Map Use ; Reading, Analysis, Interpretation [8 ed.] 9781589484696

Table of contents :
Cover
Title
Copyright
Contents
Foreword
Preface
Acknowledgments
Introduction
Part I Map reading
1 The earth and earth coordinates
2 Map scale
3 Map projections
4 Grid coordinate systems
5 Land partitioning
6 Map design basics
7 Qualitative thematic maps
8 Quantitative thematic maps
9 Relief portrayal
10 Image maps
11 Map accuracy
Part II Map analysis
12 Distance finding
13 Direction finding and compasses
14 Position finding and navigation
15 Spatial feature analysis
16 Surface analysis
17 Spatial pattern analysis
18 Spatial association analysis
Part III Map interpretation
19 Interpreting the lithosphere
20 Interpreting the atmosphere and biosphere
21 Interpreting the human landscape
22 Maps and reality
Appendix A: Digital cartographic databases
Appendix B: Mathematical tables
Glossary
Index

Citation preview

Cover image courtesy of National Geophysical Data Center and NASA.

Esri Press, 380 New York Street, Redlands, California 92373-8100 Copyright © 2016 Esri All rights reserved. Sixth edition 2009. Seventh edition 2012. Printed in the United States of America 20 19 18 17 16 1 2 3 4 5 6 7 8 9 10 ISBN 9781589484696 (e-book) The information contained in this document is the exclusive property of Esri unless otherwise noted. This work is protected under United States copyright law and the copyright laws of the given countries of origin and applicable international laws, treaties, and/or conventions. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage or retrieval system, except as expressly permitted in writing by Esri. All requests should be sent to Attention: Contracts and Legal Services Manager, Esri, 380 New York Street, Redlands, California 92373-8100, USA. The information contained in this document is subject to change without notice. US Government Restricted/Limited Rights: Any software, documentation, and/or data delivered hereunder is subject to the terms of the License Agreement. The commercial license rights in the License Agreement strictly govern Licensee’s use, reproduction, or disclosure of the software, data, and documentation. In no event shall the US Government acquire greater than RESTRICTED/LIMITED RIGHTS. At a minimum, use, duplication, or disclosure by the US Government is subject to restrictions as set forth in FAR §52.227-14 Alternates I, II, and III (DEC 2007); FAR §52.227-19(b) (DEC 2007) and/or FAR §12.211/12.212 (Commercial Technical Data/Computer Software); and DFARS §252.2277015 (DEC 2011) (Technical Data–Commercial Items) and/or DFARS §227.7202 (Commercial Computer Software and Commercial Computer Software Documentation), as applicable. Contractor/Manufacturer is Esri, 380 New York Street, Redlands, CA 92373-8100, USA. @esri.com, 3D Analyst, ACORN, Address Coder, ADF, AML, ArcAtlas, ArcCAD, ArcCatalog, ArcCOGO, ArcData, ArcDoc, ArcEdit, ArcEditor, ArcEurope, ArcExplorer, ArcExpress, ArcGIS, arcgis.com, ArcGlobe, ArcGrid, ArcIMS, ARC/INFO, ArcInfo, ArcInfo Librarian, ArcLessons, ArcLocation, ArcLogistics, ArcMap, ArcNetwork, ArcNews, ArcObjects, ArcOpen, ArcPad, ArcPlot, ArcPress, ArcPy, ArcReader, ArcScan, ArcScene, ArcSchool, ArcScripts, ArcSDE, ArcSdl, ArcSketch, ArcStorm, ArcSurvey, ArcTIN, ArcToolbox, ArcTools, ArcUSA, ArcUser, ArcView, ArcVoyager, ArcWatch, ArcWeb, ArcWorld, ArcXML, Atlas GIS, AtlasWare, Avenue, BAO, Business Analyst, Business Analyst Online, BusinessMAP, CityEngine, CommunityInfo, Database Integrator, DBI Kit, EDN, Esri, esri.com, Esri— Team GIS, Esri—The GIS Company, Esri—The GIS People, Esri—The GIS Software Leader, FormEdit, GeoCollector, Geographic Design System, Geography Matters, Geography Network, geographynetwork.com, Geoloqi, Geotrigger, GIS by Esri, gis.com, GISData Server, GIS Day, gisday.com, GIS for Everyone, JTX, MapIt, Maplex, MapObjects, MapStudio, ModelBuilder, MOLE, MPS —Atlas, PLTS, Rent-a-Tech, SDE, SML, Sourcebook•America, SpatiaLABS, Spatial Database Engine, StreetMap, Tapestry, the ARC/INFO logo, the ArcGIS Explorer logo, the ArcGIS logo, the ArcPad logo, the Esri globe logo, the Esri Press logo, The Geographic Advanta ge, The Geographic Approach, the GIS Day logo, the MapIt logo, The World’s Leading Desktop GIS, Water Writes, and Your Personal Geographic Information System are trademarks, service marks, or registered marks of Esri in the United States, the European Community, or certain other jurisdictions. CityEngine is a registered trademark of Procedural AG and is distributed under license by Esri. Other companies and products or services mentioned herein may be trademarks, service marks, or registered marks of their respective mark owners.

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Contents

Foreword Preface Acknowledgments Introduction Part I

Map reading 1 The earth and earth coordinates 2 Map scale 3 Map projections 4 Grid coordinate systems 5 Land partitioning 6 Map design basics 7 Qualitative thematic maps 8 Quantitative thematic maps 9 Relief portrayal 10 Image maps 11 Map accuracy

Part II

Map analysis 12 Distance finding 13 Direction finding and compasses 14 Position finding and navigation 15 Spatial feature analysis 16 Surface analysis 17 Spatial pattern analysis 18 Spatial association analysis

Part III Map interpretation 19 Interpreting the lithosphere 20 Interpreting the atmosphere and biosphere 21 Interpreting the human landscape 22 Maps and reality Appendix A: Digital cartographic databases

Appendix B: Mathematical tables Glossary Index

Foreword

Today, the world of creating maps is accelerating thanks to advances in digital cartography, geographic information systems (GIS), and web mapping. These technologies have become powerful engines for streamlining both the design and production of maps. They have also expanded the range of topics being mapped, as well as the actual number of maps being made. More important, they have broadened the community of people making and using maps. But have the maps themselves changed? Once primarily used as a tool to help people navigate their environment, maps today are increasingly used to help communicate and address the world’s important challenges. I personally see digital maps as a new kind of language that facilitates improved communication and collaboration in every aspect of society. Mastering this language will become an essential skill set for 21st-century living. Maps are abstractions of the earth—representations of the complexities of the world and useful for understanding the world around us. They distill and summarize information into a special type of graphic display. Creating these abstractions in such a way that they provide users with valuable information products that reflect and represent the complexities of the earth is the real challenge of those who produce the maps of today. Map Use, now in its eighth edition, is a comprehensive primer for using maps effectively. By providing the keys to unlock the codes that cartographers use to represent the world around us, this book is an invaluable companion for college-level students and instructors, for professionals in a variety of fields where maps are important, and also for casual map users. Jack Dangermond President, Esri

Preface

Many books are written on mapmaking, but because map use isn’t simply the offshoot of map-making, most of these books are of limited value to you as a map user. By contrast, this book is written for those who need to know how to use maps to build or enhance their spatial understanding of the world. Fully revised to better provide the context and demonstration of skills for reading and properly analyzing this complex form of communication, Map Use: Reading, Analysis, Interpretation, eighth edition, is a definitive resource for an introductory map use or cartography course. Map Use is also a valuable tool for people whose vocation or recreation requires knowing how to read and use maps because it takes the reader beyond visual representations and into the decision-making processes of cartographers. Academics tend to treat maps as indoor objects, rarely including in their textbooks the fact that one of the most exciting ways to use maps is in the field. Conversely, manuals and field guides on map and compass use focus narrowly on wayfinding, geocaching, and military map reading, virtually ignoring the role that maps play in how we think about and communicate environmental information. We depart from this tradition by bridging the gap between these two extremes, carefully weaving information from many fields into a coherent view of map use. This book offers readers a comprehensive, philosophical, and practical treatment of map use in three primary ways. First, we define a map as a graphic representation of an environment that shows relations between geographic features. This encompassing definition lets us include a variety of important map forms that are otherwise awkward to categorize, such as mental maps, which are discussed in the introduction, or web maps, which may exist only ephemerally. Our definition should also accommodate new cartographic forms developed in the future. This book integrates discussions of a variety of map forms, including standard planimetric maps, perspective diagrams, aerial photographs, satellite images, interactive animated maps, and others, rather than partitioning each type of map into a separate category. This integrative approach, focused on commonality rather than uniqueness, showcases the enhanced insight into the environment that the variety of mapping perspectives can inspire. It helps the reader understand that the definition of a map is fluid. This fluidity lets us view our environment from countless perspectives to glean insights that would otherwise be lost. Second, we make a clear distinction between the tangible cartographic map and the mental or cognitive map of the environment that we hold in our heads. Ultimately, it is the map in our minds, not the map in front of our eyes, that we use to make decisions. Throughout the text, we stress that cartographic maps are valuable aids for developing better mental maps. We should strive to become familiar enough with the environment that we can move through it freely, interact with it, and view it remotely, in both a physical and mental sense. Ideally, our cartographic and mental maps should merge so that our spatial understanding, communication, and behavior have the greatest chance of being tuned to the reality of the environment. Third, where appropriate, we reference commercial products and services of special interest to the map user. A few decades ago, discussing these commercial products would

have seemed strange since most mapping was done by large government agencies, but times have changed. The strong recent trend toward commercialization of all things cartographic now makes these products indispensable to people who are intent on using maps efficiently and effectively. Computer software, web apps, and digital data for mapping on home computers and mobile devices are developed and made accessible not only by government agencies at all levels, but also by private companies, academics, and even nongovernment organizations (NGOs). What you do with maps in the future will be strongly influenced by the nature of these commercial products and services. Finally, we show how map use is relevant to daily life. Whenever possible, we use examples and illustrations from popular sources and common practices. Maps touch so many aspects of our daily lives that it is simple and natural to make points and reinforce ideas with illustrations from topics of everyday interest. These illustrations are included in the text to demonstrate and reinforce basic map use principles and illustrate the universality of our relationship with maps. Other books take a simplistic, mechanical view and treat the using and making of maps primarily in an engineering and technical context. In contrast, Map Use takes a unique approach in the depth of knowledge it offers in a clear and readable style. We believe that learning to understand the intentions or goals of a map leads to making better choices in your own map use and mapmaking. Learning to use a map is a relatively easy and painless process, with an immense payoff —maps offer a look at the virtually invisible. They let us see the environment from vantage points that are distant from us in time and space. Maps also allow us to visualize aspects of our environment that are intangible, imperceptible, or purely conceptual. Most important, maps focus our attention on selected features by keeping the display free of distracting detail. Maps free us from our natural limitations, transcend our senses, and let us see the world anew.

NEW TO THIS EDITION The eighth edition of Map Use includes one new chapter and some 50 new four-color illustrations, added to the 500-plus in the previous edition. Several of the new illustrations are linked to online animated and interactive maps through QR codes (two-dimensional bar codes) printed with the illustration. The entire book is reindexed, and the glossary is enhanced to integrate the additional chapter and include acronyms. The new chapter, “Map Design Basics,” is an important addition to the eighth edition of Map Use as it focuses on an integral component of map use—how to design your own map and to comprehend when someone else designs a good map. Although many map design principles are built into modern mapping systems, the assumption is still that you are familiar with map design basics to the degree that you can make, or you understand how the mapmaker makes, wise choices among the design options available for the various components of your map. In this new chapter, we discuss the map design basics that are necessary to be a good mapmaker and that allow you to be a better map user. When you have a firm grasp of these principles, you are ready to make your own maps following the basic map design guidelines presented in chapter 6. When Map Use was first published, very little mapping was done in a computer environment. Today, not only is most mapping done with the aid of computers, but the map

user is often the one who guides the mapping process. Especially with the aid of geographic information system (GIS) software and applications, the map user is increasingly the mapmaker. Even more significant, the map user can establish insightful dialogues with maps by manipulating the digital data in various ways. To a large degree, GIS is responsible for this exciting interchange between people, maps, and environmental data, and has changed the essential nature of map use to our immense benefit. At the same time, maps contribute greatly to GIS by providing a familiar visual interface through which GIS technology’s powerful computing resources can be fully realized. Now that GIS technology is so widely available, a broader population is using it in a variety of ways. Clearly, the ability to think and communicate visually through the medium of maps is more important than ever, and this book can enhance this ability in a broad range of users. Yet for all the technological advances in map use and mapmaking, the philosophy underlying Map Use remains the same. As in earlier editions, we stress that a good map user must understand, at a basic level, what goes into the making of a map. From mapmakers, we ask for little less than a miracle. We want the overwhelming detail, complexity, and size of our surroundings reduced to a simple representation that is convenient to access. We also want abstract maps to provide us with a meaningful basis for relating to the real environment. In return, we must make a corresponding effort to become educated map users.

HOW THIS BOOK IS ORGANIZED Map Use is specifically designed and tested for use in a three-credit semester course of 15 weeks at the college freshman level. Presentation of the material is geared for the upper high school to intermediate college level. The book is aimed at both the specialized and general map user. You can use it with equal effectiveness as a basic reference work or as the textbook for a beginning map use or cartography course. We intentionally avoid the confusing terms and details that characterize so many cartographic texts. We structured the material in the book into three main parts, under the headings “Map Reading,” “Map Analysis,” and “Map Interpretation.” In most books, these terms appear with little more than vague definitions and are often used interchangeably. Here, they are defined precisely, and the relationship of each one to the other is clear. We also offer clear definitions for terms that may be new to you—these definitions are also presented in the book's extensive glossary. Part 1, “Map Reading,” helps readers develop an appreciation of how mapmakers represent the environment in the reduced, abstract form of a map. In map reading, you must mentally “undo” the mapping process. We discuss the geographic data that underpins a map, the process required to transform that information through mapping techniques, the basic principles of map design, and the importance of map accuracy. Once readers grasp the degree to which cartographic procedures can influence the appearance and form of a map, they are in a position to use maps to analyze spatial patterns and relationships in the mapped environment. Part 2, “Map Analysis,” includes chapters on distance and direction finding. Here, we also explore position finding and navigation. We examine cartometrics (making measurements on maps) to describe the properties of area, volume, shape, and more. And we look at analyses of the land surface

form, spatial patterns, and spatial associations. With each of these topics, the emphasis is on estimating, counting, measuring, analyzing, and finding patterns in map features. The results of map analysis come alive when you try to explain why the environment takes on one spatial characteristic over another. This is the subject of part 3, “Map Interpretation.” The material is divided into four chapters: “Interpreting the Lithosphere” (landforms and geology), “Interpreting the Atmosphere and Biosphere,” “Interpreting the Human Landscape,” and “Maps and Reality.” The emphasis in this final section is on environmental comprehension and understanding, for it is our surroundings, not the map, that is the real subject of map use. Two appendixes, on digital cartographic databases and relevant mathematical tables, complement material presented in the text. The glossary is expanded in this edition. It includes terms in the book that are either newly introduced or are of special importance to map use. Glossary terms appear in boldface type at the point in which they are introduced. Although a systematic development of subject matter is followed throughout the book, each part and chapter can be used independently and is cross-referenced to the rest of the material. This flexibility of design makes Map Use a versatile text, useful both for instruction in the classroom and as a reference for practicing users. The book’s organization provides a logical development of concepts and a progressive building of understanding and skills, from beginning to end. More experienced map users may focus initially on sections or chapters of special interest, and then refer to other parts of the book to refresh their memories or clarify terms, concepts, and methods. This book will serve its purpose if you finish it with a greater appreciation of maps than when you started. In even the simplest map, there is much to respect. Mapmakers have managed to shape the jumble of reality into a compact, usable form. They have done a commendable job. Now, map user, the rest is up to you. A. J. Kimerling A. R. Buckley P. C. Muehrcke J. O. Muehrcke

Acknowledgments

In previous editions of Map Use, we acknowledged the importance of contributions made by special teachers and colleagues, teaching assistants, and students at the University of Washington, University of Wisconsin, and Oregon State University. We continue to be grateful for the inspiration and assistance provided by these people. To these students and colleagues, we add the valuable contributions made by Esri staff and colleagues. Our sincere thanks to everyone at Esri Press who worked hard to make this new edition possible. To our delight, hundreds of individuals have taken the time to comment on earlier editions of this book. Some of these people are merely lovers of maps; others are professors responsible for teaching introductory courses in map reading, analysis, and interpretation; and many are students who had occasion to use the book in their studies. We are especially moved by letters from people who stumbled upon Map Use by chance at a friend’s house or library and felt compelled to let us know how pleased they were with their discovery. All these responses are gratefully received, and many were useful in crafting this expanded and improved eighth edition. This work reflects our deep love of maps and a desire to help others bring maps into their lives. We alone, of course, bear full responsibility for errors in the text or illustrations and for any controversial statements. Acknowledgments would be incomplete without expressing gratitude to our families and many friends who helped us in so many ways. Particular thanks is given to Ann Kimerling and David Sandoval for their unending encouragement and patience during the creation of this eighth edition. A. Jon Kimerling Corvallis, Oregon Aileen R. Buckley Redlands, California Phillip C. Muehrcke and Juliana O. Muehrcke Madison, Wisconsin

Map Use Reading, Analysis, Interpretation

INTRODUCTION MENTAL MAPS CARTOGRAPHIC MAPS THE MAP TRANSFORMATION PROCESS WHAT MAKES MAPS POPULAR? FUNCTIONS OF MAPS Reference maps Thematic maps Navigation maps Story and persuasive maps Maps as art MAP USE SELECTED READINGS

Introduction It should be easier to read a map than to read this book. After all, we know that a picture is worth a thousand words. Everyone, from poets to politicians, works from the assumption that maps are easy to understand and follow. The very term map is ingrained in our thinking. We use it to suggest clarification, as in “Map out your plan” or “Do I have to draw you a map?” How ironic, then, to write a book using language that is, supposedly, more complicated than the thing we are trying to explain! The problem is that maps are not nearly as simple and straightforward as they seem. Using a map to represent our detailed and complexly interrelated surroundings can be deceptive. This is not to say that maps themselves are unclear. But it is the environment, not the map, that you want to understand. A map lets you view the environment as if it was less complicated. There are advantages to such a simplified picture, but there is also the danger that you will end up with an unrealistic view of your surroundings. People who manage critical natural and human resources all too often make decisions based on maps that inherently are oversimplified views of the environment. In this book, we define a map as a spatial representation of the environment that is presented graphically. By representation, we mean something that stands for the environment, portrays it, and is both a likeness and a simplified model of it. This definition encompasses such diverse maps as those on walls; those that appear ephemerally and then are gone, as on a computer screen or in the form of holograms or virtual reality; and those held solely in the mind’s eye, known as cognitive or mental maps. You may envision the environment by using cartographic maps, which are what most people think of as traditional maps drawn on paper or displayed on computer screens. Or you can use mental maps, which are often slighted although they are the ultimate maps that you use to make decisions about the environment. Now we will look more closely at mental and cartographic maps.

MENTAL MAPS As a child, your mental map was probably based on direct experience—for example, connected pathways such as the routes from your home to school or the park. You had a self-centered view of the world in which you related everything to your own position. The cartoon in figure I.1 graphically portrays this type of mental map. As an adult, you can appreciate this cartoon because you see how inefficient the child’s mental map is. But the truth is, you will often resort to this way of visualizing the environment when thrown into unfamiliar surroundings. If you go for a walk in a strange city, you will remember how to get back to your hotel by visualizing a pathway like that in the cartoon. Landmarks will be strung like beads along the mental path. Even if you might be able to guess at a more direct route back, you may feel more comfortable following the string of landmarks to make sure that you do not get lost. Most of your mental maps are more detailed than this cartoon path, however. You take advantage of indirect as well as direct experience. You acquire information through TV, photographs, books and magazines, the Internet, and other secondary sources. You can transcend your physical surroundings and visualize distant environments, even those on the other side of the planet at different historical periods. Your mental maps become incredibly complex as they expand to encompass places and times that you have never seen and may never be able to visit. As you grow older, your self-centered view of the world is replaced by a geocentric view. Rather than relating everything to your own location, you learn to mentally orient yourself with respect to the external environment. Once you learn to separate yourself from your environment, you do not have to structure your mental map in terms of connected pathways and landmarks. You can visualize how to get from one place to another “as the crow flies”—the way you would go if you are not restricted to roads and other connected routes. It is your adult ability to visualize the “big picture” that makes the cartoon amusing. Sharing your mental map with others, either in conversation or in maps that you draw, is much easier when you use a geometric reference framework, or a framework in which you can easily describe and determine locations, distances, directions, and other geographic relationships. The system of cardinal directions (north, south, east, and west) is such a framework. You can pinpoint the location of something by stating its cardinal direction and distance from a starting location. You can say, for example, that the store is two miles north of a particular road intersection, or the police station is 200 meters west of the courthouse. This visualization of space is based on Euclidean geometry, the geometry you first learned in school. It is the geometry that says that parallel lines never cross, that the shortest distance is a straight line, that space is three-dimensional, and so on. The ability to visualize the environment in terms of Euclidean geometry is an essential part of developing a geocentric mental map. But even if you develop mental maps based on Euclidean geometry, they will only be correct over small areas. This limitation is because the earth is spherical, and the spherical geometry of the earth’s surface is inherently non-Euclidean. (You will see in chapter 1, for example, that north–south lines on the earth, called meridians, are not parallel but, rather, converge to common points at the poles.) Very few of us have well-developed spherical mental maps simply because the small portion of the environment that we experience daily appears flat from our ground perspective, unless we are astronauts.

Figure I.1. The geometry of a child’s mental map is based on direct experience and connected pathways. Courtesy of the Register and Tribune Syndicate.

Even if you are able to visualize the world geocentrically in terms of Euclidean or spherical geometry, it is hard for most people to transform their mental map to a cartographic map in a geometrically accurate manner. Try drawing, from memory, a map of the area in which you live. The hand-drawn map will tell you a great deal about the geometric accuracy of your mental map. Not only will you probably draw the places you know best with the greatest detail and spatial accuracy, you will probably draw the things that are important to your life and leave out the things you do not care about. Few people’s mental maps correspond with cartographic maps. Figure I.2 shows the distorted visual image that a person from Michigan’s Upper Peninsula might have of the country. Tongue in cheek as this map may be, it captures the fact that people visualize their own region as far more important than the rest of the world. In the same way, your mental maps emphasize your own familiar neighborhood, with distant places less well visualized. It is important to recognize these biases in your mental maps. The quality of your mental maps is crucial, because your behavior in the environment largely depends on them. You relate to your surroundings as you visualize them, not necessarily as they really are. If the discrepancy between your mental maps and the real world is great, you may act in selfdefeating or even disastrous ways. But you do not have to rely solely on mental maps, because cartographic maps are created for a multitude of places and features in the environment, and maps on mobile devices exist for nearly every place on earth.

CARTOGRAPHIC MAPS

A cartographic map is a graphic representation of the environment. By graphic, we mean that a cartographic map is something that you can see or touch. Cartographic databases or digital image files are not, in themselves, maps but are essential to current methods for the creation of maps. In a similar vein, an exposed piece of photographic film or paper does not become a photograph until it is developed into a slide, paper print, or computer screen. Cartographic maps come in many forms. Globes, physical landscape models, and Braille maps for the blind are truly three-dimensional objects, but most maps are two-dimensional representations that are cartographically enhanced. Cartographic maps have been carved, painted, or drawn on a variety of media for thousands of years, and print maps have been produced for the last five centuries. Today, maps displayed on a computer screen or mobile device are probably the most commonly seen and used maps. Maps are so intuitive and serve so many purposes that it is easy to forget that maps are one of our most sophisticated conceptual creations. They tell you as much about how people think and communicate as they do about the environment that is mapped.

Figure I.2. The United States as seen through the eyes of a resident of Michigan’s Upper Peninsula. Courtesy of Eugene S. Sinervo.

What gives a graphic representation of the environment its “mapness”? Many mapmakers say that cartographic maps have certain characteristics, with the four most important being: Maps are vertical or oblique views of the environment, not profile views such as a photograph of the front of your home taken from the street. Maps are created at a certain scale, meaning that there is a systematic reduction from ground distance to map distance, as you will see in chapter 2.

Except for 3D globes and landscape models that faithfully represent the earth’s curvature, maps are made using a map projection, which is a mathematically defined transformation of locations on the spherical earth to a map surface, as we explain in chapter 3. Maps are generalized and symbolized representations of the environment. Mapmakers select a limited number of features from the environment to display on the map, and then display these features in a simplified manner. Insignificant features are not shown, the sinuosity of linear features and area boundaries is reduced, and several small ground features may be aggregated into a single feature on the map, as explained in chapter 6. The generalized features are then shown graphically using different map symbols. The mapmaker uses different point symbols, line widths, line patterns, gray tones, colors, and pattern fills to symbolize the features, as we describe in chapters 6 through 9. Names, labels, and numbers that annotate the map are also important map symbols. A cartographic map does not need all four characteristics of maps, but it should have at least one. You can think of different types of maps as being at different places on a mapness continuum, defined by the degree to which they exhibit these four characteristics. This continuum is illustrated in figure I.3 for a gradation of map types that depict part of Crater Lake National Park in Oregon. The aspect-slope and topographic maps, which show elevations or landforms as well as a limited set of other features, in the left third of the illustration strongly reflect all four characteristics and are good examples of what most people think of as a map. The orthophotomap in the center of the illustration also has all four characteristics because topographic map symbols are printed over a geometrically corrected aerial photograph called an orthophoto, which you will learn more about in chapter 11. An orthophoto is corrected to a constant scale on a map projection surface and hence has three of the four characteristics. Finally, the aerial photograph from which the orthophoto is made is not on a map projection surface and varies in scale with elevation differences on the ground. The aerial photo has only the characteristic of being a vertical view of the environment but is still a form of cartographic map. All the other maps are also vertical views.

Figure I.3. Different types of maps lie along a mapness continuum. Their position on the continuum is defined by how many characteristics of maps they possess. All the examples in the figure are vertical views. Courtesy of the US Geological Survey.

As you can see from this illustration, there are multitudes of cartographic maps, each somewhere along the mapness continuum. The variety is so great that from now on, we will shorten the term cartographic map to simply map, in accord with what you are used to hearing these products called.

THE MAP TRANSFORMATION PROCESS Mapping, like architecture, is an example of functional design. Unlike an artist’s representation of the environment in which geometric liberties are taken to convey an idea or emotion, a map is expected to be true to the location and nature of our surroundings. Indeed, our willingness to let maps stand for the environment is because of this expected adherence to reality. Figure I.4 illustrates the process that cartographers use to transform data collected about the geographic environment, through surveys, by the Global Positioning System, from remote sensing, and more, to a map. Maintaining the highest possible fidelity in the map transformation process assures that the map is an accurate representation of reality. You, the map user, also play a part in the map transformation process through reading, analysis, and interpretation, and high fidelity must also be maintained in these activities. For example, you as a user are responsible for checking the map against reality, as features may change from the time the map is created.

WHAT MAKES MAPS POPULAR? In scrutinizing the nature of maps, the obvious question is, what accounts for their widespread popularity? There are four main factors: Maps are convenient to use. Paper maps are usually small and flat for ease of storage and handling. Maps on computer screens and mobile devices are easily accessible and often integrated into other apps, such as an Internet search engine. Thus, maps bring reality into a less unwieldy proportion for study. Maps simplify our surroundings. Without them, our world often seems a chaos of unrelated phenomena that we must organize to understand our environment. The selection of information found on a map is clear at a glance. The world becomes intelligible. Web maps provide varying levels of detail, revealing more features and information as you zoom in. Maps are credible. They claim to show how things really are. The coordination between symbol and reality seems so straightforward that we are comfortable letting maps stand for the environment. When you manipulate maps, you expect the results to apply to your surroundings. Maps, even more than the printed word, impress people as authentic. We tend to accept the information on maps without question. This blind acceptance is potentially disastrous when using maps indiscriminately—for example, using GPS to navigate by car without checking other information such as road signs or even looking at your surroundings. Maps have strong visual impact. Maps create a direct, dramatic, and lasting impression of the environment. Their graphic form appeals to our visual sense. It is axiomatic that “seeing is believing.” Web maps can dynamically show change over time and space, and they also allow you to interactively change the display to show the map you want to see.

Figure I.4. The map transformation process (after Tobler 1979) begins with the collection of data about our environment. This data is transformed into a map through selection, classification, generalization, and symbolization. The map user completes the transformation process when he or she reads, analyzes, or interprets the information in the map.

These factors combine to make a map appealing and useful. Yet these same four factors, when viewed from a different perspective, can be seen as limitations. Take convenience. It’s what makes fast food popular. When we buy processed foods, we trade quality for easy preparation. Few would argue that the result tastes like the real thing made from fresh ingredients. The same is true of maps. We gain ease of handling and storage by creating a prepared image of the environment. This representation of reality is bound to make maps imperfect in many ways. Simplicity, too, can be seen as a liability as well as an asset. Simplification of the environment through mapping appeals to our limited ability to process information, while at the same time it reduces the complexity and—potentially—the intricacies that we need to understand. By using maps, you can reduce the overwhelming and confusing natural state of reality. But the environment remains unchanged. It’s just your view of it that lacks detail and complexity. You should also question the credibility of maps. The mapmaker’s invisible (to you) hand isn’t always reliable or rational. Some map features are distortions, others are errors, and still others are omitted through oversight or design. So many perversions of reality are inherent in mapping that the result is best viewed as an intricate, controlled representation. Maps are like statistics—people can use them to show whatever they want, and maps reflect the capabilities of their maker. Once a map is made, it may last hundreds of years, although the world keeps changing. For all these reasons, a map’s credibility is open to debate.

Also, be careful not to confuse maps’ visual impact with proof or explanation. Just because a map leaves a powerful visual impression doesn’t make it meaningful or insightful. A map is a snapshot of a portion of the environment at a point in time. From this single view, it is sometimes difficult or impossible to understand the processes that caused the patterns we can see on maps. For explanations, you must look beyond maps and confront the real world (as discussed in chapters 19 to 22).

FUNCTIONS OF MAPS Maps function as media for the communication of geographic information, and it is instructive to draw parallels between maps and other communication media. You can first think of maps as a reference library of geographic information. Maps that serve this function, called reference maps, are more efficient as geographic references of the locations of different features than are maps with a certain theme. They catalog feature types and record their geographic locations. Reference maps allow you to instantly see the relative position of features and estimate directions and distances between them. Explaining these spatial relationships among features in writing could take hundreds of pages. Maps can also function as an essay on a topic. Like a well-written theme, a map can focus on a specific subject and be organized to make the subject stand out above the geographic setting. We call maps that function as geographic essays thematic maps. Some thematic maps are presented as online story maps that combine interactive maps and multimedia content to lead a reader through a story. Maps are tools for navigation, equal in utility to a compass, GPS receiver, or mobile app. When you get into your car and drive across the city, you are navigating the roads. When you find your route on a subway or bus system, you are navigating a transit system. When you hike along a trail, you are navigating through a trail network. In the first case, you use a road map; in the second case, you use a transit map; and in the last case, you use a trail map. When you step into an airplane and fly to a distant city, you must use air navigation (assuming you are the pilot). And when you motor or sail between two destinations on a body of water, you are marine navigating. In these two cases, you use navigational charts to plan your route in advance and guide you on your trip. Maps are instruments of persuasion. Like a written advertisement or television endorsement, some maps persuade you to buy a particular product, make a certain business decision, or take a targeted political action. These maps are often more sales hype or propaganda than a graphic representation of the environment, and you should view such maps with caution. Although all maps incorporate elements of graphic design, these maps are often more artistic in nature, employing art for visual influence. Graphic embellishments are also placed on maps for pure visual entertainment. And, finally, maps can be used as media in the creation of fine art. Now we will take a closer look at maps with each of these different functions.

Reference maps The earliest known maps, dating back several thousand years, are the reference type. On reference maps, symbols are used to locate and identify prominent landmarks and other pertinent features. An attempt is made to be as detailed and spatially truthful as possible so

that the information on the map can be used with confidence. These maps have a basic “Here is found . . .” characteristic and are useful for looking up the location of specific geographic features. On reference maps, no particular feature is emphasized over the others. All features are given equal visual prominence as much as possible. The topographic map and remote sensor images (orthophotomaps, orthophotos, and aerial photographs, discussed in chapter 10) in figure I.3 are excellent examples of reference maps because they show a variety of phenomena, with about the same emphasis given to each one. Reference maps are often produced in national mapping series, such as the US Geological Survey topographic map series. The topographic map segment in figure I.3 is from such a series. Topographic maps show and label natural features that are found in the physical environment, including mountains, valleys, plains, lakes, rivers, and vegetation. They also show cultural features that are created by humans, such as roads, boundaries, transmission lines, and buildings. The geographic information on reference maps makes them useful to professional and recreational map users alike. These maps are used in engineering, energy exploration, natural resource conservation, environmental management, public works design, and commercial and residential planning. Maps for outdoor activities such as biking, camping, and fishing are often variations of reference maps, in which additional features, such as bike trails, campsites, and fishing spots, are added. Globes and atlases are reference maps that show natural and cultural features in more generalized form than topographic maps. School wall maps are another form of reference map, as are the road maps and recreation guides that are produced for different states (figure I.5).

Figure I.5. Reference map examples include (A) a world globe, (B) the northwest section of a US road and recreation atlas map, (C) an Esri basemap on a handheld mobile device, and (D) a section of the Oregon State Highway map. (B) Courtesy of Benchmark Maps, (C) courtesy of the Map Shop and Esri, and (D) courtesy of the Oregon Department of Transportation.

Thematic maps Unlike reference maps, which show many types of features but emphasize none over the other, thematic maps focus on a single type of feature that is the theme of the map. Whereas reference maps focus on the locations of different features, thematic maps emphasize the geographic distribution of the theme. A climate map (figure I.6A), which shows how average annual precipitation changes continuously across the state of Oregon, is a good example. A map that shows the areal extent of Oregon vegetation provinces (figure I.6B) likewise shows the geographic distribution of physical features. Many thematic maps show the geographic distribution of concepts that don’t physically exist on the earth. One example is a map that shows Oregon’s rural population with dots and its urban population with variations in circle sizes (figure I.6C). Another example is the Oregon population density map in figure I.6D. Although you can’t actually see population

density in the physical environment, maps that show the spatial distribution of such statistical themes are useful to experts in demographics and other fields. Thematic maps ask, “What if you want to look at the spatial distribution of some aspect of the world in this particular way?” Figure I.6D, for example, asks, “What if you want to look at population density by taking the 2010 population census totals for Oregon counties as truth, dividing each total by the area of the county to get a population density, and then generalizing the densities into seven arbitrarily defined categories?” Take another look at each map in figure I.6 and note how the county outlines are superimposed on each theme. Most thematic maps have similar limited reference information to give a geographic context to the theme. Be careful not to use the contextual reference information to find specific locations or make precise measurements. Remember, that’s not the intent of thematic maps; that’s what reference maps are for. When using thematic maps, focus on their function of showing the geographic distribution of the theme.

Navigation maps As you will see in chapter 14, several types of maps are specially designed to assist you in land, water, and air navigation. Many of these maps are called charts—maps created specifically to help the navigator plan voyages and follow the planned travel route.

Figure I.6. Thematic maps for Oregon show (A) annual precipitation as colored regions of a continuous surface, (B) vegetation provinces as uniform areas, (C) rural and urban population by dots and graduated circles, and (D) population density shown by uniform colors within counties. Courtesy of the Spatial Climate Service, Oregon State University.

A topographic map, such as the segment in figure I.7A, is not only a valuable reference map but also one of the most important land navigation tools. Hikers, off-road vehicle enthusiasts, and land management professionals use topographic maps to find their way across the land. The topographic map shows ground features such as roads, trails, lakes, and streams that are both landmarks and obstacles. Contours on the maps are equally important. These contour lines connect points of equal elevation (we describe contours in chapter 9), allowing you to determine elevation changes and estimate the slopes that you’ll encounter along a route (as you’ll learn in chapter 12). This information will help you

estimate the time and physical effort it will take to complete your trip. In addition, topographic maps are drawn on map projections that allow you to measure distances and directions between locations along your route (you’ll learn how to make these measurements in chapters 12 and 13). Nautical charts, such as the southeastern corner of the San Juan Islands, Washington, chart in figure I.7B, are maps that are created specifically for water navigation. Recreational and commercial boat navigators use the detailed shoreline, navigation hazard, and water depth information on the chart to plan the “tracks” that they will follow between ports or anchorages. As you will see in chapter 14, each chart is made using a special map projection that allows you to quickly and easily measure the distance and direction of each track. Another type of nautical chart, found in a Current Atlas (see chapter 14 for an example), gives you information about the currents you must deal with on a particular day and hour. This information is of critical importance in planning your time of departure and estimating the time of arrival at your destination. Aeronautical charts are maps designed for the air navigator. Figure I.7C is part of an aeronautical chart that covers the state of Washington, with detailed information for the TriCities Airport. Air navigation involves planning and following safe routes between airfields, and the chart is filled with information that is important to safe flying. In chapter 14, you’ll learn what these map symbols mean, and you will see how pilots use the information on the chart to find distances, directions, and travel times between destinations. Air navigation also involves maintaining a safe altitude above the ground, and you’ll see that aeronautical charts show the heights of towers and other obstructions to navigation, as well as contours and special ground elevation symbols that help navigators quickly determine the minimum safe in-flight altitude.

Figure I.7. (A) Topographic maps, (B) nautical charts, and (C) aeronautical charts are important tools of land, water, and air navigation, respectively. (A) Courtesy of the US Geological Survey, (B) reproduced with permission of the Canadian Hydrographic Service, and (C) courtesy of the US National Aeronautical Charting Office.

Story and persuasive maps Maps have probably played a role in one or more of your life stories, but story maps are specifically created to tell the story of a place, event, issue, trend, or pattern in a geographic context. Online story maps combine animated and interactive maps with text, photos, video, and audio to tell the story in a moving, compelling, and unforgettable fashion. Story maps such as the Decisive Moments in the Battle of Gettysburg Esri Story Map in figure I.8 deepen our understanding of key historical events in US history. Going to the website listed

in the figure caption, or scanning the QR (quick response) code in the lower-right corner, leads you to the home page, where you can begin the animation or interactively select a map that shows a particular time or event during the bloody three-day battle that left 51,000 Union and Confederate soldiers dead, maimed, or missing. The story of this historic battle is described through a sequence of events represented chronologically along a three-day timeline. Each point in the timeline represents a noteworthy event during the battle. Decisive events for the Confederates are shown in red; blue designates decisive events for the Union Army. Narratives to the left of the map further describe each event. You can also see a panorama of the events marked with red and blue numbers. These panoramas help you analyze the visibility during the events of the battle. To better understand what was visible to soldiers from different vantage points on the battlefield, you can also explore viewshed maps that show areas that are visible from a particular vantage point or points. Maps have always played a role in decision-making, and mapmakers can deliberately try to persuade you to choose a particular product or support a certain position. Some of these persuasive maps distort or misrepresent information to such an extent that they become propaganda maps. Such propaganda is common, especially in advertising, political, and religious maps. Because all maps distort reality, what is easier than to make this distortion serve a special purpose or agenda? Unless you know enough to question every map, how would you suspect anything is wrong? Now we look at several examples of persuasive maps that either are, or border on, propaganda. Nazi Germany was a source of blatant propaganda maps after the Third Reich created a Ministry of Propaganda in 1933, with the infamous Joseph Goebbels as its minister. Maps such as the one in figure I.9 were printed in newspapers to convince the German people that demilitarized post–World War I Germany was in a perilous state, ringed by threatening neighbors with huge (highly inflated numbers on the map) armies, armor, and air forces, all aimed at the heart of a defenseless homeland. Maps like this were important to Nazi consolidation of public opinion, unifying the German people under Hitler through fear by showing such an impending threat to their homeland. Another type of propaganda involves disproportionate-size symbols as a means of persuasion. Mapmakers must make symbols overly large in relation to the size of the feature on the ground, or the symbols won’t show up at reduced map scales. In propaganda mapping, mapmakers carry this normal aspect of cartographic symbolization to extremes.

Figure I.8. Still image from Decisive Moments in the Battle of Gettysburg Esri Story Map, at http://storymaps.esri.com/stories/2013/gettysburg.

Figure I.9. Nazi propaganda map from the November 1933 issue of the Illustrite Zeitung (Leipzig).

Take the two maps of Israeli military checkpoints in the West Bank that we made from information gathered in 2002 (figure I.10). Notice on the map on the left the large soldier symbol for each checkpoint. By using such large symbols, we make the West Bank appear crowded with military checkpoints. The map on the right has the same number and placement of soldier symbols, but they are one-quarter the size of those on the left map. On the map on the right, the West Bank appears far less crowded with checkpoints. This is an interesting example, because two maps that have the same information have opposite propagandizing effects—of increasing or decreasing the sense of safety for Israelis or intimidating Palestinians to a lesser or greater degree.

Figure I.10. Disproportionately large or small map symbols may create the impression that the West Bank was more or less crowded with Israeli military checkpoints than it actually was at the time when the information for the map was collected (2002).

Presenting a misleading number of features on a map is another tool of persuasion that leans toward propaganda. Nineteenth-century railroad maps such as figure I.11 are classic examples. The map scale is selectively enlarged along the artificially straight main line from Sault Saint Marie, Michigan, to Duluth, Minnesota. The map shows several dozen stops along the main and feeder lines, some of which are towns. By mapping every stop, the railroad company is trying to persuade investors, settlers, and riders to choose it over a competing railroad. The following commentary from the Inland Printer shows how far this practice of planned map distortion went: “This won’t do,” said the General Passenger Agent, in annoyed tones, to the mapmaker. “I want Chicago moved down here half an inch, so as to come on our direct route to New York. Then take Buffalo and put it a little farther from the lake.

“You’ve got Detroit and New York on different latitudes, and the impression that that is correct won’t help our road. “And, man, take those two lines that compete with us and make ’em twice as crooked as that. Why, you’ve got one of ’em almost straight. “Yank Boston over a little to the west and put New York a little to the west, so as to show passengers that our Buffalo division is the shortest route to Boston. “When you’ve done all these things I’ve said, you may print 10,000 copies—but say, how long have you been in the railroad business, anyway?” (New York Herald 1895)

Figure I.11. In this portion of the 19th-century Duluth South Shore & Atlantic railroad map, the area along Lake Superior is deliberately enlarged to show all stops along the line.

Map simplification can also be used for persuasive purposes. Figure I.12 shows two maps of illegal West Bank settlements in the mid-1980s. Both maps are redrawn from illustrations in the same weekly newsmagazine. The map on the left that shows Israel and the vicinity depicts 16 settlements, whereas the West Bank and vicinity map on the right shows 30 settlements. Which is correct? The truth, revealed deep in the magazine’s text, is that there were 45 illegal Israeli settlements in the West Bank region when these maps were made. The legend, at the least, should provide this information. The makers of these maps, whether intentionally or not, present a picture that favors the Israeli cause by simplifying features even more when mapping a larger geographic area at a smaller map scale (a map design practice discussed in chapter 6). You should never overlook the possibility of such political bias in mapping. Whether you realize it or not, mapmakers are constantly molding your attitude. Of course, they aren’t the only people guilty of persuasion and propaganda. But the effect of map propaganda is especially insidious because so many people believe that maps are neutral and unbiased. The consequences are sometimes dramatic: a year’s vacation is ruined, or a retirement nest egg is spent, on a land parcel in the middle of a swamp.

Maps as art Think about how you would describe the look of the Monterey Bay and Canyon perspective map in figure I.13. Does it have a certain beauty that makes it a work of art as well as a factual display of land elevations, ocean depths, and landform features such as mountains and underwater canyons? You can think of maps as having several characteristics and functions of art. To begin with, art, at its most basic, is a form of communication, and maps are an important form of visual communication, with the map as a graphic representation of the environment. Mapmaking is closely tied to a subset of visual communication called graphic design. Maps are created using graphic design guidelines for the selection of colors, point symbols, line work, typography, page layout, and other components of the map. For example, the progression of colors on the Monterey Bay map, from ocean depths to mountaintops, is based on accepted color design guidelines. We explain many of these graphic design guidelines for maps in chapters 6 through 9, knowing that you may be involved in creating your own maps as well as using existing maps in your work and recreation.

