inasmuch as the problems of the rapidly developing new scientific fields are becoming more complex, and seemingly fewer new geostatistical concepts and methods are available for their solution.

With these concerns in mind, this book is an introduction to the fundamentals of modern spatiotemporal geostatistics. Modern geostatistics is viewed in the book as a group of spatiotemporal concepts and methods which are the products of the advancement of the epistemic status of stochastic data analysis. The latter is considered from a novel perspective promoting the view that a deeper understanding of a theory of knowledge is an important prerequisite for the development of improved mathematical models of scientific mapping. A spatiotemporal map, e.g., should depend on what we know about the natural variable it represents, as well as how we know it (i.e., what sources of knowledge we selected and what kinds of methods we used to process knowledge). As is discussed in the book, modern geostatistical approaches can be developed that are consistent with the above epistemic framework. The main focus of the book is the Bayesian maximum entropy (BME) approach for studying spatiotemporal distributions of natural variables. As part of the modern spatiotemporal geostatistics paradigm, the BME approach provides a fundamental insight into the mapping problem in which the knowledge of a natural variable, not the variable itself, is the direct object of study. This insight plays a central role in numerous scientific disciplines. BME’s rich theoretical basis provides guidelines for the adequate interpretation and processing of the knowledge bases available (different sorts of knowledge enter the modern geostatistics paradigm in different ways). It also forces one to determine explicitly the available physical knowledge bases and to develop logically plausible rules and standards for knowledge integration and processing. BME is formulated in a rigorous way that preserves earlier geostatistical results, which are its limiting cases, and also provides novel and more general results that could not be obtained by classical geostatistics. Indeed, a number of situations are discussed in the book in which BME’s quest for greater rigor serves to expose new, hitherto ignored possibilities. In addition, the presentation of the quantitative results, with their full technical beauty, is combined with an effort to communicate across the various fields of natural science. Finally, an attempt has been made to ensure that the case studies considered in this book involve data that are publicly accessible, so that all hypotheses made and conclusions drawn can be critically examined and improved by others (in fact, a scientific model gains authority by withstanding the criticism of other scientists). Naturally, ideas and practical suggestions on how to efficiently apply BME theory will evolve as more case studies are done.

Metaphorically speaking, the aim of modern spatiotemporal geostatistics is to integrate effectively the powerful theoretical perspective of the Reason of Plato (who proposed a conceptual framework that dominated mathematical reasoning and philosophical thinking for thousands of years) with the practical thinking of the Reason of Odysseus (who was always capable of coming up with smart solutions to all kinds of practical problems he faced during his long journey). It has been said that Plato shared his perspective with the Gods and Odysseus with the foxes. The modern geostatistician shares it with both! At this point, I must admit to using these great men as a provocative and authoritative means of setting things up and getting readers into the right mood.

In light of the above considerations, the Ariadne’s thread running throughout the book is that the modern geostatistical approach to real-world problems is that of natural scientists who are more interested in a stochastic analysis concerned with both the ontological level (building models for physical systems) and the epistemic level (using what we know about the physical systems and integrating and modeling knowledge from a variety of scientific disciplines), rather than in the pure or naive inductive account of science based merely on a linear relationship between data and hypotheses and theory-free techniques that may be useful in other areas. In this sense, modern spatiotemporal geostatistics facilitates yet another kind of integration: the horizontal integration among disparate scientific fields. By processing a variety of physical knowledge bases, the BME approach brings together several sciences which are all relevant to the aspect of reality that is to be examined. For example, BME can become an integral component of the interdisciplinary attack on fundamental environmental health systems which involve physical variables, exposure mechanisms, biological processes, human anatomy and physiology parameters, and epidemiological indicators. The subject of horizontal integration is a source of great excitement among scientists; new ideas are generated incessantly. It is expected that as the domain of modern spatiotemporal geostatistics continues to expand in search of new conquests, a variety of mapping methods aiming at horizontal integration will be added to its arsenal.

The crux of this book was projected in Spring 1986 while I was a research scientist in what is now the Mathematical Geology Section of the Kansas Geological Survey. Some results were published in a paper in Mathematical Geology in 1990. Several other research obligations prevented me from working systematically on the subject for the following six or seven years. My involvement in BME analysis has been renewed in recent years, due to increased interest in developing a new conceptual and methodological framework for geostatistics, and aided by generous funding from the Army Research Office (Grants DAAG55-98-1-0289 and DAAH04-96-1-0100). To this financial benefactor I remain grateful.