Figure I.12. The need for map simplification can easily be used to create a map that borders on propaganda. Here, West Bank settlements on maps of different scales are shown with differing numbers of dots. (These examples were redrawn from a weekly newsmagazine from the mid-1980s).

Figure I.13. Monterey Bay and Canyon, California, perspective map made in 2012 to support the National Oceanic and Atmospheric Administration’s tsunami program. Courtesy of the US National Geophysical Data Center.

Figure I.14. Downtown Chicago, Illinois, map created to promote an architects convention in the McCormick Center. © Map by Eureka Cartography, Berkeley, California.

Art can also have a propaganda function, and we described the maps in the previous section as different forms of propaganda art used to blatantly, or more subtly, influence our emotions and decision-making. Similarly, commercial art is designed to manipulate the viewer’s emotions toward buying a particular product or traveling to a particular destination using a particular type of transportation. For example, the map of downtown Chicago in figure I.14 was created to promote an architects convention in the McCormick Center. Maps

that serve as commercial art abound because maps are the “in” thing nowadays. Walking through a shopping mall, you may see maps on T-shirts, furniture fabric, cell phone covers, umbrellas, wall art, company logos, and other everyday merchandise. Art also has an entertainment function, much like art used in motion pictures and video games. Mapmakers sometimes add things to a map that are not tied to its theme and purpose but are included strictly for you to look at and enjoy. Information that further explains the map may also be surrounded with artistic embellishments—cartouches placed on reference maps from previous centuries as a classic example. A cartouche, such as the one in figure 1.15 that appeared in the lower-left corner of a 1707 reference map of Africa, is an enclosed area on the map that contains the title, author, general description, legend, scale bar, and so on. Rather plain-looking cartouches first appeared on maps in the 15th century, but by the 17th and 18th centuries, they were surrounded by ornately drawn pictures of natives, local animals and mythical beasts, fanciful landscapes, cherubs and other religious figures, and other eye-catching artistic creations not related to the content of the map. Eye-catching artistic embellishments are not limited to cartouches on maps of this period, but were also placed to fill in blank areas on the map. In figure I.16, detailed drawings of real and imaginary ships, a sea monster inserted above the ships, and two antelope-like land creatures are placed in blank areas on the land. This practice of filling blank areas on the map with fanciful drawings led Jonathan Swift of Gulliver’s Travels to quip in 1733: So geographers, in Afric-maps, With savage-pictures fill their gaps; And o’er unhabitable downs Place elephants for want of towns.

Figure I.15. Ornately embellished cartouche from the 1707 reference map of Africa by J. B. Homanno.

Figure I.16. Artistic embellishments of real and fanciful ships, a sea monster in the ocean, and antelope-like land creatures are drawn on this antique map of the California coastline and Coast Range.

Figure I.17. Matthew Cusick’s “Shauna” (2011) map collage art. Courtesy of the Pavel Zoubok Gallery.

What we call “fine art” embodies the personal functions of art such as self-expression, personal gratification, or creating an aesthetic experience for the artist and art viewers. There are artists who aren’t interested in making maps to show us how to navigate from place to place, or in helping us visualize the spatial distribution of quantitative data. These artists use maps as the physical media for their art, but the maps are more a way to explore the relationships in art between space, time, and our emotions.

Matthew Cusick’s map art (figure I.17) is a good current example. His artwork is based on mosaicking pieces of maps on a wooden board, revealing detailed recognizable images when viewed from a distance. His map pieces contain roads, rivers, municipal transit systems, and district outlines, creating an artistic collage when strategically assembled. Cusick notes that the map collage of “Shauna,” named for an adult entertainment star in the 1980s, is both an exploration of a controversial cultural archetype and of a place (Los Angeles) that attracts starry-eyed youths and transforms them through its darker culture of decadence, hopelessness, and desperation. Certain regions of Los Angeles, such as the San Fernando Valley, have become geographic centers of this darker side, according to Cusick, and the weblike intersection of the city’s highways in the collage are like arteries that extend from the hearts of American archetypes that are epitomized by Shauna.

MAP USE Map use is the process of obtaining useful information from one or more maps to help you understand the environment and improve your mental map. Map use consists of three main activities—reading, analysis, and interpretation. In map reading, you determine what the mapmakers have depicted and how they’ve gone about it. If you carefully read the maps in figure I.18, for instance, you can describe the maps as showing mortality for all types of cancer for males and females from 1980 to 1990 within data collection units called health service areas (HSAs). Reading the main map legends, you learn that the mortality rate is the number of deaths per 100,000 people, and that mortality rates are generalized into seven categories shown by a two-color sequence, with lighter to darker tones for each color. A dark-blue-green through dark-brown color progression represents low to high mortality on both maps, but the range for each category differs on the two maps. You must discover this range to understand that males have higher overall deaths from cancer. There is no explanation of how the category limits were selected, but the maps appear to have about the same number of health service areas in each category. Finally, the smaller legends tell you that areas with sparse mortality data are shown with a diagonal line pattern, and you can see a few patterned areas on both maps.

Figure I.18. White male and female 1980–1990 cancer mortality rates in the United States by health service area, adapted from maps of the same title in the Atlas of United States Mortality. Courtesy of the National Center for Health Statistics.

Part 1 of this book examines these and many other facets of map reading. You will become familiar with map scale, map projections and coordinate systems, land partitioning methods, different ways of portraying landforms, maps that show qualitative and quantitative information, and ways of expressing map accuracy. When you have a firm grasp of these map reading principles, you are ready to make your own maps following the basic map design guidelines presented in chapter 6. Learning to read the information in maps is only the first step. Your curiosity or work may lead you to go further and analyze the information in one or more maps. Part 2 of this book

is devoted to map analysis. In this stage of map use, you make measurements and look for spatial patterns. You have seen that topographic maps and navigational charts are tools for the measurement of distances, directions, and surface areas. Thematic maps are tools used to understand spatial patterns. Analysis of the spatial patterns on the maps in figure I.18 is particularly thought provoking. Although it’s comforting to believe that cancer is unpredictable, analysis of these maps shows that this belief is not true. If you focus on the patterns on these maps, you find that cancer mortality rates are far from random across the country. Regional clustering of high and low rates occurs for both males and females. For example, you see a cluster of high mortality in Kentucky and western West Virginia (the brown counties to the right of the center) and low mortality in Utah and southeastern Idaho (the blue-green counties toward the upper left). It’s unlikely that this high and low clustering of deaths occurred by chance. If you next focus on the spatial correspondence of mortality rates in the two maps, you find that eastern Kentucky and western West Virginia appear to have the highest cancer mortality for both men and women. The upper Midwest and Northeast, along with Northern California (the brown counties along the western coast), have the next highest overall male and female mortality. Some areas have high mortality rates for one sex but not the other. For instance, the Mississippi Delta region along the south-central coast has high mortality rates for males but not females. You may see clusters of high and low mortality on the two maps, but another map reader may not see things the same way. Quantitative measures of spatial patterns on a map and spatial association among patterns on two or more maps add rigor and repeatability to your map analysis, and we have devoted two chapters (chapters 17 and 18) to these important aspects of map analysis. After analyzing the maps in figure I.18 and finding spatial patterns of high and low cancer mortality, your curiosity may now be aroused still further. You may wonder how to explain the patterns and spatial correspondences between the two maps. Finding such explanations takes you into the realm of map interpretation, covered in part 3 of this book. To understand why things are related spatially, you must search beyond the maps. To do so, you may draw on your personal knowledge, fieldwork, written documents, interviews with experts, or other maps and images. In your search, you’ll find that cancer deaths are associated with many factors, including industrialized work environments, mining activities, chemical plants, urban areas, ethnic backgrounds, and personal habits of eating, drinking, and smoking. You’ll find that people die from cancer because of contaminants in their air, water, food, clothing, and building materials. Some local concentrations of high or low cancer mortality, however, don’t seem to fit this pattern, suggesting that there must be other causes or that people have migrated in or out of the area. Because maps reflect a variety of aspects of environmental knowledge, map use is intertwined with many disciplines. It is impossible to appreciate them in isolation. The more fields you study, the better you will be at using maps. Map interpretation grows naturally out of an appreciation of a variety of subjects. The reverse is also true. An appreciation of maps leads to a better understanding of the world around you—for the subject of maps, after all, is the world itself. This brings us to a final, important point. As you gain an understanding of map use, be careful not to confuse the mapped world with the real world. Remember, the reason you’re

using maps is to understand the physical and human environment. The ultimate aim of map use is to stimulate you to interact with your environment and to experience more while you do.

SELECTED READINGS Ames, G. P. 2003. “Forgetting St. Louis and Other Map Mischief.” Railroad History 188 (Spring– Summer): 28–41. Arnheim, R. 1969. Visual Thinking. Berkeley, CA: University of California Press. Balchin, W. G. V. 1976. “Graphicacy.” The American Cartographer 3 (1) (April): 33–38. Castner, H. W. 1990. Seeking New Horizons: A Perceptual Approach to Geographic Education. Montreal: McGill-Queen’s University Press. Dent, B. D. 1998. Cartography: Thematic Map Design, 5th ed. New York: McGraw-Hill. Downs, R. M., and D. Stea 1977. Maps in Mind: Reflections on Cognitive Mapping. New York: Harper & Row. Gershmehl, P. J., and S. K. Andrews. 1986. “Teaching the Language of Maps.” Journal of Geography 85 (6) (November–December): 267–70. Head, C. G. 1984. “The Map as Natural Language: A Paradigm for Understanding.” Cartographica 21:1– 32. Keates, J. S. 1996. Understanding Maps, 2nd ed. Essex: Addison Wesley Longman. Kitchin, R. M. 1994. “Cognitive Maps: What They Are and Why Study Them?” Journal of Environmental Psychology 14:1–19. Lloyd, R. 1997. Spatial Cognition: Geographical Environments. Boston: Kluwer Academic. Lynam, E. 1953. The Mapmaker’s Art. London: Patchwork Press. MacEachren, A. M. 1995. How Maps Work: Representation, Visualization and Design. New York: Guilford. Monmonier, M. 1991. How to Lie with Maps. Chicago: University of Chicago Press. Monmonier, M., and G. A. Schnell. 1988. Map Appreciation. Englewood Cliffs, NJ: Prentice Hall. New York Herald. 1895. “Some Railway Map-Making.” In Inland Printer 15:500. Pickle, L. W. 1996. Atlas of United States Mortality. Hyattsville, Md.: US Department of Health and Human Services. Robinson, A. H., and B. Bartz-Petchenik. 1976. The Nature of Maps: Essays toward Understanding Maps and Mapping. Chicago: University of Chicago Press. Swift, Jonathan. 1733. On Poetry: A Rhapsody. Tobler, Waldo. 1979. “A Transformational View of Cartography.” The American Cartographer 1 (2): 101– 6. Tufte, E. R. 1997. Visual Explanations: Images and Quantities, Evidence and Narrative. Cheshire, Conn.: Graphics Press. Wood, D. 2006. “Map Art.” Cartographic Perspectives 53:5–14.

Part I Map reading

Art, said Picasso, is a lie that makes us realize the truth. So is a map. We don’t usually associate the precise craft of the mapmaker with the fanciful realm of art. Yet a map has many of the ingredients of a painting or a poem. It is truth compressed in a symbolic way, holding meanings that it doesn’t express on the surface. And, like any work of art, it requires imaginative reading. Map reading involves determining what the mapmaker has depicted, how he or she has gone about it, and what artifacts of the cartographic method deserve special attention. To read a map, you translate its features into a mental image of the environment. The first step is to identify map symbols. The process is usually intuitive, especially if the symbols are self-evident and the map is well designed. As obvious as this step might seem, however, you should look first at the map and its marginalia (legend, text boxes, and so on), both to confirm the meaning of familiar symbols and to make sure that you understand the logic that underlies unfamiliar or poorly designed ones. This explanation is usually found in the legend, but sometimes explanatory labels are found on the map itself or in text boxes that provide the missing information. Too many people look for the symbol explanation only after becoming confused. Such a habit is not only inefficient but potentially dangerous. A better approach is to first check the marginalia, and then attempt to interpret the map. In addition to clarifying symbols, the map marginalia contains other information, such as scale, orientation, and data sources that are important to reading the map; and the marginalia sometimes includes unexpectedly revealing facts. But the marginalia is still only a starting point. The map reader must make a creative effort to translate the world that is represented on the map into an image of the real world, for there is often a large gap between the two. Much of what exists in the environment is left off the map, whereas many things on the map do not occur in reality but are instead interpretations of characteristics of the environment, such as population density or average streamflow. Thus, map and reality are not—and cannot be—identical. No aspect of map use is so obvious and yet so often overlooked. Most map reading mistakes occur because the user forgets this vital fact and expects a one-to-one correspondence between the map and reality. Because the exact duplication of a geographic area is impossible, a map is actually a metaphor. The mapmaker asks the map reader to believe that an arrangement of points, lines, and areas on a flat sheet of paper or a computer screen is equivalent to some facet of the real world in space and time. To gain a fuller understanding, the map reader must go beyond the graphic representation and study carefully what the symbols refer to in the real world. A map, like a painting, is just one interpreter’s version of reality. To understand a painting, you must have some idea of the techniques used by the interpreter—that is, the artist. You don’t expect a watercolor to look anything like an acrylic painting or a charcoal drawing, even if the subject matter of all three is identical, because the three media are different.

Therefore, the artistic techniques are varied. In the same way, the techniques used to create maps greatly influence the final portrayal. As a map reader, you must always be aware of the mapmaker’s invisible hand. Never use a map without asking yourself how it is biased by the methods that were used to make it. If the mapping process operates at its full potential, communication of environmental information takes place between the cartographer and the map reader. The mapmaker translates reality into the clearest possible picture that the map can give, and the map reader converts this picture back into a useful mental image of the environment. For such translation to occur, the map reader must know something about how maps are created. The complexities of mapping are easier to study if we break them up into simpler parts. Thus, we have divided part 1, “Map Reading,” into 11 chapters, each dealing with a different aspect of mapping. Chapter 1 examines geographic coordinate systems for the earth as a sphere, an oblate ellipsoid, and a geoid. Chapter 2 looks at ways of expressing and determining map scale. Chapter 3 introduces different map projections and the types of geometric distortions that occur with each projection. Chapter 4 focuses on different grid coordinate systems used in maps. Chapter 5 looks at land partitioning systems and how they are mapped. Chapter 6 is an introduction to how maps are designed. Chapter 7 examines various methods for mapping qualitative information, and chapter 8 does the same for quantitative information. Chapter 9 is devoted to the different methods of relief portrayal found on maps. Chapter 10 is an overview of image maps. Finally, for part 1, chapter 11 explores various aspects of map accuracy. These 11 chapters should give you an appreciation of all that goes into mapping and the ways that different aspects of the environment are shown in maps. As a result, you’ll better understand the large and varied amount of geographic information that you can glean from a map. In addition, once you realize how intricate the mapping process is, you won’t be able to help but view even the crudest map with more respect, and your map reading skill will naturally grow.

chapter one THE EARTH AND EARTH COORDINATES THE EARTH AS A SPHERE THE GRATICULE Parallels and meridians Latitude and longitude Prime meridians THE EARTH AS AN OBLATE ELLIPSOID Different ellipsoids Geodetic latitude Geodetic longitude DETERMINING GEODETIC LATITUDE AND LONGITUDE PROPERTIES OF THE GRATICULE Circumference of the authalic sphere Spacing of parallels Converging meridians Quadrilaterals Great and small circles GRATICULE APPEARANCE ON MAPS Small-scale maps Large-scale maps GEODETIC LATITUDE AND LONGITUDE ON LARGESCALE MAPS Horizontal reference datums THE EARTH AS A GEOID Vertical reference datums SELECTED READINGS

1 The earth and earth coordinates Of all the jobs that maps do for you, one stands out—they tell you where things are and allow you to communicate this information efficiently to others. This, more than any other factor, accounts for the widespread use of maps. Maps give you a superb positional reference system—a way to pinpoint the locations of things in space. There are many ways to determine the position of a feature shown on a map. All begin with defining a geometric figure that approximates the true shape and size of the earth. This figure is either a sphere (a three-dimensional solid in which all points on the surface are the same distance from the center) or an oblate ellipsoid (a slightly flattened sphere), or an ellipsoid for short, of precisely known dimensions. Once the dimensions of the sphere or ellipsoid are defined, a graticule of east–west lines called parallels and north–south lines called meridians is draped over the sphere or ellipsoid. The angular distance of a parallel from the equator and of a meridian from what we call the prime meridian (the zero meridian used as the reference line from which east and west are measured) gives us the latitude and longitude coordinates of a feature. The locations of elevations measured relative to an average gravity or sea level surface called the geoid can then be defined by three-dimensional (latitude, longitude, elevation) coordinates. Next, we look at these concepts in more detail.

THE EARTH AS A SPHERE We have known for over 2,000 years that the earth is spherical in shape. We owe this knowledge to several ancient Greek philosophers, particularly Aristotle (fourth century BC), who believed that the earth’s sphericity could be proven by careful visual observation. Aristotle noticed that as he moved north or south, the stars were not stationary—new stars appeared in the northern horizon while familiar stars disappeared to the south. He reasoned that this could occur only if the earth was curved north to south. He also observed that departing sailing ships, regardless of their direction of travel, always disappeared from view by their hull (bottom and sides) first. If the earth was flat, the ships would simply get smaller as they sailed away. Only on a sphere do hulls always disappear first. His third observation was that a circular shadow is always cast by the earth on the moon during a lunar eclipse, something that occurs only if the earth is spherical. These arguments entered the Greek literature and persuaded scholars over the succeeding centuries that the earth must be spherical in shape. Determining the size of our spherical earth was a daunting task for our ancestors. The Greek scholar Eratosthenes, head of the then-famous library and museum in Alexandria, Egypt, around 250 BC, made the first scientifically based estimate of the earth’s circumference. The story that has come down to us is of Eratosthenes reading an account of a deep well at Syene near modern Aswan, about 500 miles (800 kilometers) south of Alexandria. The well’s bottom was illuminated by the sun only on June 21, the day of the summer solstice. He concluded that the sun must be directly overhead on this day, with rays perpendicular to the level ground (figure 1.1). Then he reasoned brilliantly that if the sun’s rays are parallel and the earth is spherical, a vertical column such as an obelisk should cast a shadow in Alexandria on the same day. Knowing the angle of the shadow would allow the earth’s circumference to be measured if the north–south distance from Alexandria to Syene could be determined. The simple geometry involved here is that if two parallel lines are intersected by a third line, the alternate interior angles are equal. From this supposition, he reasoned that the shadow angle at Alexandria equals the angular difference (7°12´) at the earth’s center between the two places.

Figure 1.1. Eratosthenes’ method for measuring the earth’s circumference.

The story continues that on the next summer solstice, Eratosthenes measured the shadow angle from an obelisk in Alexandria, finding it to be 7°12´, or 1/50th of a circle. Hence, the distance between Alexandria and Syene is 1/50th of the earth’s circumference. He was told that Syene must be about 5,000 stadia south of Alexandria because camel caravans traveling at 100 stadia per day took 50 days to make the trip between the two cities. From this distance estimate, he computed the earth’s circumference as 50 × 5,000 stadia, or 250,000 stadia. A stadion is an ancient Greek unit of measurement based on the length of a sports stadium at the time. A stadion varied from 200 to 210 modern yards

(182 to 192 meters), so his computed circumference was somewhere between 28,400 and 29,800 modern statute miles (45,700 and 47,960 kilometers), 14 percent to 19 percent greater than the currently accepted circumference distance of 24,874 statute miles (40,030 kilometers). We now know that Eratosthenes’ error was because of an underestimate of the distance between Alexandria and Syene, and because the two cities are not exactly north–south of each other. However, his method is sound mathematically and was the best circumference measurement until the 1600s. Equally important, Eratosthenes had the idea that careful observations of the sun allowed him to determine angular differences between places on earth, an idea that you shall see was expanded to other stars and recently to the Global Navigation Satellite System (GNSS), commonly known as the Global Positioning System (GPS). GPS is a “constellation” of earth-orbiting satellites that make it possible for people to pinpoint geographic location and elevation with a high degree of accuracy. GPS uses ground receivers that digitally process electronic signals from the satellite constellation to provide the position and exact time at a location (see chapter 14 for more on GPS).

THE GRATICULE Once the shape and size of the earth were known, mapmakers required a system for defining locations on the surface. We are again indebted to ancient Greek scholars for devising a system for placing reference lines on the spherical earth.

Parallels and meridians Astronomers before Eratosthenes placed horizontal lines on maps to mark the equator (forming the circle around the earth that is equidistant from the North and South Poles) and the Tropics of Cancer and Capricorn (lines that mark the northernmost and southernmost positions in which the sun is directly overhead on the summer and winter solstices, respectively. (As we mentioned earlier, Syene is located almost on the Tropic of Cancer.) Later, the astronomer and mathematician Hipparchus (190–125 BC) proposed that a set of equally spaced east–west lines called parallels be drawn on maps (figure 1.2). To these lines, he added a set of north–south lines called meridians that are equally spaced at the equator and converge at the North and South Poles. We now call this arrangement of parallels and meridians the graticule. Hipparchus’s numbering system for parallels and meridians was, and still is, called latitude and longitude. We now look at latitude and longitude on a sphere, sometimes called geocentric latitude and longitude. Later in this chapter, we explain how geocentric latitude and longitude differ from geodetic latitude and longitude on an oblate ellipsoid.

Figure 1.2. Parallels and meridians.

Latitude and longitude Latitude on the spherical earth is the north–south angular distance from the equator to the place of interest (figure 1.3). The numerical range of latitude is from 0° at the equator to 90° at the poles. The letters N and S, such as 45° N for Fossil, Oregon, are used to indicate north and south latitudes, respectively. Instead of the letter S, you may see a minus sign (−) for south latitudes; however, a plus sign (+) is not used for north latitudes. Longitude is the angle, measured along the equator, between the intersection of the reference meridian, called the prime meridian, and the point at which the meridian for the feature of interest intersects the equator. The numeric range of longitude is from 0° to 180° east and west of the prime meridian, twice as long as parallels. East and west longitudes are labeled E and W, so that Fossil, Oregon, has a longitude of 120° W. As with south latitude, west longitude may also be indicated by a minus sign, but a plus sign is not used for east longitude.

Figure 1.3. Latitude and longitude on the sphere allow explicit identification of the positions of features on the earth.

Putting latitude and longitude together into what is called a geographic coordinate (such as 45° N, 120° W or 45°, −120°) pinpoints a place on the earth’s surface. There are several ways to write latitude and longitude values. The oldest is the Babylonian sexagesimal system of degrees (°), minutes (´), and seconds (˝), where there are 60 minutes in a degree and 60 seconds in a minute. For example, the latitude and longitude of the capitol dome in Salem, Oregon, is 44°56´18˝ N, 123°01´47˝ W (or 44°56´18˝, −123°01 ´47˝). Latitude and longitude can also be expressed in decimal degrees (DD) through equation (1.1):

where dd is the number of whole degrees, mm is the number of minutes, and ss is the number of seconds. For example, in equation (1.2):

Decimal degrees are often rounded to two decimal places, so the location of the Oregon state capitol dome is written in decimal degrees as 44.94, −123.03. If you can accurately define a location to the nearest one second of latitude and longitude, you can specify its location to within 100 feet (30 meters) of its true location on the earth.

Prime meridians The choice of prime meridian (the 0° meridian used as the reference from which longitude east and west are measured) is entirely arbitrary, because there is no physically definable starting point like the equator. In the fourth century BC, Eratosthenes selected Alexandria, Egypt, as the starting meridian for longitude; and in medieval times, the Canary Islands off the west coast of Africa were used because they were then the westernmost outpost of western civilization. In the 18th and 19th centuries, many countries used their capital city as the prime meridian for their national maps (see appendix B, table B.6 for a listing of historic prime meridians). You can imagine the confusion that must have existed when trying to locate places on maps from other countries. The problem was eliminated in 1884 when the International Meridian Conference selected as the international standard the British prime meridian, defined by the north–south optical axis of a telescope at the Royal Observatory in Greenwich, a suburb of London. This standard is called the Greenwich meridian. You may occasionally come across a historical map that uses one of the prime meridians listed in appendix B, at which time knowing the angular difference between the prime meridian used on the map and the Greenwich meridian becomes useful information. As an example, you might see in an old Turkish atlas that the longitude of Seattle, Washington, is 151°16´ W (based on the Istanbul meridian), and you know that the Greenwich longitude of Seattle is 122°17´ W. You can determine the Greenwich longitude of Istanbul through subtraction, in equation (1.3):

The computation is done more easily in decimal degrees, as described earlier, in equation (1.4):

THE EARTH AS AN OBLATE ELLIPSOID Scholars assumed that the earth was a perfect sphere until the 1660s, when Sir Isaac Newton developed the theory of gravity. Newton thought that gravity should produce a perfectly spherical earth if it was not rotating about its polar axis. The earth’s 24-hour

rotation, however, introduces outward centrifugal forces that are perpendicular to the axis of rotation, or the equatorial axis. The amount of force varies from zero at each pole to a maximum at the equator. Newton noted that these outward centrifugal forces counteract the inward pull of gravity, so the net inward force decreases progressively from the pole to the equator causing the earth to be slightly flattened. Slicing the earth in half from pole to pole would then reveal an ellipse with a slightly longer equatorial radius and slightly shorter polar radius; we call these radii the semimajor and semiminor axes, respectively (figure 1.4). If we rotate this ellipse 180° about its polar axis, we obtain a three-dimensional solid that we call an oblate ellipsoid.

Figure 1.4. The form of the oblate ellipsoid was determined by measurements of degrees at different latitudes, beginning in the 1730s. Its equatorial radius was about 13 miles (21 kilometers) longer than its polar radius. This figure is true to scale, but our eye cannot see the flattening because the difference between the north–south and east–west axis is so small.

The oblate ellipsoid is important because parallels are not spaced equally as they are on a sphere but decrease slightly in spacing from the pole to the equator. This variation in the spacing of parallels is shown in figure 1.5, a cross section of a greatly flattened oblate ellipsoid. We say that on an oblate ellipsoid the radius of curvature (the measure of how curved the surface is) is largest at the pole and smallest at the equator. Near the pole, the ellipse is flatter than near the equator. You can see that there is less curvature on the ellipse with a larger radius of curvature and more curvature with a smaller radius of curvature. The north–south distance between two points on the ellipsoidal surface equals the radius of curvature times the angular difference between the points. Because the radius of curvature on an oblate ellipsoid is largest at the pole and smallest at the equator, the north– south distance between points that are a degree apart in latitude should be greater near the pole than at the equator. In the 1730s, scientific expeditions to Ecuador and Finland measured the length of a degree of latitude at the equator and near the Arctic Circle, proving Newton to be correct. These and additional measurements in the following decades for other parts of the world allowed the semimajor and semiminor axes of the oblate ellipsoid to be computed by the early 1800s, giving about a 13-mile (21-kilometer) difference between the two, only one-third of 1 percent.

Different ellipsoids During the 19th century, better surveying equipment was used to measure the length of a degree of latitude on different continents. From these measurements, slightly different oblate ellipsoids, varying by only a few hundred meters in axis length, best fit the measurements. Table 1.1 is a list of these ellipsoids, along with their areas of usage. Note the changes in ellipsoid use over time. For example, the Clarke 1866 ellipsoid was the best fit for North America in the 19th century and was used as the basis for latitude and longitude on topographic and other maps produced in Canada, Mexico, and the United States from the late 1800s to about the late 1970s. By the 1980s, vastly superior surveying equipment, coupled with millions of observations of satellite orbits, allowed us to determine oblate ellipsoids that are excellent average fits for the entire earth. Satellite data is important because the elliptical shape of each orbit monitored at ground receiving stations mirrors the earth’s shape. The most recent of these ellipsoids, called the World Geodetic System of 1984 (WGS84), replaced the Clarke 1866 ellipsoid in North America and is used as the basis for latitude and longitude on maps throughout the world. In table 1.1, the WGS84 ellipsoid has an equatorial radius of 6,378.137 kilometers (3,963.191 miles) and a polar radius of 6,356.752 kilometers (3,949.903 miles). On this ellipsoid, the distance between two points that are one degree apart in latitude between 0° and 1° at the equator is 110.567 kilometers (68.703 miles), shorter than the 111.699-kilometer (69.407-mile) distance between two points at 89° and 90° north latitude. Hence, on equatorial- and polar-area maps of the same scale, the spacing between parallels is not the same, but the difference is small.

Figure 1.5. This north–south cross section through the center of a greatly flattened oblate ellipsoid shows that a larger radius of curvature at the pole results in a larger ground distance per degree of latitude relative to the equator. Table 1.1 Historical and current oblate ellipsoids

Geodetic latitude Geodetic latitude is defined as the angle made by the horizontal equator line and a line perpendicular to the ellipsoidal surface at the parallel of interest (figure 1.6). Geodetic latitude differs from latitude on a sphere because of the unequal spacing of parallels on the ellipsoid. Lines perpendicular to the ellipsoidal surface pass through the center of the earth only at the poles and the equator, but all lines perpendicular to the surface of a sphere pass through its center. This centricity is why the latitude defined by these lines on a sphere is called geocentric latitude.

Figure 1.6. Geocentric and geodetic latitudes of 45°. On a sphere, circular arc distance b–c is the same as circular arc distance c–d. On the greatly flattened oblate ellipsoid, elliptical arc distance b–c is less than elliptical arc distance c–d. On the WGS84 oblate ellipsoid, arc distance b–c is 4,984.94 kilometers (3,097.50 miles) and arc distance c–d is 5,017.02 kilometers (3,117.43 miles), a difference of about 32 kilometers (20 miles).

Geodetic longitude There is no need to make a distinction between geocentric longitude on the sphere and geodetic longitude on the ellipsoid. Both types of longitude are defined as the angular distance between the prime meridian and another meridian passing through a point on the earth’s surface, with the center of the earth as the vertex of the angle. Because of the geometric nature of the ellipsoid, the angular distance turns out to be the same as for geocentric longitude on the sphere.

DETERMINING GEODETIC LATITUDE AND LONGITUDE The oldest way to determine geodetic latitude is with instruments for observing the positions of celestial bodies. The essence of the technique is to establish celestial lines of position (east–west and north–south) by comparing the predicted positions of celestial bodies with their observed positions. A handheld instrument, called a sextant, was the tool historically used to measure the angle (or altitude) of a celestial body above the earth’s horizon (figure 1.7). Before GPS, it was the tool that nautical navigators used to find their way using the moon, planets, and stars, including the sun. Astronomers study and tabulate information on the actual motion of the celestial bodies that help pinpoint latitude. Because the earth rotates on an axis defined by the North and South Poles, stars in the Northern Hemisphere’s night sky appear to move slowly in a circle centered on Polaris (the North Star), which lies almost directly above the North Pole. A navigator needs only to locate Polaris to find the approximate North Polar axis. In addition,

because the star is so far away from the earth, the angle from the horizon to Polaris is approximately the same as the latitude (figure 1.8).

Figure 1.7. A sextant is used at sea to find latitude from the vertical angle between the horizon and a celestial body such as the sun, planet, or star. Courtesy of Dr. Bernie Bernard.

However, Polaris is actually 0.75 degrees from directly above the North Pole, circling the pole in a small circle that is 1.5 degrees in diameter. Precisely determining the spot near Polaris that is directly above the North Pole requires accurately knowing the time and using Polaris position data from the Astronomical Almanac, published jointly by the US Naval Observatory and Her Majesty’s Nautical Almanac Office in the United Kingdom. Errors in determining latitude can otherwise approach three-quarters of a degree, which translates into a ground position error of around 50 miles (80 kilometers). In the Southern Hemisphere, latitude is harder to determine by celestial measurement because there is no equivalent to Polaris directly above the South Pole. Navigators instead use a small constellation called Crux Australis (the Southern Cross) to serve the same function (figure 1.9). Finding south is more complicated because the Southern Cross is a collection of five stars that are part of the constellation Centaurus. The four outer stars form a cross, while the fifth, much dimmer star (Epsilon) is offset about 30 degrees below the center of the cross.

Figure 1.8. In the Northern Hemisphere, it is easy to determine your approximate latitude by observing the height of Polaris above your northern horizon. For example, at 50° N latitude, you will see Polaris at 50 degrees above the north horizon.

Longitude can be determined on land or sea by careful observation of time. In previous centuries, accurately determining longitude was a major problem in both sea navigation and mapmaking. It was not until 1762 that a clock that was accurate enough for longitude finding was invented by Englishman John Harrison. This clock, called a chronometer, was set to the time at Greenwich, England, now called Greenwich Mean Time (GMT), before departing on a long voyage. As you saw earlier, the prime meridian at 0° longitude passes through Greenwich, England. Therefore, each hour difference between your time and that of Greenwich is equivalent to 15 degrees of longitude from Greenwich. The longitude of a distant locale was found by noting the GMT at local noon (the highest point of the sun in the sky, found with a sextant). The time difference was simply multiplied by 15 to find the longitude.

Today, you can find your longitude in the field simply by looking at the value computed by your GPS receiver or GPS unit in your smartphone. The details of how GPS receivers find your latitude and longitude are found in chapter 14, but you can rest assured that the GPS coordinates are more accurate than the most carefully measured time differences made in the past using Harrison and similar chronometers.

PROPERTIES OF THE GRATICULE Circumference of the authalic sphere When determining latitude and longitude, we sometimes use the spherical approximation to the earth’s shape rather than the oblate ellipsoid. Using a sphere leads to simpler calculations, especially when working with small-scale maps of countries, continents, or the entire earth (see chapter 2 for more on small-scale maps). On these maps, differences between locations on the sphere and the ellipsoid are negligible. Cartographers use a value of the earth’s spherical circumference called an authalic sphere. The authalic (meaning “area preserving”) sphere is a sphere that has the same surface area as the oblate ellipsoid being used. The equatorial and polar radii of the WGS84 ellipsoid are what we use to calculate the radius and circumference of the authalic sphere that is equal to the surface area of the WGS84 ellipsoid. The computations involved are moderately complex and best left to a short computer program, but the result is a sphere of radius of 3,958.76 miles (6,371.017 kilometers) and circumference of 24,873.62 miles (40,030.22 kilometers).

Figure 1.9. The Southern Cross is used for navigation in the Southern Hemisphere. To approximately locate the point in the sky directly above the South Pole, extend the long (Gacrux to Acrux) axis of the Southern Cross 4.5 times the length of the

axis, and then draw in the sky a line perpendicular from the end of this line to the left that is the length of the short (Becrux to Delta) axis. Courtesy of Dr. Yuri Beletsky.

Spacing of parallels As you saw earlier, on a spherical earth the north–south ground distance between equal increments of latitude does not vary. However, it is important to know how you want to define that distance. As you will see next, there are different definitions for terms that you may take for granted, such as a “mile.” Using the authalic sphere circumference based on the WGS84 ellipsoid, latitude spacing is always 24,873.62 miles ÷ 360°, or 69.09 statute miles per degree. Statute miles of 5,280 feet in length are used for land distances in the United States, whereas nautical miles are used around the world for maritime and aviation purposes. The original nautical mile was defined as one minute of latitude measured north–south along a meridian. The current international standard for a nautical mile is 6,076.1 feet (about 1.15 statute miles) so that there are 60.04 nautical miles per degree of latitude on the authalic sphere. The metric system is used to express distances in countries other than the United States. The kilometer (1,000 meters) is, like the nautical mile, closely tied to distances along meridians, because the meter was initially defined as one 10-millionth of the distance along a meridian from the equator to the North or South Pole. There are exactly 1,609.344 meters in a statute mile and 1,852 meters in a nautical mile (see table B.1 in appendix B for these and additional metric and English unit distance equivalents). Expressed in metric units, there are 111.20 kilometers per degree of latitude on the authalic sphere. You have seen that parallels on an oblate ellipsoid are not spaced equally as they are on the authalic sphere, but instead decrease slightly from the pole to the equator. You can see in table B.2 in appendix B the variation in the length of a degree of latitude for the WGS84 ellipsoid, measured along a meridian at one-degree increments from the equator to the pole. The distance per degree of latitude ranges from 69.407 statute miles (111.699 kilometers) at the pole to 68.703 statute miles (110.567 kilometers) at the equator. The graph in figure 1.10 shows how these distances differ from the constant value of 69.09 statute miles (111.20 kilometers) per degree for the authalic sphere. The WGS84 ellipsoid distances per degree are about 0.3 miles (0.48 kilometers) greater than the sphere at the pole and 0.4 miles (0.64 kilometers) less at the equator. The ellipsoidal and spherical distances are almost the same in the mid-latitudes, somewhere between 45 and 50 degrees.

Converging meridians A quick glance at any world globe (or figure 1.2) shows converging meridians—that is, the meridians progressively converge from the equator to a point at the North and South Poles. You will also see that the length of a degree of longitude, measured east–west along parallels, decreases from the equator to the pole. The spacing of meridians on the authalic sphere at a given latitude is found by using equation (1.5):

Figure 1.10. Distances along the meridian for one-degree increments of latitude from the equator to the pole on the WGS84 ellipsoid and authalic sphere.

For example, at 45 degrees north or south of the equator, cosine(45°) = 0.7071. Therefore, the length of a degree of longitude is found in equation (1.6):

This spacing of meridians is roughly 20 miles (32 kilometers) shorter than the 69.09-mile (111.20-kilometer) spacing at the equator.

Quadrilaterals Many navigational maps cover quadrilaterals, which are bounded by equal increments of latitude and longitude. Because meridians converge toward the poles, the shapes of the quadrilaterals vary from a square on the sphere at the equator to a narrow spherical triangle at the pole, such as the 15° × 15° quadrilaterals from equator to pole in figure 1.11. The equation cosine of center latitude, or cos(center latitude), gives the aspect ratio (width to height) of a quadrilateral. A quadrilateral centered at the equator has an aspect ratio of 1.0, whereas a quadrilateral at 60° N has an aspect ratio of 0.5. A map that covers 1° × 1° or any other quadrilateral extent looks long and narrow at this high latitude, whereas a quadrilateral map at the equator is essentially square in shape.

Figure 1.11. Aspect ratios for 15° × 15° quadrilaterals on the earth from the equator to the North Pole. The cosine of the mid-latitude for each quadrilateral gives its aspect ratio.

Great and small circles A great circle is the largest possible circle that can be drawn on the surface of the spherical earth and also the shortest distance between two points on the earth’s surface. Its circumference is that of the sphere and its center is the center of the earth, so that all great circles divide the earth into halves. Notice in figure 1.3 that the equator is a great circle that divides the earth into the Northern and Southern Hemispheres. Similarly, the prime meridian and its antipodal meridian, situated at the opposite side of the earth at 180°, form a great circle that divides the earth into the Eastern and Western Hemispheres. All other pairs of meridians and their antipodal meridians are also great circles. Because a great circle is the shortest route between any two points on the earth, great circle routes are fundamental to long-distance navigation, as you will see in chapter 14.

Any circle on the earth’s surface that intersects the interior of the sphere at any location other than the center is called a small circle, and its circumference is smaller than a great circle. You can see in figure 1.3 that all parallels other than the equator are small circles. The circumference of a particular parallel is given by equation (1.7):

For example, the circumference of the 45th parallel is 24,874 × 0.7071, or 17,588 statute miles (40,030 × 0.7071, or 28,305 kilometers).

GRATICULE APPEARANCE ON MAPS Small-scale maps Small-scale world or continental maps such as globes and world atlas sheets normally use geocentric latitude-longitude coordinates based on an authalic sphere. There are several reasons for using the authalic sphere. Prior to using digital computers to make these types of maps numerically, it was much easier to construct them from spherical coordinates. Equally important, the differences in the plotted positions of geocentric and corresponding geodetic parallels become negligible on maps that cover so much area on a small page. For example, mathematicians have computed the maximum difference between geocentric and geodetic latitude as 0.128 degrees at the 45th parallel. If you draw parallels at 45° and 45.128° on a map scaled at one inch per degree of latitude, the two parallels are drawn a very noticeable 0.128 inches (0.325 centimeters) apart. Now imagine drawing the parallels on a map scaled at one inch per 10 degrees of latitude, a scale that corresponds to a world wall map approximately 18 inches (46 centimeters) high and 36 inches (92 centimeters) wide. The two parallels are now drawn 0.013 inches (0.033 centimeters) apart, a difference that is not even noticeable considering the width of a line on a piece of paper.

Large-scale maps Parallels and meridians are shown in different ways on large-scale maps that show small areas in great detail (see chapter 2 for more on large-scale maps). Large-scale topographic maps, which show elevations and landforms as well as other ground features, use tick marks to show the location of the graticule (see chapters 2, 9, and 19 for more on topographic maps). For example, all US Geological Survey 7.5-minute topographic maps have graticule ticks at 2.5-minute intervals of latitude and longitude (figure 1.12). The latitude and longitude coordinates are printed in each corner, but only the minutes and seconds of the intermediate edge ticks are shown. Note the four plus sign symbols used for the interior 2.5-minute graticule ticks in the figure.

Figure 1.12 Graticule ticks on the Corvallis, Oregon, 7.5-minute topographic map.

The graticule is shown in a different way on nautical charts for marine navigation. The chart segment in figure 1.13 shows that alternating white and dark bars spaced at the same increment of latitude and longitude line the edge of the chart, along with ticks at one-minute increments. These ticks are used to find the latitude and longitude of mapped features to within a fraction of a minute. Because of the convergence of meridians, the spacing between ticks, which shows equal increments of latitude, is longer within the vertical bars on the left and right edges of the chart than the spacing between ticks within the horizontal bars at the top and bottom. The one exception is charts of areas along the equator, in which the spacing between ticks is the same on all edges. The more closely spaced ticks beside each horizontal bar are placed every 10th of a minute.

Figure 1.13. Graticule bars with ticks on the edges of a nautical chart segment. Reproduced with permission of the Canadian Hydrographic Service.

Aeronautical charts display the graticule in yet another way. The chart segment for a portion of the Aleutian Islands in Alaska (figure 1.14) shows that parallels and meridians are drawn at 30-minute latitude and longitude intervals. Ticks are placed at one-minute increments along each graticule line, allowing features to be located easily to within a fraction of a minute.

GEODETIC LATITUDE AND LONGITUDE ON LARGE-SCALE MAPS You will often find parallels and meridians of geodetic latitude and longitude on detailed maps of small areas. Geodetic coordinates are used to make the map a close approximation to the size and shape of the part of the ellipsoidal earth that it represents. To see the perils of not doing this step, you need only examine one-degree quadrilaterals at the equator (from 0° to 1° in latitude) and at the North or South Poles (from 89° to 90° in latitude).

Figure 1.14. Graticule ticks on a small segment of an aeronautical chart. Courtesy of the National Aeronautical Charting Office.

You can see in table B.2 in appendix B that the ground distance between these pairs of parallels on the WGS84 ellipsoid is 68.703 and 69.407 miles (110.567 and 111.699 kilometers), respectively. If the equatorial quadrilateral is mapped at a scale so that it is 100 inches (254 centimeters) high, the polar quadrilateral mapped at the same scale is 101 inches (256.5 centimeters) high. If both quadrilaterals are mapped using the authalic sphere with 69.09 miles (111.20 kilometers) per degree, both quadrilaterals are 100.6 inches (255.5 centimeters) high. Having both maps several 10ths of an inch (or around a centimeter) longer or shorter than they should be seems a small error, but it can be an unacceptably large error for maps that are used to make accurate measurements of distance, direction, or area. Yet you can see in figure 1.10 that the height differences at the equator and the poles are the extremes and that there is little difference at mid-latitudes.

Horizontal reference datums To further understand the use of different types of coordinates on detailed maps of smaller extents, you must first look at datums—the collection of accurate control points (points with known positional accuracy) that surveyors and others use to georeference map data so that it aligns with the geodetic latitude and longitude coordinate system (see chapter 5 for more on control points and georeferencing). Surveyors determine the precise geodetic latitude and longitude of horizontal control points spread across the landscape. You may have seen a horizontal control point monument (also called a survey marker, survey mark, or sometimes a geodetic mark), a fixed object established by surveyors when they determine the exact position of a point. Monuments, such as the one in figure 1.15, may be placed in the ground on top of a hill or other prominent feature, but in less prominent locations such as a road or sidewalk, they may be buried so that they will not be hit by a car or bicycle. From the 1920s to the early 1980s, these control points were surveyed relative to the Clarke 1866 ellipsoid, together forming what was called the North American Datum of 1927 (NAD27). Topographic maps, nautical and aeronautical charts, and many other large-scale maps of this period have graticule lines or ticks based on this datum. For example, the southeast corner of the Corvallis, Oregon, topographic map first published in 1969 (figure 1.16) has a NAD27 latitude and longitude of 44°30´N, 123°15´ W.

Figure 1.15 Horizontal control point marker cemented in the ground. Courtesy of the National Oceanic and Atmospheric Administration.

Figure 1.16 Southeast corner of the Corvallis, Oregon, topographic map showing the difference between its NAD27 and NAD83 positions. Courtesy of the US Geological Survey.

By the early 1980s, better knowledge of the earth’s shape and size and far better surveying methods led to the creation of a new horizontal reference datum, the North American Datum of 1983 (NAD83). The NAD27 control points were corrected for surveying errors where apparent, and then were added to thousands of more recently acquired points. The geodetic latitudes and longitudes of all these points were determined relative to the Geodetic Reference System of 1980 (GRS80) ellipsoid, which was essentially identical to the WGS84 ellipsoid when GRS80 was devised. The change of horizontal reference datum from NAD27 to NAD83 meant that the geodetic coordinates for control points across the continent changed slightly in 1983, and this change had to be shown on large-scale maps published earlier but still in use. On topographic maps, the NAD83 position of the map corner is shown by a dashed plus sign, as in figure 1.16. Often, the shift is in the 100-meter range and must be taken into account when plotting on older maps the geodetic latitudes and longitudes obtained from GPS receivers and other modern position-finding devices. Europe in the early 1900s faced another problem—separate datums for different countries that did not mesh into a single system for the continent. Military map users in World War II found different latitudes and longitudes for the same ground locations on topographic maps along the borders of France, Belgium, the Netherlands, Spain, and other countries in which major battles were fought. The European Datum of 1950 (ED50) was created after World War II as a consistent reference datum for most of western Europe; although Belgium, France, Great Britain, Ireland, Sweden, Switzerland, and the Netherlands continue to retain and use their own national datums. Latitudes and longitudes for ED50 were based on the International Ellipsoid of 1924. Users of GPS receivers will find that, moving westward through Europe from northwestern Russia, the newer European Terrestrial Reference System 1989 (ETRS89) longitude coordinates based on the WGS84 ellipsoid gradually shift to the west of those based on the 1924 International Ellipsoid. In Portugal and western Spain, the WGS84 longitudes are approximately 100 meters to the west of those found on topographic maps based on ED50. Moving southward, WGS84 latitudes gradually shift northward from those based on ED50, reaching a maximum difference of around 100 meters in the Mediterranean Sea. As noted, Great Britain and Ireland are examples of countries that continue to use datums based on ellipsoids defined in the 19th century to best fit that region. Topographic maps in both nations use the Airy 1830 ellipsoid as the basis for the Ordnance Survey Great Britain 1936 (OSGB36) datum for geodetic latitude and longitude coordinates. Along the south coast of England, WGS84 latitudes are about 70 meters to the south of those based on OSGB36. This southward shift gradually diminishes to zero near the Scottish border, and then becomes a northerly difference that reaches a maximum value of around 50 meters at the northern extremes of Scotland. In Ireland, WGS84 longitudes are around 50 meters to the east of their OSGB36 equivalents and gradually increase to a maximum difference of around 120 meters along the southeast coast of England (go to http://www.colorado.edu/geography/gcraft/notes/datum/datum_f.html for a detailed list of horizontal datums and reference ellipsoids used in foreign countries throughout the world).

THE EARTH AS A GEOID

When the earth is treated as an authalic sphere or oblate ellipsoid, mountain ranges, ocean trenches, and other surface features that have vertical relief are neglected. There is justification for this treatment, as the earth’s surface is truly smooth when you compare the surface undulations to the 7,918-mile (12,742-kilometer) diameter of the earth based on the authalic sphere. The greatest relief variation is the approximately 12.3-mile (19.8-kilometer) difference between the summit of Mount Everest (29,035 feet or 8,852 meters) and the deepest point in the Mariana Trench (36,192 feet or 11,034 meters). This vertical difference is immense on the human scale, but it is only 1/640th of the earth’s diameter. If you look at the difference between the earth’s average land height (2,755 feet or 840 meters) and ocean depth (12,450 feet or 3,795 meters), the average roughness is only 1/2,750th of the diameter. In fact, if the earth was reduced to the diameter of a bowling ball, it would be smoother than the bowling ball. The earth’s global-scale smoothness aside, knowing the elevations and depths of features is important to us. Defining locations by their geodetic latitude, longitude, and elevation gives you a simple way to collect elevation data and display this information on maps. The top of Mount Everest, for example, is located at 27°59´ N, 86°56´ E, 29,035 feet (8,852 meters), but what is this elevation relative to? This question leads us to another approximation of the earth called the geoid, which is a surface of equal gravity used as the reference datum for elevations.

Vertical reference datums Elevations and depths are measured relative to what is called a vertical reference datum, an arbitrary surface with an elevation of zero. The traditional datum used for land elevations is mean sea level (MSL) (see chapter 9 for more on mean sea level). Surveyors define MSL as the average of all low and high tides at a particular starting location over a Metonic cycle (a period of approximately 19 years or 235 lunar months, at the end of which the phases of the moon begin to occur in the same order and on the same days as in the previous cycle). Early surveyors chose this datum because of the measurement technology of the day. Surveyors first used the method of leveling, in which elevations are determined relative to the point at which mean sea level is defined, using horizontally aligned telescopes and vertically aligned leveling rods. A small circular monument was placed in the ground at each surveyed benchmark elevation point. A benchmark is a permanent monument that establishes the exact elevation of a place. Later, surveyors could determine elevation by making gravity measurements at different locations on the ground and relating them to the strength of gravity at the point used to define MSL. Gravity differences translate into elevation differences because the strength of gravity changes with elevation. MSL is easy to determine along coastlines, but what about inland locations? It requires extending mean sea level across the land. Imagine that mean sea level is extended across the continental land masses on the ellipsoidal surface, which is the same thing as extending a surface that has the same strength of gravity as mean sea level (figure 1.17). However, this imaginary equal-gravity surface doesn’t form a perfect ellipsoid, because differences in topography and earth density affect gravity’s pull at different locations, and thus the shape of the surface. The slightly undulating, nearly ellipsoidal surface that best fits mean sea level for all the earth’s oceans is called a global geoid. The global geoid varies approximately 100 meters above and below the oblate ellipsoid surface in an irregular fashion. World maps

that show land topography and ocean bathymetry use land heights and water depths relative to the global geoid surface. In the conterminous United States, geoid heights range from a low of −51.6 meters in the Atlantic Ocean to a high of −7.2 meters in the Rocky Mountains (figure 1.18). Worldwide, geoid heights vary from −105 meters just south of Sri Lanka to 85 meters in Indonesia.

Figure 1.17 The geoid is the surface at which gravity is the same as at mean sea level. Elevations on maps are measured relative to the global geoid, but modern GPS-determined heights are relative to the WGS84 ellipsoid.

The mean sea level datum based on the geoid is so convenient that it is used to determine elevations around the world and is the basis for the elevation data found on nearly all topographic maps and nautical charts. But be aware that the local geoid used in your area is probably slightly above or below (usually within two meters of) the global geoid used for world maps. This difference is because the mean sea level at one or more nearby locations is being used as the vertical reference datum for your nation or continent, rather than the average sea level for all the oceans.

Figure 1.18 Geoid heights in the United States and vicinity. Courtesy of the National Geodetic Survey GEOID2009 model.

In the United States, for example, you may see elevations relative to the National Geodetic Vertical Datum of 1929 (NGVD29) on older topographic maps. This datum was defined by the observed heights of mean sea level at 26 tide gauges, 21 in the United States and 5 in Canada. It also was defined by the set of elevations of all benchmarks resulting from over 60,000 miles (96,560 kilometers) of leveling across the continent, totaling over 500,000 vertical control points. In the late 1980s, surveyors adjusted the 1929 datum with new data to create the North American Vertical Datum of 1988 (NAVD88). Topographic maps, nautical charts, and other cartographic products for the United States, Canada, Mexico, and Central America made from this time forward use elevations based on NAVD88. Mean sea level for the continent was defined at one tidal station on the Saint Lawrence River at Rimouski, Quebec, Canada. NAVD88 was a necessary update of the 1929 vertical datum because about 400,000 miles (650,000 kilometers) of leveling was added to the NGVD since 1929. Additionally, numerous benchmarks were lost over the decades, and the elevations at others were affected, by vertical changes because of rising land elevations since the retreat of glaciers at the end of the last ice age or subsidence from the extraction of natural resources such as oil and water. GPS has created a second option for measuring elevation rather than basing elevation on a vertical datum (see chapter 14 for more on GPS). GPS receivers calculate what is called the ellipsoidal height h, the distance above or below the surface of the WGS84 ellipsoid along a line from the surface to the center of the earth (see figure 1.17). An ellipsoidal height is not an elevation, because it is not measured relative to the mean sea level datum for your local geoid. Therefore, you must convert GPS ellipsoidal height values to mean sea level datum elevations H before you can use them with existing maps. You do this by subtracting the geoid height N at each point from the ellipsoid height h measured by the GPS receiver using equation (1.8):

Ellipsoidal height is the same thing as height above the ellipsoid (HAE), and this is the elevation most often found by GPS receivers. The lookup table to make this conversion is usually stored in your GPS receiver’s computer.

SELECTED READINGS Greenhood, D. 1964. Mapping. Chicago: University of Chicago Press. Iliffe, J. C. 2000. Datums and Map Projections for Remote Sensing, GIS and Surveying. Caithness, Scotland: Whittles. La Condamine, C.M. de. 1747. A Succinct abridgement of a Voyage Made with the Inland Parts of South America, from the Coasts of the South-Sea, to the Coasts of Brazil and Guiana, down the River of Amazons: As It Was Read in the Public Assembly of the Academy of Sciences at Paris. London: E. Withers, at the Seven Stars. Maling, D. H. 1992. Coordinate Systems and Map Projections, 2nd ed. New York: Pergamon. Maupertius, P. L. M. de. 1738. The Figure of the Earth Determined from Observations Made by Order of the French King, at the Polar Circle (translation from French). London. Meade, B. K. 1983. “Latitude, Longitude, and Ellipsoidal Height Changes NAD-27 to Predicted NAD-83.” Surveying and Mapping 43:65–71. Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. “Basic Geodesy.” Chap. 4 in Elements of Cartography, 6th ed. New York: John Wiley & Sons. Smith, J. R. 1988. Basic Geodesy: An Introduction to the History and Concepts of Modern Geodesy without Mathematics. Rancho Cordova, CA: Landmark Enterprises. Snyder, J. P. 1987. “Map Projections: A Working Manual.” US Geological Survey Professional Paper 1395. Washington, DC: US Government Printing Office. Sobel, D. 1995. Longitude: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time. New York: Walker. Wallis, H. M., and A. H. Robinson, eds. 1987. Cartographical Innovations. London: Map Collector. Wilford, J. N. 1981. The Mapmakers. New York: Alfred A. Knopf.

chapter two MAP SCALE EXPRESSING SCALE Representative fraction Word scale Scale bar LARGE- AND SMALL-SCALE MAPS Topographic map series at different scales USGS map series Canadian map series Multiscale maps CONVERTING SCALE DETERMINING MAP SCALE Determining map scale from a known ground feature Determining map scale from the equator Determining map scale from reference material Determining map scale from ground resolution and pixel density Determining map scale from the spacing of parallels and meridians SELECTED READINGS

2 Map scale Maps are always smaller in size than the environment they represent. The amount of size reduction is known as the map scale, which tells you the relationship between distances on the map and their corresponding ground distances. To use maps effectively, you must convert measurements from map units to ground units. As you might expect, an understanding of map scale is central to performing this task. In this chapter, we explore what you need to know about map scale to become a skilled map user.

EXPRESSING SCALE Map scale is almost always given in the form: “This distance on the map represents this distance on the earth’s surface.” The relationship between map and ground distance can be expressed in a number of ways—most commonly as a representative fraction, word scale, or scale bar.

Representative fraction A common way to describe scale is to use a representative fraction (RF), which is the ratio between map and ground distance. The RF simplifies calculations involving scale. An RF is written as either 1/x or 1:x. The numerator is always 1 and represents map distance, while the denominator x indicates distance on the ground in the same units of measurement as the distance on the map. Therefore, in equation (2.1): 1/x = map distance / ground distance,

and

The advantage of having identical units on the top and bottom of the fraction is that map measurements can be made in centimeters, inches, or whatever distance unit you choose. For example, an RF of 1/24,000 or 1:24,000 means that one inch on the map represents 24,000 inches on the ground, one centimeter on the map represents 24,000 centimeters on the ground, and so on. You can also say that the scale reduction is 24,000 to 1.

Word scale Another familiar way to express scale is to use a descriptive word scale. The word scale expresses large RF denominator values in more familiar (larger) units of measurement. We express the scale in words as so many “inches to one mile” or “centimeters to one kilometer.” An RF of 1:24,000 can be expressed using a word scale as “1 inch to 2,000 feet” because 2,000 feet equals 24,000 inches, and an RF of 1:100,000 can be expressed as “one centimeter to one kilometer” because there are 100,000 centimeters in a kilometer. These expressions are also sometimes seen as “inches to the mile” or “centimeters to the kilometer” statements. Word scales can also be expressed as so many “miles to one inch” or “kilometers to one centimeter.” Word scales for commonly used map scales are given in table 2.1, along with examples of maps published at each scale. At first, it may be confusing to find that one map indicates scale as “one centimeter to one kilometer” and another as “four miles to one inch.” This lack of consistency should cause little trouble, however, because the smaller unit of measurement (inches or centimeters) always refers to the map whereas the larger measurement unit refers to the ground. Many word scales are only close approximations to the RF and should not be used in mathematical calculations. An RF of 1:250,000 is really “3.9457 miles to 1 inch,” but the word scale is much easier to remember if it is rounded to four miles. The reason that the values can be rounded is because of the way in which they are used. Word scales are often used with rulers to determine the length of a feature. Because the divisions of a ruler are

fairly large, to state the map scale using multiple decimal places is no more useful than stating the rounded values. However, if you want to indicate more precision, you can use more decimal places in the word scale.

Scale bar A third way to show map scale is to use a scale bar (also called a bar scale). A scale bar looks like a small ruler that can be placed on the map to assist with map distance measurements. You’ll usually read the scale bar from left to right, beginning at zero. Sometimes the scale bar is extended to the left of the zero point, using smaller markings than on the right (figure 2.1). This scale bar extension allows you to determine distance not only in whole units, but also in fractions of units such as 10ths of a mile or kilometer. The marks on the scale bar are arranged to provide whole numbers or fractions of miles or kilometers of ground distance. This method of marking means that the marks don’t necessarily represent whole numbers of centimeters or inches—some fraction will almost always be left over. In other words, although the scale bar looks like a ruler, its markings do not coincide with those on your ruler. Rare exceptions are a scale of 1:100,000, because at this scale one centimeter on the map is exactly one kilometer on the ground, and 1:63,360, because one inch on the map is exactly 63,360 inches, or one mile, on the ground. The scale bar has three features that make it especially useful. First, if the map is enlarged or reduced using photocopying or screen display, the scale bar changes size in direct proportion to the physical size of the map. However, the word scale and representative fraction are incorrect when the map size changes. Second, both kilometers and miles can be shown conveniently on the same scale bar (see bottom of figure 2.1). This is called a stacked scale bar because the two scale bars using different units are stacked one on top of the other. And finally, the scale bar is easy to use when figuring distances on a map, as you’ll see in chapter 12. Table 2.1 Commonly used ways of expressing map scale

Figure 2.1. Scale bars in kilometers, statute miles, and nautical miles taken from maps identical in scale, as well as a stacked scale bar that shows both kilometers and miles. The scale bar extensions to the left of the zero point are used when making more precise distance measurements.

Figure 2.2. Variable scale bar that might be seen on a map using the Mercator projection.

On one special map, the standard scale bar is sometimes replaced with a variable scale bar. Figure 2.2 shows an example of this type of scale bar. The only type of map in which it is appropriate to use a variable scale bar is a map in which scale varies systematically in the north–south direction (along the parallels) and the projection is conformal so that distortion in any direction is the same at a given latitude (see chapter 3 for more on map projections). The only map that meets these qualifications is a map made using the Mercator projection. If you have seen a variable scale bar on a map with any other type of map projection, it was added to the map incorrectly. When you use a variable scale bar, you are working with a scale bar that is stretched to match the local map scale. First, decide at what latitude you want to use the variable scale bar, and then find the scale bar for this latitude on the variable scale bar. If the latitude falls between two of the scale bars, you can add a scale bar for the latitude as a horizontal line that you place in the correct vertical position on the variable scale bar. For example, you draw the scale bar for 45° latitude as a horizontal line halfway between the 40° and 50° scale bars. You then use the scale bar as a ruler in the manner described in chapter 14.

LARGE- AND SMALL-SCALE MAPS Because map distance is always stated in the numerator of the RF as 1, it follows that the smaller the denominator, the closer to a 1:1 ratio and, consequently, the larger the scale will be. Thus, a map scale of 1:20,000 is twice as large as a scale of 1:40,000. If that sounds backward, remember that the terms large scale and small scale come from the numeric value of the representative fraction 1/x. The number 1/1,200 is much larger than 1/100,000,000. Thus, the Ordnance Survey’s Landplan Map, at a scale of 1:1,250, is a large-scale map, in which small areas are shown in great detail, and a world map at a scale of 1:100,000,000 is a small-scale map. Common as it is to classify maps by their scales, there is no general agreement on where the class limits should be set for large-, medium-, and small-scale maps. If you sort maps into two groups—large and small scale—1:500,000 is a likely dividing point. One reason for dividing maps into large and small scale is that you can use large-scale maps to make

accurate distance and area measurements, but you cannot do so on small-scale maps. World atlas and wall maps of continental coverage then fall into the small-scale group, whereas topographic maps, city street guides, and other detailed maps of small areas are in the large-scale class. If a more detailed three-way grouping is used, maps with scales of 1:1,000,000 and smaller are probably classed as small scale, and maps with scales of 1:250,000 and larger are large scale. Maps that range in scale between these extremes are then referred to as medium scale. The last column in table 2.1 shows the progression of scales from large through medium to small. Any such classification, of course, is arbitrary and should not be given meaning beyond the organizational convenience that it provides. When you use a map, be sure to note whether it is a large-, medium-, or small-scale product. Check that the features of interest are displayed at the correct scale for your purposes. If you want to study a small ground area in great detail with little generalization of features, you need a large-scale map. When you are required to make accurate distance, direction, and area measurements, you should use only large-scale maps (see chapter 6 for more on generalization). The change in scale across the surface of a large-scale map is negligible, so you can trust the map as a geometrically exact representation of the small piece of the earth it covers. If you are more interested in a generalized presentation of a large area, such as a state, country, continent, or the entire globe, choose a small-scale map. The scale changes continuously across small-scale maps, so the RF or scale bar printed on these maps gives the scale at a particular point or along a given line but not for the entire map (unless, as noted, there is a variable scale bar on a map that is using the Mercator projection).

Topographic map series at different scales You may have noticed in table 2.1 that national mapping agencies create map series for their country at standard map scales. In this section, we focus on the topographic map series currently used in the United States and Canada. These maps are produced by the US Geological Survey (USGS) and Natural Resources Canada (NRCAN). Topographic maps show water bodies and rivers; forested areas; urban and populated areas; and roads, railways, and other human-made features. These maps are available in paper or digital form directly from government mapping agencies, such as the US Geological Survey and NRCAN, and from private vendors. They are used not only by hikers, cyclists, and other recreationists, but also in field studies carried out by geographers, geologists, natural resource managers, mining scientists, civil engineers, and others. USGS map series The US Geological Survey has traditionally produced three major national map series that cover the conterminous United States (the lower 48 states). The 1:250,000 scale topographic series is the smallest in scale, at about one inch to four miles (1 centimeter to 2.5 kilometers). Each of the 489 maps in the series is a 1° × 2° quadrilateral called a quadrangle that is given the name of a prominent town or physical feature in the mapped area. Washington state, for example, is covered by 18 quadrangles (figure 2.3, top index map) placed at one-degree latitude increments (46° N, 47° N, and so on) and two-degree longitude increments of (118° W, 120° W, and so on). The Yakima map sheet, for example, covers an area from 46° to 47° N and 120° to 122° W.

The second USGS topographic map series is at the 1:100,000 scale (1 inch to 1.5 miles or 1 centimeter to 1 kilometer). The nearly 2,000 maps in this series are 30 × 60 minute quadrangles with parallels of latitude spaced 30 minutes apart and meridians of longitude spaced at 60-minute (one-degree) increments. The 52 map sheets that cover the state of Washington are shown in the middle index map in figure 2.3. Each map is named in a similar manner as the 1:250,000-scale series on the basis of a prominent geographic feature. Four 1:100,000-scale maps nest in each quadrangle of the 1:250,000-scale map series (for example, Mount Rainier, Yakima, Mount Adams, and Toppenish are quarters of the Yakima 1:250,000-scale map). Also, the Mount Rainier 1:100,000-scale map sheet centered on the peak is a more detailed representation of the area than the 1:250,000-scale map, showing the names and locations of glaciers and other physical features on the mountain. The largest scale and best-known USGS map series is at the 1:24,000 scale (1 inch to 2,000 feet or 1 centimeter to 240 meters). Nearly 57,000 topographic maps cover the lower 48 states, Hawaii, and parts of Alaska. Each map is a quadrangle bounded by parallels of latitude and meridians of longitude that are spaced 7.5 minutes apart (1/64th of a 1° × 1° quadrilateral). Each map is named in a similar manner as the smaller scale map series. The bottom index map in figure 2.3 shows that 32 1:24,000-scale topographic maps are nested within a 1:100,000-scale map sheet. The Mount Rainier West quadrangle, a portion of which is shown in figure 2.3, shows a highly detailed representation of Mount Rainier’s peak, including contour lines showing elevation. You can use the areas of quadrilaterals in table B.5 in appendix B to find the number of square miles or kilometers covered by topographic maps at different latitudes. For example, at the southern tip of Florida, near 25° N, a 7.5-minute quadrangle covers an area of about 67 square miles (174 square kilometers). Each quadrangle covers 49 square miles (127 square kilometers) at the Canadian border along the 49° N parallel. The Mount Rainier West quadrangle covers an area of approximately 51 square miles (131 square kilometers). Canadian map series NRCAN developed the National Topographic System (NTS) to provide generalpurpose topographic maps of the country at 1:50,000 and 1:250,000 scales. The NTS divides Canada into three zones defined by latitude extent. The Southern Zone includes topographic maps between 40° N and 68° N, the Arctic Zone includes maps between 68° N and 80° N, and the High Arctic Zone includes maps between 80° N and 84° N (figure 2.4).

Figure 2.3. Index maps showing the 1:250,000-scale, 1:100,000-scale, and part of the 1:24,000-scale USGS topographic maps that cover the state of Washington. Representative map segments showing the summit of Mount Rainier at the three scales are to the right of each index map. Go to http://nationalmap.gov/ustopo for more information on USGS topographic map products. Topographic map segments courtesy of the US Geological Survey.

A map in this series is identified first by a numbered area—the first element in an NTS map number. Each of these numbered areas covers four degrees of latitude. Numbered areas in the Southern and Arctic Zones cover eight degrees of longitude, whereas those in

the High Arctic Zone cover 16 degrees of longitude. In the NTS map numbers in figure 2.4, numbered area 1 is a quadrilateral that extends from 44° N to 48° N latitude and 48° W to 56° W longitude. Numbered areas in the Southern Zone are divided uniformly into 16 1:250,000-scale map areas that cover one degree of latitude by two degrees of longitude (see, for example, numbered areas 1 and 10 in the large map in figure 2.4). In the Arctic and High Arctic Zones, they are divided into eight uniform areas that span 1° × 4° and 1° × 8°, respectively (for example, numbered areas 29 and 120). The 1:250,000-scale topographic maps within a numbered area are lettered A through P, beginning with A in the southeast corner and zigzagging to P in the northeast corner. This letter is the second element in an NTS map number. In figure 2.4, map 1M extends from 47° N to 48° N latitude and 54° W to 56° W longitude.

Figure 2.4. NTS includes topographic maps for the country at 1:50,000 and 1:250,000 scales. Map sheets are identified by a number and letter code as illustrated here. Go to http://www.nrcan.gc.ca/earthsciences/geography/topographic-information/maps/9765 for additional details.

Each lettered 1:250,000-scale map area is divided into 16 1:50,000-scale map sheets, numbered from 1 through 16 in the same zigzag pattern used to identify lettered areas. This

number is the final element in an NTS map number. For example, 1:50,000-scale map sheet 1M16 is in the upper-right corner of map area 1M, spanning latitude 47°45´ N to 48° N and longitude 54° W to 54°30´ W. The 1:250,000-scale map areas in the lower-right map in figure 2.4 appear to dramatically increase in size from south to north and differ in shape. Variations in size and shape exist among map sheet areas, but the tremendous variation in this figure is because of the use of the Mercator map projection that greatly enlarges areas on the ground near the polar areas (see chapter 3 for more on the Mercator projection). Figure 2.5 shows the true relative sizes and shapes of map areas 10P, 16P, 29H, and 120H. Map area 10P is wider than map area 120H, even though maps in the Southern Zone span two degrees of longitude, whereas those in the High Arctic Zone span eight degrees. This fact may puzzle you until you recall from chapter 1 the progressive converging of meridians from equator to pole. Convergence of meridians from 68° N to 80° N also explains the similar appearance of maps 16P and 29H, despite map 16P spanning two degrees of longitude and map 29H spanning four degrees. Table B.3 in appendix B gives more information about the amount of convergence, indicating that the length of a degree of longitude is 25.988 miles (41.823 kilometers) at 68° N and only 12.051 miles (19.394 kilometers), about half as long, at 80° N.

Figure 2.5. True relative sizes and shapes of selected 1:250,000-scale map sheet areas within the Southern, Arctic, and High Arctic Zones.

Multiscale maps You may have used commercial applications, such as ArcGIS Earth or ArcGlobe from Esri, or Google Maps, to see an area at increasingly larger or smaller scales while zooming in or out. The maps you view are ArcGIS Pro multiscale maps, which are a series of maps at varying scales that each have an amount and type of information that is appropriate to the scale being displayed. As you zoom in on the map, more geometric detail, and often

additional themes of information and labels, are shown. These maps are seamless, allowing you to pan across the entire map extent without breaks. The six maps in figure 2.6, for example, show the different levels of detail for a hydrologic map of an area in the state of Texas.

Figure 2.6. A multiscale hydrologic map at a series of scales as indicated by the representative fraction. As the map scale increases, so does the amount of detail. Courtesy of Esri.

The maps are designed so that when you zoom in or out, the content remains legible and clear. You see a progressive increase in detail at larger scales, because the cartographer

has designed the maps to change the degree of generalization at each map scale. For example, different types of roads are displayed at different scales, from freeways only at the smallest scale to freeways, highways, arterials, and the local street network at the largest scale. Additionally, streams that are lines at smaller scales become channels that are shown as progressively wider polygons at larger scales. The amount of labeling of the streams in the maps in figure 2.6 increases as the map scale increases, until almost all streams are named at the largest scale. In figure 2.6, the roads become more geometrically precise as the map scale increases, although major roads and streets are artificially widened on the map to emphasize their importance in the road network.

CONVERTING SCALE The scale depicted on the map you are using may be in the wrong form to best serve your purpose. Therefore, you may need to make conversions between a word scale, RF, and scale bar. Scale conversions are based on applying common distance equivalents such as 1 foot = 12 inches, 1 statute mile = 5,280 feet, 1 statute mile = 63,360 inches, or 1 kilometer = 100,000 centimeters. Table B.1 in appendix B is a more complete list of English and metric distance equivalents. For instance, you may have a map with a word scale and want to know the RF. The first thing to remember with any scale conversion is that the ratio 1/x is always map distance (numerator) to ground distance (denominator). Suppose that the word scale is “3 inches to 10 miles.” In converting to an RF, the ratio is equation (2.2):

But you can’t have an RF with different units in the numerator and denominator. So you must convert miles to inches—the math is no problem if you remember that there are 63,360 inches in a statute mile. So you have equation (2.3):

The miles in the denominator cancel out, as do the remaining inches in the numerator and denominator, leaving equation (2.4):

Remember that the numerator of an RF is always 1. So in this case, you must also reduce the numerator from 3 to 1 by dividing the numerator and denominator by 3. To solve for x, you invert the fractions to get the correct map scale of 1:211,200 in equation (2.5):

The conversions are even easier with metric units. If the word scale is “four centimeters to one kilometer,” you have equation (2.6):

To convert kilometers to centimeters, first convert the denominator knowing that there are 100,000 centimeters in a kilometer, in equation (2.7):

In this case, the kilometers in the denominator cancel out, as do the remaining centimeters in the numerator and the denominator, leaving equation (2.8):

To solve for x, invert the fraction and divide 100,000 by 4 to get the correct map scale of 1:25,000, in equation (2.9):

Sometimes you may find yourself in the opposite situation. You know the RF but want to know how many “inches to a mile” (or “miles to an inch”) or “centimeters to a kilometer” (or “kilometers to a centimeter”) the map scale represents. If it is a “miles to an inch” word scale that you want, merely divide the number of inches in a mile, or 63,360, by the denominator of the RF. To convert an RF of 1:200,000 to an “inches to a mile” word scale, the conversion is equation (2.10):

If it is a “miles to an inch” word scale that you want, divide the denominator of the RF by the number of inches in a mile, or 63,360. For example, to convert an RF of 1:200,000 to a “miles to an inch” word scale, use equation (2.11):

As noted earlier, this number is often rounded off so that the word scale is “three miles to one inch.” To convert an RF of 1:200,000 to a word scale of “centimeters to a kilometer,” use equation (2.12):

This calculation gives a word scale of “0.5 centimeters to 1 kilometer.” If you want a “kilometers to one centimeter” word scale, divide the denominator of the RF by 100,000. To convert an RF of 1:200,000 to a word scale of “kilometers to a centimeter,” use equation (2.13):

The word scale is then “two kilometers to one centimeter.” Sometimes you may want to create a scale bar from a word scale or an RF. Imagine that you have a map at a scale of three-quarters of a mile to an inch, or 1:47,520. A map at this scale is considered a large-scale map, so your scale bar should have one-mile (or onekilometer) increments because it will facilitate ground distance measurements. Because an inch represents ¾ mile, each mile covers inches on the map. So you mark off -inch intervals on your scale bar to indicate the mile increments. To make it more useful universally, you can also show kilometers.

DETERMINING MAP SCALE It’s good to know that no matter what sort of map scale (word scale, RF, or scale bar) you encounter, you can change it to the type of scale you want. But what if you come across a map that has no scale depicted at all, or an RF or word scale that seems incorrect because the map appears to have been enlarged or reduced after it was created? Lack of scales on maps or incorrect scales occur more often than you might expect. For instance, you may want to know the scale of a photocopied portion of a map in which no scale is shown. Digital maps that you copy from the Internet are another good example, because they often are scans of original maps that do not include an RF or scale bar. Even if the RF is visible, it may not be correct because the map is displayed at a different pixel density than when it

was created. For example, a map created at 150 pixels per inch but displayed on your computer monitor at a default density of 72 pixels per inch is slightly more than twice as large in scale as the RF indicates. You can figure out the map scale if you know the ground distance between any two points on the map. Simply measure the distance between these two points on the map. The ratio of map to ground distance, in the same units of measurement, is the map’s scale. But how do you find the ground distance between the two points? One method is to use some ground feature whose length is known, as described in the next section. Keep in mind that scale may vary across the extent of the map, so for small-scale maps in particular, the scale you come up with is only accurate around the ground feature that you selected.

Determining map scale from a known ground feature Some features have standard lengths. If you can identify one of them on your map, you can easily figure out the map scale. A regulation US football field, for example, is 100 yards long. If the map distance of the field is 0.5 inch, 0.5 inch on the map represents 100 yards on the ground. To determine the map scale, convert yards to inches and reduce the numerator to 1 using equation (2.14):

This calculation gives an RF of 1:7,200. In Canada, a football field is 110 yards, so you must rework the calculations for the different length. This method works best for large-scale maps because most features of standard lengths are short, and their lengths and widths are accurately shown only on large-scale maps.

Determining map scale from the equator For a world map, you can make use of the fact that the earth’s circumference is approximately 25,000 statute miles (you saw in chapter 1 that the actual figure is 24,901.46 statute miles or 40,075.017 kilometers for an authalic sphere using the semimajor axis of the WGS84 ellipsoid). If, for example, the equator is shown in its entirety and is measured on the map as eight inches long, you can find the map’s RF using equation (2.15):

Rounding off this number gives an RF of 1:197,219,600, to seven significant digits. The word scale for “miles to an inch” is calculated as equation (2.16):

which can be rounded to “3,113 miles to 1 inch.” In metric units, 8 inches is 20.32 centimeters, and the earth’s authalic sphere circumference is 40,075.017 kilometers, so the calculation is in equation (2.17):

As you can see, this calculation gives the same RF of 1:197,219,600, to seven significant digits. To determine the word scale in metric units, use equation (2.18):

or “1,972 kilometers to 1 centimeter.” Remember that this scale is only valid at the equator because the scale may be much larger or smaller at other places on a small-scale map.

Determining map scale from reference material If you can’t find a feature of standard length on your map, you can still determine scale if you turn to other reference material, such as road maps, topographic sheets, atlases, or other maps of a similar scale to yours. From these sources, you should be able to find the distance of something on your map—for instance, the distance between two prominent road intersections or two towns or two mountain peaks. Using this information, you can compute the map scale using the calculations described in this section for determining map scale from a known ground feature. For example, on a map with a scale of 1:100,000, you might measure the map distance between two road intersections as 0.7 centimeters, and as 0.5 centimeters on your map. Because the ratio of measured distances equals the ratio of the two RFs, you can use the proportion in equation (2.19):

This proportion reduces to equation (2.20):

giving an RF of 1:140,000.

Determining map scale from ground resolution and pixel density You may be faced with finding the scale of an image map that is made from imagery obtained by one of the remote-sensing devices described in chapter 10. You can compute the image map scale if you know the ground resolution (GR) of a pixel in the image and the pixel density (PD) of the image map on your device. Say that your image map is made from satellite imagery with a 10-meter ground resolution, meaning that each pixel in the image covers a 10 × 10 meter square on the ground. You find that the image map is displayed on your computer monitor at 50 pixels per centimeter. Then use equation (2.21):

where GR and PD are both in the same units of measurement. In this example, a ground resolution of 10 meters per pixel is the same as 1,000 centimeters per pixel, or equation (2.22):

The PD is 50 pixels per centimeter, so you have equation (2.23):

The RF for your image map is 1:50,000.

Determining map scale from the spacing of parallels and meridians It isn’t always convenient, or even possible, to find a known ground feature or to use other reference materials to determine map scale. But on many maps, especially those of small scale, parallels and meridians are shown. You can determine map scale by finding the ground distance between these lines. Finding the ground distance between parallels is simple because a degree of latitude varies only slightly from pole to equator. Small-scale world maps usually use the authalic sphere as the approximation to the earth, so parallels are equally spaced at 69.093 miles (111.195 kilometers) per degree of latitude. From table B.2 in appendix B, you see that on the WGS84 ellipsoid, the first degree of latitude is 68.703 statute miles (110.567 kilometers) north or south of the equator, whereas the distance from the 89th parallel to the adjacent pole is 69.407 miles (111.699 kilometers). Thus, the variation in a degree of latitude from equator to pole is only 0.704 miles or 1.132 kilometers. This difference is so small that you can safely ignore it for small-scale maps. Say, for example, that on a small-scale world map you find two parallels separated by two degrees of latitude, and there is a map distance of five inches between the parallels. To find the ground distance between these parallels, multiply the number of degrees separating them by the length of a degree, in equation (2.24):

Then you can find the ratio between map distance and ground distance, in equation (2.25):

This calculation gives an RF of 1:1,751,020. In metric units, 5 inches is 12.7 centimeters, so you have equation (2.26):

Again, you can find the ratio between map distance and ground distance, in equation (2.27):

This calculation gives an RF of 1:1,751,180, which is nearly the same value—the difference lies in rounding the numbers. However, two parallels are not always shown on a map. If two meridians are shown, you can still compute the map scale. The complication is that, because meridians converge at the poles, the distance between a degree of longitude varies from 69.09 miles (111.20 kilometers) along the equator to zero at either pole (see table B.3 in appendix B). The distance between meridians, called the longitudinal distance, depends on the mapped area’s latitude. If you again assume an authalic sphere for the earth’s shape in making a small-scale map, there’s a simple functional relationship between latitude and longitudinal distance: longitudinal distance decreases by the cosine of the latitude. To find the longitudinal distance for a degree of longitude at a given latitude, multiply the length of a degree of latitude (69.093 miles or 111.195 kilometers) by the cosine of the latitude. This relationship is written mathematically as equation (2.28): Longitudinal distance = cos(latitude) × 69.093 miles

or

So how can you use this equation to determine map scale? The procedure is straightforward if viewed as a series of simple steps. Begin by measuring the map distance between two meridians. For example, on the Aleutian Islands aeronautical chart segment in figure 2.7, you find that at 53° N, meridians that span 0.5 degrees (30 minutes) of longitude on the map are 2.66 inches (6.75 centimeters) apart.

Next, determine the ground distance between these two meridians. To do that, use a calculator to find the cosine of 53° and multiply that by the length of a degree of latitude (69.093 miles), in equation (2.29):

Figure 2.7. You can begin to compute the RF of this Aleutian Islands aeronautical chart segment by measuring 0.5 degrees of longitude as 2.66 inches long on the chart (latitude and longitude ticks are spaced one minute apart on the chart). Courtesy of the National Aeronautical Charting Office.

This calculation gives a longitudinal distance of 41.58 miles. For kilometers, use equation (2.30):

So the longitudinal distance is 66.92 kilometers. You now know that one degree of longitude at 53° N covers a ground distance of 41.58 miles or 66.92 kilometers. But because the measured map distance spans one-half rather than one degree of longitude, the ground distance in this problem is half the computed value, or equation (2.31):

You now have all the information necessary to find the map scale—the distance between two meridians on the map and the corresponding ground distance. Simply compute the ratio between map and ground distance, in equation (2.32):

This calculation gives an RF of 1:495,208, which is rounded to 1:495,000. In metric units, 2.66 inches equals 6.756 centimeters, so you have equation (2.33):

This calculation gives an RF of 1:495,234, which is also rounded to 1:495,000.

SELECTED READINGS ASCE Committee on Cartographic Surveying. 1983. Map Uses, Scales, and Accuracies for Engineering and Associated Purposes. New York: American Society of Civil Engineers. Dickinson, G. C. 1969. Maps and Air Photographs, 99–107, 140–48. London: Edward Arnold. Espenshade, E. B. 1951. “Mathematical Scale Problems.” Journal of Geography 50 (3): 107–13. Greenhood, D. 1964. Mapping, 39–53, 180–84. Chicago: University of Chicago Press. Hodgkiss, A. G. 1970. Maps for Books and Theses, 37–43, 169–70, 239–42. New York: Pica. Quattrochi, D. A., and M. F. Goodchild. 1997. Scale in Remote Sensing and GIS. Boca Raton, FL: Lewis. Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. Elements of Cartography, 6th ed. New York: John Wiley & Sons. Thompson, M. M. 1979. Maps for America: Cartographic Products of the US Geological Survey and Others. Reston, VA: US Geological Survey. US Army. 1969. “Scale and Distance.” Chap. 4 in Map Reading, FM 21–26, sections 4–1 to 4–4. Department of the Army Field Manual.

chapter three MAP PROJECTIONS GLOBES VERSUS FLAT MAPS THE MAP PROJECTION PROCESS MAP PROJECTION PROPERTIES Scale Completeness Correspondence relations Continuity MAP PROJECTION FAMILIES Projection families based on geometric distortion Distance Shape Direction Area Projection families based on developable surfaces Planar Cylindrical Conic MAP PROJECTION PARAMETERS Tangent and secant case Aspect Other map projection parameters COMMONLY USED MAP PROJECTIONS Planar projections Orthographic Stereographic Gnomonic Azimuthal equidistant Lambert azimuthal equal area Cylindrical projections Equirectangular World Mercator Web Mercator Gall-Peters Transverse Mercator Conic map projections Lambert conformal conic

Albers equal-area conic Pseudocylindrical and other projections Mollweide Sinusoidal Homolosine Robinson Winkel tripel Oblique-perspective projections MAP PROJECTIONS ON THE SPHERE AND ELLIPSOID SUMMARY NOTE SELECTED READINGS

3 Map projections A map projection is a geometric transformation of the earth’s spherical or ellipsoidal surface onto a flat map surface. Much has been written about map projections, yet people still find this subject one of the most bewildering aspects of map use. Many people readily admit that they don’t understand map projections. This shortcoming can have unfortunate consequences, as it hinders their ability to understand how people, plants and animals, and other features are actually distributed across the earth in our global society. It also makes them easy prey for politicians, advertisers, special-interest groups, and others who, through lack of understanding or by design, use map projections in potentially deceptive ways. There is an infinite number of map projections, but some are better suited than others for particular uses. How, then, do you go about distinguishing one projection from another and choosing among them? One way is to organize the wide variety of projections into a limited number of map projection families on the basis of shared attributes. Two approaches commonly used include classifying the projections into families based on their geometric distortion properties (relating to distance, shape, direction, and area) and examining the nature of families based on the surface used to construct the projection (a plane, cone, or cylinder), which, in turn, helps you understand the pattern of spatial distortion over the map surface. The two approaches work together, because the map user is concerned, first, with what spatial properties are preserved (or lost) and, second, with the pattern and extent of distortion. This chapter closely examines the distortion properties of map projections and the surfaces used in the creation of map projections. Before we discuss map projection properties and families, we can help clarify the issue of map projections with a discussion of the logic behind them, and of why projections are necessary. We begin our discussion with globes, a form of map that you have probably looked at since childhood. Your familiarity with globes makes it easy for us to compare them with flat maps.

GLOBES VERSUS FLAT MAPS Of all maps, a globe (a sphere on which a map of the earth is depicted) gives the most realistic picture of the earth as a whole. Basic geometric properties such as distance, direction, shape, and area are preserved because the globe is the same scale everywhere (figure 3.1). However, globes have a number of disadvantages. They don’t let you view all parts of the earth’s surface simultaneously—the most you can see is a hemisphere (half of the earth). Neither are globes useful for seeing the kind of detail you might find on a roadmap that you might keep in your car or a topographic map that you might carry when hiking. Globes also are bulky and don’t lend themselves to convenient handling and storage like the maps you use on your mobile device, such as your smartphone. You wouldn’t have as many handling and storage problems if you used a baseball-size globe. But such a tiny globe is of little practical value for map reading and analysis because it would have a scale of approximately 1:125,000,000. Even a globe that is two feet (60 centimeters) in diameter is still at a scale of 1:20,000,000. It would take a globe about 40 to 50 feet (12 to 15 meters) in diameter—the height of a four-story building—to provide a map of the scale used for state highway maps. A globe that is nearly 1,800 feet (about 550 meters) in diameter—the length of six US football fields—would be required to provide a map of the same scale as the standard 1:24,000-scale topographic map series in the United States.

Figure 3.1. A typical world globe. Courtesy of Replogle Globes Inc.