In carrying out the project, I have benefited from comments made by my colleagues Patrick Bogaert, Dionissios Hristopulos, Ricardo Olea, John Davis, Jürgen Pilz, Tom Jones, and Hyemi Choi. Also my students, Marc Serre, Kyung-mee Choi, Alexander Kolovos, and Jordan Kovitz read my class notes and recommended improvements. In some cases, it was indeed my confrontation with youth that prompted a fresh look at the basis of geostatistics. These students continue to work in the field of modern geostatistics and are expected to contribute significantly to its further advancement. Finally, I am deeply indebted to Jo Anne DeGraffenreid, IAMG editor of the Oxford monograph series, Studies in Mathematical Geology; her continuous encouragement and editorial acumen have proven invaluable.

It has been said that in science, the quality of a scientist’s work is closely related to the quality of those thinkers with whom he/she disagrees. I have personally benefited greatly from discussions, criticisms, and exchanges of ideas with theoretical opponents for whom I have the greatest respect. If this book is somehow critical of some of their ideas, it is because healthy disagreement—and not imitation—is the deepest sign of an abiding appreciation.

The book is dedicated to the unknown Pythagorean. He was a student at the famous Pythagorean school of ancient Greece (ca. 6th century B.C.), a young man who for the first time in history proved an irrefutable truth by the power of reason alone.* For his astonishing discovery that challenged the views of the Establishment of the time he was cursed and declared blasphemous, and then—legend has it—he was thrown into the sea. Thus perished in the dark waters of the Mediterranean, remaining forever unknown, the young man who brought us the light of reason…

George Christakos

Chapel Hill, 1999

---

*He proved that there is no quotient of integers whose square equals two (see, e.g., the presentation of the Pythagorean School in Omnés, 1999).

TABLE OF CONTENTS

PREFACE

CHAPTER 1: Spatiotemporal Mapping in Natural Sciences

Mapping Fundamentals

The Epistemic Status of Modern Spatiotemporal Geostatistics: It Pays to Theorize!

Why Modern Geostatistics?

Scientific content

Indetermination thesis

Spatiotemporal geometry

Sources of physical knowledge

The non-Procrustean spirit

Bayesian Maximum Entropy Space/Time Analysis and Mapping

BME features

The Integration Capability of Modern Spatiotemporal Geostatistics

The Knowledge-Map Approach

CHAPTER 2: Spatiotemporal Geometry

A More Realistic Concept

The Spatiotemporal Continuum Idea

The Coordinate System

Euclidean coordinate systems

Non-Euclidean coordinate systems

Metrical Structure

Separate metrical structures

Composite metrical structures

Some comments on physical spatiotemporal geometry

The Field Idea

Restrictions on spatiotemporal geometry imposed by field measurements and natural media

Restrictions on spatiotemporal geometry imposed by physical laws

The Complementarity Idea

Putting Things Together: The Spatiotemporal Random Field Concept

Correlation analysis and spatiotemporal geometry

Permissibility criteria and spatiotemporal geometry

Effect of spatiotemporal geometry on mapping

Some Final Thoughts

CHAPTER 3: Physical Knowledge

From the General to the Specific

The General Knowledge Base

A mathematical formulation of the general knowledge base

General knowledge in terms of statistical moments

General knowledge in terms of physical laws

Some other forms of general knowledge

The Specificatory Knowledge Base

Specificatory knowledge in terms of hard data

Specificatory knowledge in terms of soft data

Summa Theologica

CHAPTER 4: The Epistemic Paradigm

Acquisition and Processing of Physical Knowledge

Epistemic Geostatistics and the BME Analysis

Prior stage

Meta-prior stage

Integration or posterior stage

Conditional Probability of a Spatiotemporal Map and its Relation to the Probability of Conditionals

Material and strict map conditionals

Other map conditionals

The BME Net

CHAPTER 5: Mathematical Formulation of the BME Method

A Pragmatic Framework of the Mapping Problem

The Prior Stage

Map information measures in light of general knowledge

General knowledge-based map pdf

General knowledge in the form of random field statistics (including multiple-point statistics)

General knowledge in the form of physical laws

Possible modifications and generalizations of the prior stage

The Meta-Prior Stage

The Integration or Posterior Stage

The Structure of the Modern Spatiotemporal Geostatistics Paradigm

The Two Legs on Which the BME Equations Stand

CHAPTER 6: Analytical Expressions of the Posterior Operator

Specificatory Knowledge and Single-Point Mapping

Posterior Operators for Interval and Probabilistic Soft Data

Posterior Operators for Other Forms of Soft Data

Discussion

CHAPTER 7: The Choice of a Spatiotemporal Estimate

Versatility of the BME Approach

The BMEmode Estimate

Statistics—Hard and soft data

Physical laws—Hard and soft data

The West Lyons Porosity Field

Other BME Estimates

A Matter of Coordination

CHAPTER 8: Uncertainty Assessment

Mapping Accuracy

Symmetric Posteriors

Asymmetric Posteriors

The Equus Beds Aquifer

The study area

Data collection

The water-level elevation model

BME water-level elevation mapping

Optimal decision making

Doing Progressive Guesswork

CHAPTER 9: Modifications of Formal BME Analysis

Versatility and Practicality

Functional BME Analysis

General formulation

The support effect

Multivariable or Vector BME Analysis

General formulation

Physical laws

Transformation laws

Decision making

Multipoint BME Analysis

Multipoint BME estimation

Multipoint BME uncertainty assessment

BME in the Context of Systems Analysis

Risk analysis of natural systems

Human-exposure systems

Associations between environmental exposure and health effect

Bringing Plato and Odysseus Together

CHAPTER 10: Single-Point Analytical Formulations

The Basic Single-Point BME Equations

Ordinary Covariance and Variogram-Hard and Soft Data

Particulate Matter Distributions in North Carolina

Generalized Covariance-Hard and Soft Data

Some Non-Gaussian Analytical Expressions

Theory, Practice, and Computers

CHAPTER 11: Multipoint Analytical Formulations

The Basic Multipoint BME Equations

Ordinary Covariance-Hard and Soft Data

Ordinary Variogram-Hard and Soft Data

Other Combinations

Spatiotemporal Covariance and Variogram Models

Separable models

Nonseparable models

And Still the Garden Grows!

CHAPTER 12: Popular Methods in the Light of Modern Spatiotemporal Geostatistics

The Generalization Power of BME

Minimum Mean Squared Error Estimators

Kriging Estimators

Simple and ordinary kriging

Lognormal kriging

Nonhomogeneous/nonstationary kriging

Indicator kriging

Other sorts of kriging

Limitations of kriging techniques-Advantages of BME analysis

Random Field Models of Modern Spatiotemporal Geostatistics

A unified framework

The class of coarse-grained RF

The class of S/TRF-ν/μ models in heterogeneity analysis

The class of space/time fractal RF models

The class of wavelet RF

The Emergence of the Computational Viewpoint in BME Analysis

Modern Spatiotemporal Geostatistics and GIS Integration Technologies

CHAPTER 13: A Call Not to Arms but to Research

Unification and Distinction

The Formal Part

Interpretive BME and the Search for Rosebud

The Argument of Modern Spatiotemporal Geostatistics

The Ending as a New Beginning

BIBLIOGRAPHY

INDEX

1

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

Mapping Fundamentals

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).

Figure 1.1. Maps of estimated maximum hourly ozone concentration (ppm) over the eastern U.S. on (a) July 15, 1995, and (b) July 16, 1995. From Christakos and Vyas (1998).

(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.

Figure 1.2. Maps of the Earth’s core and mantle. The top row shows cutaway maps below the Atlantic (left) and the Pacific (right) oceans to a depth of 550 km. The bottom row shows cutaway maps of the Atlantic and Pacific oceans to a depth of 2,890 km. While the Atlantic maps reveal cold, dense, sinking material, the Pacific maps represent hot, buoyant, rising material. [From Dziewonski and Woodhouse, 1987; © 1987 by AAAS, reproduced with permission.]

(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

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.)

Figure 1.3. Map of oil-phase isopressure contours for an anisotropic intrinsic permeability field (pressure given in units of entry pressure). From Christakos et al. (2000b).

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

Figure 1.4. Map of the predicted maximum daily temperatures (°Celsius) over Belgium for one day of the year 1990. The equidistance between contours is 0.5°; the lowest level contour is 4.5° (SE part). From Bogaert and Christakos (1997).

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

Figure 1.5. Maps of radioactivity present in the atmosphere following the Chernobyl accident (U.S. Air Force weather data and computer simulation by Lawrence Livemore National Laboratory; see Enger and Smith, 1995).

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

Figure 1.6. A health damage indicator map (number of representative receptors affected/km²) showing damage due to ozone exposure in the New York City–Philadelphia area on July 20, 1995 (from Christakos and Kolovos, 1999).

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

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).

Figure 1.7. A simulated spatiotemporal cell field for a target organ. The affected cells are white and the normal cells are black; some repair is taking place, as well (from 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

Figure 1.8. Genetic map of populations from the Near East to the European continent. [From Menozzi et al., 1978; ©1978 by AAAS, reproduced with permission.]

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 north-eastern 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

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