Another problem with globes is that the computations and techniques that are suited for measuring distance, direction, and area on spherical surfaces are relatively difficult to use. Computations on a sphere are far more complex than on a plane surface. (For a demonstration of the relative difference in difficulty between making distance computations from plane and spherical coordinates, see chapter 12.) Finally, globe construction is laborious and costly. High-speed printing presses have kept the cost of flat-map reproduction to manageable levels but have not yet been developed to work with curved media. 3D printers can now be used to make small plastic single-color globes, but at relatively high cost. Therefore, globe construction is not suited to the volume of map production required for modern map use. How about virtual globes? Virtual globes are not solid 3D objects, but rather a series of maps that look three-dimensional but are really earthlike map projections on a 2D computer

screen or mobile device. In the future, we may be able to display and analyze virtual globes projected on a real 3D display surface. It would be ideal if the earth’s surface could be mapped undistorted on a flat medium, such as a sheet of paper or computer screen. Yet the spherical earth is not what is known as a developable surface, defined by mathematicians as a surface that can be flattened on a plane without geometric distortion. The only developable surfaces in our threedimensional world are cylinders, cones, and planes, so all flat maps necessarily distort the earth’s surface geometrically. To transform a spherical surface that curves away in every direction from every point into a plane surface that doesn’t curve in any direction from any point means that the earth’s surface must be distorted on the flat map. Map projection affects basic properties of the representation of the earth on a flat surface, such as scale, continuity, and completeness, as well as geometric properties that relate to direction, distance, area, and shape. What you want to do is minimize these distortions or preserve a particular geometric property at the expense of others. This conundrum is the map projection problem.

THE MAP PROJECTION PROCESS The concept of map projections is somewhat more involved than implied in the previous discussion. Not one but a series of geometric transformations is required. The irregular topography of the earth’s surface is first defined relative to a much simpler 3D surface, and then the 3D surface is projected on a plane. This progressive flattening of the earth’s surface is illustrated in figure 3.2. The first step is to define the earth’s irregular surface topography as elevations of land (topography) and depths of water (bathymetry) relative to a more regular surface known as the geoid, as discussed in chapter 1. The geoid is the surface that would result if the average level of the world’s oceans (mean sea level) was extended under the continents. It serves as the vertical reference datum, or starting reference surface, for elevation data on maps. (See chapter 1 for more on datums.) The second step is to project the slight geoidal undulation (the difference between the geoid and the ellipsoid) on the more regular oblate ellipsoid surface. (See chapter 1 for further information on oblate ellipsoids.) This new surface serves as the basis for the geodetic control points determined by surveyors and as the datum for the geodetic latitude and longitude coordinates found on maps. (See chapter 1 for more on geodetic latitude and longitude.)

Figure 3.2. The map projection process involves a number of steps: (1) the land elevation or sea depth of every point on the earth’s surface is defined relative to the geoid surface; (2) the elevations and depths on the geoid are projected on the more regular ellipsoid surface to define the positions with geodetic latitude and longitude; (3) optionally, geodetic latitudes and longitudes may be converted into spherical coordinates for only small-scale maps (the small difference between the ellipsoid and sphere is not apparent at this scale); (4) finally, the ellipsoidal or spherical coordinates are transformed into planar (x,y) map coordinates through map projection equations.

An optional step in making a small-scale flat map or globe is to mathematically transform geodetic coordinates into geocentric coordinates on a sphere, usually equal in surface area to the ellipsoid—that is, an authalic sphere. (See chapter 1 for more on the authalic sphere.) The last step involves projecting the ellipsoidal or spherical surface on a plane through the use of map projection equations that transform geographic or spherical coordinates into planar (x,y) map coordinates. The greatest distortion of the earth’s surface geometry occurs in this step.

MAP PROJECTION PROPERTIES Now that you understand the map projection process, you can see that it ultimately leads to distortion of the earth’s geometry on the flat map. So now we look at how the properties of scale, completeness, correspondence relations, and continuity, which are a result of this distortion, are affected by the map projection process.

Scale Because of the stretching and shrinking that occurs in the process of transforming the spherical or ellipsoidal earth surface into a plane, the stated map scale (see chapter 2 for more information on scale) is true only at selected points or along particular lines called points and lines of tangency, which we talk about later in this chapter. Everywhere else, the scale of the map is actually smaller or larger than the stated scale. To grasp the idea of scale variation in map projections, you first must realize that there are, in fact, two map scales. One is the actual scale–the scale that you measure in the vicinity of any point on the map; it differs from one location to another. Variation in actual scale is a consequence of the geometric distortion that results from flattening the earth. The second is the principal scale of the map. You can think of the principal scale as the constant scale of what cartographers call a generating globe—a globe that is reduced to the scale of the desired flat map. You can then visualize this globe as being transformed into a flat map (figure 3.3), knowing that the actual transformation is through map projection equations. The constant scale of the generating globe is the principal scale stated on the flat map, which is correct only at the points or lines at which the generating globe touches the projection surface. To understand the relationship between the actual scale and the principal scale at different places on the map, we compute a ratio called the scale factor (SF), which is defined in equation (3.1):

We use the representative fractions of the actual and principal scales to compute the SF. An actual scale of 1:50,000,000 and a principal scale of 1:100,000,000 thus give an SF as follows, in equation (3.2):

If the actual and principal scales are identical, the SF is 1.0, leading to unity. But, as you saw earlier, because the actual scale varies from place to place, so does the SF. An SF of 2.0 on a small-scale map means that the actual scale is twice as large as the principal scale (see figure 3.3). An SF of 1.15 on a small-scale map means that the actual scale is 15 percent larger than the principal scale. On large-scale maps, the SF should vary only slightly from 1.0, following the general rule that the smaller the area being mapped, the less the scale distortion.

Completeness Completeness refers to the ability of map projections to show the entire earth. You’ll find the most obvious distortion of the globe on “world maps” that don’t actually show the whole world. Such incomplete maps occur when the equations used for a map projection can’t be applied to the entire range of latitude and longitude. The Mercator projection world map (see figure 3.19) is a classic example. In this map projection, y-coordinates for the North and South Poles are positive and negative infinity1, respectively, so the map usually extends to only the 80th parallel north and south. Omitting these high latitudes may be acceptable for maps that show political boundaries, cities, roads, and other cultural features; but the data for physical phenomena such as average temperatures, ocean currents, or landforms is usually global and should therefore be shown on a map of the entire earth. The gnomonic projection, discussed later in this chapter, is an even more extreme example (see figure 3.15), because it is limited mathematically to covering less than a hemisphere (the SF is infinitely large 90 degrees away from the projection center point). So one of the considerations when choosing a map projection is the ability to show your complete area of interest.

Figure 3.3. Scale factors (SFs) greater than 1.0 indicate that the actual scales are larger than the principal scale of the generating globe. The principal scale is identical to the actual scale at the equator, where the map projection surface touches the generating globe.

Correspondence relations You might expect that each point on the earth is transformed into a corresponding point in the map projection. The agreement between points on the earth and points on the projected surface is called correspondence. Point-to-point correspondence allows you to shift attention with equal facility from a point on the earth to the same point on the map, and vice versa. However, this desirable property can’t be maintained for all points in many world map projections. As figure 3.4 shows, one or more points on the earth may be transformed into straight lines or circles on the boundary of the map projection, most often at the North and South Poles. You may notice that the SF in the east–west direction must be infinitely large at the poles in this projection, yet it is 1.0 in the north–south direction. Other map projections, such as the orthographic and Lambert azimuthal equal area (see figures 3.13 and 3.17), discussed later in the chapter, allow for point-to-point correspondence.

Continuity To represent an entire spherical surface on a plane, the continuous spherical surface must be interrupted at some point or along some line. These breaks in continuity (preservation of spatial proximity at all locations) form the map border in a world projection. Where the mapmaker places the discontinuity is a matter of choice. On some maps, for example, opposite edges of the map are the same meridian (see figure 3.4). Using the same meridian for opposite edges of the map means that features next to each other on the ground are found on opposite sides of the map. This blatant violation of proximity relations is a source of confusion for many map users. Similarly, a map may show the North and South Poles with lines as long as the equator, as in the map in figure 3.4. Maps of individual continents or nations almost always show these areas without breaks in continuity. To do otherwise needlessly complicates reading, analyzing, and interpreting the map.

MAP PROJECTION FAMILIES As we mentioned previously, it is sometimes convenient to group map projections into families based on common properties so that you can distinguish and choose among them. The two most common approaches to grouping map projections into families are based on geometric distortion and the projection surface.

Figure 3.4. The point-to-point correspondence between the earth and map projection may become point-to-line at some locations as along the top and bottom edges of the map on` the right. There may also be a loss of continuity, in which the same line on the globe forms two edges of the map, such as the left and right edges of the map on the right, both of which are the 180° meridian.

Projection families based on geometric distortion You can gain an idea of the types of geometric distortions that occur in map projections by comparing the graticule (latitude and longitude lines) on the projected surface with the same lines on a globe (see chapter 1 for more on the graticule). Figure 3.4 shows how you might make this comparison (keeping in mind that, in the figure, the drawing of a globe on the left is actually a map projection of a hemisphere). Ask yourself several questions: To what degree do meridians converge? Do meridians and parallels intersect at right angles? Do parallels shorten with increasing latitude? Are the areas of quadrilaterals in the projection the same as on the globe? You can use these observations to better understand how cartographers assign the types of geometric distortion you see in the map projection into the categories of distance, shape, direction, and area. Now we look at each type of distortion, beginning with variations in distance. Distance The preservation of spherical great-circle distance in a map projection, called equidistance, is, at best, a partial achievement of distance preservation (see chapter 1 for more on great circles). For a map projection to be truly distance-preserving (equidistant), the scale must be equal in all directions from every point, which is geometrically impossible to achieve on a flat map. Because the scale varies continuously from location to location in a map projection, the great-circle distance between the two locations on the globe must be distorted in the projection.

Equidistant projections are used to show the correct distance between a selected location and any other point in the projection. Although the correct distance will be shown between the selected point and all others, distances between all other points are distorted. As a consequence, no flat map can preserve both distance and area. For some equidistant projections such as the azimuthal equidistant projection that you will learn more about later in this chapter, cartographers make the SF a constant 1.0 radially outward from a single point such as the North Pole (see figure 3.16). Great-circle distances are correct along the lines that radiate outward from that central point only. Long-distance route planning is based on knowing the great-circle distance between the starting point, usually at the center of the map, and the ending point; and equidistant maps are excellent tools for determining these distances on the earth. One obvious drawback is that the map projection must be changed every time the starting point is moved. Shape When angles on the globe are preserved at individual points on the map (thus preserving shape), the projection is called conformal, meaning “correct form or shape.” Unlike the property of equidistance, conformality can be achieved at all points in conformal projections. To attain the property of conformality, the local map scale must be the same in all directions from a point. A circle on the globe thus appears as a circle on the map. But to achieve conformality, it is necessary to either enlarge or reduce the scale by a different amount at each location on the map (figure 3.5). This change in scale means, of course, that the map area around each location must also vary. Tiny circles on the earth always map as circles, but their sizes will differ on the map. Because tiny circles on the earth are projected as circles, all directions from the center of the circles are correct in the projected circles. The circles must be tiny because the constant change in scale across the map means that they would be projected as ovals if they were hundreds of miles in diameter on the earth. You can also see the distortion in the appearance of the graticule—although the parallels and meridians intersect at right angles, the distance between parallels varies. Typically, there is a smooth increase or decrease in scale across the map. Conformality applies only to directions or angles at—or in the immediate vicinity of—points; thus shape is preserved only in small areas. Conformality does not apply to areas of any great extent— the shape of large regions can be greatly distorted. Conformal maps are best suited for tasks that involve plotting, guiding, or analyzing the motion of objects over the earth’s surface. Thus, conformal projections are used for aeronautical and nautical charts, topographic quadrangles, and meteorological maps. They are also used when the shape of features is a matter of concern. Direction It’s sometimes important to preserve directions globally. Projections that preserve global directions are called true-direction projections. You will also sometimes see these referred to as azimuthal projections. As you will see in the next section, this term is also used instead of “planar” to describe projections. To avoid confusion, we use the terms true direction and planar in this book. No projection can represent correctly all directions from all points on the earth as straight lines on a flat map. But scale across the map can be arranged so that certain types of direction lines are straight, as is the case in which all meridians radiating outward from the pole are correct directionally. All true-direction projections correctly show the azimuth or

direction from a reference point, usually the center of the map (see chapter 13 for more on azimuths). Hence, great-circle directions on the ground from the reference point can be measured on the map. True-direction projections are used to create maps that show the great-circle routes from a selected point to a desired destination (and thus, the shortest distance on the earth’s surface). Special true-direction projections for maps, such as the gnomonic projection (see figure 3.15) discussed later in this chapter, are used for long-distance route planning in air and sea navigation. Area When the relative size of regions on the earth is preserved everywhere on the map, the projection is said to be equal area and have the property of equivalence. The demands of achieving equivalence are such that the SF can be the same in all directions only along one or two lines, or from, at most, two points. Because the SF and, hence, angles around all other points will be deformed, the scale requirements for equivalence and conformality are mutually exclusive. No projection can be both conformal and equal area—only a globe can be. And, as we also stated earlier, no flat map can preserve both area and distance.

Figure 3.5. On a map that has a conformal projection, identical tiny circles (greatly magnified here) centered at points on the generating globe are projected to the flat map as different-size circles according to the local SF. The appearance of the graticule in all the projected circles is the same as on the globe.

Adjusting the scale along meridians and parallels so that shrinkage in one direction from a point is compensated by exaggeration in another direction creates an equal-area map projection. For example, a small circle on the globe with an SF of 1.0 in all directions may be projected as an ellipse with a north–south SF of 2.0 and an east–west SF of 0.5 so that the area of the ellipse is the same as the circle. Equal-area world maps, therefore, compact, elongate, shear, or skew circles, as well as the quadrilaterals of the graticule on the generating globe (figure 3.6). The distortion of shape, distance, and direction is usually most pronounced toward the map’s margins. Despite the distortion of shapes that is inherent in equal-area projections, they are the best choice for tasks that call for area or density comparisons from region to region. Examples of geographic phenomena that are best shown on equal-area map projections,

like the Mollweide projection in figure 3.6, include world or continental maps of population density, per capita income, literacy, poverty, and other human-oriented statistical data.

Projection families based on developable surfaces So far, we’ve categorized map projections into families based on the types of geometric distortions that occur in them, realizing that particular geometric properties can be preserved under special circumstances. We can also categorize map projections based on the different surfaces used to construct them. As a child, you probably played the game of casting hand shadows on the wall. You can think of the surface that your shadow was projected on as the projection surface. You discovered that the distance and direction of the light source relative to the position of your hand influenced the size and shape of the shadow you created. But the surface on which you projected your shadow also had a great deal to do with it. A shadow cast on a corner of the room or on a curved surface such as a beach ball or lampshade is quite different from one cast on the flat wall. Now think of the generating globe as your hand and the map projection as the shadow on the wall. To visualize this shadow casting, imagine a transparent generating globe that has the graticule and continent outlines drawn on it in black. Then consider placing this globe at various positions relative to a light source and a surface that you project its shadow on (figure 3.7). So now you have the generating globe (your hand) being projected as a map (casting a shadow) on the projection surface (the wall). In the case of maps, the projection surface is always flat. In cartography, this flat surface is called the developable surface. There are three basic projection families based on developable surfaces: planar, conic, and cylindrical projections. Depending on which type of developable surface you use and where your light source is placed, you will end up with different map projections cast by the generating globe. All projections that you can create in this light-casting way are called true-perspective projections.

Figure 3.6. Tiny circles on the generating globe (greatly enlarged here) are projected as ellipses of the same area but a different shape on a map using an equal-area projection. This distortion in shape skews the directions of the major and minor axes for each ellipse. Similarly, quadrilaterals are skewed and sheared in an equal-area projection.

Strictly speaking, very few projections actually involve casting light on planes, cones, or cylinders in a physical sense. Most projections are not purely geometric, and all use a set of mathematical equations that transforms latitude and longitude coordinates on the earth into x,y coordinates on the projection surface. The x,y coordinates for a coastline, for example,

are the endpoints for a sequence of short, straight lines drawn on paper or displayed on a computer monitor. You see the sequence of short lines as a coastline. Projections that don’t use developable surfaces are more common, because they can be (1) designed to serve any desired purpose; (2) made conformal, equal area, or equidistant; and (3) readily produced through computing. Yet even nondevelopable surface projections can usually be thought of as variations of using one of the three basic developable surfaces. Planar Planar projections (also called azimuthal projections) can be thought of as being made by projecting the generating globe on a flat plane. The plane can either touch the generating globe at a point—called the point of tangency—or slice through the generating globe, which results in a line of tangency that is a small circle. If you again consider your childhood shadow-casting game, recall that the projection surface was only one of the factors that influences the shape of the shadow cast on the wall. Another influence is the distance of the light source from the wall. Thus, there are basic differences in the planar projection family, depending on where the imaginary light source is located. This family includes three commonly used true-perspective planar projections (figure 3.8): orthographic (light source at infinity), stereographic (light source on the surface of the generating globe opposite the point of tangency), and gnomonic (light source at the center of the generating globe). Other projections, such as the azimuthal equidistant and Lambert azimuthal equal-area projections (described later in this chapter), are mathematical constructs that cannot be created geometrically and must be built from a set of mathematical equations; however, they can still be thought of as variations on using a plane as the developable surface. Cylindrical Cylindrical projections can be thought of as being made by projecting the generating globe on a cylinder that touches or slices through the generating globe. A cylinder that touches the generating globe results in one line of tangency, usually at the equator. If the cylinder slices through the generating globe (usually at the same latitude north and south of the equator), the result is two lines of tangency.

Figure 3.7. Planes, cones, and cylinders are used as developable surfaces for trueperspective map projections.

As with planar projections, cylindrical projections can also be distinguished by the location of the light source relative to the projection surface. The cylindrical projection family includes two commonly used true-perspective cylindrical projections (figure 3.9): central cylindrical (light source at the center of the generating globe) and cylindrical equal area (linear light source, akin to a fluorescent light with parallel rays, along the polar axis). These projections are described in more detail later in this chapter. The whole world can’t be projected on the central cylindrical projection because the polar rays will never intersect the cylinder. For this reason, this projection is often cut off at some specified latitude north and south.

Figure 3.8. You can visualize three true-perspective planar projections (clockwise, from left: orthographic, gnomonic, and stereographic) as being constructed by changing the location of the light source relative to the generating globe.

Figure 3.9. You can visualize the two true-perspective cylindrical projections (central cylindrical and cylindrical equal area) as being constructed by placing the light source at the center of or outside the generating globe, respectively.

Conic The last family of map projections based on the surface used to construct the projection is conic projections, which can be thought of as projecting the generating globe on a cone that touches or slices through the generating globe. As with a cylinder, a cone that touches the generating globe has one line of tangency, and a cone that slices into the generating

globe has two. We usually select mid-latitude parallels as the line or two lines of tangency. The conic projection family includes one true-perspective conic projection: the central conic projection (light source at the center of the generating globe). To fully understand map projection families based on the projection surface, you must also understand various projection parameters, including case and aspect, among others.

MAP PROJECTION PARAMETERS Tangent and secant case As we have said before, the developable surface can either touch or slice through the generating globe so that the projection surface has either a tangent or secant relationship with the generating globe, called the case of the projection. A tangent case projection surface touches the generating globe at either a point (called the point of tangency) for planar projections or along a line (called a line of tangency) for conic or cylindrical projections) (figure 3.10). The SF is 1.0 at the point of tangency for planar projections or along the line of tangency for cylindrical and conic projections. The SF increases outward from the point of tangency or perpendicularly away from the line of tangency. A secant case planar projection surface intersects the generating globe along a smallcircle line of tangency (figure 3.10). Secant-case conic projections have two small-circle lines of tangency, called standard parallels, usually at the mid-latitudes. As mentioned earlier, secant-case cylindrical projections have two small-circle lines of tangency that are equidistant from the parallel at which the projection is centered. For example, a secant-case cylindrical projection centered at the equator might have the 10° N and 10° S latitudes as lines of tangency. Secant-case conic and cylindrical projections have an SF of 1.0 along the lines of tangency. Between the lines of tangency, the SF decreases from 1.0 to a minimum value halfway between the two lines, while outside the lines, the SF increases from 1.0 to a maximum value at the edge of the map. So the SF is slightly smaller than the stated scale in the middle part of the map and slightly larger at the edges. For secant-case planar projections, the SF increases outward from the circle of tangency and decreases inward to a minimum value at the center of the circle of tangency. The advantage of the secant case is that it minimizes the overall scale distortion on the map more than in the tangent case. The SF is 1.0 along a circle instead of at a single point (for a secant-case planar projection) or two lines instead of one (for secant-case conic or cylindrical projections). As a result, the central part of the projection has a slightly smaller SF than the stated scale, and the edges don’t have SFs as large as with the tangent case, thus creating a map with less overall scale distortion. Distortion is minimal around the point or line of tangency. This minimal distortion explains why earth curvature may often be ignored without serious consequence when using flat maps for measurement or analysis of a local area, but only if the line or lines of tangency are positioned within the area being mapped. As the distance from the point or line of tangency increases, so does the scale distortion. By the time a projection has been extended to include the entire earth, scale distortion may have greatly impacted the earth’s appearance on the map away from the point or line(s) of tangency. Secant-case planar, conic, and cylindrical projections are used more often than tangent case projections, whose small area of minimal scale distortion limits their practical value.

Secant-case cylindrical projections are best suited for world maps because they have less overall scale distortion than tangent-case projections—this distortion is minimized in the mid-latitudes, which is also where the majority of the earth’s population lives. Secantcase conic projections are best suited for maps of mid-latitude regions, especially those elongated in an east–west direction. The United States and Australia, for example, meet these qualifications and are frequently mapped using secant-case conic projections. You will find, however, that although secant-case planar projections have less overall scale distortion, the tangent case is often used instead for equidistant projections.

Figure 3.10. Tangent and secant cases of the three basic developable surfaces.

Aspect

Map projection aspect refers to the location of the point or line(s) of tangency on the projection surface (figure 3.11). A projection’s point or line(s) of tangency can, in theory, touch or intersect anywhere on the developable surface. When a point or line of tangency is at or along the equator, in the tangent case, or equidistant from the equator, in the secant case, the resulting projection is in equatorial aspect. When the point of tangency is at, or the line or lines of tangency encircle, either pole, the projection is in polar aspect. With cylindrical projections, the term transverse aspect is also used. Transverse aspect occurs when the line or lines of tangency for the equatorial aspect projection are shifted 90 degrees so that they follow a meridian, in the tangent case, or a pair of meridians, in the secant case (figure 3.12). Any other alignment of the point or line(s) of tangency to the generating globe is in oblique aspect. The aspect that has historically been used the most for the planar, conic, and cylindrical projection families is the normal aspect. The normal aspect of planar projections is polar —the parallels of the graticule are concentric around the center, and the meridians radiate from the center to the edges of the projection (figure 3.11). In the normal aspect for conic projections, parallels are projected as concentric arcs of circles, and meridians are projected as straight lines that radiate at uniform angular intervals from the apex of the cone (see figure 3.10). The normal aspect of cylindrical projections is equatorial. The graticule appears entirely different on normal-aspect and transverse-aspect cylindrical projections (figure 3.12). You can recognize the normal aspect of cylindrical projections by horizontal parallels of equal length, vertical meridians of equal length that are also equally spaced, and right-angle intersections of meridians and parallels. Transverse-aspect cylindrical projections look quite different. The straight-line parallels and meridians in the normal-aspect projection become curves in the transverse aspect. These curves are centered on the vertical line of tangency in the tangent case or halfway between the two vertical lines of tangency in the secant case. As you can see in figures 3.11 and 3.12, the choice of projection surface aspect leads to different-looking appearances for the earth’s land masses and the graticule. Yet the distortion properties of a given projection remain unaltered when the aspect is changed from polar to oblique or equatorial. The map on the left in figure 3.12 is a good example. In the normal aspect of the tangent case, the SF for the cylindrical projection is 1.0 at the equator and increases north and south at right angles to the equator. In the transverse aspect of the tangent case (figure 3.12, right), the SF is 1.0 along the pair of meridians that form the line of tangency (that is, a selected meridian and its antipodal meridian [see chapter 1 for more on antipodal meridians]). The SF again increases perpendicularly to the line of tangency. For a normal-aspect projection, a particular SF value can be found on horizontal lines that are the same distance above and below the equator. In a transverseaspect projection, SF values are the same on vertical lines that are equally spaced to the left and right of the meridian that is the line of tangency.

Figure 3.11. Tangent case equatorial, oblique, and polar aspects for planar projections. Scale distortion increases radially away from the point of tangency, no matter where the point is located on the globe.

Figure 3.12. Tangent-case normal (left) and transverse (right) aspects for cylindrical projections. Scale distortion increases perpendicularly away from the line of tangency, despite where the line is located on the globe.

Other map projection parameters Many other map projection parameters are important because they can affect the definition, appearance, and appropriate use of map projections. The central meridian (also called the longitude of origin or, less commonly, the longitude of center) defines the origin of the x-coordinates. This meridian is usually placed in the center of the area being mapped. The central parallel (also called the latitude of origin or, less commonly, the latitude of center) defines the origin of the y-coordinates. It is commonly placed in the

center of the mapped area with a false northing used to avoid negative y-coordinates. Conic projections, however, often are designed with a lower central parallel so that the origin of ycoordinates is below the mapped area, which makes all y-coordinates positive.

COMMONLY USED MAP PROJECTIONS As you have seen, one way that map projections are commonly grouped into families is on the basis of the developable surface—a plane, cone, or cylinder. We use these family groupings to structure our discussion of commonly used map projections.

Planar projections Orthographic The orthographic projection is how the earth would appear if viewed from a distant planet. Because the light source for this true-perspective projection is at an infinite distance from the generating globe, all rays are parallel (figure 3.8). This projection appears to have first been used by astronomers in ancient Egypt, but it came into widespread use during World War II, with the advent of the global perspective provided by the age of air travel. It is even more popular in today’s space age, often used to show land cover and topography data obtained from remote-sensing devices (see figure 3.13). In fact, the generating globe and half-globe illustrations in this book are orthographic projections, as is the day/night globe on the front cover of the book. The main drawback of the orthographic projection is that only a single hemisphere can be seen at a time. In the past, showing the entire earth required two maps, often of the Northern and Southern or Western and Eastern Hemispheres. Nowadays, animated rotating globes, such as the one you can see on the website or by scanning the QR code in figure 3.13, are created by combining a series of maps of satellite imagery or topography in orthographic projections, each with a slightly different point of tangency. Stereographic Projecting a light source from the antipodal point on the generating globe to the point of tangency creates the stereographic projection (see figure 3.8). This projection is conformal, so shape is preserved in small areas. The Greek scholar Hipparchus, credited with proposing the system of parallels (see chapter 1), is also credited with inventing this projection in the second century BC. It is now most commonly used in its polar aspect and secant case for maps of polar areas (figure 3.14). It is the projection surface for the universal polar stereographic grid coordinate system for polar areas, as you will see in chapter 4. A disadvantage of the stereographic projection is that it is generally restricted to one hemisphere. If it is not restricted to one hemisphere, the distortion near the edges increases to such a degree that the geographic features in these areas are basically unrecognizable. In past centuries, it was used for atlas maps of the Western or Eastern Hemisphere.

Figure 3.13. The orthographic projection is often used to show the spherical shape of the earth. You can view an animated rotating earth made from orthographic projections of satellite images at http://eoimages.gsfc.nasa.gov/images/imagerecords/57000/57760/rotate_320. mpg or by scanning the QR code below the still image of the rotating earth. Rotating globe from http://eoimages.gsfc.nasa.gov/images/imagerecords/57000/57760/rotate_32.

Gnomonic Projecting the generating globe on a planar surface with the light source at the center of the generating globe produces the gnomonic projection (see figure 3.8). One of the earliest map projections, the gnomonic projection was first used by the Greek scholar Thales of Miletus in the sixth century BC for showing different constellations on star charts, which are used to plot planetary positions throughout the year. The position of constellations in the sky over the year was used as a calendar, telling farmers when to plant and harvest crops and when floods would occur. Horoscopes and astrology also began with the ancient Greeks over 2,000 years ago. Many believed that the position of the sun and the planets had an effect on a person’s life and that future events in their lives could be predicted on the basis of the location of celestial bodies in the sky. The gnomonic projection has the distinction of being the only projection with the useful property of all great circles on the globe being shown as straight lines on the map (figure 3.15). Because a great-circle route is the shortest distance between two points on the earth’s surface (see great-circle directions in chapter 13), the gnomonic projection is especially valuable as an aid to navigation. The gnomonic projection is also used for plotting the global dispersal of seismic and radio waves. Its major disadvantages are extreme distortion of shape and area away from the point of tangency and the inability to project a complete hemisphere. Azimuthal equidistant

The azimuthal equidistant projection in its polar aspect has the distinctive appearance of looking like a dart board—equally spaced parallels and straight-line meridians that radiate outward from the pole (figure 3.16). This arrangement of parallels and meridians results in all straight lines drawn from the point of tangency being great-circle routes. Equally spaced parallels means that great-circle distances are correct along these straight lines because the north–south SF along meridians is 1.0. The ancient Egyptians apparently first used this projection for star charts, but during the age of air travel it also became popular for use by pilots planning long-distance air routes. In the days before electronic navigation, the flight planning room in major airports had a wall map of the world that was in an oblique-aspect azimuthal equidistant projection centered on the airport. You will also find these maps in the public areas of some airports. All straight lines drawn from the airport show the correct distances (that is, they are correctly scaled great-circle routes). This planar projection is one of the few that can show the entire surface of the earth, although the distortion in shape and area toward the edges of the map is extreme because the east–west SF increases to infinity at the South Pole, which is shown as a circular line.

Figure 3.14. Polar stereographic projection of the Northern Hemisphere. Because this is a conformal projection, tiny circles on the generating globe are projected as circles of the same size at the point of tangency to four times as large at the equator on the edge of the map.

Figure 3.15. Polar gnomonic projection of the Northern Hemisphere from 15° N to the pole. All straight lines on the projection surface are great-circle routes. This projection severely distorts shape compared with the polar stereographic projection in figure 3.14, despite the fact that less than a hemisphere is shown.

Lambert azimuthal equal area In 1772, the mathematician and cartographer Johann Heinrich Lambert published equations for the tangent-case planar Lambert azimuthal equal-area projection, which, along with other projections he devised, carries his name. This equal-area projection is usually restricted to a hemisphere, with polar and equatorial aspects used historically in commercial atlases (figure 3.17). More recently, this projection is used for statistical maps of continents and countries that are basically circular in overall extent, such as North America

and Africa. You will also see the oceans shown on maps that use the equatorial or oblique aspects of this projection. The Lambert azimuthal equal-area projection is particularly well suited for maps of the Pacific Ocean, which is almost hemispheric in extent.

Figure 3.16. Polar-aspect azimuthal equidistant projection for a map of the world. Great-circle distances are correct along straight lines outward from the point of tangency at the pole.

Figure 3.17. The Lambert azimuthal equal-area projection is often used for maps of continents that have approximately equal east–west and north–south extents. This map of North America in the box is part of an oblique aspect of the projection centered at 45° N, 100° W.

Cylindrical projections Equirectangular The equirectangular projection is also called the plate carrée or equidistant cylindrical projection (also recently called the geographic projection). This simple map projection, nearly 2,000 years old, is attributed to Marinus of Tyre, a Greek geographer, cartographer, and mathematician credited with founding mathematical geography, who is thought to have constructed the projection in about AD 100. Parallels and meridians are mapped as a grid of equally spaced horizontal parallels and vertical meridians, with the parallels twice as long as the meridians (figure 3.18). The equal spacing of parallels means that the projection is equidistant in the north–south direction, with a

constant SF of 1.0. In the east–west direction, the SF increases steadily from a value of 1.0 at the equator to infinity at each pole, which is projected as a straight line (and therefore has point-to-line correspondence). You may see world maps that show elevation data or satellite imagery made in this projection. This choice of projection is not based on any geometric advantage, but rather on the simplicity of projecting the spherical earth on a flat map, particularly when map projections were drawn by hand. World Mercator Invented by Gerhardus Mercator in 1569, the Mercator projection is a tangentcase cylindrical conformal projection. As with all conformal projections, shape is preserved in small areas. This projection offers a classic example of how a single projection can be used both poorly and well. Looking at the projection (figure 3.19), you can imagine Mercator starting the construction of his projection with a horizontal line to represent the equator, and then adding equally spaced vertical lines to represent the meridians. Mercator knew that meridians on the globe converge toward the poles so that meridians he drew as parallel vertical lines must become progressively more widely spaced toward the poles than they would be on the generating globe. He progressively increased the spacing of parallels away from the equator so that the increase matched the increased spacing between the meridians. As a result of this extreme distortion toward the poles, he cut off his projection at 80° N and S. This spacing of parallels produced a conformal map projection. More importantly, it produced the only projection in which all lines of constant compass direction, called rhumb lines, are straight lines on the map. Navigators who used a magnetic compass immediately saw the advantage of plotting courses on maps using the Mercator projection, because any straight line they drew was a line of constant compass bearing, or the azimuth of a track. This geometric property meant that they could plot a course on the map and simply maintain the associated bearing during passage to arrive at the plotted location. It is not hard to understand why navigators preferred a map on which compass bearings appeared as straight lines (see chapter 13 for more on rhumb line plotting). The Mercator projection has been used ever since for nautical charts, such as small-scale navigational charts of the oceans. Large-scale nautical charts used for coastal navigation can be thought of as small rectangles cut out of a world map made in the Mercator projection. Of course, using these maps for navigation is not really as simple as plotting a single straight line and then maintaining this heading. Recall that the gnomonic projection is the only projection in which all great circles on the generating globe are shown as straight lines on the flat map. Lines drawn in a Mercator projection show constant compass bearing, but they are not the same as the great-circle route, which is the shortest distance between two points on a globe. Therefore, the Mercator projection is often used in conjunction with the gnomonic projection to plot navigational routes, as we explain in chapter 13.

Figure 3.18. Equirectangular world map projection.

Figure 3.19. Mercator projection shown with rhumb lines between selected major world cities.

The use of the Mercator projection in navigation is an example of a projection used for its best purpose. An example of a poor use of the Mercator projection is for wall maps of the world. You saw earlier that this projection cannot cover the entire earth, and is often cut off at 80° N and S. Cutting off part of the world creates a rectangular map with a height-to-width ratio that fits walls well. The problem, of course, is the extreme scale enlargement and consequent area distortion at higher latitudes. The area exaggeration of North America, Greenland, Europe, and Russia gives many people an erroneous impression of the size of the land masses in the Northern Hemisphere. Web Mercator The web Mercator projection, a variation of the Mercator world projection, is widely used for online maps. Fundamentally, it is the Mercator world projection with latitude limits of 85.05° N and S, bounding parallels that make the projection a square. It is mathematically similar to the Mercator world projection on the authalic sphere but uses latitude-longitude coordinates based on the WGS84 ellipsoid for map production. These ellipsoid coordinates, coupled with slightly different mathematical formulas, make the web Mercator projection not precisely conformal. The difference between the web and world Mercator projections is imperceptible on small-scale world maps, but on large-scale maps of local areas, the difference becomes noticeable. The discrepancy becomes progressively larger farther north or south of the equator, and can reach as much as 35 kilometers (about 22 miles). Gall-Peters The Gall-Peters projection is a variation of the cylindrical equal-area projection. Its equations were published in 1885 by Scottish clergyman James Gall as a secant case of the cylindrical equal-area projection that lessens shape distortion in higher latitudes by placing lines of tangency at 45° N and 45° S. Arno Peters, a German historian and journalist, devised a map based on Gall’s projection in 1967 and presented it in 1973 as a “new invention” superior to the Mercator world projection (figure 3.20). The “new” projection generated intense debate because of Peters’ assertion that it was the only “nonracist” world map (not to mention the fact that he did not “invent” it). Peters claimed that his map showed the sizes of developing countries more faithfully than the Mercator projection, which distorts and dramatically enlarges the size of Eurasian and North American countries.

Figure 3.20. Gall-Peters projection world map.

Although the relative area of land masses is maintained in the Gall-Peters projection, their shapes are distorted. According to prominent cartographer Arthur H. Robinson, discussed later in the chapter, the Gall-Peters map is “somewhat reminiscent of wet, ragged long winter underwear hung out to dry on the Arctic Circle.” Although several international organizations adopted the Gall-Peters projection when it was introduced, it was later realized that other equal-area world projections distort the shapes or land masses far less. Maps based on the Gall-Peters projection continue to be published, although few major map publishers use the projection today. Transverse Mercator The same year that Lambert constructed his azimuthal equal-area projection, in 1772, he also constructed the transverse Mercator projection, along with the Lambert conformal conic projection described later in this chapter. In Europe, the transverse Mercator projection is called the Gauss-Krüger projection, in honor of the mathematicians Carl Gauss and Johann Krüger who later worked out formulas describing its geometric distortion and equations for making it on the ellipsoid. Lambert’s idea for the transverse Mercator projection was to rotate the Mercator projection by 90 degrees so that the line of tangency becomes a pair of meridians—that is, any selected meridian and its antipodal meridian (see figure 3.12, bottom right). The resulting projection is conformal, as is the Mercator projection, but rhumb lines are no longer straight lines. Along the line of tangency, the SF is 1.0, and the scale increases perpendicularly away from the line of tangency. You’re likely to see the transverse Mercator projection used to map north–south strips of the earth called gores (figure 3.21), which are used in the construction of globes because narrow north–south strips of the earth are projected with no local shape distortion and little

distortion of area. Because printing the earth’s surface directly on a spherical surface is difficult, a map of the earth is printed as flat, elongated gores that are cut out along their edge meridians, and then pasted on a spherical base surface. The narrow, six-degree-wide zones of the universal transverse Mercator grid coordinate system (described in chapter 4) are based on a secant-case transverse Mercator projection. North–south trending zones of the US state plane coordinate system (also explained in chapter 4) are also based on secant cases of the projection. Most 1:24,000-scale USGS topographic maps are projected on these state plane coordinate system zones.

Figure 3.21. Two gores of the globe in transverse Mercator projections that are 30 degrees wide at the equator and centered at 90° W and 120° W. To make a world globe, the gores are trimmed to their edge meridians, and then pasted on the sphere used to make the globe.

Conic map projections Lambert conformal conic The Lambert conformal conic projection is another widely used map projection devised mathematically by Lambert in 1772. It is a secant-case normal-aspect conic projection with its two standard parallels placed to minimize the map’s overall scale distortion. The standard parallels for maps of the conterminous United States are placed at 33° N and 45° N to minimize scale distortion within the mapped area with a maximum of less than 3 percent at the map’s edges (figure 3.22). Although the transverse Mercator projection is used as the basis for the state plane coordinate system zones in north–south trending states in the United States, the Lambert conformal conic projection is used as the basis for east–west zones, such as Oregon and Wisconsin. One major use of the Lambert conformal conic projection is for aeronautical charts in midlatitude countries—all US 1:500,000-scale sectional charts are in this projection. Recall that navigators (like aviators) prefer navigational charts that use conformal projections, which preserve local shapes and directions. Equally important is that straight lines drawn on the charts are close to being great-circle routes on the earth’s surface (figure 3.22). The Mercator projection, of course, is also conformal, but the Lambert conformal conic has the advantage of much smaller scale distortion in the mid-latitudes (and no distortion along the two standard parallels). Albers equal-area conic Mathematically devised in 1805 by the German mathematician Heinrich C. Albers, the Albers equal-area conic projection was first used in 1817 for a map of Europe. This projection is most often used for statistical maps of the conterminous United States and other mid-latitude east–west trending regions such as Europe and Australia. For example, you will see this projection used for statistical maps created by the US Census Bureau and other federal agencies in the United States. The reason for the widespread use of the Albers projection is simple—people looking at maps that use this projection can assume that areas on the map are true to their areas on the earth.

Figure 3.22. Lambert conformal conic projection used for a map of the conterminous United States. Straight lines are close to being great-circle routes, particularly on north–south paths.

Figure 3.23. Albers equal-area conic projection of the conterminous United States with standard parallels at 29.5° N and 45.5° N, resulting in maximum scale distortion of 1.25 percent.

The secant-case version that has been used for nearly 100 years for the conterminous United States has standard parallels placed at 29.5° N and 45.5° N (figure 3.23). This placement reduces the scale distortion to less than 1 percent at the 37th parallel in the middle of the map, and to 1.25 percent at the northern and southern edges of the country. It also gives the minimum average distortion in shape for the area mapped, as measured by what cartographers call angular deformation. Maps of Alaska use standard parallels of 55° N and 65° N, and maps of Hawaii use standard parallels of 8° N and 18° N. Some USGS products, such as the national tectonic and geologic maps, also use the Albers projection, as do reference maps and satellite image maps of the country at a scale of around 1:3,000,000.

Pseudocylindrical and other projections You have seen that map projections can be classed into families on the basis of the nature of the surface used to construct the projection, resulting in planar, cylindrical, and conic projections. We also noted that map projections can be classified on the basis of their geometric distortion properties relating to distance, shape, direction, and area. For many purposes, a projection that “looks right” is more important than a projection that rigidly provides fidelity to area, distance, shape, or direction. A “right-appearing” projection is called orthophanic, a compromise projection that is neither equal area nor conformal nor equidistant, but rather is an attempt to balance these geometric properties. Now we look at a last set of projections that don’t strictly fit into one or both of the families we’ve already described. For example, pseudocylindrical map projections are

similar to cylindrical projections in that parallels are horizontal lines and meridians are equally spaced. The difference is that all meridians except the vertical-line central meridian are curved instead of straight. We also examine a few map projections that are of special interest for a variety of reasons, including orthophanic projections. Mollweide If you’ve seen a world map in the shape of an ellipse, most likely you were looking at a map made using the Mollweide projection. This projection was invented in 1805 by the German mathematician Carl B. Mollweide as an elliptically shaped equal-area projection with the equator as the standard parallel and the prime meridian as the central meridian (figure 3.24). Parallels are projected as horizontal lines, but they are not equally spaced as on the generating globe. Instead, the distance in spacing is reduced toward the poles. The elliptical shape of this projection makes it look more “earthlike,” and the overall distortion in shape is less than in other equal-area world projections such as the GallPeters.

Figure 3.24. Mollweide equal-area projection used for a world map.

You’ll find the Mollweide projection used for world maps that show a wide range of global phenomena, from population to land cover to major diseases. Cartographers have devised other versions of the projection by adjusting the central meridian to better show the oceans or to center attention on a particular continent. The projection also looks “earthlike,” making it one of the most popular projections for showing human-oriented statistical data for the world. Sinusoidal According to some sources, Jean Cossin of Dieppe, France, appears to be the originator of the sinusoidal projection, which he used to create a world map in 1570. Others suggest that Mercator devised it since it was included in later editions of his atlases. Like the Mollweide, it is an equal-area projection for the world with straight parallels and curved meridians, but the distinctive feature of this projection is the pointed shape of the poles (figure 3.25). It was used by Nicholas Sanson (circa 1650) of France for atlas maps of

the world and continents, and by John Flamsteed (1729) of England for celestial maps. Hence, you may also see it called the Sanson-Flamsteed projection.

Figure 3.25. Sinusoidal equal-area projection for a map of the world.

In addition to correctly portraying the relative areas of continents and countries, the sinusoidal world projection has an SF of 1.0 along the central meridian, and the east–west SF is 1.0 anywhere on the map. This projection is equidistant in the east–west direction and in the north–south direction along the central meridian, but only in these directions. Severe shape distortion is also evident at the edges of the map. Homolosine World map projections can also be made by combining different projections along certain parallels or meridians. One example is the uninterrupted Goode homolosine projection, or more commonly called the homolosine projection (figure 3.26, top). Constructed in 1923 by American geography professor J. Paul Goode, it is a composite of two equal-area pseudocylindrical projections. The sinusoidal projection is used for the area from 40° N to 40° S latitude, while Mollweide projections are used for areas from 40° N and 40° S to the respective poles. Because the two component projections are equal area, the homolosine projection is also equal area. Compare the maps in figures 3.24 and 3.25 to the map in figure 3.26—the shapes of the continents look less distorted in the homolosine projection than in the Mollweide and sinusoidal projections, particularly in polar areas. An interrupted projection is one in which the generating globe is segmented to minimize the distortion within any lobe (section) of the projection. Shape distortion at the edges of the map can be lessened considerably by interrupting the composite projection into lobes that are pieced together along a central line, usually the equator. With interruption, the less geometrically distorted parts of the projection are used within each lobe. The interrupted Goode homolosine projection (figure 3.26, bottom), sometimes shortened to the interrupted homolosine projection, was created in 1923 by Goode from the uninterrupted homolosine projection. It is an interrupted pseudocylindrical equal-

area composite map projection used for world maps. The projection is a composite of 10 Mollweide and sinusoidal projection pieces that form six interrupted lobes. The two lobes at the top and four at the bottom are Mollweide projections from 40° N or 40° S to the adjacent pole, each with a different central meridian. The four interior regions from the equator to 40° N and 40° S are sinusoidal projections, each with a different central meridian. If you look carefully along the edges of the lobes, you can see a subtle discontinuity at the 40th parallels. The two northern sections are usually shown with some land areas repeated in both regions to show the Greenland land mass without interruption. You will find the interrupted homolosine projection used historically for maps in commercial world atlases, most notably the Goode’s World Atlas used in schools throughout the United States, to show a variety of global information. It is a popular projection for showing physical information about the entire earth, such as land elevations and ocean depths, land cover and vegetation types, and satellite image maps. In this projection, either land areas or water areas can be shown in their entirety, depending on the placement of the central meridians for each of the lobes. When viewing these maps, remember that the continuity of the earth’s surface is lost by interrupting the projection, which may not be apparent if the graticule is left off the map.

Figure 3.26. Top, uninterrupted homolosine projection and, bottom, interrupted Goode homolosine equal-area world projection.

Robinson In 1963, prominent American academic cartographer and professor of geography Arthur H. Robinson constructed a pseudocylindrical projection, the Robinson projection, that is neither equal area nor conformal but is a compromise projection that makes the continents “look right.” For this orthophanic projection (figure 3.27), Robinson visually adjusted the horizontal parallels and curving meridians until they appeared suitable for use as a world map projection for atlases and wall maps. To make this adjustment, Robinson represented the poles as horizontal lines that are a little over half the length of the equator. You may have seen this projection on world maps or wall maps of the world that show the shape and area of continents with far less distortion than maps made in the Mercator world projection.

Figure 3.27. Robinson pseudocylindrical compromise projection for a world map.

Winkel tripel Map projections may also be constructed as mathematical combinations of two or more projections. Perhaps the best-known example is the Winkel tripel projection constructed in 1921 by the German cartographer Oswald Winkel. The term tripel is not someone’s name, but rather a German word that means a combination of three elements. Winkel used the term to emphasize that he had constructed a compromise projection that was neither equal area nor conformal nor equidistant, but rather that it minimized all three forms of geometric distortion. The Winkel tripel projection looks similar to the Robinson projection, but if you look closely you will see that parallels are not the straight, horizontal lines characteristic of pseudocylindrical projections. Rather, they are slightly curving, nonparallel lines (figure 3.28). The Winkel tripel projection was not used widely until 1998, when the National Geographic Society announced that it was adopting the projection as its standard for maps of the entire world. As a result, use of the Winkel tripel projection has increased dramatically. One advantage of using this projection for web maps is that the aspect ratio (the width-to-height ratio) is convenient for on-screen display.

Figure 3.28. Winkel tripel projection used to create a world map.

Oblique-perspective projections True-perspective projections, such as the orthographic, stereographic, and gnomonic projections described earlier, are vertical-perspective projections because the center point of projection is the point of tangency of the projection surface. Any other alignment produces an oblique-perspective projection of the earth. In figure 3.29, the vertical-perspective developable surface is a horizontal plane that is tangent to the generating globe at the North Pole, the polar-aspect projection point of tangency. The mapped area extends outward from the point of tangency to the parallel at which rays from the light source are seen on the generating globe as being on the horizon (tangent to the globe).

Figure 3.29. Geometry of vertical- and oblique-perspective map projections. From Map Projections: A Working Manual, p. 170.

An oblique-perspective polar-aspect projection has the same light source above the North Pole, but the developable surface is tilted and secant to the generating globe. In figure 3.29, the developable surface intersects the globe at the North Pole and at about 37° N. The rays from the light source to the parallels on the generating globe intersect the verticalperspective developable surface in a symmetrical manner to the right and left of the North Pole (the ever closer spacing of rays intersecting the developable surface is the same). Now examine how the rays from the light source intersect the oblique-perspective developable surface to the right and left of the North Pole. The rays to the right of the pole are more closely spaced on the developable surface than those to the left of the pole. We use the term foreshortening for this compression and extension of the projected positions of features equally spaced on the generating globe. Foreshortening is what gives oblique-perspective projections the appearance of viewing part of the earth obliquely from a point above the surface of the earth. Tilting the oblique-perspective developable surface even more and lowering the light source closer to the globe creates an obliqueperspective map of a smaller area, such as California and Nevada as seen in figure 3.30. Web apps such as ArcGlobe and ArcGIS Earth use oblique-perspective projections to show the globe as it appears from space. In these apps, you can interactively pan and zoom, flying over the earth from the perspective of a spacecraft or an airplane.

MAP PROJECTIONS ON THE SPHERE AND ELLIPSOID Mapmakers have a general rule that small-scale maps can be projected from a sphere, but large-scale maps must always be projected from an ellipsoidal surface using a datum such

as WGS84. You saw in chapter 1 that small-scale world or continental maps normally use coordinates based on a spherical approximation of the ellipsoidal earth, because it is much easier to construct these maps from spherical geocentric coordinates. Equally important, the differences in the plotted positions of spherical and corresponding geodetic coordinates are negligible on small-scale maps. Large-scale maps are projected from an ellipsoidal surface because, as you saw in chapter 1, the spacing of parallels decreases slightly but significantly from the poles to the equator. We note in chapter 1 that on the WGS84 ellipsoid, the distance between two points that are one degree apart in latitude at the equator, between 0° and 1°, is 110.567 kilometers (68.703 miles), a little more than a kilometer shorter than the 111.699-kilometer (69.407-mile) distance between two points at the North or South Poles between 89° and 90°. Now we examine the differences in length when you project three lines that span one degree of geodetic latitude using the transverse Mercator projection, one along a meridian at the North Pole, the second at the 45th parallel, and the third at the equator (figure 3.31). At a scale of 1:1,000,000, a line the length of a degree of latitude from 89° to the adjacent pole is projected as slightly over one millimeter longer than a line that is one degree of latitude in length from the equator to 1°. The difference in these two lengths may seem minimal, but on the ground it represents slightly over a kilometer (0.62 miles), as you would expect from the distances listed in the previous paragraph.

Figure 3.30. Oblique-perspective projection for a map of California and Nevada. Courtesy of Esri ArcGlobe.

At larger map scales, this slightly greater than a kilometer difference in ground length for a degree of latitude at the equator and the poles becomes more noticeable in the transverse Mercator projection—for instance, slightly over one centimeter on polar and equatorial maps at a scale of 1:100,000. Topographic and other maps at this scale and larger are projected from an ellipsoid so that accurate distance and area measurements can be made. The same holds true for large-scale equidistant projections such as the polar-aspect azimuthal equidistant projection, because the spacing of parallels in the projection must be slightly lessened from the pole outward to reflect their actual spacing on the earth.

Figure 3.31. Transverse Mercator map projections for lines one degree of geodetic latitude in length on the WGS84 ellipsoid between 89° and 90°, 44° and 45°, and 0° and 1°.

You saw in chapter 1 and in table B.2 in appendix B that the only place on the earth where the 69.09 miles (111.20 kilometers) per degree spacing of parallels on the authalic sphere is the same as on the WGS84 ellipsoid is in the mid-latitudes close to the 45th parallel. Imagine examining two maps that straddle the 45th parallel made at the same scale in the transverse Mercator map projection, one based on the authalic sphere and the second based on the WGS84 ellipsoid. Lines that are one degree of latitude in length on the earth are projected as north–south lines of the same length in the two projections.

SUMMARY Every map projection has its virtues and limitations. You can evaluate a projection only in light of the purpose for which a map is to be used. You should not expect that the best projection for one use is the most appropriate for another. If the cartographer does a good job of considering projection properties and you are careful to take projection distortion into consideration in the course of map use, the map projection problem effectively vanishes. At the same time, however, there is the real possibility that the mapmaker, through ignorance or lack of attention, chose an unsuitable projection or projection parameters. Therefore, it is of utmost importance to map users to understand the map projection concepts discussed in this chapter to make sure that the maps they use are in the appropriate projection. Often, map use takes place at the local level, in which earth curvature isn’t a big problem, so global map projections aren’t a great concern for many users. With regions as small as those covered by topographic map quadrangles, the main projection-related problem is that, although the individual map sheets match in a north–south direction, they don’t fit together in an east–west direction. Yet even this difficulty isn’t a serious handicap unless map users try to create a large map composed of many small map images, which is difficult and unrealistic. You no longer need to “make do” with inappropriate map projections. You can modify projections for almost any use. Furthermore, it is now practical to create your own projections, although there are few who can actually carry out the required mathematical computations. That said, with mapping software, such as ArcMap software from Esri, you can easily modify existing map projections and their parameters to meet nearly any map use. Knowing how to modify the parameters is contingent on your understanding of map projections and their properties. Therefore, a better approach is to manipulate existing map projections to create the ideal projection for your map use problem. Of course, this manipulation is only possible if you know enough about map projections to take advantage of the opportunities presented. Being projection savvy also helps you evaluate maps published in magazines and newspapers and on websites. A surprising number of maps continue to be created with inappropriate projections. World maps in the popular media often lack the latitude and longitude graticule. Lack of the graticule masks the extreme spatial distortion in the map projections used and gives the impression that shapes or areas are faithfully represented. Thus, you are well advised to reconstruct the missing graticule mentally as a first step in map reading. Understanding the map projection concepts described in this chapter will go a long way toward helping you make the right decisions when using and choosing map projections. There are also books and websites dedicated to map projections, so be sure to check them out if your map use or mapmaking needs warrant it. National mapping agencies such as the US Geological Survey and the Swiss Federal Office of Topography (swisstopo), the Swiss national mapping agency, also offer useful map projection information on their websites and in their publications.

NOTE 1. The equation for the sphere is equation (3.3):

where R is the earth’s radius, and φ is the latitude in radians (there are 2π radians in a 360° circle, so that 1 radian is approximately 57.295 degrees). The value of equation (3.3) at 90° (π/2 radians) is infinity and at −90° (−π/2 radians) is minus infinity.

SELECTED READINGS American Cartographic Association. 1986. Which Map Is Best? Projections for World Maps. Bethesda, MD: American Congress on Surveying and Mapping. ———. 1988. Choosing a World Map—Attributes, Distortions, Classes, Aspects. Bethesda, MD: American Congress on Surveying and Mapping. ———. 1991. Matching the Map Projection to the Need. Bethesda, MD: American Congress on Surveying and Mapping. Bugayevskiy, L. M., and J. P. Snyder. 1995. Map Projections: A Reference Manual. London: Taylor & Francis. Canters, F., and H. Decleir. 1989. The World in Perspective: A Directory of World Map Projections. New York: John Wiley & Sons. Hsu, M. L. 1981. “The Role of Projections in Modern Map Design.” Cartographica 18 (2): 151–86. Maling, D. H., 1992. Coordinate Systems and Map Projections, 2nd ed. New York: Pergamon. Pearson, F. 1990. Map Projections: Theory and Applications. Boca Raton, FL: CRC. Richardus, P., and R. K. Adler. 1972. Map Projections: For Geodesists, Cartographers and Geographers. New York: American Elsevier. Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. Elements of Cartography, 6th ed., 59–90. New York: John Wiley & Sons. Snyder, J. P. 1987. “Map Projections: A Working Manual.” Professional Paper 1395. Washington, DC: US Geological Survey. ———. 1993. Flattening the Earth: A Thousand Years of Map Projections. Chicago: University of Chicago Press. Snyder, J. P., and H. Steward, eds. 1988. Bibliography of Map Projections. Bulletin 1856. Washington, DC: US Geological Survey. Snyder, J. P, and P. M. Voxland. 1989. “An Album of Map Projections.” Professional Paper 1453. Washington, DC: US Geological Survey. Tobler, W. R. 1962. “A Classification of Map Projections.” Annals of the Association of American Geographers 52:167–75.

chapter four GRID COORDINATE SYSTEMS CARTESIAN COORDINATES GRID COORDINATES Universal transverse Mercator system Universal polar stereographic system State plane coordinate system US state grids Other grid coordinate systems GRID COORDINATE DETERMINATION ON MAPS Grid coordinate system appearance on maps Grid orientation Grid coordinate determination GRID CELL LOCATION SYSTEMS Military Grid Reference System US National Grid Ordnance Survey National Grid Proprietary grids Map publishers Amateur radio operators SELECTED READINGS

4 Grid coordinate systems There are several ways to pinpoint locations on maps. You saw in chapter 1 that the latitude-longitude graticule has been used for over 2,000 years as the worldwide locational reference system. Geocentric latitude and longitude coordinates on the sphere or geodetic latitudes and longitudes on the oblate ellipsoid, still key to modern position finding, are not as well suited for making measurements of length, direction, and area on the earth’s surface. The basic difficulty is that latitude-longitude is a coordinate system that gives positions on a rounded surface. It would be much simpler if we could designate location on a flat surface, using horizontal and vertical lines spaced at regular intervals to form a square grid. We could then simply read coordinates from the square grid of intersecting straight lines. In chapter 3, we explain that most maps are created by projecting the earth’s spherical surface on a flat surface, such as a sheet of paper or a computer screen. The advantage of the flat-map projection surface is that you can locate something by using a simple two-axis coordinate reference system. This coordinate system is the basis for the square grid of horizontal and vertical lines on the map. A coordinate system mathematically placed on a flat-map projection surface is called a grid coordinate system. To devise such a system for large areas, you must deal somehow with the earth’s curvature. From chapter 3, you know that transferring something spherical to something flat always introduces geometric distortion. But you also know that map projection distortion caused by the earth’s ellipsoidal shape is minimal for fairly small regions. If you superimpose a square grid on flat maps of small areas, you can achieve positional accuracy that is good enough for many map uses. All grid coordinate systems are based on Cartesian coordinates, invented in 1637 by renowned French philosopher and mathematician René Descartes.

CARTESIAN COORDINATES If you superimpose a square grid on the map, with divisions on a horizontal x-axis and a vertical y-axis in which the x- and y-axes cross at the system’s origin, you have established the familiar Cartesian coordinate system (figure 4.1). You can now pinpoint any location on the map precisely by giving its two coordinates (x,y). The Cartesian coordinate system is divided into four quadrants (I to IV) on the basis of whether the values along the x- and y-axes are positive or negative. Mapmakers use only quadrant I for grid coordinate systems so that all coordinates are positive numbers relative to the (0,0) grid origin (see figure 4.1). To use only the first quadrant, they position the origin to the southwest of the entire mapped area, as you’ll see for the various systems described in this chapter.

GRID COORDINATES Defining map positions using Cartesian coordinates has a definite advantage over using the spherical graticule to define positions. Measuring x- and y-coordinates from horizontal and vertical axes that have equally spaced distance increments greatly simplifies locating environmental features because you do not have to deal with the decreasing separation between meridians that converge toward the poles. Grid coordinate systems based on the Cartesian coordinate system are especially handy for such map analysis procedures as finding the distance or direction between locations or determining the area of a mapped feature such as a lake. We examine in depth one type of grid coordinate system—universal transverse Mercator (UTM)—that is found on many topographic maps, military maps, and navigational charts, among other types of maps. We also look in detail at the state plane coordinate system, which is found on many large-scale maps of the United States.

Figure 4.1. The structure of the Cartesian coordinate system. The basic notation for this system includes appropriate use of a negative sign for the x- and ycoordinates in quadrants II, III, and IV. Cartographers use only quadrant I (shaded) on maps because it is desirable to have all coordinates as positive numbers.

Universal transverse Mercator system A grid coordinate system can be used worldwide if enough zones are defined to ensure reasonable geometric accuracy. A zone is a well-defined region on the earth between definite limits, especially between two meridians of longitude or parallels of latitude. The best-known grid coordinate system of international scope is the universal transverse Mercator (UTM) system. The UTM system was developed by the French military at the end of World War I and was soon adopted worldwide. The accuracy of UTM grid coordinates determined from topographic maps sufficed for most military applications during World War II and later conflicts, but today GPS receivers provide highly accurate geodetic latitude and longitude positions in the field that are easily converted to UTM grid coordinates (see chapter 14 for more on GPS). The UTM grid system extends around the world from 84° N to 80° S. Sixty north–south zones are used, each six degrees in longitude (figure 4.2). Each zone has its own central meridian and uses a secant-case transverse Mercator projection centered on the zone’s central meridian for each of the 60 zones (see chapter 3 for more on this map projection). This projection makes it possible to achieve a scale distortion (see chapter 3 for more on scale distortion) of one part in 2,500 (0.04 percent) or less because scale factors, or SFs (see chapter 3 for more on scale factors), range from 0.9996 to 1.0004 within each zone.

Figure 4.2. The 60 zones of the universal transverse Mercator (UTM) grid coordinate system.

Zones are numbered from west to east, beginning with Zone 1 from 180° W to 174° W. Zones 10 through 19 cover the conterminous United States, Zones 49 through 56 cover Australia, and so on. Each zone has separate origins for the Northern and Southern Hemispheres. To understand how the origins are specified, it is useful to first understand a few terms and concepts. You read and record x-coordinates to the east of the zone’s origin and the y-coordinates to the north of the origin, giving rise to the use of the terms “eastings” and “northings” for the x- and y-coordinates, respectively. An easting is simply the x-coordinate in a grid coordinate system—that is, the distance east from the origin. In both the Northern and Southern Hemispheres, an easting value of 500,000 meters (written 500,000mE) is assigned to the central meridian of each UTM zone. This value, called the false easting, is added to all x-coordinates so that there are no negative eastings in the zone. Similarly, a northing is the y-coordinate in a grid coordinate system. In the Northern Hemisphere, a northing value of 0mN is assigned to the equator so that all northings are positive numbers. Because no false northing value is added, a UTM northing is simply the distance in meters north of the equator. In the Southern Hemisphere, the equator is given a false northing of 10,000,000mN. There are no negative y-values in the southern UTM zone, because this large false northing value places the origin of the zone very close to the South Pole. Taking Zone 10, which covers much of the West Coast of the United States, as an example (figure 4.3), you can examine the origin for the Northern and Southern Hemispheres. The x-axis aligns with the equator. The central meridian for this zone is 123° W, and the longitude range is 120° W to 126° W. The origin lies on the equator 500,000 meters west of the central meridian at 123° W. In the Southern Hemisphere, the zone’s origin lies 500,000 meters west and 10,000,000 meters south of the intersection of the equator and the central meridian at 123° W. As noted earlier, these origins were selected to make all UTM eastings and northings positive numbers.

Because the equator has different northings for the Northern and Southern Hemispheres, every location that lies exactly on the equator has two UTM grid coordinate pairs. For example, the coordinates for the intersection of the equator central meridian of Zone 10 are as follows: 500,000mE, 0mN in the Northern Hemisphere (for Zone 10 North)

and 500,000mE, 10,000,000mN in the Southern Hemisphere (for Zone 10 South).

Knowing these terms and concepts, you can now designate a location in the UTM grid coordinate system. Because north is conventionally at the top of the map, it may be helpful to remember a simple rule—always read coordinates right-up, or right and then up. You give the easting in meters, the northing in meters, the zone number, and the zone hemisphere (north or south). Thus, you designate the location of the Capitol dome in Madison, Wisconsin, as 305,900mE, 4,771,650mN, Zone 16 North.

The near-global extent (84° N to 80° S) of the UTM grid makes it a valuable worldwide referencing system. The UTM grid is found on most foreign topographic maps and on all recent USGS quadrangles in the topographic, orthophotomap, and orthophoto quad series (see chapter 10 for a description of orthophotomaps and orthophoto quads). All GPS manufacturers have UTM grid coordinates as an option with their receivers. UTM coordinates differ when different horizontal datums are used (see chapter 1 for more on datums), so you should check the datum information on the map and in the GPS receiver to assure that the coordinates are being recorded in the correct datum.

Figure 4.3. The complete universal transverse Mercator grid for Zone 10.

Because meridians and not administrative boundaries delimit UTM zones, it usually takes more than one UTM zone to cover an administrative area completely. For example, the state of Oregon falls into Zones 10 and 11, and Wisconsin falls into Zones 15 and 16 (see figure 4.2).

Universal polar stereographic system As mentioned earlier, UTM grid zones extend from 84° N to 80° S. The UTM grid system was not extended to each pole because the 60 zones converge at the poles, meaning that a new zone would be encountered every few miles. To have complete global coverage in the

remaining polar areas, a complementary grid coordinate system called the universal polar stereographic (UPS) system was created. Virtually all large-scale maps of these high latitudes, such as topographic sheets for Antarctica, are based on the UPS grid. The UPS grid coordinate system consists of a north zone and a south zone. Each zone is superimposed on a secant-case polar stereographic projection that covers a circular region over each pole (see chapter 3 for more on this map projection). The north zone (figure 4.4) extends from 83°30´ N to the pole, at which the UPS coordinate at the grid center is 2,000,000mE, 2,000,000mN. These large numbers were selected so that all eastings and northings are again positive numbers. The south zone extends from the pole to 79°30´ S (80 degrees, plus an additional 30 minutes of latitude extending into the UTM grid). As with the north zone, the grid center is 2,000,000mE, 2,000,000mN, to assure that all coordinates are positive.

State plane coordinate system The state plane coordinate system (SPCS), sometimes shortened to SPC, was created in the 1930s by the land surveying profession in the United States as a way to define property boundaries that simplifies computation of land parcel perimeters and areas. The idea was to completely cover the United States and its territories with grids superimposed on appropriate map projections so that the maximum scale distortion does not exceed one part in 10,000. Thus, a distance measured over a 10,000-foot course is accurate to within a foot of the true measure. This level of accuracy cannot be achieved if there was only one grid covering the whole country, because the area is too large. The solution is to divide each state into one or more zones and make a separate grid for each zone. The United States was originally divided into 125 zones, each having its own map projection based on the Clarke 1866 ellipsoid and NAD27 geodetic latitudes and longitudes (see chapter 1 for further details on this datum). Most states have several zones, as figure 4.5 shows. Secant-case Lambert conformal conic projections are used for zones of predominantly east–west extent, and secant-case transverse Mercator projections are used for zones of greater north–south extent. The only exception is the Panhandle of Alaska, which, because of its geographic orientation, uses an oblique Mercator projection. For states with more than one zone, the names North, South, East, West, and Central are used to distinguish various zones. For example, Texas has North, Central, and South Zones, but also two additional zones—North Central and South Central. In California, which has six zones, roman numerals are used instead.

Figure 4.4. North zone of the UPS grid coordinate system.

The SPCS was originally based on NAD27 and used feet as its primary unit of measure. Because of technological advancements in measuring the earth’s shape and size, some states have converted to NAD83 and meters. In this conversion process, three states (Montana, Nebraska, and South Carolina), Puerto Rico, and the Virgin Islands chose to each have a single zone, replacing what were several zones in the original SPCS. Additionally, California chose to merge two of its zones, resulting in six zones for the state. The logic of the SPCS zones is simple. Zone boundaries follow county boundaries because surveyors must register land surveys within a county. The map projection for each zone has its own central meridian roughly in the center of the zone. An origin is established to the west and south of the zone. Although different for each zone, the origin is always at a parallel to the south of the zone to ensure that all y-coordinates (northings) are positive numbers. The zone center is usually 2,000,000 feet west of the central meridian for Lambert conformal conic zones and 500,000 feet west of the central meridian for transverse Mercator zones (figure 4.5). So the central meridians usually have an x-coordinate (false

easting) of either 500,000 feet for the transverse Mercator or 2,000,000 feet for the Lambert conformal conic zones. These large false eastings for zone centers were selected so that all x-coordinates are positive numbers.

Figure 4.5. Zones of the state plane coordinate system for the conterminous United States. The transverse Mercator projection is used for zones that are oriented north– south, and the Lambert conformal conic projection is used for zones oriented east– west. Multiple zones within a state are identified by North, South, Central, East, and West (and their combinations in Texas and Wyoming), and by roman numerals in California. States without letters have only one zone.

To further illustrate how zones are defined, look at the case of Oregon. The Lambert conformal conic is the map projection for Oregon’s north and south SPCS zones (figure 4.6). The central meridian is the same for both zones, with an easting of 2,000,000 feet in the original SPCS. The x-axis for the origin in each zone, called the latitude of origin, is a parallel that is just south of the counties in the zone (see chapter 3 for more on this and other projection parameters). The intersection of this parallel and the central meridian has a northing of 0 feet and an easting of 2,000,000 feet. You read and record SPCS coordinates in the same manner as UTM coordinates—first to the east and then to the north of the zone’s origin. Specifically, the correct form of SPCS notation is to give the easting in feet, the northing in feet, the state, and the zone name. For example, the location of the state Capitol dome in Madison, Wisconsin, in its abbreviated form is 2,164,600 ft E, 392,280 ft N, Wisconsin, South Zone. As mentioned earlier, the SPCS was modernized by switching to NAD83 and the GRS80 ellipsoid. Zones were redefined in metric units, so the intersection of the Oregon central meridian and the latitude of origin now has an easting of 2,500,000 meters and a northing of 0 meters. These changes in values and the switch from English to metric units make it easy

to tell whether the map is based on NAD27 or NAD83. Because American surveyors apparently could not handle using metric units, almost all states have redefined their NAD83 SPCS units into feet. Consequently, you should check the map legend to see whether old SPCS metric- or new English-unit grid coordinates are used on the map.

Figure 4.6. North and south zones of the Oregon SPCS.

The SPCS served the needs of the states when it was created, and state plane coordinates are still widely used for public works and land surveys. However, the SPCS is now largely obsolete as far as surveyors and other professional map users are concerned. One reason is that the accuracy of one part in 10,000 in locating points is now easily exceeded using modern surveying methods. Also, each SPCS zone is a separate entity with its own grid definition—a fact that frustrates and discourages use across zone boundaries. Nevertheless, the SPCS grid is useful for some map analysis applications, such as distance and direction finding (see chapters 12 and 13 for more on finding distance and direction on maps). You can also enter SPCS zone parameters into your GPS receiver as a user-defined grid (see chapter 14 for more information on GPS and maps).

US state grids

The widespread use of computer mapping and geographic information systems or GIS (automated, spatially referenced systems for the capture, storage, retrieval, analysis, and display of data about the earth) in state and local government has kindled the desire for grid coordinate systems tailored to each state’s needs. States that fall into two UTM zones often create a special state grid by shifting the central meridian of a UTM zone to the center of the state. For example, Wisconsin routinely records and reports data in the Wisconsin transverse Mercator (WTM) system (figure 4.7). The UTM and WTM grid coordinate systems have the same geometric accuracy, but the WTM system avoids the problem of two UTM zones covering the state. The same approach can be taken when a state has two state plane zones. The Oregon Lambert system is a good example (figure 4.8). A single grid has replaced Oregon’s two SPCS zones based on a Lambert conformal conic projection. The new grid coordinate system has the same central meridian but a different latitude of origin. The easting value at the central meridian (400,000mE) is entirely different from the SPCS value, and the standard parallels are different. Georgia, Illinois, Kentucky, and Pennsylvania also have developed single-zone grid coordinate systems for use by their state agencies that deal with land and resource management.

Other grid coordinate systems Many nations have a national grid coordinate system that covers their territory, including, for example, Australia, Belgium, Finland, Great Britain, Ireland, Italy, the Netherlands, New Zealand, Sweden, and Switzerland. Many systems are based on a single transverse Mercator map projection that covers the country with minimal scale distortion. The national grid system used in Great Britain is one of the first systems to be implemented.

Figure 4.7. The Wisconsin transverse Mercator grid coordinate system shifts the UTM zone to the center of the state.

Figure 4.8. The Oregon Lambert system combines two SPCS grids into a single Lambert conformal conic grid coordinate system that covers the entire state.

Great Britain‘s national mapping agency, the Ordnance Survey (OS), devised the Ordnance Survey National Grid (OSNG) coordinate system, which is heavily used by land surveyors as well as for maps based on surveys made by the Ordnance Survey or commercial map firms. Grid coordinates that cover Great Britain, including its outlying islands, are also used in publications such as Admiralty nautical charts, guide books, and government planning documents. The grid is based on the Ordnance Survey of Great Britain 1936 (OSGB36) datum, described in chapter 1. A transverse Mercator projection of the 1830 Airy ellipsoid, on which the datum is based, is used with its origin set at 49° N, 2° W (figure 4.9) so that the 2° W central meridian is a vertical line (2° W is used because it is more central to the entire area mapped). The map projection is superimposed on the national grid so that the central meridian has an easting of 400,000mE and the northing at 49° N is given a value of −100,000mN. Because all eastings and northings must be positive numbers, the grid coordinate system actually starts a little south of the 50° N parallel, at which the northing is 0mN.

Figure 4.9. The Ordnance Survey National Grid coordinate system covers all of England, Scotland, and Wales.

GRID COORDINATE DETERMINATION ON MAPS Grid coordinate system appearance on maps You have seen that UTM grid coordinates appear on US and foreign topographic maps and that state plane coordinates are printed on quadrangles of the USGS topographic series. The way the grid looks varies among map series, so we focus first on US large-scale topographic maps. USGS topographic maps created for use by the military have UTM grid

lines superimposed at one-kilometer intervals, whereas civilian maps show both UTM and SPCS grids by grid ticks along the edges of each map.

Figure 4.10. Appearance of 1,000-meter and 10,000-foot grid ticks in the left corner (top) and left corner (bottom) of a USGS 1:24,000-scale topographic map. Courtesy of the US Geological Survey.

Figure 4.10 shows how state plane and UTM grid ticks appear on two sections of the Madison West 1:24,000-scale topographic map. Black ticks along the outer margin indicate SPCS 10,000-foot grid lines. The northings or eastings of these ticks are given by the value

at one tick on each edge of the map (for example, a 370,000-foot northing at the tick on the upper-left edge of the Madison West quadrangle in figure 4.10). Other ticks are spaced every 10,000 feet on the 1:24,000-scale topographic maps. UTM grid ticks are spaced at 1,000- or 10,000-meter intervals, depending on the map scale. On USGS quadrangles of the 1:24,000-scale topographic series, 1,000-meter grid ticks are printed in blue. These ticks are labeled, with their easting and northing values in black, along the map margin. The principal digits—those that show the eastings and northings in kilometers—are printed in larger type, with the three trailing digits (000) dropped from all but one label along each edge of the map that gives the full coordinate value. UTM grid coordinates are shown in different ways on foreign topographic maps, as you can see in figure 4.11. The use of single digits to identify the 10,000-meter easting and northing increment on the Vancouver, British Columbia, Canada, 1:250,000-scale map (left) is similar to how easting and northing ticks are labeled on USGS topographic maps at the same scale. However, light-blue grid lines, not grid ticks, are drawn on the Canadian map. Grid lines are shown prominently in black on the Sonneberg, Germany, 1:50,000-scale topographic map (right). The UTM easting and northing are given for each grid line, with the three trailing digits (000) dropped in all numbers except those near each map corner, which indicate the full coordinate value. This style of showing the UTM grid is identical to that used for US military topographic maps, which is not surprising when you realize that the German map was reprinted by the US Army with the grid likely added to the original map.

Grid orientation Grid lines are rarely oriented parallel to the edges of topographic and other quadrilaterally formatted maps. You can see why by looking at UTM Zone 10 in figure 4.3. Horizontal UTM grid lines intersect the zone edges perpendicularly only at the equator. Moving toward either pole, the converging edge meridians intersect the horizontal grid lines at increasingly acute angles to a maximum of slightly less than three degrees from perpendicular at the top and bottom of the zone. Now look at the southwest and southeast corners of USGS 1:100,000scale topographic maps that are at the same 43° N latitude but at the center and east edge of UTM Zone 10 (figure 4.12). On the map on the left, at the 123° W central meridian for Zone 10, notice that the UTM grid lines are vertical and horizontal (the vertical 500,000mE line at the central meridian is not shown on the map because it is the same line as the meridian). There is a slight clockwise rotation (two degrees at this latitude) of the east UTM edge meridian to vertical on the map to the right. This slight rotation of grid lines from horizontal at zone edges is termed grid convergence.

Figure 4.11. UTM grid coordinates are shown differently on these corner pieces of Canadian 1:250,000-scale (left) and German 1:50,000-scale (right) topographic maps. Reproduced by permission from the Minister of Public Works and Government Services Canada, 2007, and courtesy of National Resources Canada. German map courtesy of the US Defense Mapping Agency.

Figure 4.12. UTM grid lines at the zone center (left) are vertical and horizontal but are slightly rotated clockwise to the east-edge meridian of the zone (right) when they are drawn as vertical lines on the map. These segments of USGS 1:100,000-scale

topographic maps in UTM Zone 10 are of Diamond Lake (left) and Christmas Valley, Oregon (right). Courtesy of the US Geological Survey.

Grid convergence at zone boundaries is, therefore, zero at the equator and maximum at the top and bottom of the UTM zone. You may wonder how to deal with grid coordinates on a map such as the one in figure 4.13, which spans UTM grid Zones 15 and 16. Measuring half your coordinates in each zone is confusing and difficult to work with if you want to calculate lengths, directions, and areas from the coordinates. The solution is to extend the zones outward to cover the entire map, which allows you to choose one of the two zones for your map work. UTM zones, for example, can be overlapped with acceptable distortion of up to 30 minutes of longitude within their neighboring zones. For the same reason, SPCS zones also extend above and below or to each side of the US counties they cover.

Grid coordinate determination Although the structure of grid coordinate systems is relatively easy to understand, it may take practice to gain skill using the coordinates on maps. Sometimes, you’ll want to determine a feature’s SPCS or UTM coordinates. At other times, you’ll want to find the position on the map of a feature whose coordinates are given.

Figure 4.13. UTM grid lines usually are not parallel with graticule lines, so they intersect at a slight angle. Maps that span two grid zones always have grid convergence, a problem solved by extending one or both grids across the entire map.

Figure 4.14. To determine the UTM coordinates of the gravel pit or to plot the gravel pit’s location from UTM coordinates, follow the steps outlined in the text. Courtesy of the US Geological Survey.

For instance, say you want to determine the UTM coordinates of the gravel pit on the 1:24,000-scale topographic map segment in figure 4.14. If UTM grid lines aren’t printed on the map, you can use the marginal grid ticks and a straightedge to construct the grid lines that lie immediately to the south, north, east, and west of the gravel pit. First, note the coordinate values of the grid lines to the west and south of the gravel pit (297,000mE and 4,764,000mN, respectively). Next, measure the map distance from these lines to the gravel pit (3.73 and 1.00 centimeters, respectively) and form ratios between these values and the grid interval distance in map units (4.17 centimeters per kilometer for a 1:24,000-scale map). Multiply these proportions (0.894 and 0.240) by the grid interval distance in ground units (1,000 meters), and add the results to the west and south grid line values. Thus, the UTM coordinates of the gravel pit are found in equations (4.1) and (4.2):

and

Now imagine that you want to plot the location of a feature for which you know the grid coordinates. The problem is essentially the reverse of determining the UTM coordinates of a

mapped feature, such as the gravel pit in figure 4.14, which you now know is at 297,894mE, 4,764,240mN. First, determine the UTM northing and easting for the grid lines that fall immediately below and to the left of the coordinate, respectively (4,764,000mN, 297,000mE). If these grid lines aren’t drawn on the map, use the grid ticks and a straightedge to draw them. Next, subtract the grid line value immediately west of the easting from the easting (297,894 m − 297,000 m = 894 m), and subtract the grid line value immediately south of the northing from the northing (4,764,240 m − 4,764,000 m = 240 m). Form proportions between these differences and the 1,000-meter grid interval distance (0.894 and 0.240, respectively). Then multiply these proportions by the grid interval in map units (4.17 centimeters per kilometer) to obtain the easterly and northerly differences in map units (3.73 cm and 1.0 cm, respectively). Finally, plot these distances from the grid lines to the west and south. The two plotted lines will intersect at the gravel pit. If you’ll be working with a single coordinate system over and over on maps of a certain scale, it may pay to construct a simple measurement aid called a roamer. Commercially available roamers are clear plastic devices that have calibrated rulers etched into their surface (figure 4.15). The rulers match standard topographic map scales. Check your local map or outdoor recreation store to get one of these handy ready-made products. You can also easily construct your own roamer by marking off the grid interval distance along each edge of a piece of transparency, starting at the corner. Then divide these distances into units that are fine enough for the precision of your measurements. Millimeters or tenths of inches normally suffice. You can also copy the roamer in figure 4.15 to a transparency and use it in your map work, but be sure it is first correctly scaled to your map.

Figure 4.15. Roamers for determining SPCS and UTM coordinates on 1:24,000-scale topographic maps.

By aligning the roamer with the north–south and east–west grid lines, you can determine the coordinates of map features and plot coordinate locations quickly and accurately (figure 4.16). Of course, you’ll need a separate roamer for each map scale and grid system. Therefore, you may want to put UTM and SPCS roamers for standard maps in opposite corners of the same transparency.

GRID CELL LOCATION SYSTEMS The grid coordinate systems that we’ve described so far all use x,y coordinates. Other reference systems can also be found on maps, including grid cell location systems which are based on grid cell location—that is, the location of a feature inside a cell within

the grid. These grid cell location systems consist of columns and rows that are identified by alphanumeric codes. In this section, we explore several grid cell location systems.

Military Grid Reference System The US Military Grid Reference System (MGRS), developed by the US Army for the entire earth, is a grid cell location system used by militaries in North American Treaty Organization (NATO) nations. The US Army devised this system to minimize the confusion of using the long numeric coordinates (up to 15 digits for small features) and numeric grid zone specifications in the UTM and SPCS grid coordinate systems. The army extended the UTM system by substituting single letters for several numerals and basing the system on cells rather than point locations. The MGRS is extended from the UTM and UPS grids, although we discuss only the UTM version of the MGRS in detail because the majority of map users have little call for locating features in polar regions. In the MGRS, each of the 60 UTM zones is divided into 19 quadrilaterals covering 8 degrees and one (the northernmost) covering 12 degrees of latitude. Quadrilaterals are assigned the letters C through X consecutively on the UTM grid, beginning at 80° S latitude (figure 4.17). Letters A, B, Y, and Z are used for quadrilaterals on the south and north UPS grids, and the letters I and O are omitted to avoid possible confusion with the numerals 1 and 0. Each quadrilateral is identified by an alphanumeric code that refers first to the UTM zone number and then to the quadrilateral letter. Madison, Wisconsin, for example, is located in quadrilateral 16T (UTM Zone 16, quadrilateral T).

Figure 4.16. You can use a roamer to determine rectangular coordinates for a feature or plot a feature’s position from known coordinates, such as the gravel pit on this topographic map. Courtesy of the US Geological Survey.

Figure 4.17. In the MGRS, each of the 60 UTM zones is first divided into eightdegree quadrilaterals of latitude and lettered from south to north. Each of these quadrilaterals is divided into 100,000-meter squares that are given a two-letter code.

Figure 4.18. A map sheet miniature shows the 100,000-meter-square cell identification letters found in the marginal grid reference box on maps that use the MGRS. Courtesy of the US Geological Survey.

Next, each quadrilateral is divided into 100,000-by-100,000-meter squares, known as 100,000-meter-square cells, and each cell is identified by a two-letter code. The first letter is the column designation, and the second letter is the row designation. The letters I and O are again omitted to avoid possible confusion with numerals. For example, the 100,000-metersquare cell that contains Madison, Wisconsin, is designated 16TCC. To assist the map user, the 100,000-meter-square cell identification letters for each map sheet are generally shown in the sheet miniature, which is part of the grid reference box found in the lower margin of the map (figure 4.18). For more precise designations of grid cells, the MGRS uses the standard UTM numerals. Thus, the regularly spaced lines that make up the UTM grid on any large-scale map are divisions of the 100,000-meter-square cell. These lines are used to locate a point using the desired precision within the cell. Dividing the cell by 10 adds a pair of single-digit numerals, which designate a cell of 10,000 meters on a side (figure 4.19). For example, Wisconsin’s Capitol dome is located in the 10,000-meter cell designated 16TCC07. Further division by 10 requires a pair of two-digit numbers, yielding the designation 16TCC0571 for the 1,000meter cell that contains the Capitol. The process can be continued until the desired level of precision is reached. In each instance, the coordinate pair designates the southwest corner of the grid cell at the specified level of precision.

Figure 4.19. MGRS designations within 100,000-meter-square cells progressive subdivisions by 10 and the use of standard UTM numerals.

involve

MGRS cells are printed on 1:25,000-, 1:50,000- and 1:250,000-scale military topographic maps. The 10,000- or 1,000-meter cell row and column numbers are printed over the UTM grid lines for easy identification, along with the two-letter code for the 100,000-meter square. This designation allows grid cells such as the two 10,000-meter squares in the corner of the McDermitt 1:250,000-scale military topographic map in figure 4.20 to be easily identified as 11TMS23 and 11TMS24.

US National Grid In 2005, the US Department of Homeland Security (DHS), a department of the federal government charged with protecting the United States from terrorist attacks and responding to natural disasters, proposed that the US National Grid (USNG) be used for all geographic referencing and mapping. This proposal was put forward in response to the ever increasing use of portable GPS devices, GPS-enhanced cell phones, and automated vehicle location (AVL) technology. Users of these devices require a nationally uniform grid reference system for accurate and consistent identification, communication, and mapping of ground coordinates. Identifying ground locations to 10- or

1-meter resolution is important in high-accuracy navigation, as well as in scientific studies and land management activities, especially in urban areas. Consequently, GPS receivers— from recreational to survey-grade instruments—calculate and display geographic positions to this level of precision in USNG format (see chapter 14 for more on GPS and maps). A single national grid reference system is easier to use because it is not necessary to convert between different grid systems such as UTM and state plane coordinates.

Figure 4.20. MGRS grid cell boundaries are printed on military topographic maps, along with column and row identifiers. The full MGRS identifiers (11TMS23 and

11TMS24) have been added to two 10,000-meter squares. Courtesy of the National Geospatial Intelligence Agency.

The USNG (officially known as the US National Grid for Spatial Addressing) is a grid cell location system that is based on the UTM coordinate system and the alphanumeric grid square referencing system used in the MGRS. As mentioned earlier in the chapter, 10 UTM zones, numbered 10 through 19, span the conterminous United States, from 126° to 66° west longitude. The MGRS divides each UTM zone into quadrilaterals that are eight degrees in latitude, and the letters R, S, T, and U identify the four quadrilaterals from 24° to 56° north latitude. The land area of the conterminous United States falls within 32 of the 40 quadrilaterals shown in figure 4.21.

Figure 4.21. Two-letter codes for the 40 6° × 8° quadrilaterals from the Military Grid Reference System that form the US National Grid.

To better understand the USNG, look at a particular quadrilateral in a UTM grid zone, such as quadrilateral 17R which covers the Florida peninsula. As with the MGRS, this quadrilateral is divided into 100,000-by-100,000-meter cells, and each cell is identified by a two-letter code. The first letter is the column designation, and the second is the row designation. For example, Saint Lucie Inlet, approximately 30 miles (50 kilometers) north of West Palm Beach, Florida, on the southeast coast lies in 100,000-meter cell NL, or 17RNL in USNG notation (figure 4.22). You can subdivide cell NL into 10,000-by-10,000-meter and then 1,000-by-1,000-meter cells in the manner shown for the MGRS in figure 4.19. Here’s how to give the MGRS and USNG coordinates for finer subdivisions of 1,000-by-1,000-meter cell 17RNL7513 in the northwest corner of the Saint Lucie Inlet, Florida, topographic map in figure 4.22. Reading right and then up, the six-digit coordinate for the 100-by-100-meter cell in the upper-left corner of the cell is 17RNL750139, the eight-digit coordinate for the 10-by-10-meter cell in the lower-left corner is 17RNL75001300, and the 10-digit code for the 1-by-1-meter cell in the upper-right corner is 17RNL7599913999.

Ordnance Survey National Grid The Ordnance Survey National Grid, described earlier in this chapter, is commonly used in England, Scotland, and Wales. A similar grid cell location system, used in Ireland and Northern Ireland, is the Irish Grid reference system. OSNG references are commonly given for geographic features and locations cited in publications such as guide books and government planning documents for Great Britain.

Figure 4.22. Northwest corner of the Saint Lucie Inlet, Florida, 1:24,000-scale USGS topographic map, annotated with coordinates for 100-by-100-meter, 10-by-10-meter, and 1-by-1-meter cells in the USNG. Courtesy of the US Geological Survey.

Figure 4.23 shows that the OSNG divides the land area covered by England, Scotland, and Wales into 56 100,000-by-100,000-meter (100-by-100-kilometer) cells identified by a two-letter code. For the first letter, the grid is divided into 500,000-by-500,000-meter (500by-500-kilometer) cells. Four of these cells contain significant land area within Great Britain: S, T, N, and H. (The “O” square contains a tiny area of Yorkshire coastline, mostly below the mean high tide line.) Each 500-by-500-kilometer cell is divided into 25 100-by-100-kilometer cells that have a second letter code from A to Z (omitting I), starting with A in the northwest corner to Z in the southeast corner. For example, cell NA is in the northwest corner of large cell N, while square NZ is in the southeast corner.

Figure 4.23. The Ordnance Survey National Grid divides the area covered by England, Scotland, and Wales into 56 100,000-by-100,000-meter cells identified by a two-letter code.

Proprietary grids In recent years, mapping has become more and more commercialized. Proprietary grids are developed and marketed by private companies for use with their products. These grid cell reference systems may be map related, or they may be related to the use of maps in conjunction with specialized instruments. This section looks at a few examples. Map publishers

Publishers of maps and atlases often develop their own version of a grid cell location system. You have probably seen arbitrary grid cell systems superimposed on city street, state highway, recreational, or atlas maps (figure 4.24). It is common for these grids to have lettered columns and numbered rows (or vice versa), so that a cell might have a column and row identification such as C-2. The grid cells are usually keyed to a place-name index. If you are looking for a particular street, simply look it up in the index, in which its grid cell identification can be found. Finding a particular feature within a grid cell can be difficult if the density of names is great or the cell is large in relation to the size of the feature you are trying to find. It can also be difficult to find a feature that spans many cells, such as an interstate highway, river, or mountain range. In these cases, the location is sometimes identified by the cell locations at which the feature starts and ends on the map. The problem of searching for a difficult-to-find feature is even greater when only margin ticks indicate cell boundaries. Remember, too, that each grid cell is specific to the map for which it is drawn. If people ask you where a street is and you tell them C-2, the information won’t be useful unless they have the same map you do. Keep this tip in mind when you use maps that include insets. Because each inset may have its own grid cell locator system, the same place can be represented in several grid cells (see chapter 6 for more on inset maps). Some brands of GPS equipment use their own brand of arbitrary grid cell system for locating features. For example, the well-known map publisher Thomas Brothers Maps has created the Page and Grid reference system for use with its three scales of maps that cover the continental United States. Through special arrangement with Trimble Navigation, a vendor of GPS receivers, Thomas Brothers Maps supplies a grid-related product called Thomas Guides. Trimble, in turn, markets a coordinate extension called Trimble Atlas (supplied by Thomas Brothers Maps), which is designed to increase the locational precision of the Page and Grid system. This proprietary grid reference system assigns a unique page number to each geographic area. Map grids are half-mile squares, ranging from Grid A1 through Grid J7, and the grids always appear at the same location on each page, so you can quickly and easily estimate distances and locate cities, streets, addresses, and points of interest.

Figure 4.24. An arbitrary grid cell system identifies streets by column letter and row number. In this part of a North Albany, Oregon, street map, the location of Palestine Avenue is given as B-2. Courtesy of the US Geological Survey.

The more specific the zone reference, the larger the scale of the map referenced. The result is a tailored package. A GPS receiver locates the user on a map, providing the ability to zoom in or out to see the desired level of detail. As more vendors market electronic navigation systems that display maps on a portable computer screen or mobile device, such linking of zone coordinates to map features may become more common. Amateur radio operators Amateur (ham) radio operators use a different form of proprietary grid. This reference system is based on the Maidenhead grid system, which partitions the earth into progressively smaller quadrilaterals of latitude and longitude. The first two letters in the reference divide the earth into 20° × 10° fields. Pairs of numbers designate 2° × 1° squares within these fields. Two more letters are used to define 5 × 2.5 foot squares within each cell. Thus, a six-character code can locate any place on earth within a rectangular zone of up to 5½ × 3 miles. As with the Page and Grid system, Trimble has extended the Maidenhead grid system so that it provides more precise spatial referencing. This extension makes the grid more suitable for use with its GPS receivers. The extension, called the Trimble Grid Locator, adds a pair of numbers and a pair of letters to the six-character Maidenhead code.

SELECTED READINGS Defense Mapping Agency. 1989. The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS). DMA Technical Manual 8358.2 (September). Washington, DC: Defense Mapping Agency. http://earth-info.nga.mil/GandG/publications/tm8358.2/TM8358_2.pdf.

———. 2006. Datums, Ellipsoids, Grids, and Grid Reference Systems. DMA Technical Manual 8358.1 (June). Washington, DC: Defense Mapping Agency. http://earthinfo.nga.mil/GandG/publications/tm8358.1/toc.html. Maling, D. H. 1992. Coordinate Systems and Map Projections, 2nd ed. New York: Pergamon. Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. “Scale, Reference, and Coordinate Systems.” Chapter 6 in Elements of Cartography, 6th ed., 92–111. New York: John Wiley & Sons. Stem, James E. 1990. State Plane Coordinate System of 1983. NOAA Manual NOS NGS 5. Rockville, MD: National Office of Atmospheric Administration. http://www.ngs.noaa.gov/PUBS_LIB/ManualNOSNGS5.pdf. US Department of the Army. 2006. “Grids.” Chapter 4 in Map Reading and Land Navigation. FM 3–25.26. Washington, DC: Department of the Army.

chapter five LAND PARTITIONING IRREGULAR SYSTEMS Metes and bounds French long lots Spanish and Mexican land grants Donation land claims REGULAR SYSTEMS Centuriation system NATIONAL PUBLIC LAND SURVEY SYSTEMS US Public Land Survey System Townships and sections Fractional divisions Government lots and platted lots Survey irregularities PLSS appearance on maps Canada’s Dominion Land Survey LAND RECORDS Types of land records Subdivision plats The cadastre Cadastral maps Engineering plans Land information systems SELECTED READINGS

5 Land partitioning In chapter 4, we examine grid coordinate systems, which allow you to use numeric values to identify the location of a point or cell on the surface of the earth. Many people confuse land partitioning systems with grid coordinate systems, and often experience less than satisfactory results because of this confusion. In this chapter, we clarify the differences between these systems through an exploration of a variety of land partitioning systems. Since the beginning of recorded history, people have created spatial referencing systems that are convenient for land partitioning. Land partitioning is the division of property into parcels, also called tracts, which are areas that have some implication for landownership or land use. You may also see the term lot, which is a special type of parcel within a legal subdivision (a piece of land divided from a larger area) that is recorded on a map. One of the first steps in the management of an area of land is to divide it into parcels that are then recorded on plats—maps that are drawn to scale to show the lots into which the area is divided. A platted subdivision is therefore a subdivision that is mapped to show the subdivided lots. The map is then placed on public record. Landownership, zoning, taxation, and resource management are just a few uses of such a system. The system must be conceptually simple enough to be generally understood and technically simple enough to be readily implemented in the field at the time of land settlement or development. Land partitioning systems are used throughout the world. For example, when European settlers first arrived in the United States, they brought with them a host of land partitioning methods. As a result, both irregular and regular land partitioning systems were soon in vogue in the colonies. Geometrically irregular schemes, such as the metes-and-bounds system used in Great Britain and the rest of Europe, prevailed since the late 17th century in the colonial states, Texas, and portions of Louisiana. Other irregular systems, such as French long lots, Spanish and Mexican land grants, and donation land claims have also been used in the United States. Regular systems of laying out the land were also developed, such as the geometrically systematic town plans that are common in New England. In this chapter, we look at all these systems in more detail, starting with irregular systems and then moving on to the more regular systems that are still in use today.

IRREGULAR SYSTEMS Metes and bounds Much of the land in the United States that was colonized before 1800 was characterized by widely scattered settlements. This settlement pattern was particularly true in the Southeast, where the climate was moderate and agriculture was widely practiced. These conditions encouraged people to settle far apart rather than cluster together, as they did in the Northeast. In the Southeast, a family was granted land of a certain size, often 400 acres (about 162 hectares), through a gift or purchase. The family was then permitted to select their 400-acre parcel from anywhere within the remaining unsettled area. English settlements in the relatively humid Northeast and Southeast were located mainly with respect to soil and timber resources. Consequently, the shape of parcels was decided mainly by the geography of the local setting—people naturally picked the best land they could, regardless of its shape. The convenience of the land parcel description was only an afterthought. An irregular parcel, such as the one shown in figure 5.1, is harder to describe than a simple geometric figure such as a square. The problem was to define these irregular parcels well enough to make clear whose land was where. The solution was to start from an established point, called the point of beginning (POB), and then describe a connected path around a parcel’s boundaries, noting landmarks along the way. A parcel might be described using only natural and anthropogenic features. The parcel in figure 5.1A, for instance, can be given the following metes-and-bounds description:

Figure 5.1. Metes-and-bounds descriptions may consist of (A) landmarks and approximate directions or (B) precise distance and direction readings. “Parcel beginning at the point where Pine Road crosses Beaver Creek, thence due east to the big rock pile, thence northeasterly to the confluence of Beaver Creek and Pine Creek, thence northwesterly to the south corner of Tom Smith’s fence, thence back to the point of beginning.”

This form of land description is called metes and bounds, referring to the boundaries of a property. With this method, minimal skill in surveying (measuring angles and distances on the ground so that they can be accurately plotted on a map) was required to delineate a property boundary. The legal property description was tied to geographic features and remained useful as long as neighbors agreed with the place-names and accepted the boundaries, and as long as the landmarks still existed in the landscape. Landmarks are features that are easily seen and recognized from a distance that allow people to establish their location. Some landmarks, such as trees and rocks, may be missing when the parcel description is later retraced. After land survey methods were developed, it became less common to use geographic features in land parcel descriptions. Descriptions became more mathematical, based on surveyed distances and directions from an established point. The land survey form of the feature-based parcel description shown in figure 5.1B might read:

“Parcel beginning at the NE corner of the Pine Road bridge over Beaver Creek, thence along a compass sighting of 90° for 500 yards, thence along a compass sighting of 10° for 500 yards, thence along a compass sighting of 300° for 950 yards, thence along a compass sighting of 170° for 1,000 yards back to the point of beginning.” The description would also likely include the parcel’s size in common areal units used in the measurement of land, such as acres (1/640th of a square mile, or 43,560 square feet) or hectares (10,000 square meters or 2.471 acres). Areas that are surveyed by metes and bounds are easily identified on topographic maps and aerial photographs (figure 5.2). The characteristic pattern of irregular boundaries, for fields, and winding roads intersecting at oblique angles tells you that this is a metes-andbounds survey area.

French long lots French settlements in Quebec and Louisiana, from the 1630s until around 1760, had a unique land parcel arrangement that resulted from the French feudal system of giving land grants, called seigneuries, to soldiers and other elite citizens. Settlement usually took place along rivers or lakes, which provided the chief source of transportation and communication for the French. Therefore, people given a seigneury usually divided the block of land into a number of narrow long lots used for farming. Boundaries ran back from the waterfront as roughly parallel lines, creating narrow ribbons or long lots, also called arpent sections or French arpent land grants (an arpent is a French measurement that equals approximately 192 feet or 58 meters in length or 0.84 acres or 0.34 hectares in area). This land division allowed the settler to have a dock on the river, a home on the natural levee formed by the river, and a narrow strip of farmland that often ended at the edge of a marsh or swamp. Through subsequent subdivision, the parcels often became so narrow that they were no longer practical to farm, but their boundaries still exist legally and are plotted on maps. In the United States, long lot boundaries are often shown on USGS topographic maps (figure 5.3). Long lots are particularly apparent along the Mississippi River in Louisiana, but you can also see them in other areas of early French settlement, such as Detroit, Michigan; Green Bay, Wisconsin; and Vincennes, Indiana.

Figure 5.2. Topographic map segment and air photo overlay of an area just south of Easton, Maryland, showing the appearance of metes-and-bounds surveys. Irregular field boundaries and winding roads reflect the original settlement pattern described in the metes-and-bounds land survey. Courtesy of the US Geological Survey.

Figure 5.3. French long-lot boundaries and lot numbers like these along the Mississippi River in Louisiana, are shown on 1:24,000-scale USGS topographic maps. Courtesy of the US Geological Survey.

Spanish and Mexican land grants A land grant is an area of land whose title was given to its owner before the territory was part of the United States. After the territory was acquired by the US government, the title was confirmed officially. From the late 1600s to around 1850, when much of the southwestern United States was part of Spain and, later, the Mexican Republic, three types of land grants were issued by these governments: Pueblo grants issued to communities of Native Americans were among the earliest, and today Indian reservations in New Mexico and other states are often based on these grants. Private grants of property by the government that could be sold by the owner were made to individuals as a reward for service to the government. Community grants were made to groups of settlers. Individuals in the group were given small tracts to settle and cultivate, but most of the grant was held in common for grazing, timber, and other purposes. To receive a land grant, you had to (1) physically step on the land, (2) run your fingers through the soil, and (3) make a commitment to live on the land, cultivate it, and defend it with your life, if necessary. Land grant boundaries were made easy to recognize. Physical features, such as hilltops, rivers, and arroyos (dry creeks or streambeds), always defined boundaries. Settlers were

concerned with water resources, so many grants straddle rivers and lakes. Descendants of the original settlers still live on these lands, and the boundaries of large grants appear on 1:24,000-scale USGS topographic maps, as in figure 5.4.

Donation land claims The Donation Land Claim Act of 1850 granted 320 acres (almost 130 hectares) of federal land to any qualified settler who, before 1851, resided on public lands for four years or more in the Oregon Territory (Idaho, Oregon, Washington, and western Montana). Settlers claiming public lands between 1851 and 1854 were awarded a smaller, 160-acre (about 65 hectares) donation land claim (DLC). By law, both the 320-acre and 160acre parcels had to be surveyed with north–south and east–west boundaries. These boundaries had to conform to the survey in the Public Land Survey System (described later in this chapter) if the survey was already made. DLCs were numbered and shown on US General Land Office (GLO) maps. The GLO was an independent agency of the US government that was responsible for public-domain lands. Created in 1812, it became the US Bureau of Land Management (BLM) in 1946 to administer public lands in the United States. DLC boundaries and parcel numbers appear on large-scale topographic maps of land in the former Oregon Territory, as shown in figure 5.5.

Figure 5.4. Spanish and Mexican land grant boundaries are shown on 1:24,000scale USGS topographic maps as red lines with a dash-dot pattern and the words “grant boundary,” in all capital letters, inside the grant. This Mexican land grant is near Socorro, New Mexico. Courtesy of the US Geological Survey.

The land partitioning systems we’ve discussed so far are unsystematic partitioning schemes, in which the land was allotted before systematic surveys were made. These unsystematic systems encouraged inefficient partitioning of land—at least from the government’s point of view. The first people into a region had a virtual monopoly over the choicest lands. The only land left available to settle later often had swamps, steep slopes, or poor soils. In many areas, fragments of poor land remained unowned and unwanted long after a region was “fully” settled. Furthermore, land claims were often found to overlap, and boundary descriptions were often in error. In homogeneous environments, it was hard to establish accurate borders in the first place. In any setting, boundary mistakes naturally occurred as the land passed through various owners and environmental changes. Good fences, as Robert Frost pointed

out in his “Mending Wall” poem (1914), did make good neighbors, because they clarified the location of the irregular boundaries. It used to be the custom for neighboring landowners to walk together around their lots each spring. The excursion was far more than a social outing —it ensured that all owners agreed on the borders between their lands. Problems arose, however, when later generations ignored the boundary-walking tradition. Many markers were shifted or destroyed through time—trees died or were cut down, lakes dried up, streams shifted, rock piles were moved, and fences fell down. The result is a long history of legal battles over property boundaries of irregularly settled land.

Figure 5.5. Donation land claims south of Corvallis, Oregon, are shown with dashed red boundary lines and claim numbers, always greater than 36 (36 is the largest section number in a US Public Land Survey System township, which is discussed later in this chapter). Claims 58 and 59 have north–south and east–west boundaries. Courtesy of the US Geological Survey.

REGULAR SYSTEMS The alternative to the scattered settlement of irregular parcels was to survey and divide the land systematically before settlers arrived. Surveying land into geometrically regular parcels in a systematic manner is an ancient practice. The pattern of agricultural land subdivision in both ancient Egypt and China was a grid, and cities in these—and succeeding—cultures often were planned as a rectangular grid of streets and lots. Another example is the Roman Centuriation system, which we explain in more detail here as a model of an early regular system.

Centuriation system In Roman times, agricultural land in northern Italy and elsewhere throughout the empire was subdivided into a grid of square parcels using the Centuriation system of land surveying. Field surveyors called agrimensores divided large tracts of land into square centuria—in modern terms, about 132 acres (53.5 hectares) in area. Each centuria was further divided into 100 square heredia, each of which was allotted to a family. Agrimensores used a rudimentary surveying instrument called a cross-staff (which measured the angle between the North Star and the horizon) to lay out the vertical and horizontal boundaries of centuria. The grid of lines began in the middle of the area, with a north–south line called the cardo maximus and an east–west line called the decumanus maximus. Main roads often were constructed along these two initial lines and narrower public lanes followed centuria boundary lines. Today, you can still see, on topographic maps and satellite images, remnants of centuriation in certain parts of Italy and other Mediterranean locales that were Roman colonies. For example, the square road pattern for the agricultural land near Ravenna, Italy, shown in the satellite image and road map segments in figure 5.6, outlines centuria surveyed during Roman imperial times. In the United States, rectangular grids of streets and lots were used for planning towns in many of the colonies, occasionally in the South but mostly in the North. Town sites were laid out and surveyed, and plat maps of the parcels were prepared and recorded, all prior to settlement. These plat maps were part of the foundation of the US Public Land Survey System, the dominant public land partitioning system in the United States. This system served as the model for the Dominion Land Survey in central and western Canada. Both systems are described in more detail later in this chapter. It is possible that the Roman centuriation system influenced US President Thomas Jefferson’s original proposal for the Public Land Survey System, although there is no direct historical evidence to support this claim.

Figure 5.6. The road pattern seen in this satellite image and road map segment from ArcGlobe of an agricultural area near Ravenna, Italy, is a remnant of Roman centuriation land subdivision. ©2007 Esri, TANA, and i-cubed.

NATIONAL PUBLIC LAND SURVEY SYSTEMS US Public Land Survey System In 1783, the US Congress of the brand-new confederation of 13 states was faced with an urgent need for a national land policy. It had to devise a way to manage the vast lands east of the Mississippi River that were ceded by Great Britain to the United States after the Revolutionary War. Quick action was important for several reasons—land was promised to Revolutionary War soldiers, a source of income was necessary to run the new country, and future states had to be carved out of the wilderness. By the time the lands west and north of the Ohio River (called the Northwest Territory, covering all of modern-day Ohio, Indiana, Illinois, Michigan, and Wisconsin and

the northeastern part of Minnesota) were opened to settlement in the late 1700s, the newly formed US government had come up with what seemed to be an orderly way to transfer land to settlers. The solution was called the Land Ordinance of 1785, which established the US Public Land Survey System (USPLSS or, more commonly, PLSS). Otherwise known as the Township and Range System, this plan called for regular, systematic partitioning of land into easily understood parcels prior to settlement. The legal description for each parcel was carefully recorded, and settlers were able to select surveyed parcels to their liking. The PLSS was first implemented in the Northwest Territory and subsequently in the even vaster territories acquired by the United States over time through the Louisiana Purchase, Red River Cession, Florida Cession, Oregon Country Cession, Mexican Cession, Gadsden Purchase, and Alaska Purchase. To establish the land partitioning system in these areas, the first step was to arbitrarily select an initial point. You can actually visit the locations of initial points, such as the survey marker for the intersection of the Willamette meridian and its baseline (figure 5.7) near Portland, Oregon. Government land surveyors then determined the parallel and meridian that intersected at the initial point. The parallel was called the baseline or geographer’s line (surveyors were called geographers in those days), and the meridian was called the principal meridian. Each principal meridian was given a number or name (although the baselines were not), which was used to identify surveyed parcels within the region.

Figure 5.7. Initial points were surveyed and defined physically with survey markers. This is the current marker at the intersection of the Willamette principal meridian and baseline in Portland, Oregon. The original marker was placed in this spot on June 4, 1851. Courtesy of Richard Garland.

Thirty-five principal meridians and baselines were established within the conterminous United States (figure 5.8), and five were established in Alaska. The testing ground was Ohio, in which seven methods were experimented with before the final structure was developed using the 1st principal meridian and baseline in the northwest corner of the state. The system was then used to survey the rest of the nation to the west and south. At first, smaller areas east of the Mississippi River were surveyed from the initial point. Later, surveys in the West covered larger areas, and often a territory was subsequently divided among one or more states (for example, see Oregon and Washington in figure 5.8). The small surveys you see in the West were for Indian reservation lands. Texas has its own system, which is based on early Spanish land grants. In figure 5.8, it appears as if the entire area covered by a principal meridian and baseline was surveyed completely, but this complete coverage is not the case. In fact, some land has

never been surveyed under the PLSS, mainly because it was deemed unsuitable for settlement or reserved for national forests, Indian reservations, or other government use. Townships and sections Once an initial point was established, range lines were surveyed along meridians at sixmile intervals east and west of the principal meridian. Township lines were surveyed along parallels at six-mile intervals north and south of the baseline (figure 5.9). The six-bysix-mile quadrilaterals bounded by these intersecting township and range lines were called survey townships (sometimes called congressional townships and often shortened to townships). Each survey township is identified with a township and range designation. Township designations indicate the distance north or south of the baseline (in units of six miles), and range designations indicate the distance east or west of the principal meridian (in units of six miles). The township notation indicates township or range, then notes the number, and finally adds a suffix to indicate the direction away from the principal meridian or baseline. So T3N, R5W identifies the township in the third row of townships north of a baseline and in the second column of townships west of a principal meridian. The word “township” is used in three ways: to identify surveyed lines, as the shortened name for sixby-six-mile surveyed areas, and as the term for the distance north or south of a baseline (in units of six miles).

Figure 5.8. Principal meridians (P.M. and M. on the map) and baselines (B.L. on the map) of the Public Land Survey System for public land surveys within the conterminous United States. An initial point marks the intersection of these two lines. Areas surveyed from each initial point are shown with different colors. Areas that are not colored were never surveyed under the PLSS.

The surveyors encountered a problem, however. Because meridians converge toward the North Pole, the east–west distance between the meridians decreases (see chapter 1 for more on converging meridians). Thus, the township grid could not be extended indefinitely north or south from the initial survey point without townships becoming distorted in shape and area. To reduce the problem of unequal township dimensions, correction lines were established at every fourth township (24 miles or 38.6 kilometers from the baseline or other correction line), along which new range lines were surveyed to the east and west of the principal meridian at six-mile intervals. The effect of this pattern of surveying is seen in the map at the top of figure 5.9. The length of township lines north of the baseline decreases progressively until the correction line is reached, at which point they are resurveyed and hence appear offset. This systematic decrease in length means that township boundaries are not truly six miles on a side. Nor are they exactly 36 square miles in area, because their areas progressively decrease northward from the baseline and correction lines.

Figure 5.9. The structure and notation of the PLSS provides a systematic means of identifying land parcels as small as 10 acres (four hectares).

Each township is further subdivided into 36 parcels called sections, which are approximately one square mile (640 acres or 2.59 square kilometers) in area. Sections are numbered from 1 to 36, depending on their position in the township (see figure 5.9). A rowby-row zigzag method of numbering the sections was adopted, beginning with section 1 in the upper-right corner and ending with section 36 in the lower-right corner. This method of numbering is used so that every section shares a common side with the section with the

number before and after it (in figure 5.9, for example, section 24 shares a common side with sections 23 and 25). Fractional divisions A section can be divided successively into halves or into smaller units—for example, half sections (one-half of one section) and quarter sections (one-half of one-half of a section.) Half or quarter sections can be further subdivided to produce even smaller units. For example, a quarter-quarter section is 1/16th of a section (one-quarter of onequarter of a section). An easy way to correctly identify divisions of sections is to think of a compass placed at the center of the section. The compass direction from the center of the section to the center of each quarter section is northeast, southeast, southwest, or northwest. With this system, each land parcel’s legal description is unique and unambiguous. In figure 5.9, for example, parcel 4 (the 10-acre or 0.04-square-kilometer piece of land in section 24) is described in abbreviated form as follows: “NE¼, NW¼, SE¼, Sec. 24, T.2N, R.7E, 6th P.M.” In expanded form, the description reads as follows: “The northeast quarter of the northwest quarter of the southeast quarter of section 24 of township 2 north, range 7 east, 6th principal meridian.”

To locate a parcel from its legal description, the trick is to read backward, beginning with the principal meridian and working through the township and range to the section, and then to the fractional division. To give the parcel’s legal description, simply reverse the preceding procedure and work outward from the smallest division to the principal meridian. It is useful to understand the structure and notation of the system because the PLSS is the basis for property abstracts (legal documents that chronicle transactions associated with a particular parcel), deeds, and other landownership documents in much of the United States. Government lots and platted lots There are some exceptions to the use of the PLSS fractional land division in areas covered by the system. One exception occurs when most of a small parcel (less than 10 acres or four hectares) borders a body of water so that the parcel is not a complete square. In this case, a subdivision of the section called a government lot is assigned instead. These lots may be regular or irregular in shape, and their acreage may vary from that of regular fractional divisions of a section. A government lot is designated by its lot number, such as Lot 3 in figure 5.10. Parcels in a platted subdivision within a section, called platted lots, are also specified by lot numbers. Other exceptions are parcels that are often, but not necessarily, small in size, which are given descriptions in metes and bounds, rather than in PLSS notation, because of their irregular shapes.

Figure 5.10. Land divisions for small parcels, such as (A) government lots and (B) platted lots, are designated with lot numbers rather than PLSS fractional division identifiers.

Survey irregularities In practice, the idealized structure of the PLSS frequently broke down in areas in which it was used. Inaccuracies in the original survey occurred because of the relatively crude instrumentation of the period, rugged terrain, or, in some cases, sloppy work by surveyors who were paid on the basis of total miles surveyed (figure 5.11). These inaccuracies have persisted, primarily because historical boundaries as originally surveyed hold legal precedence over newer surveys of these boundaries.

Figure 5.11. Survey inaccuracies are evident in this actual segment of the PLSS section grid.

Variations in the shape and size of sections resulting from survey inaccuracies weren’t the only obstacles to partitioning the land into square-mile sections. Convergence of meridians toward the poles on the ellipsoidal earth made the surveyor’s job of laying out a regular grid of exactly one-square-mile sections within a township an impossible task. To systematize the distribution of shape and area variations from the ideal, surveyors determined section corners in a standardized order so that variations accumulated along the western and northern tiers of sections within each township. Another source of irregularity in the PLSS grids is the point at which surveys that start from different initial points meet. The section lines from one survey will not likely mesh with those of another. These junctures result in a mismatch of the grids of the two surveys. An irregular pattern of township and section lines also occurs at the boundary between the areas surveyed from different principal meridians, particularly when the boundary is an irregular feature such as a river. For example, the boundary between areas surveyed from the Michigan principal meridian and the 4th principal meridian follows the Michigan– Wisconsin state line, so you will find misalignment and partial townships and sections all along the Montreal River boundary between the two states (figure 5.12).

Figure 5.12. Irregularities in PLSS township and section boundaries may be because of the convergence of surveys from different principal meridians.

Another source of deviation from the ideal pattern of the PLSS grid is that in some areas townships were not surveyed systematically outward from the initial point. A good example is the surveying of townships in Oregon along the baseline for the Willamette principal meridian. The earliest townships were surveyed outward from the initial point in Portland, but independent surveys also were completed at the coast and from three separate starting points along the baseline in eastern Oregon. These other starting points were calculated so that in theory they would mesh perfectly with the surveys from the initial point. But inherent inaccuracies in surveys of these early periods made this perfect meshing impossible to achieve. The many independently surveyed parts of PLSS grids result in a mismatch at the points or lines where they come together. Along the boundary where these independent surveys converge, some confusing land partition descriptions can occur. This confusion is especially true for two land parcels that straddle the boundary, and is complicated by the fact that two property deeds are required. A similar problem may occur when the PLSS grid converges with earlier land partitioning systems—this type of survey boundary mismatch can happen in regions that were not surveyed at the time of settlement or in which boundaries were defined in an older system such as DLCs or Spanish land grants. The earlier Spanish land grant boundary in figure 5.4

is a classic example—both section lines and the principal meridian stop at the land grant boundary.

Figure 5.13. Appearance of PLSS township and section information on a typical USGS topographic map (northeast corner of Cannon Beach, Oregon, 1:62,500 quadrangle). Courtesy of the US Geological Survey.

PLSS appearance on maps The PLSS grid is found on many different types of maps. On maps made by government agencies, for example, townships and sections are often included because they are so closely associated with the boundaries of civil townships, counties, and other political units. The PLSS grid probably finds its fullest expression on large-scale printed USGS topographic maps. On these maps, township and section lines, section corners, section numbers, and marginal township-range notations are all printed in red (figure 5.13). Dashed red lines indicate section lines with doubtful locations, whereas red lines with a dash-dot pattern show land grants, as you saw earlier in this chapter. Township and section lines are also found on a host of government-produced maps that deal with land resources, such as US Forest Service (USFS) and BLM ownership maps. Privately produced road and recreation atlases, some available in digital as well as paper form, often include the PLSS grid. Surveys of city and rural lots in PLSS areas also are tied to section or fractional-division corners. These corners are commonly shown on land subdivision plat maps and tax assessor cadastral maps (described in more detail later in this chapter), both of which are used as land records.

As you’ve seen, the PLSS grid shown on these topographic and land resource management maps partitions a region into easily defined area parcels. You can also use it as a grid cell location referencing system (see chapter 4 for more on grid cell location systems) by stating the location of features relative to the corner or center of a standard PLSS parcel. You can often pinpoint features this way because the cultural landscape has been aligned to the PLSS grid in many parts of the country.

Canada’s Dominion Land Survey The Dominion Land Survey (DLS) is the public land survey system used to divide most of Canada’s prairie and western provinces into townships and sections for agricultural settlement and other purposes. In 1869, two years after the Dominion of Canada was created from British North America, Canada purchased what was called Rupert’s Land from the Hudson’s Bay Company. The Canadian government wanted to survey this massive territory, including much of what is now the Northwest Territories, Nunavut, Alberta, Saskatchewan, and parts of Manitoba, in a systematic manner before immigrants settled it. After British Columbia joined the Dominion of Canada in 1871, the government used the DLS to divide the “Railway Belt” of land on either side of the Canadian Pacific Railway main line and the Peace River Block in the northeast corner of the province. Surveying began in 1871, nearly a century after the initial surveys of the PLSS, and over the decades the DLS expanded to cover roughly 800,000 square kilometers (300,000 square miles) of western Canada. Although based on the PLSS, the DLS differs in the way that townships and sections are surveyed and identified numerically. By the late 1870s, seven principal meridians and baselines were surveyed (figure 5.14), the first just west of Winnipeg, Manitoba, and the second through seventh at increments of four degrees of longitude, from 102° W to 122° W. All seven principal meridians have the same 1st baseline—the boundary with the United States at the 49th parallel. The 2nd, 3rd, and subsequent baselines to the north are spaced at 24-mile (about 39-kilometer) intervals (figure 5.15). Starting at the initial point (at the intersection of a principal meridian and baseline) and progressing to the west (and also to the east of the 1st principal meridian), nearly square six-by-six-mile (9.8-by-9.8-kilometer) townships were surveyed so that there are two tiers of townships to the north and south of each baseline. The west and east borders of each township were surveyed every six miles as meridians of longitude that converge toward the North Pole, as in the PLSS. Therefore, the northern border of every township is slightly shorter than the southern border, and the southern border is six miles in width only along each baseline.

Figure 5.14. Principal meridians and the 1st baseline of the Canadian DLS.

Halfway between each pair of baselines are correction lines, at which the wider townships to the south of the 2nd baseline meet the narrower townships to the north of the 1st baseline. The townships surveyed north and south from each baseline converge toward these correction lines. The boundaries of the townships are offset to the east or west of each other along the correction lines, and on maps, you can see these offsets by east–west jogs every 24 miles (the length of four six-mile townships between correction lines). On the ground, you can see them in roads, property boundaries, and other human-made features that follow these survey lines. Townships in the DLS are identified by township and range numbers. Township 1 is the first row north of the 1st baseline, and Range 1 is the first column west of the principal meridian. Township numbers increase to the north. Range numbers begin in the first column west of each principal meridian. Range numbers increase to the west (except that they also are numbered east of the 1st principal meridian). On maps, township numbers are given in arabic numerals, and roman numerals are often used for range numbers. However, in written descriptions of land parcels, arabic numerals are used for both townships and ranges. For example, the notation for townships highlighted in the map at the top of figure 5.15 is “township 7, range 23, west of the 3rd principal meridian,” abbreviated as “7-23-W3.”

As with the PLSS, each township is divided into 36 sections, each approximately one square mile in area. But sections are numbered in reverse order to PLSS sections, beginning in the lower-right corner of the township and progressing northward row by row in zigzag fashion. For example, the notation for township 24 highlighted in township 7, range 23 in figure 5.15 is

“Section 24, township 7, range 23, west of the 3rd principal meridian,” abbreviated as “24-7-23W3.”

Each section can be divided into other fractional divisions such as quarter sections. Division into four northwest, northeast, southeast, and southwest quarter sections is used primarily for agricultural lands. The written description of the upper-left quarter of section 24 shown in figure 5.15 is “The northwest quarter of section 24, township 7, range 23, west of the 3rd principal meridian,”abbreviated as “NW-24-7-23-W3.”

A section can also be divided into as many as 16 legal subdivisions (LSDs). Although LSDs may be square, rectangular, and sometimes even triangular, a half-quarter section of roughly 80 acres (32 hectares) is most common. LSDs of 40-acre quarter-quarter sections, numbered from south to north in a zigzag manner, are shown in figure 5.15 for section 24. The oil and natural gas industry in Canada uses LSDs for describing the locations of wells and pipelines.

LAND RECORDS Today, everyone uses and relies on land records, which are a publicly owned and managed system, defined by statewide or provincial standards, that records real estate ownership, transfers, taxation, and development. Traditionally, the basic spatial unit for this record keeping is the land parcel. As mentioned earlier, a parcel results from the division of property into areas with some implication for landownership or land use. The land parcel is the smallest unit of ownership or, as in the case of an agricultural field, unit of uniform use. Physical features, resource reserves, market value, ownership, improvements, accessibility, and restrictions on land use are but a few of the attributes recorded for land parcels.

Figure 5.15. Townships, ranges, sections, and fractional divisions in Canada’s DLS.

Types of land records Subdivision plats Land surveyors carry out the subdivision of PLSS fractional lots and other parcels. The boundaries of most lots and parcels are still determined using plane surveying methods, in which the earth’s surface is assumed to be a “flat” plane, similar to the metes-and-bounds survey described in figure 5.1B. Property boundary corners are defined by the distance and bearing from the previous corner, with the first corner, or POB, defined by the distance and direction from a previously surveyed point, such as a PLSS section corner, called a control or reference point. The corner-to-corner survey of the property lines for the entire parcel is called a traverse. A traverse is generally classified as either open or closed (figure 5.16). An open traverse ends at a survey station whose position was not known previously. A closed traverse forms a closed loop, with the first point surveyed, or POB, also being the last— these types of traverses provide checks against errors in the distances and bearings because they must ultimately return to the original starting point. The requirement of ending

at the starting point allows surveyors to check the accuracy of their work and make adjustments to all the points along the traverse to force an exact fit. This accuracy check makes the extra effort of closing the traverse worthwhile.

Figure 5.16. A typical closed traverse—the starting point, or POB, is determined relative to a known control point, in this case a PLSS section corner. The bearings and distances for each property line are shown.

Land subdivision surveys are recorded on a subdivision plat, which is a map that shows subdivided lots. Plats must be legally recorded in the city or county surveyor’s office, or an equivalent land management bureau. A typical subdivision plat map (figure 5.17) shows each property line, along with its distance and, sometimes, its bearing from the previous survey point. In addition to these property line measurements, the area of each lot is often shown, along with the names and dimensions of proposed and existing streets. Building setback lines (the distance from a lot line beyond which building or

improvements may not extend without permission from an authority) and easements (a right held by one person to make limited use of another person’s property) are also shown, making the subdivision plat the best source of detailed information about any land parcel you are interested in purchasing. The geometry of the lots shown on the subdivision plat is translated to the actual ground locations of lot boundaries by the placement of survey markers or monuments (see chapter 1 for more on monuments) at corners of the subdivision, lot corners, and street intersections during the original survey of the subdivision.

Figure 5.17. Subdivision plats show detailed land survey information and survey markers or monuments for individual lots. Courtesy of Michael D. Lemke.

The cadastre The written records kept on land parcels form the cadastre. There are different types of cadastres. Fiscal cadastres of property valuation and land taxation are among the oldest, which is not surprising considering that governments have taxed property since the beginning of recorded history. Fiscal cadastres include the owner’s name and address, parcel description and size, the parcel’s assessed value and any improvements, and the current tax levy. In the United States, the fiscal cadastre for rural areas is the responsibility of the county government and is housed in the county treasurer’s office in the county seat (often the courthouse). The fiscal cadastre for city property is the responsibility of the associated municipal government office for property ownership and taxation. Another type of cadastre is the legal cadastre, which consists of records that concern proprietary interests in land parcels. These records contain the current owner’s name and address, legal description of the property, deed, title, abstract, and legal encumbrances (such as easements, mineral rights, and transfer restrictions). In the United States, some of this information is held by the property-taxing authority, but the rest may be scattered among several government agencies and nongovernment entities, such as property abstract and title companies that are in business to help people track down these sometimes elusive records that are essential to providing title insurance for a property. In practice, fiscal and legal cadastres have a great deal of overlap and duplication; and they are of limited value to land administrators, managers, and planners who make decisions involving natural resources, land use, and infrastructure (fire, police, ambulance, disaster relief services). These land information specialists often rely on a third type of cadastre, called the multipurpose cadastre, an integrated land information system that comprises database layers containing legal (ownership and taxation), physical (land slope and aspect, soil characteristics, drainage, vegetation cover), and cultural (land use, number of residents, building construction, access road width and surface material, utility service, zoning restrictions, and so on) information about land parcels. Each layer contains a set of map features that are georegistered to other database layers through a common coordinate system, such as geodetic latitude and longitude or state plane grid coordinates. Cadastral maps A major problem for the cadastre is that land partitioning systems developed to accommodate early settlers aren’t best suited for the cadastre’s primary function—land records management. The PLSS is a case in point. It was an excellent system when the government’s main concern was selling or giving away land parcels as quickly and efficiently as possible. It also served land transfer needs fairly well in subsequent years. But for purposes of land management, the PLSS is an administrative nightmare. The problem is that, because of technological barriers faced at the time of the original PLSS surveys (which were mostly performed during the 19th century), the locations of section corners were not originally defined by their latitude and longitude. Rather, these locations are known only in reference to other points and boundaries in the system (that is, prime meridians and baselines, and subsequently section corners). So there is no convenient way in the original cadastre to determine which parcel contains a particular location or what resources are found in a specific parcel. In the past, this information could only be found on archived print maps and written records, but obtaining these documents

was not a convenient and seamless process. For example, learning whose land was damaged by a flood involved rescaling and overlaying several maps, as well as searching through diverse text and tabular records. This tedious document searching and manual map overlaying has changed now that the cadastre is represented digitally in land information systems, described at the end of this chapter. Cadastres contain two complementary parts. One part contains the written record, or register, with information about landownership. It may include all types of documents, forms, official seals, and stamps of approval that characterize administrative activities. These written records have traditionally been scattered among government offices, each with a different mission and authority. Assembling the register material can therefore be frustrating, time consuming, and costly. The second part of most cadastres, thoroughly cross-referenced with the first part, contains detailed geographic descriptions of each parcel. These descriptions may be in the form of the original subdivision plat or of cadastral maps made from the subdivision plats—these maps contain detailed information about property in the form of land, called real property (figure 5.18). In either case, you can determine the location and areal extent of each parcel from these records.

Figure 5.18. A cadastral map shows information about real property, including property boundaries and tax lot numbers that are tied to the fiscal and legal cadastre and, increasingly, the multipurpose cadastre. Courtesy of the County of Fairfax, Virginia.

A key function of the cadastral map is to provide the foundation for a system of land rights transfer. A parcel’s boundaries must be known before it can be “conveyed” without ambiguity from one owner to the next. Land conveyance, or the transfer of the title to land by one or more persons to another person or persons, involves more than a geographic description of a parcel, of course, because such a description says nothing about possible restrictions or encumbrances on the property. It is for this reason that the cadastral map is cross-referenced with the register, and it is important to consult both forms of land records when transferring property rights. Engineering plans City engineering or public works departments in local governments usually maintain a series of large-scale maps called engineering plans. For example, a city might be mapped on a series of 1:1,200-scale map sheets (one inch equals 100 feet), each covering a 2,000-by3,000-foot (609-by-914-meter) rectangle defined by state plane coordinates. Each sheet can be compiled to include features such as property boundaries, streets and sidewalks, building footprints and street addresses, and detailed elevation contours. Additionally, the local government (that is, the lowest tier of public administration) that manages these plans may also be responsible for the management of utilities such as water and electrical. In these cases, there is also information about the related facilities, such as water and sewer lines, and telephone and power poles can also be added to the compiled map. A typical engineering plan (figure 5.19) includes several, but usually not all, of these features. The variety of features that can be shown in an engineering plan are often stored as separate data layers in a GIS. All data layers in a GIS database are georeferenced. As noted in chapter 1, georeferencing is the procedure used to align data layers via known ground location control points or, alternatively, the procedure of aligning a map or data layers with the earth’s surface via a common coordinate system. The result is that for all georeferenced layers, every location in one layer is precisely matched to its corresponding locations in all the other layers. Once the layers are georeferenced, they can be used in overlay operations. Overlay is the process of superimposing two or more maps or map themes to better understand the relationships between the geographic features of interest and their attributes. The overlay capabilities of GIS allow the compilation on demand of engineering plans, and indeed other maps, to show the types of things that are important in land parcel management and maintenance. Increasingly, you will find city engineering and public works departments providing websites in which engineering plan overlays can be viewed and downloaded to computers or mobile devices not only by city employees but also by the public.

Figure 5.19. An engineering plan, such as this 1:1,200-scale segment for Corvallis, Oregon, shows property boundaries, building outlines, elevations to the nearest tenth of a foot (the small numbers), and features such as telephone poles that are critical to public works management. Courtesy of the City of Corvallis, Oregon, Public Works Department.

Land information systems The cadastre is an important part of the geographic information database used in a modern land information system (LIS)—a special type of GIS used for cadastral and land-use mapping, typically by local governments. An LIS also incorporates powerful analytic and graphic tools to help users analyze data, generate and manage land records, produce maps and reports, and make better land management decisions. An LIS is a special form of GIS in two key respects: An LIS is a large-scale mapping and analysis system used primarily by local governments to deal with land records. A land record is any legal document recorded on paper or in electronic format that affects the title to real property. The basic geographic unit of an LIS is the land parcel, not x,y coordinates, because the sale and ownership of land, taxation, and land-use decisions are done

by land parcel. Land parcel boundaries and positions of other features are given in an appropriate coordinate system. Latitude-longitude coordinates are ideal because they are universal, but grid coordinates, such as state plane, are also acceptable if the zone boundary problems they introduce (see chapter 4 for more on these coordinate systems and their limitations) are considered. If several spatial reference systems are used in recording data, they should at least be transformable from one to the other. Land information systems are an important part of local government activities because they provide current, reliable land information that is necessary for many public programs and activities, such as land-use planning, infrastructure development and maintenance, environmental protection and resource management, emergency services, social service programs, and so on. They also provide basic information for land markets, land development, and other local economic activity.

SELECTED READINGS Crossfield, J. K. 1984. “Evolution of the United States Public Land Survey System.” Surveying and Mapping 44 (3): 259–65. Estopinal, S. V. 1989. A Guide to Understanding Land Surveys. Eau Claire, WI: Professional Education Systems. Hart, J. F. 1975. “Land Division in America.” In The Look of the Land, 45-66. Englewood Cliffs, NJ: Prentice-Hall. Johnson, H. B. 1976. Order upon the Land: The US Rectangular Land Survey and the Upper Mississippi Country. London: Oxford University Press. National Research Council. 1980. Need for a Multipurpose Cadastre. Washington, DC: National Academy Press. ———. 1982. Modernization of the Public Land Survey System. Washington, DC: National Academy Press. ———. 1982. Procedures and Standards for a Multipurpose Cadastre. Washington, DC: National Academy Press. Robillard, Walter G., Donald A. Wilson, Curtis M. Brown, and Winfield Eldridge. 2006. Evidence and Procedures for Boundary Location, 5th ed. Hoboken, NJ: Wiley. Thrower, N. J. W. 1972. “Cadastral Survey and County Atlases of the United States.” The Cartographic Journal 9 (1) (June): 43–51. Thrower, Norman Joseph William. 1972. Maps and Civilization: Cartography in Culture and Society, 25. Chicago: University of Chicago Press. US National Atlas. 2008. The Public Land Survey System (PLSS). Last modified April 29, 2008. http://www.atlas.usgs.gov/articles/boundaries/a_plss.html. Ventura, S. J. 1991. Implementation of Land Information Systems in Local Government: Steps toward Land Records Modernization in Wisconsin. Madison, WI: Wisconsin State Cartographer’s Office. Vonderohe, A. P., R. F. Gurda, S. J. Ventura, and P. G. Thum. 1991. Introduction to Local Land Information Systems for Wisconsin’s Future. Madison, WI: Wisconsin State Cartographer’s Office. Von Meyer, N. 2004. GIS and Land Records. Redlands, CA: Esri Press. Wilson, D. A. 2006. Interpreting Land Records. Hoboken, NJ: Wiley.

chapter six MAP DESIGN BASICS BASIC CHARACTERISTICS OF MAPS CARTOGRAPHIC ABSTRACTION Cartographic selection Cartographic generalization Vector generalization operations Raster generalization operations Cartographic classification Cartographic symbolization MAP DESIGN CONSIDERATIONS Purpose and audience Map projection use Level of generalization Map symbolization Matching symbols to the data Legibility Symbol size Visual contrast Effective color use Color systems Color depth Color contrast Color schemes Figure-ground organization Visual hierarchy Marginalia Visual balance Text Size and readability Visual contrast Text placement Map critique WEB MAP DESIGN Why web maps are special Web map design considerations Size and resolution Geographic extent and map scale Map projection Symbols and text

SELECTED READINGS

6 Map design basics Mapmaking is a skill that combines art, science, and technology. The first five chapters present basic map use topics that are also fundamental to mapmaking. These topics are of importance to modern mapping, in which you the map user can easily be a mapmaker. Mapping software, such as ArcGIS, including ArcMap and ArcGIS Pro, and graphics software, such as Adobe Illustrator, provide powerful tools for making professional maps. Interactive web apps, in which the user interface runs a browser such as ArcGIS Online, Google Map Maker, or the US Census Data Mapper, allow you to load data and generate maps instantly. Such apps give you the ability to create and share content: you can create maps that you can print or use in a publication, view in a browser or mobile device, embed in a website, and share via email or social media websites. The production of maps that use these tools has become much less expensive and much faster, but it does not automatically result in well-designed maps that communicate your message clearly and accurately. Although map design principles are built into modern mapping apps and software to some extent, there is still an assumption that you are familiar enough with the map design basics that you can make wise choices among the design options for the various components of your map. In this chapter, we discuss the map design basics that are necessary to be a good mapmaker as well as an effective map user. The systematic process of arranging and assigning meaning to elements on a map for the purpose of communicating geographic knowledge in a pleasing format marks good map design. But first we examine the basic characteristics of maps that we listed in the introduction to see how maps differ from other representations of the environment, such as architectural drawings, paintings, and photographs.

BASIC CHARACTERISTICS OF MAPS In the introduction, we talked about the basic characteristics of maps that give them their “mapness.” In this chapter, we review these basic characteristics because map design begins by shaping these characteristics to best fit the map’s theme and purpose. Maps are a graphic representation of a geographic setting—they are a collection of symbols used to represent a portion of the earth. All maps are concerned with two primary elements: locations and attributes. Maps are reductions of reality and therefore require scale conversions (see chapter 2 for more on scale). Maps are transformations of space that involve applications of map projections and coordinate systems (see chapters 3 and 4 for more on these concepts, respectively). Maps also are abstractions of reality and so must maintain their connection to—and present an accurate representation of—that reality. Maps require generalization of the geographic information. In addition, maps use signs and symbols, through a process called cartographic symbolization. As you can imagine, designing a map that takes all these characteristics into account can be difficult, or even at times impossible, without sacrificing some degree of quality. This selective process is called cartographic abstraction.

CARTOGRAPHIC ABSTRACTION Cartographic abstraction is the process of transforming data that is collected about the environment into a graphic representation of features and attributes that are relevant to the purpose of the map. Much of the power of maps lies in cartographic abstraction—salient aspects of the environment are preserved so that you can focus on patterns that may not be apparent when there is too much complexity or an overwhelming amount of information. To promote understanding about the information that is shown graphically, maps often portray the basic or universal character of the environment rather than individual features and unique attributes. To transform geographic data into a map, cartographers rely on cartographic selection, generalization, classification, and symbolism. They decide what type of—and how much— information to portray (selection). In addition, they eliminate or de-emphasize unwanted or unneeded detail (generalization and classification). Finally, they make appropriate choices about how the information is shown graphically (symbolization). Of course, the operations of cartographic selection, classification, generalization, and symbolization are more interrelated than it sounds—map-making is more of an iterative process than a linear one. Cartographers continually refine the results they obtain through the cartographic abstraction processes. Now we look at each of these operations in more detail.

Cartographic selection Cartographic selection refers to deciding which classes of features to show on the map. Based on the map’s purpose and scale, the cartographer chooses the relevant information to include on the map and determines what should be left out. Selection decisions relate to the themes of information on the map (hydrography, transportation, boundaries, physiography, cultural features, and so on), as well as the features within the themes. For example, on a large-scale topographic map, you will usually find perennial streams (those that flow all year long), intermittent streams (those that flow a large portion of the time but cease to flow occasionally or seasonally), and ephemeral streams (those that flow only during and immediately after periods of rainfall or snowmelt). Each of these features is symbolized differently, with the perennial streams shown with darker or wider lines and the other streams shown with thinner or dashed lines, as shown in figure 6.1. On smaller scale maps, the hydrographic theme is generally included, but ephemeral and even intermittent steams may be eliminated. Special-case features are retained or eliminated on a case-by-case basis. For example, at a particular scale, all intermittent streams might be eliminated; however, a stream that has cultural significance might be retained. You can imagine that an intermittent stream that passes through the center of a town and around prominent cultural features might be retained on a map on which all other intermittent streams are omitted. Similarly, physiographic features such as hills or mountains below a certain elevation might be eliminated, except ones that have some special cultural significance. Roads of a certain class might be eliminated except those that provide some important connection for motorists. These examples show that it is useful to be aware of exceptions to the decisions that relate to the selection of whole classes of features. Cartographic selection for general reference maps is much different from selection for thematic maps. For reference maps, the challenge that the mapmaker faces is to decide which classes of features are of greatest interest and use to a wide variety of readers. For

example, many people use topographic maps for a variety of purposes, so these maps must carry a lot of information, although only some of it may be relevant for a particular map use task. With thematic maps, the challenge for the mapmaker is to decide which features to include as locational reference information without detracting from the theme. Only the locational information that is relevant to the theme is included because superfluous information may distract you or cause you to misinterpret the map’s message. For example, showing prominent administrative boundaries, contours, and other line features on a map with a hydrographic theme makes it hard to distinguish rivers and streams from related features such as dams, waterfalls, lakes, and reservoirs. Although it is the responsibility of the mapmaker to choose the themes and features wisely, it is the map reader’s responsibility to understand that only a limited selection of all possible features is shown on any map.

Figure 6.1. The degree of generalization is proportional to a feature’s spatial detail and inversely proportional to map scale. Curvilinear features are smoothed as the map scale is reduced from 1:24,000 to 1:62,500. Rectilinear features exhibit less sensitivity to generalization through scale change. Courtesy of the US Geological Survey.

Cartographic generalization Cartographic generalization refers to reducing the amount of information on a map through a change in the geometric representation of features. As you learned earlier, all maps are reductions of reality. It is impossible to show all the real-world detail on a map. If we did, the map would essentially have a scale factor of 1:1, which is the same as the world we live in. The ability to show detail on a map is determined by its scale because scale dictates the amount of space available. The degree of generalization is proportional to a feature’s spatial detail, and inversely proportional to map scale (figure 6.1). Recall from chapter 2 that map scale is a ratio between distances on the map and their corresponding distances on the earth. When ground features are displayed on a map page, distances between the symbols and lengths of features must be reduced. Features on the

map may become too small to see or differentiate from other features. Patterns may disappear as features coalesce. There is less space to add additional helpful information such as labels. Visual chaos can result. Cartographers must make careful decisions about the kind, number, and representation of features on the map. These decisions are governed, in large part, by the map scale (the map purpose, audience, and technical constraints also play a role). Ideally, the level of detail on a map is appropriate to the scale. Because all maps are abstractions of reality, they require a reduction of information content for features to be legible. When the world around you is reduced to the scale of a map, the challenge is to remove unnecessary detail while preserving the basic geometric form and spatial positioning of the simplified features. Even environmental features that are well defined, such as roads, rivers, and coastlines, don’t get mapped in all possible detail because a map cannot carry all that information in the space allowed. Linear features and edges of areal features are smoothed on the map. This loss of detail is called line generalization for features such as boundaries, rivers, and roads. We look at other types of generalization as well in this section. Generalization can also cause displacement of features as smoothing cuts off natural irregularities or causes features to overlap. For example, as a map is reduced in scale, two linear features that are next to each other, such as a river and a road, begin to coalesce. Since you know that a map should graphically communicate the geographic setting, the real message to the map reader should be that the features are next to each other in reality, not on top of each other. To preserve the integrity of the spatial relationships on the map, the cartographer must displace one or both of the features.

Figure 6.2. The most common vector generalization operations are illustrated in the examples in the figure. From Slocum et al. 2009.

Generalization in mapping is a complex topic, and entire books have been written on the subject. Because there is so much literature available on generalization and there are so many ways to approach the subject, we severely limit our discussion here and instead provide a broad overview of generalization as it relates to map design. Generalization can relate to either raster data (grid cell or pixel) or vector data (point, line, and polygon)— we look at both types, starting with vector generalization.

Vector generalization operations There is a variety of vector generalization methods, and no standard exists to supply you with a complete and agreed upon set. The set of vector operations shown in figure 6.2 is fairly comprehensive, and most mapmakers would agree that it includes the most commonly used operations. To see how these methods can be used, we look at a study of a map that was reduced in scale from 1:5,000 to 1:50,000. The map at the top of figure 6.3 shows the features with all their detail at the source scale (the scale at which the data was originally derived from its sources). The map at the larger, 1:5,000 scale requires many generalization operations to still be readable at the smaller, 1:50,000 scale (figure 6.3, middle). Streets originally represented by casings (lines of a different color than the fill) must be collapsed to a single line. Complex building polygons must be replaced with regular polygons, such as squares or rectangles. Some polygons must be collapsed to point features. And groups of similarly shaped polygons must be replaced with a simpler representation, such as a set of similar point symbols or a simple polygon. The final result is a map with an appropriate amount of detail to show at a reduced size related to the smaller scale (figure 6.3, bottom).

Figure 6.3. These examples show vector generalization operations that can be used to reduce the amount of detail on a map at a smaller scale. Top, the map shows the features with full detail at the scale of the source map; middle, the map at the larger scale shows a number of generalization operations used to make it readable at the smaller scale; bottom, the map has the correct amount of detail to show at a reduced size at the smaller scale. These maps are not to scale; they are resized to fit the page. Maps and data courtesy of the Institut Cartogràfic de Catalunya and Dan Lee of Esri.

Raster generalization operations Generalization operations for raster data are used to either clean up small errors in the data or reduce unnecessary detail. The errors may be unclassified data that originates from a

satellite image, unnecessary lines or text that originates from a scanned paper map, or artifacts from the conversion of the data from some other raster format. Figures 6.4, 6.5, and 6.6 illustrate how raster generalization operations were used to clean up errors or eliminate detail in a classified remotely sensed image. Classification of an image into categories, such as water, residential, hardwoods, conifers, and so on, results in a jagged, unrealistic representation of the boundaries. This poor representation is often the result of various limitations of classification. For example, in the classified image, a single cell may be misclassified as different from the cells surrounding it, when, in reality, the cell belongs in the category of one of the cells that surrounds it. These types of errors can be smoothed out using raster generalization operations, as shown in figure 6.5. Single, misclassified cells can be generalized by assigning them the value that appears most often in their immediate neighborhood using a majority filter function. The boundaries between zones (cells with the same value) can be smoothed. Then, larger zones can be manipulated to invade smaller zones by expanding and shrinking their boundaries. It may also be desirable to eliminate groups of cells that are smaller than a certain size. The smallest allowable size for a group of cells to be seen as a single entity in a raster image is called the minimum mapping unit (MMU). The group of cells that are below the specified MMU may be misclassified or too small to be represented as a group in the image. Eliminating these groups by combining them with the other classes results in a more generalized land-cover map (figure 6.6). The result is much easier to interpret than the original raw satellite image or even the original classified image.

Cartographic classification Classification can be defined as ordering, scaling, or grouping features into classes that simplify the features and their attributes. The goal is to capture the essential characteristics of the features and their attributes. In the process, the representation of the original features or attributes may be modified, or they may even be replaced entirely with a different representation. We talk about classification in detail in chapter 8, so we point out only a few important aspects of classification as they relate to cartographic generalization here.

Figure 6.4. The raw satellite image (top) was first classified into an image with a select set of land-cover classes (bottom). Notice the many small, isolated single cells or groups of cells throughout the image. Courtesy of Esri.

Classification of qualitative attributes is achieved by grouping together features that are similar. In figure 6.2, you can see that this qualitative classification can involve the aggregation of points into polygon features or the amalgamation of areas into composite polygon features. Classification of quantitative attribute values for map features can be done by numerical or statistical classification, which we talk about in chapter 8 relative to choropleth maps. For mapmakers, classifying features and attributes meaningfully requires

a high level of expertise. Mapmakers make better judgments about classification if they understand quantitative distributions, statistical measures, and classification methods.

Figure 6.5. After applying raster generalization operations, many of the smaller groups of cells disappear. Single cells are eliminated using a majority filter or by shrinking and expanding classes of cells. Courtesy of Esri.

Figure 6.6. Raster generalization operations were used to create this generalized land-cover map from the satellite image at the top of figure 6.4. Courtesy of Esri.

Cartographic symbolization Cartographic symbolization is the fourth part of the cartographic abstraction process. Once cartographers have selected, classified, and generalized the environmental data, they symbolize it so that the map communicates “reality” to map users. Symbolization is the use of visual variables to represent feature attributes (see chapters 7 and 8 for more on visual variables). Cartographers generalize by symbolization in two ways. First, they may reduce the level of measurement (see chapter 7) of the attribute value. For example, ratio-level rainfall data can be reduced from classes of 0–2 inches, 2–4 inches, and 4–8 inches to ordinal-level classes of light, medium, and heavy. Second, they may change the conception of the feature’s dimensionality. For example, a city represented as a polygonal feature might be reduced to a point symbolized by a circle for a map at smaller scales (see chapter 7 for more on mapping point features). Cartographers have devised a number of mapping methods to display qualitative and quantitative attribute data for features conceived of as points, lines, areas, and continuous surfaces. These mapping methods are discussed in detail in chapters 7 through 9.

MAP DESIGN CONSIDERATIONS Maps are meant to reveal something meaningful, interesting, or useful by displaying the results of spatial data manipulated to expose essential characteristics of the geographic phenomena they represent. You probably have looked at two or more maps and thought to

yourself that one was better designed than the other. You might be hard pressed to explain this general impression of design quality without a list of map design criteria, such as the one that follows, firmly in mind. Toward that end, a well-designed map is easily understood, has a clear message, gives an accurate representation of the data, does not mislead, attracts the reader’s attention to the most important information, is well presented and attractive, fits the output format and intended use, and can stand by itself without further explanation. The following design considerations will help you make sure that the map you create, or determine if the map you use, is well designed and communicates its message clearly and accurately.

Purpose and audience Before you start making a map, think about its purpose. What is the theme of the map? What information should it convey? What geographic area will it cover? Is the available data fit for the purpose? For example, a map of a walking tour is designed differently from a map for street maintenance. This variation in design is because the maps have different purposes and audiences. One is meant to encourage exploration and provide access to amenities and entertainment. The other is meant to find and manage infrastructure features and their characteristics. Once you identify the map’s purpose, think about the map’s audience. Who is the target audience? What is the context in which the map will be used? Are there any accessibility constraints such as color-blind map users? Who will read the map once it is published? Will it be subject matter experts or a general audience? A walking tour map will be used by the general public as they navigate through an area of interest that is new to them. A street maintenance map will be used by professionals with special training about the map’s features and who likely already possess knowledge of the geographic area. Finally, consider the map situation. In what setting will the map be used and displayed? Will it be printed on paper, viewed on a computer, or seen in a presentation at a conference? Are there technical constraints such as black-and-white map printing on paper of a certain size? Regarding the walking tour map and street maintenance map comparison, the walking tour map might be printed in black and white on letter-size paper—a fairly inexpensive way to produce a lot of maps for a large audience. The street maintenance map might be produced as a web map that can be accessed on a smart device while in the field so that crews can record information about the infrastructure features and feed the data back to the main office. Although making both of these maps requires thinking about the same design considerations, the solutions will be different. Before starting the mapmaking process, you’ll save time and effort if you can first clearly identify the spatial data you will use, what the data tells you (the geographic distribution), and any special things to note, such as data outliers, high or low values, or missing values.

If you discover that there is no story to tell or that the story cannot be told by a map, don’t make the map! Some topics are just not good candidates for mapping and might be better communicated in a graph or data table. There is no point in mapping the intended data if the data does not relate to the message you want to communicate, the data cannot be mapped in a way that the intended audience will be able to read effectively, the data has no locational references, there is no significant variation in the data, and there is not enough space available to present the data so that it can be properly read and understood.

Figure 6.7. When using the Lambert conformal conic map projection for the conterminous United States (A) centered correctly at 96 degrees west longitude, a map of the Pacific Northwest cropped from the national map (B) appears tilted. Changing the central meridian of the map projection (C) to 120 degrees west longitude (the approximate east–west center of the states of Washington and Oregon) fixes the problem.

Map projection use You saw in chapter 3 that map projections are designed for particular purposes and have certain properties, so choosing the right projection is critical in mapmaking. Choosing which projection to use is often one of the first decisions to make—a wrong choice can result in a bad map, no matter what other decisions you make. Often, this factor is a moot point because the projection has been decided by mapping standards, precedence set in other cases, or client requirements. When you are the one choosing a map projection, you will want to consider certain characteristics of the projections. The first is the geometric distortion properties that relate to distance, shape, direction, and area. The second is understanding how the surface used to construct the projection determines the pattern of

spatial distortion over the map surface. If you must choose a map projection, refer to the discussion of the various map projections and their uses in chapter 3. You should not only choose an appropriate map projection for the type of map you are making, but also be ready to modify the projection to make it serve the purpose of the map. A good example of modifying a projection is to redefine the central meridian (the origin of the x-coordinates in the map projection) to the center of the mapped area. The US map in figure 6.7A is positioned correctly with the central meridian for the conterminous United States, but the Pacific Northwest map cropped from the national map (figure 6.7B) appears tilted, with north at the upper left of the page. Simply rotating the projection by redefining the central meridian as vertically at, or near, the center of the mapped area (figure 6.7C) takes care of the problem.

Level of generalization Another consideration in map design is the level of generalization of the data you plan to use for your map. The level of generalization is closely tied to the map scale you select, as you saw in chapter 2. Small-scale maps (that cover a larger area) look better when the data is more generalized. Large-scale maps (that show a smaller area) require more detailed data. Generalization also relates to the set of map symbols you use. For example, thinner lines tend to reveal more about the true geometry of the features. Thicker lines can mask unrealistically jagged edges and sliverlike areas in which features don’t line up exactly—a trick that cartographers use to fix such problems with the map data. Clues that will help you tell whether your data is at the right level of generalization can be found when you examine a curvilinear line plotted from the data, such as a shoreline, river, or boundary that follows a natural feature such as a mountain ridge. Does the line collapse in on itself, causing portions of the line to appear as two-dimensional closed areas rather than one-dimensional lines (figure 6.8A)? This apparent change in the basic geometry of features is a good indicator that the data is too detailed for the selected map scale and line width. Conversely, lines that are supposed to appear curvilinear but have sharp angles (figure 6.8D) indicate that the data is too generalized. Often, the best way around both problems is to replace the data with a more appropriate dataset.

Figure 6.8. These four maps are drawn with lines of the same width—only the extent and level of detail in the data vary. Curvilinear lines can help you determine whether your data is too detailed for the map (A) or too general for the map (D). Examples (B) and (C) are at an appropriate level of generalization for the map scale. However, if you zoom in on map B or out on map C, the results will look like A and D, respectively.

Map symbolization The symbols you select for different features on the map are critical to effective map design. If your map readers cannot tell what a symbol is or what it means, your map is unintelligible or misleading and, in either case, likely useless. There are three things you can do to make the set of symbols more easily understood by your map readers: use familiar symbols, use

intuitive symbols, and provide good explanations for symbols that are not familiar or intuitive. From their previous map use experience, many map readers will already be familiar with some symbols such as a blue line for a river or a green polygon for a patch of vegetation. For features that do not have a familiar representation, try picking a symbol that your readers will intuitively understand, such as a pictographic symbol of a picnic table to indicate a picnic area, as in figure 6.9 (see chapter 7 for more on pictographic symbols) or a pattern of scattered trees for a forest. For symbols that are not familiar or intuitive, provide a good explanation in a legend or through explanatory text on the map—for example, a label for a feature next to its symbol (for example, Watchman Overlook in figure 6.9).

Matching symbols to the data To choose the right symbols for features on your map, first determine whether the data for the features is qualitative or quantitative. In chapter 7, we show that appropriate visual variables (properties of symbols that you can alter to change their appearance) for qualitative data include variations in shape, color hue, arrangement, and orientation. For example, on USGS topographic maps (discussed in chapter 2), variations in color hue (what we commonly think of as color, such as red, green, and blue) and shape are used to create standard line and area symbols (figure 6.10). The same hue is used for features of the same category, such as brown for earth landform features, blue for hydrographic features, and green for vegetation. These colors were selected as familiar hues that map users tend to associate with the colors of the features in the environment. The arrangement of shapes in a symbol can provide additional clues. The regular spacing of the symbols in the orchard symbol is different from the random pattern of the leaves in the mangrove symbol. Similarly, many of the patterns contain shapes that resemble the appearance of the features on the ground—for example, shapes in the marsh symbol mimic irregularly spaced tufts of vegetation. Other methods used to display qualitative data on maps are examined in chapter 7.

Figure 6.9. The small black dot for an overlook point in Crater Lake National Park is not an intuitive symbol, so the dot is labeled by the feature’s name (Watchman Overlook). Courtesy of the National Park Service.

In chapter 8, we point out that when mapping quantitative data, the appropriate visual variables to use are size, color value (the lightness or darkness of a color), hue, and color saturation (the intensity of the hue) because the eye will intuitively see larger, darker, or more intensely colored symbols as having greater magnitude values. For example, the map of the San Francisco Bay Area county seat population in figure 6.11 uses size, lightness, and saturation differences in combination to give maximum visual emphasis to differences in population among the county seats. Chapter 8 is devoted to the variety of mapping methods that you can use to display quantitative point, line, and area data.

Figure 6.10. Shape, color hue, and arrangement variations used by USGS cartographers for qualitative line and area symbols on topographic maps. Courtesy of the US Geological Survey.

Words, numbers, and other types of text are also important symbols on a map, and the rules for symbolizing qualitative and quantitative features also apply to text. The color hue helps distinguish different categories of qualitative features, such as blue river labels, brown contour values, and black anthropogenic feature names. Larger and often bolder text is used to label quantitative features that have more of something—for example, people in a city or traffic on a road.

Figure 6.11. Quantitative visual variables of size, color lightness, and color saturation are used together to enhance the magnitude message in this map of the population of county seats in the San Francisco Bay Area.

Figure 6.12. Point symbols that are too small (A) cannot be seen. Once large enough to be seen (B), they must also be understood.

Legibility Legibility is the degree to which something can be read and deciphered. Legibility of symbols on a map depends on how well they can be seen and on how well they are understood. Cartographers promote legibility by selecting symbols that are familiar and by choosing appropriate symbol sizes so that the results are effortlessly seen and easily understood (figure 6.12). For example, geometric symbols are easier to read at smaller sizes, whereas more complex symbols require larger amounts of space to be legible. The legibility of map symbols depends on a number of factors, including not only the size of map symbols, but the degree of visual contrast between symbols and their background (discussed later in this chapter), the map reader’s familiarity with the map theme and the area being mapped, the reader’s vision limitations, and the map viewing conditions. Symbol size Whether a map symbol can be seen and recognized from a distance depends on the visual angle. The visual angle is the angle between the light rays from the two ends of the viewed object as they hit the eye, as shown in figure 6.13. Vision research shows that under perfect viewing conditions, a person with 20/20 vision can distinguish objects at a visual angle of one minute (1/60th of a degree). Because this guideline assumes perfect viewing conditions and visual acuity, cartographers often assume a higher minimum visual angle—two minutes is probably more realistic. The values in table 6.1 are minimum readable map symbol sizes for different map viewing distances, based on a two-minute minimum visual angle. If you are making a wall map or a map that is projected onto a screen in a room, remember that the distance between the audience and the screen is not the same as the distance between you and your computer screen. For example, you must consider whether the person in the back of a 30-foot-long (9-meter-long) room can see the point symbol on your computer display map projected on a screen at the front of the room. The corresponding entry in table 6.1 tells you that the projected symbol must be at least 1.57 inches or 39.9 millimeters high for easy readability from the back of the room. Additionally, the resolution quality of the projected map is lower

than for the print map, so the print symbol can be slightly smaller (1.15 inches or 29.3 millimeters).

Figure 6.13. Geometric relationships among visual angle, viewing distance, and map symbol size. Table 6.1 Recommended minimum symbol sizes

Visual contrast Visual contrast relates to how map features differ from each other and their background. To understand how visual contrast works, consider your inability to see well in a dark environment. Your eyes do not receive much reflected light so there is little visual contrast among features, and you cannot easily distinguish objects from one another or from their surroundings. Add more light, and you are now able to distinguish features from their background and each other. Visual contrast is a key element in map design (figure 6.14). A map with visual contrast is seen as crisp, clean, and sharp. Conversely, when the map has little to no visual contrast, it

appears flat and uninteresting, and readers have a difficult time distinguishing the features that make up the map theme from the background. For qualitative data, using variations of a single hue such as red does not provide as much visual contrast as using a variety of hues, such as the orange, red, and green hues for Prince Edward Island and two adjacent Canadian provinces in figure 6.14. When mapping quantitative data, there must be enough visual contrast between colors or symbol sizes on the map for the reader to easily distinguish among different classes. The use of color for mapping quantitative data is discussed in more detail in chapter 8.

Figure 6.14. Prince Edward Island, in light orange, contrasts with the two Canadian provinces, New Brunswick and Nova Scotia, which are shown with slightly darker colors.

Figure 6.15. The primary colors and their combinations for the subtractive (left) and additive (right) color systems.

Effective color use Color systems Color is one of the things that can have the biggest impact on map design, so knowing about the system used to specify colors will help you make better color choices. A color system specifies colors numerically according to their individual components. Two systems used widely in cartography are the subtractive color system, for specifying print map colors, and the additive color system, for specifying on-screen colors. Print map colors consist of tiny dots of cyan, magenta, yellow, and black translucent inks. Cyan, magenta, and yellow are the primary colors of the subtractive color system (figure 6.15, left), which involves pigments subtracting (absorbing) different wavelengths of light. Overprinting cyan and magenta ink pigments on the page produces blue, cyan and yellow gives green, and yellow plus magenta is perceived as red. The combination of the three primary colors is seen as black, but printing black ink instead is easier and gives a stronger impression of black. Today, almost all maps are produced in color, whether they are printed or viewed on desktop computers or mobile devices. On a desktop computer screen, colors are created by mentally combining tiny closely spaced dots of red, green, and blue light in different intensities. Red, green, and blue are the primary colors of the additive color system (figure 6.15, right). Combining red and green light gives yellow, blue and green produces cyan, and red plus blue is seen as magenta. White is produced by combining red, green, and blue light at full intensity. Color depth On-screen color can be described not only in terms of the system used and its primary components, but also by the number of bits (binary digits) used for each component of a single pixel, called color depth. Maps on desktop computers are usually shown in 24-bit color (also called true color), so that each color is specified according to the intensity of

its red, green, and blue (RGB) components. Each component is represented by an 8-bit byte, with a 0–255 intensity range, for a total of 256 × 256 × 256, or 16,777,216, color variations. The RGB intensity values for the subtractive and additive primary colors in figure 6.15, along with a few other basic colors used in map design, are specified in figure 6.16.

Figure 6.16. RGB and brightness equation values for additive and subtractive primary colors and other basic colors used in map design.

Color contrast The ability to distinguish a foreground color, as in a map symbol or text, from its background color depends on the color contrast. A color contrast metric called the brightness difference is a good guide for color selection that promotes contrast. A standard measure of color brightness, the brightness equation, can be computed to determine the brightness difference between map symbols and their background color. The brightness equation from the World Wide Web Consortium is in equation (6.1):

Brightness equation values for additive and subtractive primary colors and other basic colors are listed in the far-right column in figure 6.16.

Figure 6.17. Brightness difference values for different combinations of text and background colors for the colors shown in figure 6.16. A standard brightness difference of 125 or greater is the professionally agreed on minimum acceptable value for color contrast.

The diagram in figure 6.17 shows brightness differences for different combinations of foreground and background colors. The diagram illustrates the brightness difference with text and thin lines. Combinations that have greater brightness difference values are more legible. We used a professionally agreed on brightness difference of 125 as the threshold for acceptable symbol legibility. Most color combinations in the figure are below the brightness difference threshold, and hence should not be used. Black symbols on a white background have the greatest brightness difference (255). The reverse is also true because white text on a black background has the same brightness difference. Combinations of many of the colors on a white or yellow background also have acceptable brightness differences. It is not surprising that many road and other signs use these color combinations. Color schemes The brightness difference values in the preceding section are for fully saturated map symbol and background colors, but mapmakers often use variations of hues to create more visually appealing maps. These arrangements or combinations of colors used for map data are called color schemes, or sometimes color progressions. In chapter 7, you learn how colors that vary in hue are used to show different classes of qualitative data. These color classifications are called categorical color schemes. In chapter 8, you see that progressions of colors that vary in value and saturation, from light to dark, are used to show classes of quantitative data on choropleth and dasymetric maps. Sequential color schemes are used for data values that range from low to high.

These color schemes are a light to dark progression of gray tones or a single color hue (figure 6.18). Diverging color schemes are used to show data that has a critical value in the midrange of the distribution from which other values differ progressively. Colors in this scheme have variations in two hues that range from light to dark, with a light gray often used for the midrange class (figure 6.19). To help select appropriate colors for their map data, cartographers often rely on the ColorBrewer app (http://colorbrewer2.org), which guides users through the color scheme selection process.

Figure 6.18. Grayscale (top) and single-hue (bottom) sequential color schemes for maps that show Oregon county population density in 2010.

Another common practice that you’ll see in chapters 7 and 8 is to use a lighter color for large background areas and a darker color of the same or a different hue for smaller map symbols that are superimposed on the background.

Figure-ground organization Figure-ground organization is a perceptual phenomenon in which our mind and eye work together to spontaneously organize what we are viewing into two contrasting impressions—the figure, on which our eye settles, and the amorphous ground below or behind it. Cartographers use this design principle to help map readers easily find the areas of the map to focus on. Without figure-ground organization, readers find it hard to distinguish the foreground from the background (figure 6.20A). There are several ways to promote good figure-ground organization. If the area of focus is a closed form such as an island, one way to promote figure-ground organization is simply to show only the closed form (figure 6.20B). Using a color for the figure that contrasts well with the ground forces the eye to focus on the closed form. Maps of countries, states, counties, or cities often use this technique. When mapping an area that is not a closed form or to show geographic context for the area of focus, use color value (lightness of a color) to help promote figure-ground organization (figure 6.20C). As with the closed-form example, using different color values works whether you darken the figure or the ground, as long as the difference in color value is clearly visible.

Figure 6.19. Diverging color scheme for a map that shows Oregon county population change from 2011 to 2012.

If the area of focus includes both land and water, use color hue or value differences to help your map readers see either the land or the water as the figure, depending on what is important for the map. If color is an option as in figure 6.20D, blue is a familiar color as a water symbol on maps. In most cases, the land will be seen as the figure and the water as ground. Of course, certain maps, such as nautical charts for marine navigation, promote the water as figure and the land as ground by using a shade of blue for the water and a light tint of yellow or tan for the land (figure 6.21).

Figure 6.20. Maps of Bangladesh with (A) no figure-ground organization, (B) the country as a closed form, (C) contrast in color value, (D) land-water differentiation,

(E) more detail in the country, and (F) a combination of C, D, and E to strengthen the figure-ground perception.

Areas on a map that have more detail are also seen as the figure if the ground is relatively empty. For example, adding Bangladesh city point symbols and labels to the map in figure 6.20E further focuses attention on the country, particularly if familiar places that map readers recognize immediately are included. Of course, you can employ several of these techniques together, as in figure 6.20F, in which color value, land-water differentiation, and the use of detail are combined to enhance figure-ground organization.

Visual hierarchy Visual hierarchy is the graphic structuring of the features that make up a map. This visual layering of information is fundamental to people’s ability to read a map. You can think of visual hierarchy as the separation of map features into planes of information, from most to least visually prominent. Visual hierarchy within the map helps readers focus on what is important to see first and then use what is in the visual background as locational reference information. Hierarchical organization on reference maps that show a variety of physical and cultural features, such as terrain, roads, boundaries, and settlements, works differently than on thematic maps that show the distribution of a single theme. For reference maps, no single feature should be more important than another, and all features should be on the same visual plane. On these maps, visual hierarchy is usually more subtle, and the map reader brings different features to the visual forefront by focusing attention on them. For thematic maps, the theme is more important than the base that provides geographic context and hence should be in the visual forefront.

Figure 6.21. Water is promoted as the figure on nautical charts. Land is the background so it is shown using a lighter color, often gray or tan. Courtesy of the National Oceanic and Atmospheric Administration.

The two thematic maps in figure 6.22 show the vast difference in appearance and readability when visual hierarchy is totally lacking (top) and strongly developed (bottom). Both maps show the 50 largest cities in the United States by population, but the bottom map is organized into a number of visual layers, with the proportional circles (see chapter 8) for each city population in the top layer. The mapmaker achieves this hierarchy by filling each circle with dark gray, covering the other map symbols and making the circles stand out visually above everything else on the map. Circles for smaller cities are superimposed on the circles for larger cities to create a visual hierarchy within this class of features. This visual hierarchy gives the city population distribution a 3D appearance. Lakes were chosen as the second visual layer, overlaid on all features except the population circles. The mapmaker then ordered the remaining visual layers as state boundaries, US and foreign land areas, graticule lines, and finally the oceans, superimposing each layer over those lower in the hierarchy. This visual hierarchy enhances the figure-ground organization by creating a strong land-water distinction; using a light-yellow color for the United States; and

using population circles, state boundaries, and lakes as detail that strengthens the map’s figure.

Figure 6.22. Thematic maps of the 50 largest US cities. The top map lacks a visual hierarchy, in contrast to the bottom map, which has a well-developed visual hierarchy.

Marginalia Maps have two basic components: the mapped area and additional information, commonly called map marginalia, or marginalia for short, displayed within the mapped area or in the margin of the map that helps explain or support the map. Map marginalia that is often

placed within the mapped area includes titles, legends, scale bars or other indicators of scale, north arrows or other indicators of direction, and sometimes inset or locator maps, which are explained later in this chapter. Information about the data, map projection, map author, and publication date can be placed either within the mapped area or outside the map’s margin (figure 6.23). Other marginalia may include text blocks, graphs, charts, and other elements that support the map theme and provide additional explanation for content on a map. For topographic maps and various charts, additional marginalia may include content that further describes the mapped area or the mapmaking process (figure 6.24).

Figure 6.23. Thematic map of America’s 50 largest cities with map marginalia, including title, city population legend, scale bar, map projection, data source, map author, and publication date.

Figure 6.24. Marginalia at the bottom of this USGS topographic map provides additional information about the mapped area, the data sources used to make the map, the datums and coordinate systems on the map, and the revisions made to the map. Courtesy of the US Geological Survey.

Figure 6.25. The areas covered by locator maps can be identified on the main map in several ways.

All maps need a title. A legend should be used for symbols that may be unclear or confusing, especially if the map is for a general or international audience. A legend was needed for the map in figure 6.23 because the relationship between circle size and city population is unknown otherwise. Whether to include a scale bar or other indicator of map scale depends on how much area is shown on the map. In the map in figure 6.23, there is

enough north–south scale variation that the scale bar had to be clarified as being strictly valid only at the map’s center. If your map covers a large area or is in tilted rather than vertical perspective (see chapter 3 for more on vertical-perspective maps), you should not include a north arrow because north orientation varies significantly across the map. Instead, including the latitude-longitude graticule (see the inset map in figure 6.26) is a good alternative that also graphically illustrates variations in scale. Knowing who made the map, when it was published, and what data was used to make it helps users assess the accuracy and currency of the information on the map (see chapter 11 for more on map accuracy and currency). If the location on the earth of the mapped area is not obvious, as in figure 6.20, you may need a locator map to show the position of the area within its state, country, continent, or other area with which readers are more familiar. Locator maps can be created in several ways, as shown in figure 6.25, but should always be a smaller, simple depiction of the location shown on the main map. Inset maps show areas that are part of the main map but are geographically distant from the area shown on the main map or are close-ups of areas that are too congested to easily see on the main map. For example, cartographers often make thematic maps of the 50 US states by creating a separate inset map showing Alaska and Hawaii, usually made at different scales and in different map projections better suited to the geographic location and extent of these two distant states (figure 6.26).

Visual balance Visual balance involves the harmonious organization of the mapped area and any marginalia on the page or desktop computer screen. Good visual balance gives the map a pleasing “look and feel.” Imagine that the center of the map page or desktop computer screen is balanced on a fulcrum. The factors that will “tip” the map in a particular direction include the relative location, shape, and size of elements on the page. For example, maps A through F in figure 6.27 place Africa in different positions on the map page. All but map F look off-balance, tipped toward the edge of the boxed area in different directions. Visual balance is achieved in F by centering Africa horizontally on the page and placing it slightly above the center of the page.

Figure 6.26. Inset map (on an orthographic map projection) showing the location of the conterminous United States, Alaska, and Hawaii on the globe.

The bottom three map pages in figure 6.27 show different ways of positioning marginalia to promote visual balance. Pages G and H are examples of placing the map’s title, legend, and a locator map to fill white space (empty space on the map) and create what is called informal balance, whereas on page I, the same marginalia are arranged symmetrically to create formal balance on the page. On the three bottom pages, the map position is shifted slightly to accommodate the location of the marginalia.

Text The words and numbers on a map identify point features such as cities that are represented as dots, delineate the extent of line features such as rivers and roads, and show the extent of area features such as states and lakes. These text labels can also be used to reduce

or eliminate ambiguity in the other symbols on a map, but the text must be placed with readability in mind. Size and readability Similar to other map symbols, for map text to be recognized from a distance, it must be seen at a one-minute or greater visual angle. The required minimum text sizes for different viewing distances listed in table 6.2 follow the guidelines for minimum symbol size shown in table 6.1 but are about 50 percent larger because of the increased visual complexity of text. Because letter forms are more complex, a general rule of thumb is that any text on a print map that is read at a normal viewing distance (about 18 inches or 46 centimeters) should not be smaller than six points, about 1/12 inch or 0.21 centimeters in size. The result of going below this minimum-size guideline is illustrated in figure 6.28. This lack of readability becomes even more apparent when maps designed for viewing at one distance are used indiscriminately at another distance. For example, a person with 20/20 vision seated at the back of a 30-foot room cannot clearly see text projected on a screen at the front of the room unless the map’s text is 2¼ inches high or greater on the desktop computer screen. Table 6.2 Recommended minimum text sizes

Figure 6.27. Which of the top six maps, (A) through (F), seems most balanced? It should appear that (F) has visual balance, usually achieved by placing the map’s figure (Africa) slightly above center on the page. Maps (G) and (H) are examples of placing map marginalia to fill white space and create informal balance, whereas in map (I) the same marginalia are arranged in a symmetrical manner to create formal balance.

Figure 6.28. (A) Using small text can make map text illegible without a magnifying glass. (B) Text on print maps should be six points or larger to be readable with normal vision.

Visual contrast Visual contrast for map text is achieved by varying text font size, form, boldness, and color. You will often want to vary text to show differences in feature attribute values (for example, larger text size to show larger cities) or different fonts to show different categories. Based on cartographic research, you should try to use one font (for example, Arial) and different font forms, such as upright or roman and italic, in several sizes to create enough visual contrast to differentiate among different classes of features on the map. For text sizes between 5 and 15 points, it takes at least a 2-point difference to distinguish between text sizes. For text sizes over 15 points, differences should be at least 15 percent, but 25 percent is more desirable. Often, the default choices in mapping software reflect these basic size-difference guidelines. Using different color hues for text labels that annotate different types of features also promotes visual contrast and helps readers distinguish among classes of features. This distinction is especially true if familiar hues are used for both the feature and text label, such as blue text labels for blue rivers. Using a combination of upright and italic letter forms can serve the same purpose, such as upright or roman text for names of cities and other anthropogenic features and italic or slant text for rivers, lakes, and other hydrologic features. You can also use the font type, such as serif versus sans serif, to make these kinds of distinctions. Serifs are the small lines or decorations added to the ends of the main strokes of a character that help the letters flow and lead the eye across text during reading. A common serif font is Times New Roman, whereas Arial is a widely used sans serif font. Serif fonts are often used for natural features, such as rivers and lakes or mountain ranges and peaks. Sans serif fonts are often used for cities, roads, political boundaries, and other human constructs in the landscape. Visual contrast between text and the background is promoted by having a large brightness difference between the text and its background color hue. You saw in figure 6.17 that black text on a white map background (or white text on a black background) gives the greatest contrast. Refer to this figure for other brightness differences for text.

Text placement The placement of text on the map affects the overall “look” and readability as much as the choice of text fonts, sizes, boldness, forms, and colors. The general idea is that text labels should clearly relate to the features they identify because poorly placed text looks sloppy and amateurish and can lead to confusion when reading the map. Cartographic researchers, large government mapping agencies, and private mapping firms have developed a number of text label placement guidelines, the most important being: Text labels should have visual precedence over other map symbols—that is, they should be in the map’s figure at the top level of the visual hierarchy. As much as possible, text labels for point features should be placed slightly above and to the right of the symbol. If this positioning is not possible, labels should be placed as shown in figure 6.29. Text labels on large-scale maps should be oriented parallel with the rectangular map border, with most labels placed horizontally. On small-scale maps, text should be placed parallel to the parallels, such as the placement of city names in figure 6.23. Text labels can be placed vertically to align with the vertical edge of the map but should never be placed upside down. Text labels for linear features should be aligned with the line symbols and placed slightly above if possible, as in figure 6.30. Labels should be spaced to fill larger areas (for example, states or countries), as in figure 6.30, but the spacing between letters should not be noticeably wide. Labels with more than one word (for example, Mississippi River) should be spaced so that the words are seen as a unit, as in figure 6.30, and not as widely spaced entities. Text labels should be placed entirely on land or on water, although the area mapped may make this a challenge, as in figure 6.23.

Map critique Once you create a map, check that you have followed the guidelines for good map design. Check online for a map critique checklist. It will help you identify errors, omissions, or oversights. A good last step is to ask a colleague or two to review your map, providing you with a map critique, or an external review of the design of your map. A new set of eyes can often find small errors—or even glaring ones—that may have become invisible to you during the mapmaking process. Ask your colleagues to tell you what story they see in the map. This will help you know if you have met your goal of making a meaningful map. Also, if the map is ambiguous, confusing, or visually unappealing, a good friend will tell you that. It’s much better to get this feedback before the map is published rather than after.

Figure 6.29. First through tenth choices for where to place a text label next to a point symbol on the map.

WEB MAP DESIGN Web maps have special characteristics that make them different from print maps or other on-screen maps. A web map includes not only the map presented in an online environment but also related content with an appropriate user interface, as described later in this chapter, and optional functionality for queries and reports. Sharing a map on the web gives the ability to add functionality such as pop-up windows, which appear in the foreground, or animation, which is impossible on a printed map. The web also makes it easier for maps to reach far more people. Knowing how to design maps specifically for the web will help you create maps that online readers will find useful, interesting, and notable.

Figure 6.30. Text labels should be letter spaced to fill larger areas. Text for rivers and other linear features should be aligned with the line symbol for the feature and placed slightly above the symbol.

Why web maps are special Web users typically have relatively short attention spans and high expectations. They do not focus long on content or tasks before becoming distracted, so a web map should not only display quickly, but its purpose should be immediately understood. Web users often expect a minimum level of interaction, such as panning and zooming, so web map functions should respond rapidly. Web users often expect that what they are viewing is of immediate and personal use to them. These characteristics challenge web mapmakers to design maps that possess high levels of graphic and information clarity. Because the web environment is well suited for interaction, information that is normally displayed on a print map can be revealed in a timely manner using pop-up windows, tool tips, side-panel information boxes, and hyperlinks. A web map can have fewer labels and detailed features and still convey the intended information because users can zoom in to

see greater detail, if desired. The map can be linked to databases that report attribute information, display still or animated images, play sounds when users click related map features, or perform analyses by accessing GIS geoprocessing functions, to manipulate or analyze GIS data. Web maps can also be portals (specially designed websites for downloading or uploading geographic or other content). Users will likely have certain expectations for web map content. As with print maps, data should be complete, consistent, and authoritative. Users expect current data and sometimes continuously updated data, as on maps that show traffic, weather, or disasters. Web map users also expect interactive maps that sometimes support query, analysis, and user customization. They may even expect the data that is used to make the map to be downloadable and free. For large-scale maps, users expect detail and realism. For smallscale maps, they expect higher levels of detail (density and quantity of information) at higher zoom levels (for example, 21 fixed levels of magnification, ranging from level 0 for a global map at a scale of 1:600,000,000 to level 20 for a neighborhood map at scale 1:128). See chapter 2 for more on multiscale maps.

Web map design considerations The workflow for making web maps encompasses five primary activities: compiling the information to be shown on the map, designing the map, designing the user interface, designing the user experience, and promoting the finished web map. The first two steps are essentially the same as for print maps. However, when designing the map, you must also consider how the user interface (UI) can be used to help promote the map’s message and support the information being presented. When designing the user experience (UX) (the usability and functionality presented to the user), consider how users will navigate the web page and its elements, interact with the map and its related information, and discover the information they are looking for. Once the map is finished, promote the map not only to its intended audience but also other potential audiences to maximize its value. Size and resolution When designing a web map, you must determine the size and resolution (the number of pixels the screen can display) of the display. From there, you can determine the map scale or scales. Web maps are usually designed for 17- or 19-inch (43- or 48-centimeter) desktop computer monitors. However, web maps can also be viewed on other display devices, such as laptops, tablets, notebooks, smartphones, or other mobile devices. Table 6.3 lists common display devices and screen sizes. Table 6.3 Common display devices and screen sizes, along with resolution and viewing distance

In general, computer display or mobile device resolution is low when compared with print maps. For desktop computers, it is common to design for a resolution of 96 dots per inch (dpi) because all computer monitors support this resolution. Newer monitors typically have a pixel density of 120 dpi or 144 dpi. Some of the newest devices have retinal displays with pixel resolutions so high (300 dpi) that it is impossible to distinguish a single pixel in the display. Common resolutions are listed in table 6.3, along with viewing distances. Design your web map for the type of display that your target audience will mostly likely use. Low resolution impacts the design of a web map. Because screen displays are in pixels, lines other than vertical or horizontal and sharp edges appear jagged. These jagged edges can be softened by using antialiasing, adding pixels of intermediate color between the object and the background, which fools the eye into seeing a jagged edge as a smooth one. Responsive design is an approach aimed at an optimal viewing and interaction experience across a wide range of devices. Good responsive design results in easy reading with a minimum of resizing, panning, and scrolling of the user interface. Geographic extent and map scale Another determination for web map design is what geographic extent (amount of area) to show. Because users can pan and zoom, the geographic extent of the map can be greater than what is shown initially on the screen. Sometimes it is useful—and necessary— to restrict the map extent, as with a map of a site such as a park or building interior. Other times, it makes more sense to provide a more expansive view, as with a thematic map of a country or the world. The geographic extent and size of display relate directly to the map scale or scales. If readers can zoom in and out, the map scale is variable and you must create a multiscale map (see chapter 2 for more on multiscale maps). In a multiscale map, a separate map should be compiled for each zoom level or a small range of zoom levels to ensure that the zooming experience appears seamless. Map projection

The map projection you use depends on whether the map will be mashed up with other web maps. For example, if you want your map to overlay other maps in ArcGIS Online or Google Maps, use the web Mercator projection, which is described in chapter 3. If you use a different projection, anyone who wants to use your map in a mashup will have to use that same projection. If you do not think anyone will use your map in a mashup, consider an alternate projection that is more appropriate to the geographic extent and map theme, such as the Winkel tripel projection for world maps or the Lambert conformal conic or Albers equal-area conic projections for maps of the conterminous United States (also see chapter 3 for more on these map projections). Symbols and text Web maps on the desktop are normally viewed at a distance of between 20 and 40 inches, or 1.66 to 3.33 feet (50 to 100 centimeters), from the eye to the front surface of the computer screen, as shown in table 6.3. Table 6.1 shows that the recommended minimum point symbol size for this reading distance range is 7 to 12 point (0.09 to 0.12 inches or 0.23 to 0.30 centimeters). Text is more visually complex than point symbols and hence has larger minimum recommended sizes. In table 6.2, you can see that for the 20- to 40-inch normal viewing range, the text on your computer screen should be between 13 and 18 points (0.18 to 0.25 inches or 0.46 to 0.64 centimeters). Using the values in tables 6.1, 6.2, and 6.3, you can determine the minimum point symbol and text size for various viewing conditions. When possible, use text fonts designed for the web. Good web fonts have a generous amount of space within and between characters, as shown in figure 6.31. A tall x-height (the distance between the baseline of a line of type and the top of the main body of lowercase letters) also opens up the space within a character. These properties make fonts more legible on the computer screen. A recent study identified Arial (or Helvetica on Apple’s Mac), Verdana, Georgia, Trebuchet, and Century Gothic (all installed on Windows systems), and Lucinda Grande and Palatino (installed on most systems) as the most popular fonts for web design.

Figure 6.31. Characteristics of good web fonts.

With the exception of Georgia and Palatino, the fonts are sans serif fonts, as described earlier in this chapter. Serif fonts are popular in print; however, many designers and cartographers believe that sans serif fonts are more suitable for web map design because serifs compromise the space between characters. This preference holds true for small blocks of text (labels on maps, titles, legend text), but for large blocks of text, serif fonts are still easier to read.

SELECTED READINGS Brewer, C. 2015. Designing Better Maps: A Guide for GIS Users, 2nd ed. Redlands, CA: Esri Press. Dent, B. D., J. S. Torguson, and T. H. Hodler. 2009. Cartography: Thematic Map Design, 6th ed. Boston, MA: WCB-McGraw Hill. Krygier, J., and D. Wood. 2011. Making Maps: A Visual Guide to Map Design for GIS, 2nd ed. New York, NY: Guilford. MacEachren, A. M. 1994. Some Truth with Maps: Primer on Symbolism and Design. Washington, DC: Association of American Geographers. Muehlenhaus, I. 2014. Web Cartography: Map Design for Interactive and Mobile Devices. Boca Raton, FL: Taylor & Francis Group. Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. Elements of Cartography, 6th ed. New York, NY: John Wiley & Sons. Slocum, T. A., R. B. McMaster, F. C. Kessler, and H. H. Howard. 2009. Thematic Cartography and Geographic Visualization, 3rd ed. Upper Saddle River, N.J.: Pearson Prentice Hall. Tyner, J. A. 2010. Principles of Map Design. New York, NY: Guilford.

chapter seven QUALITATIVE THEMATIC MAPS POINT, LINE, AND AREA INFORMATION Functions of points on maps Data collection for point features Functions of lines on maps Data collection for line features Functions of areas on maps Data collection for area features HOMOGENEITY PRINCIPLES OF SYMBOLIZATION Graphic marks Levels of measurement Visual variables SINGLE-THEME MAPS Point feature maps Line feature maps Area feature maps MULTIVARIATE MAPS Multivariate point symbols Multivariate line symbols Multivariate area symbols MAPPING QUALITATIVE CHANGE Change in attributes over time Small multiples Change maps Superimposed maps Change in location over time Point symbols Line symbols Area symbols DYNAMIC QUALITATIVE-CHANGE MAPS SELECTED READINGS

7 Qualitative thematic maps In the introduction, you saw that maps can be divided into four broad categories on the basis of their purpose or function. For example, topographic maps and world atlas sheets are reference maps, which are used to locate features and learn the basic geography of a region. Nautical and aeronautical charts and road maps are navigational maps used to plan travel routes. Story and persuasive maps help convince you to accept an idea or take an action. In this chapter, we focus on the fourth category of maps—thematic maps. Thematic maps emphasize a single theme or a few related themes. A map of different climate zones is a good example, as are maps of land cover, soils, per capita income, and race and ethnicity. Basic geographic reference information appears on a thematic map to provide the locational base for the reader, but the theme stands out visually as the most important message of the map. Thematic maps can show both qualitative and quantitative information. In chapter 8, we examine the methods that mapmakers use to portray quantitative information, or information that portrays a magnitude message, such as the population of a city, annual rainfall, and measures of streamflow. In this chapter, we focus on the ways that mapmakers show qualitative information that has classes that vary in type but not quantity. Examples include landownership, zoning, and soils. First, we look at the functions that point, line, and area features have on maps. Then, we look at the basic principles of symbolization so you can understand how point, line, and area features are symbolized on maps. Next, we examine the ways that mapmakers show a single theme about point, line, or area features. We also explore examples of how mapmakers create multivariate thematic maps that show the geographic relationships between two or more themes. Finally, we explore dynamic qualitative thematic maps that focus on changes in feature locations and attributes over time.

POINT, LINE, AND AREA INFORMATION In the foreword to this book, we noted that the essence of mapping is to let something “stand for” something else. We let marks on the map stand for features in the environment. If you don’t understand what the marks mean, you can’t read, analyze, or interpret the map. Therefore, we start by exploring how point, line, and area marks on the map function graphically. We also examine how data for features in the environment is collected and represented as points, lines, and areas.

Functions of points on maps Points serve several functions on maps. They can be used to represent features that exist at points, features that are referenced to points, or features that are conceived of as points for mapping purposes. The first function is to represent point features—zero-dimensional entities without width or area that are defined solely by their geographic location. The horizontal survey control points discussed in chapter 1 and PLSS section corners discussed in chapter 5 are examples. Think about the control points. Although the metal marker is actually a feature that takes up some area (albeit very little) on the ground, truly zero-dimensional geographic coordinates define the control point. A second function is to represent features that are referenced to points. An example is the mean center of population. You can think of this point as the place where a flat, rigid map would balance perfectly if all the people in the dataset weighed the same. Figure 7.1 shows the mean center of population for the United States, from 1790 to 2010. Even though the theme is about population for the nation, the features on the map are represented as points. The third function of point features is to represent a two-dimensional feature that is conceived of as a point for mapping purposes. In this case, features that are really areas on the ground are shown as points on a map. You have seen many cases of this, even if you don’t know it. Think of how many times you have seen a map that shows cities as points that are labeled. The cities are, of course, areas, but the symbols on the map are points. This representation is a common way to deal with the generalization of area features when mapping them at smaller scales (see chapter 6 for more on generalization).

Figure 7.1. A map of the center of population for the United States shows the mean center as points. To see the center of population mapped every 10 years, go to http://www.census.gov/2010census/data/center-of-population.php. Courtesy of the US Census Bureau.

Figure 7.2. Pictorial point symbols for trees can be used to map data captured as either points or areas.

To understand these functions of points, consider how data for a single theme—in this case, trees—might be captured and used for mapping. A city parks department might collect data for all the trees that it or other municipal departments manage. One way that it might capture the tree data is to map each tree as an area that is as large as its crown (the extent of the branches and leaves). Alternatively, each tree might be captured at the point at which the center of the trunk is located. Where there are many trees that are hard to distinguish individually, the area in trees might be represented by a line around the extent of the vegetation. Another approach might be to represent trees by the administrative unit—for example, a city park or an orchard in a parcel that contains the trees. So you can see that a single theme—trees—can be captured as a point, line, or area feature type. These three feature types are commonly used in GIS. Information about the trees is also likely to be captured. Qualitative attributes might include the species, the date on which the tree was last trimmed, or the department that is responsible for its maintenance. Quantitative attributes might include the diameter of the

trunk, radius of the crown, height, or age of the trees. Both the feature and attribute data can be managed using GIS. How a map is made about the theme starts with the type of data that is used. For example, a map can be made that shows trees as area features—crowns filled with colors that represent different species types. A different approach might be to map the crown area data as points in the center of the features—these central features are called centroids, and they can easily be created from the area features using GIS. In this case, the points are locations to which the tree features are referenced. These centroids can be mapped using point symbols of different shapes and colors to indicate the different tree species. The advantage of using this approach is that pictorial point symbols can be used rather than simple colors to fill the area features (figure 7.2). This example shows that features with areal extent can be mapped either as points or areas, depending on the map’s scale and purpose. Data collection for point features Data about the locations and attributes of geographic features is collected in a variety of ways. In our tree mapping example, a GPS receiver may have been used to survey (see chapter 5 for more on surveying) the location of each tree, and the species type may have been determined by field observation (observing features in situ). It is also possible that the location of each tree was mapped in a field sketch (a drawing produced to help support data collected in a field study), and the species type was determined through an interview with a park department employee. Geographic data is also collected through image interpretation, using air photos or satellite images to identify objects and record their attributes (see chapter 10 for more on image maps). Any of these methods can be used to create a point feature in a GIS database, using an x,y coordinate to identify its location and an attribute that gives the species information. Modern mobile device technology is changing the face of point, line, and area data collection. People use their smartphones, tablets, and other devices to locate their positions, track their travels, and save photos of these locations using cloud-based photosharing programs such as Flickr or Instagram. Many apps, including those in ArcGIS Online, also allow you to digitize (record digitally) point, line, and area features while in the office or out in the field. For example, it’s not uncommon to use field tablets with GPS receivers and GIS software to record the locations of features and the attributes that are used to manage, analyze, and map these features. It is also possible to send the fieldcollected information to computers in the office, where staff can work instantly with the data. The accuracy of the information about the location and attributes of point features is an important consideration for you as the map user. In our tree mapping example, if a professional botanist trained in using GPS receivers or a GIS professional trained in the identification of tree species collected the data, great faith can be placed in the accuracy of the point data in terms of both the location and the species attribute of the trees. If the same botanist used field observations to record the location and species, you can still be confident of the species attribute, but you should assume that the mapped position of each tree is only approximately correct. Alternatively, if the GIS professional was not adequately trained to recognize different tree species, you can assume that the location of each tree is correct but you might suspect the species attribute data. A photogrammetrist (a person trained to compile reliable data from aerial photographs) may have determined the location and species of each tree from large-scale

air photos. In this case, different tree species are usually determined with the aid of tree identification keys that show the typical appearance of each species in an aerial photo. Although photogrammetrically compiled locations can be precise, identifying different types of trees from the typical appearance in aerial photos is usually less accurate than direct field observation. In this case, you can place more faith in the mapped position of the trees than in the species attributes.

Functions of lines on maps As with point features, line features serve several functions on maps. They can represent truly one-dimensional features that have length and direction. Surveyed lines such as property, political, and administrative boundaries are true one-dimensional features. Most lines that you see on maps, however, represent features that have width but are portrayed graphically as one-dimensional lines—that is, they represent features that are conceived of as lines for mapping purposes. For example, roads have standard design widths, and these features vary in width on the ground. In large-scale engineering plans, the road width is shown as the edge of pavement, and on some maps, it’s shown as the rightof-way (real property designated for a specific purpose). However, roads are often shown by one-dimensional lines on maps. Surface composition, such as asphalt, concrete, gravel, or dirt, is an example of a qualitative attribute for these features. On smaller scale maps, mapmakers use line symbols that vary in width, color, pattern, or combinations of these visual variables to represent the location and attributes (such as width or surface composition) of roads. Lines also serve other graphic functions on maps, including representing paths of movement, the structure of linear networks, boundaries between areas, and lines of equal value (isolines). Data collection for line features As with point feature data, line feature information is commonly collected through a variety of methods, including surveys, field observation, and image interpretation. Consider a map of roads that shows differences in surface composition. Road locations may have been digitized from the original engineering plans by converting the maps from paper to digital format. The road surface type may be entered as an attribute of the digitized features. Alternatively, a person could drive each road with a GPS receiver in the car to digitize road segments and use field observations to note whether the road surface composition is asphalt, concrete, gravel, or dirt. A photogrammetrist could digitize the roads as centerlines (either the painted line in the center of the road or the interpolated center from the edges of the pavement) from air photos, and then use a road cover interpretation key to record the surface composition. As with our tree mapping example, the data collection method has an impact on both the locational accuracy of the features and their attributes. City and county public works or engineering departments often store road data collected using one or more of these methods in their GIS database. Other road attributes such as type (interstate, US or state highway, ramp, and primary or secondary road) and direction (two-way or one-way) are also recorded. Mapmakers regularly use this digital data to create a variety of maps, including those designed specifically to show road surface composition.

Functions of areas on maps

Similar to points and lines, area features serve the function of representing truly twodimensional features or features that are conceived of as areas for mapping purposes. Additionally, they can represent things that are referenced to areas. An example is a standard deviational ellipse, which is a common way of measuring the trend for a set of points or areas to see if the distribution of features is elongated and therefore has a particular orientation. The ellipse is centered on the mean center for all the features, the orientation represents the directional trend, and the length and width give you an idea of how concentrated the features are. The map in figure 7.3 shows standard deviational ellipses for each year of zebra mussel sightings, from 1986 to 2011. The ellipses allow you to see if the distribution of features is elongated and therefore has a directional trend. Comparing the size, shape, and overlap of ellipses or the ellipses from year to year gives you an indication of how the zebra mussels are spreading over time. Data collection for area features Mapmakers use qualitative area feature data collected by ground survey, image interpretation, or other methods such as census taking to determine the category for each data collection area. A census is a survey that collects data from all the members of a population, whether it’s people, animals, businesses, or other entities, within a defined area at a specified time. Often, a variety of characteristics are recorded for each member of the population. For a census of human populations, demographic, economic, and social attributes are recorded.

Figure 7.3. Standard deviational ellipses represent the directional trend in a set of points or areas, such as zebra mussel sightings in water bodies. Courtesy of the US Geological Survey and Esri.

HOMOGENEITY

Mapmakers commonly use two-dimensional area features to represent areas that have a common attribute. The area features are considered to be homogeneous—that is, uniform in structure or composition throughout. When data is collected for line and area features, the features are also homogeneous. For example, a stand of trees might prove to be entirely of the same species. The stand is a completely homogeneous qualitative area feature because all its defining objects (trees) are of the same category (species) within the homogeneous area (stand). The interesting thing about features mapped as though they are homogeneous is that most features, actually, are not. In our tree example, although the trees within the stand may be the same species, there may be an understory of shrub, brush, or grass. At the edges of the stand, the trees may mix with other species to create a mixed-species transition zone between two single-species stands. Mapmakers deal with this real-world problem in several ways. They may define the boundary between the two single-species stands as the middle of the transition zone. Now neither of the stands are homogeneous area features, although they are mapped as being so. Mapmakers may add a transition zone to the classification so that the map now includes a mixed class as well as the two single-species classes. This solution increases the complexity of the map, but it allows the single-species stands and transition zone to be mapped as homogeneous areas that more accurately reflect what’s on the ground. They can alternatively add a note to the map explaining that they have included the transition zone as part of one of the stands. Now we look at an example of homogeneity within line features. In our road example, many roads are probably completely asphalt or totally concrete. When these homogeneous features are shown on the map as asphalt or concrete roads, the maps agree with what’s on the ground. However, other features may have more than one surface type, such as a forest road that’s mostly gravel but with short sections of asphalt and other sections that are dirt. In this case, the composition of the majority of the road can be used as the category for the entire road segment. An alternative is segmenting the features by splitting them at locations in which the attribute—for example, road surface type—changes. The feasibility of this approach relates to the map scale (see chapter 2 for more on map scale). For example, you don’t analyze, navigate, or manage road segments on the basis of what would be minute variations at a small map scale. However, roads that have many surface types but were classified into one category on a larger scale map may cause travel and other problems for the unwary map reader. Without the appropriate surface composition information, map users may find themselves unable to plan for associated changes in access, traffic, or speed.

PRINCIPLES OF SYMBOLIZATION Knowing what the marks on maps mean and how the data for them is collected is one step to successfully reading a map. But you must also know something about the principles of symbolization (the use of symbols to represent features and their attributes) used by the cartographer to manipulate and display the data graphically in a way that reveals something interesting or useful about the mapped environment, as we noted in the foreword to this book. Cartographic symbolization is based on three basic building blocks: graphic marks, levels of measurement, and visual variables. Cartographers use point, line, and area

graphic marks on the map to represent geographic features. The graphic marks are symbolized using what mapmakers call visual variables (see chapter 6 for more on visual variables). The choice of visual variable depends on the level of measurement.

Graphic marks The most common graphic marks on a map are points, lines, and areas (also called polygons). However, pixels are also often used to represent terrain and other surfaces (see chapter 9 for more on pixels). A pixel is a cell in an array, which commonly forms a surface or image. You can think of it as a minute area of ink on a page or a tiny spot on a display screen which, when viewed with the other pixels in the array, creates an image. You’ll learn more about pixels in chapters 9 and 10. Cartographers and spatial scientists are also starting to explore the use of voxels (volume pixels) to represent three-dimensional data. A voxel is the three-dimensional equivalent of a pixel and the tiniest distinguishable element of a three-dimensional object. It is a volume element (usually a cube) that represents a specific cell value in 3D space. As you have already seen, the graphic mark on the map does not need to have the same dimensionality as the feature it represents. For example, a city, which is a two-dimensional area feature, can be mapped as a zero-dimensional point graphic mark, or a river, which has width and length, can be mapped as a one-dimensional line graphic mark. It is important when you read a map to understand how the graphic mark relates to the feature it represents. For cartographers, the choice of graphic mark dictates, to a large extent, how they symbolize the feature on the map.

Levels of measurement Qualitative information tells you only what types of things exist, while quantitative information gives you the magnitudes of things, such as how large, wide, fast, or high they are. This simple dichotomy between qualitative and quantitative information pervades our descriptions of the environment. But to categorize all information as either qualitative or quantitative is needlessly restrictive because scientists think of data as being at one of four basic levels of measurement, which is a classification used to describe the nature of information about features. The nominal level is associated with qualitative information, whereas quantitative data can be at the ordinal, interval, or ratio measurement levels (see chapter 8 for details about the three quantitative measurement levels). Nominal-level data tells you simply what type of thing a feature is. Features are often assigned to categories or classes. The term categorical data is often used for nominallevel data with categories that distinguish different types of features, whereas the term classed data is often used when numerical data is grouped into classes with ranges of values. Features within a category are assumed to be relatively similar, whereas differences between categories should be distinct. Nominal-level data collected for point, line, and area features can be attributes, which carry the descriptive information for the geographic features in a GIS (see chapter 5 for more on GIS). For example, zoning and ownership are common nominal-level attributes used to describe parcel features in a dataset. The categories for the zoning attribute might include industrial, commercial, multifamily, residential, and so on. Sometimes the categories are defined by numerals, such as the Land Capability classes in figure 7.4. These roman numerals do not represent a quantitative measure, but rather nominal-level information about the features. Can you see how it would be inappropriate to work with these numbers if you assume they are quantities?

Visual variables Visual variables are the properties of a symbol that can be changed to alter its appearance and meaning. Certain visual variables work well for qualitative data; others work well for quantitative data. Symbols that work well for qualitative data impart a message of types or categories, but not how much of something there is in a category. Symbols that work well for quantitative data impart a magnitude message. Symbols on maps are easiest to read if the mapmaker has assigned the correct visual variables to the point, line, or area graphic marks for the features they represent.

Figure 7.4. Land capability classes are designated by roman numerals that represent qualitative rather than quantitative differences between the categories. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

The visual variables that evoke qualitative differences among features are shape, orientation, arrangement, and color hue (figure 7.5). A shape can also be repeated along a line graphic mark or within an area graphic mark to create a pattern—we sometimes call this a line pattern or area pattern. Likewise, orientation and arrangement can be used to create patterns within the graphic marks—this is called pattern orientation and pattern arrangement. For example, a set of point symbols might be designed as circles with lines in them that are oriented in different directions. An example of area arrangement is the use of small tree symbols within an area fill. A regular arrangement, such as a square-grid dot or equilateral triangle pattern, gives the impression of a human-created landscape, such as an orchard or a tree plantation; an irregular arrangement gives the impression of a more natural landscape. Understanding the visual variables and how they are used in symbolizing the graphic marks on maps helps you correctly interpret the map.

Figure 7.5. The visual variables that naturally evoke qualitative differences among features are shape, orientation, and color hue. A shape repeated across an area or along a line creates the element of pattern. Orientation can also be used to create patterns within graphic marks.

Keep in mind, however, that maps are not always designed as well as they might be. Thus, some other visual variable (size, texture, color value, or color saturation) that is more appropriate for showing quantitative differences (see chapter 6 for more on color hue) may be used for nominal-level data. When maps use the wrong visual variables, your first reaction may be to think that quantitative rather than qualitative information is being shown, or vice versa. The only way to keep from being misled by incorrectly made maps is to read the legend to see what the symbols actually represent. For example, on the map in figure 7.6, the mapmaker properly used color hue to differentiate most of the habitat classes (see chapter 6 for more on hue). Nevertheless, for some categories, the color value (lightness) varies, which can easily give you the impression that the categories differ in some quantitative way. For this map, the legend clarifies any ambiguities in the symbols. The different categories may consist of purely one feature (“Upland Aspen Forest”) or of two or more intermixed features found in certain regions (“Westside Lowland Conifer– Hardwood Forest”). You can see that the mapmaker can define inherently homogeneous areas in many ways, so you should carefully read the map legend to understand what each category means.

Figure 7.6. Color hue is used to differentiate most habitat classes on this map of northeastern Oregon, but color value (lightness) is also used (note the symbols for the first few classes in the legend). It is useful to refer to the legend to understand that the map shows only qualitative differences. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

SINGLE-THEME MAPS The simplest and most common qualitative thematic maps show a single theme. We examine these single-theme maps on the basis of whether they show point, line, or area features.

Point feature maps Point feature maps have symbols that show the location of where something exists or has occurred. The word “point” has a loose interpretation here. It isn’t only being used in the strict mathematical sense of a zero-dimensional figure, but rather, as you saw earlier, also as the point location to which something is referenced or the location that represents a feature conceived of as a point for mapping purposes.

Figure 7.7. Qualitative point symbols range from geometric (A) to pictographic (D).

Qualitative point features are usually represented by point symbols that are somewhere on a continuum between geometric and pictographic. Geometric symbols use simple shapes, such as squares, circles, triangles, and stars, to represent features (figure 7.7A). Because they are so simple, they usually require a legend to be interpreted correctly (figure 7.8). At the other end of the continuum are pictographic symbols, which are designed to look like miniature versions of the features they represent. Pictographic symbols are often used on landmark maps that require a reader to identify specific buildings or other landmarks at first sight. On these maps, small sketches of the buildings can sometimes be used to allow the reader to immediately distinguish individual buildings or other features. These symbols are more like example D in figure 7.7.

Figure 7.8. Simple geometric shapes (in this case, stars) are used to show different episodes of major conflict on this map. Without the legend, it would be impossible to tell what the symbols represent. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Somewhere in the middle of the continuum is a mimetic symbol—one that “mimes” the thing it represents. Mimetic symbols are often created as a combination of geometric shapes, such as a square with a triangle on top to represent a house; or they can be more complex, such as a small cartoon of a particular type of building, such as a ranger station or museum. Because these symbols are intuitive, they are popular for mapping point features or labeling area or line features. You will find mimetic symbols on tourist maps, recreation maps, children’s maps, and maps on the web. A good example is the pushpin symbol you see used in Google Maps. A special type of point symbol is the standard symbol, which is a symbol that is used as a standard within a mapping company or agency or for a mapping product. Sets of standard symbols have been created to show different categories of transportation, recreation, and other activities. The US National Park Service (NPS) uses standard symbols to depict visitor amenities and facilities on its national park maps (figure 7.9). Standard symbols have a more professional look, but they still suffer problems associated with pictographic symbols in general. They must be relatively large for details to be apparent in crowded spaces or at small scales. Also, only a limited number of environmental features can be successfully symbolized because readers become confused if there are too many symbols and if they must refer constantly to the legend to decipher the

map. What, for example, is an obvious icon for a vista or an overlook? And what if there are 10 different cryptic symbols on the map? Although these symbols are intended to be intuitive at a glance, you’ll often have to check the map legend to determine what is being symbolized (figure 7.10). Mapmakers sometimes cleverly change the orientation or hue of a symbol to show two or more attributes of the point feature. For example, in figure 7.5, the orientation and hue of the mimetic symbol for a coniferous tree are changed to show the tree as alive (vertical and green) or dead (horizontal or brown). As in this example, mapmakers sometimes use both orientation and hue differences to make sure that you see the attributes of each feature. The drawbacks of more complex point symbols are overcome by using geometric shapes, such as circles, squares, triangles, and so on (see figure 7.7A). Although these symbols may look abstract, they can be read correctly, even when small. They can also be used for categories with larger numbers of individual features. Their small size allows mapmakers to pack more information into the map than they can do with larger, more complex point symbols. Furthermore, since the correspondence between real-world features and their geometric symbols is strictly arbitrary, any point feature can be represented this way. The greater level of abstraction embodied in geometric symbols increases their flexibility as symbols. It also requires close study of the legend to determine what is being symbolized.

Figure 7.9. Standard symbols are used on this map to show various attractions and facilities within Crater Lake National Park in Oregon. Courtesy of the National Park Service.

Figure 7.10. The legend for the map of Crater Lake. Courtesy of the National Park Service.

Line feature maps Mapmakers make line feature maps using qualitative line symbols that show different categories of linear features, such as roads, streams, or boundaries. As with point symbols, the word “line” isn’t used here in the strict mathematical sense of a one-dimensional figure. Although line features may be truly one-dimensional (such as boundary lines), line feature maps can also be used to show features that are conceived of as lines for mapping purposes (such as a river, which, in reality, has width but is shown with a line on a map), lines to which things are referenced, paths of movement, networks, boundaries between areas, and lines of equal value. The lines used to symbolize linear features on maps have width as well as length, and the symbol width rarely corresponds directly with the feature width on the ground. For example, if you measure the width of the lines that represent railroads on the map in figure 7.11, the map lines are much wider than the width of the corresponding feature on the earth when the feature’s map width is converted to ground distance. Lines on a map are often drawn wider to show the category using the appropriate visual variables—color hue, line shape (usually as combinations of dashes and dots), and the arrangement or orientation of the pattern within the line. The mapmaker must widen the lines on the map for these visual variables to be seen.

Figure 7.11. Different hues are used to distinguish ownership of railroad lines on this map. The lines on this map do not accurately reflect their width on the ground. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

You have likely seen lines of different color hues on maps. Some hues are standardized for certain features. For example, water features are usually shown with blue lines, boundaries are depicted with red lines, roads and railroads are drawn in black, and contour lines are shown in brown. At other times, hue carries a categorical message, as on the map of railroad ownership in figure 7.11. To read these kinds of maps correctly, you must refer to the legend. Line pattern is also commonly used to distinguish different categories of line features (see figure 7.5). The pattern is created as a repetition of shapes, usually combinations of dashes and dots, along the line. The individual marks repeated along the line are usually geometric, but they can also be mimetic. Mimetic shapes may be easier to discern than abstract geometric shapes, but they’re more difficult to miniaturize and repeat along a line. Therefore, most patterned line symbols use repetitions of geometric shapes. Again, some standards apply. For example, administrative boundaries are often shown using a variety of dashed-line symbols, and railroads are commonly shown using a solid line with crosshatches that mimic the railroad ties. On many maps, color hue and line pattern are used together to show a greater variety of lines. For example, the lines on a map can be used to represent features that are linear

(fences or walls), as well as networks (roads and railroads), and the edges of area features (administrative boundaries). In addition, there may be lines to show the graticule, the edges of the mapped area, and other bounded areas on the page such as the legend or title. Having a variety of line symbols to choose from helps the cartographer create a map that you can more easily decipher.

Area feature maps Qualitative area feature maps use area (polygon) symbols to portray homogeneous areas. On areal thematic maps, area features are best symbolized using the same visual variables that give the impression of differences in type or kind as for point and line features —that is, color hue, pattern shape, pattern arrangement, and pattern orientation. For example, a mapmaker might use two different hues to show the states won by the Democratic Party (blue) and Republican Party (red) presidential candidate (figure 7.12). Alternatively, the mapmaker could use a pattern of donkeys as mimetic symbols to fill the Democratic states and a pattern of elephants for the Republican states because those symbols are used for the political parties. The mapmaker could also use a combination of pattern, shape, and hue (red elephants and blue donkeys) to make sure that you correctly see the category for each region. The habitat map in figure 7.6 and the election map in figure 7.12 exemplify two basic types of qualitative thematic maps that differ in the kind of data collection areas being mapped, although they both use color hue as the primary visual variable. The presidential election map in figure 7.12 is based on legislatively defined data collection areas (states). On the map, states are given one of two hues, depending on which candidate received the most votes. The areas are correctly portrayed as homogeneous, because the candidate with the most votes receives all the state’s Electoral College votes. This type of map is often called a categorical map—a map that has polygons that enclose areas that are assumed to be uniform or areas for which a single description can apply.

Figure 7.12. Areal symbols can be used to distinguish regions from each other on the basis of attributes such as states won by the Democratic or Republican Party presidential candidates.

Figure 7.13. Color hue is used to distinguish Level III ecoregions, and color lightness and color intensity are used to show the minor classes within the ecoregions. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

The habitat map in figure 7.6 is an example of mapping area features that are inherently homogeneous in some way, such as having the same vegetation. This map is compiled by letting the environmental data determine the boundary between classes—differing markedly from a boundary that is compiled using already defined administrative or other boundaries that bear no natural relation to the data.

MULTIVARIATE MAPS Most qualitative thematic maps show a single variable—that is, attribute—of a feature. But sometimes mapmakers show several attributes of the feature on a multivariate map. There are two methods for symbolizing multivariate information. One method is to use a different visual variable to show each attribute within a single symbol. In theory, mapmakers can show five different qualitative attributes at once by varying the symbol’s shape, color hue, pattern shape, pattern arrangement, and pattern orientation. In practice, symbols that show even two or three feature attributes can be difficult to read if they are not created carefully. You’ll usually have little trouble telling when multivariate information is mapped, because the symbols appear more complex than those that show a single attribute. In the second method of symbolizing multivariate information, mapmakers show a concept as defined by a composite of attributes. At first glance, some qualitative thematic maps may seem to show a single theme, when they are actually showing multivariate information. This use of multivariate information is the case with the ecoregion map in figure 7.13. An ecoregion is defined by temperature and precipitation ranges, as well as by typical vegetation, soils, geology, human influences on the landscape, and more. You may think of ecoregions as a single theme, but each defining component plays a crucial part in their makeup. Similarly, the concept of a soil class is defined by a set of soil attributes, including slope, depth, drainage, color, and texture. The legend for the map in figure 7.14 gives you a clue about this multivariate information because it indicates both soil type and slope class. You can probably think of a number of other examples of environmental phenomena that are determined by a composite of attributes. Reading these kinds of maps can be tricky because the symbols on multivariate thematic maps may look exactly like the symbols on single-theme maps. Only the nature of the information symbolized, not the form of the symbols, is changed. Therefore, it is essential to check the legend as your initial step in map reading.

Figure 7.14. Classes on a soil map signify areas in which certain combinations of landscape attributes exist, such as soil type and slope class. Courtesy of the US Soil Conservation Service.

Multivariate point symbols Mapmakers like to use multivariate point symbols to show multivariate attributes of a point feature because they can pack information into each symbol by combining several visual variables. The resulting symbol is sometimes referred to as a glyph. Some multivariate point symbols are pictographic, but most are geometric. For example, the symbols for landfills and dumps shown in figure 7.15 are circles and squares. Arrows in two opposite directions indicate the status of operation, and different hues represent ownership.

Mapmakers sometimes use the same visual variable more than once to create multivariate symbols. For instance, they may use two shapes together, such as a star within a circle to show a city that is also a capital, which is a common way to show capital cities. More often, they will show two different attributes of a feature with two different qualitative visual variables, as in figure 7.15. Although mapmakers should use visual variables with qualitative connotations to construct these symbols, they don’t always do so. So a good rule of thumb is to check the legend to make sure you understand what the symbols mean.

Multivariate line symbols You probably won’t find many examples of maps that show multivariate qualitative linear data. One reason is that the options for linear visual variables are limited to using primarily color hue and line pattern. But occasionally you’ll find such maps. For example, road maps may be enhanced with multivariate data. It’s common to see US highway maps with red lines for freeways, black lines for state highways, and gray lines for US or state highways. Sometimes these single-variable line symbols are augmented to show a second variable. For instance, dot patterns may be added alongside scenic routes, which can comprise any of the various road types. You may also see multivariate line feature maps with line symbols that are wide enough to have different hues and line patterns within the two bounding lines. Cased line symbols have an interior line bounded by a casing shown in a different color. You can find these symbols on many kinds of maps. For example, on USGS topographic maps, solid or dashed cased lines show primary or secondary highways, unimproved roads, roads under construction, and road tunnels (figure 7.16).

Figure 7.15. A great deal of attribute information can be shown on a map through the use of multivariate point symbols.

Multivariate area symbols Mapmakers can make qualitative multivariate maps for area features in several ways. One common method is to overlap two types of area symbols that are appropriate for nominal data. The most common visual variables used are color hue for one attribute and pattern shape for the other attribute. For example, the map in figure 7.17 shows the type of ocean bottom off the Oregon coast by using light gray for sand, dark brown for bedrock, and light brown for silt and mud. Vertical and horizontal dashed-line patterns within the areas are overlaid to show where crab or shrimp are harvested. The area patterns for type of shellfish overlap to literally form plus signs to show areas in which both crab and shrimp are harvested. This map lets you see the geographic relationship between the type of ocean bottom and the kind of shellfish harvested there. On other maps, two different hues, such as yellow and cyan, are used for two different categories so that their area of overlap is seen as the hue combination—in this case, green (figure 7.18). However, the overlap area may go unrecognized, because green is normally seen as a separate hue, not as a mixture of yellow and cyan. Another problem with this approach occurs if two hues whose combination is not easily recognized are used. Can you predict what hue to use to show the overlap between an area symbolized with purple and

another shown in green? In such cases, it’s necessary to refer to the map legend to see what each hue, including the overlap hue, represents. An alternative is to use alternating bands of the two hues within the overlap area—in this example, stripes of purple and green. These symbols are more easily seen as areas in which both attributes are found.

Figure 7.16. Roads and related features legend for USGS topographic maps. Courtesy of the US Geological Survey.

Figure 7.17. Two different types of area symbols (color hue and pattern shape, in this example) can be overlaid to create a multivariate map that shows areas in which different categories overlap. From P. L. Jackson and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press.

Figure 7.18. Cyan and yellow are used for two different feature categories (states with one or more NFL or NBA teams) so that their area of overlap (states with teams in both leagues) is the combination color—green.

MAPPING QUALITATIVE CHANGE Portraying changes in the environment has long challenged mapmakers. Until recently, changing phenomena weren’t attractive candidates for mapmaking because the data is often difficult to find, the maps are more time consuming to create, and the maps are more challenging for the map reader to understand. These limitations will become more evident as we discuss the approaches used to map qualitative change. There are two types of qualitative-change maps: those that show change in the nominal-level attributes of features at a location over time, and those that show change in the location of a feature over time. For both types of change maps, several different mapping methods can be used. We start by looking at maps that show change in qualitative attributes over time.

Change in attributes over time Small multiples To show qualitative change over time, mapmakers often use a method called small multiples, also called constant-format displays, in which the same basemap is used in a series but the data shown on the basemaps changes (figure 7.19). This method can be

used to map change in both a nominal-level attribute over time and change in location over time. Small multiples, or small multiple maps, allow you to easily become familiar with the geographic region shown on the basemap, so you can focus on the changes in the data between time periods.

Figure 7.19. These small multiple maps are used to show qualitative attribute changes in the same location over time. Historic building use in Portland, Oregon, is shown using the same basemap, allowing you to focus on changes in the number of buildings and their use. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Small multiple maps require you to compare the distributions among the maps to see the change. To use these maps appropriately, you must first be able to decipher each map in the series, and then accurately interpret the changes among the maps. The opportunity for misinterpretation increases with every map added to the series and, of course, with the complexity of the maps. Change maps An alternative to small multiples is for the mapmaker to explicitly show the locations in which change occurs. Change maps show locations in which attributes have changed over time. For instance, the red areas in figure 7.20 represent land-cover change from nonurban to urban, from 1986 to 2002 in the Minneapolis–St. Paul, Minnesota, metropolitan

area. From this figure, it’s clear that most of the nonurban to urban change occurred on the fringe of the core urban area. Change maps can show changes for many categories simultaneously, in which case each category uses a separate map symbol. In our land-cover example, only one change is uniquely symbolized—change from nonurban to urban. The categories in the map can be expanded to include changes between other categories as well, such as forest to agriculture or forest to urban. Each of these change categories would then have a unique symbol (color hue, in this case). To help readers, these maps are often accompanied by before and after maps, or maps of separate categories. As with small multiples, appropriate use of these maps requires that each map be interpreted correctly.

Figure 7.20. This qualitative-change map depicts land-cover change from nonurban to urban between 1986 and 2002 in the Minneapolis–St. Paul metropolitan area. Courtesy of the University of Minnesota’s Remote Sensing and Geospatial Analysis Laboratory.

Figure 7.21. Dam construction over time is shown in this series of small multiples by superimposing the data for one time period over data for all previous periods. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Superimposed maps Mapmakers can also show qualitative change by laying one map over another to create a superimposed map. For example, a map of one time period may be laid over another of an earlier period. You can then see what changes have taken place during that time. The maps in figure 7.21 show dam construction over time. Although the data for each period can be shown using a different symbol, in this series only the most recent period is symbolized uniquely and all previous periods are aggregated. This series also makes use of small multiples. Superimposed maps require that the mapmaker use clear symbology so that the reader can understand where the symbols are superimposed. For a succession of time periods, you must be able to see each discrete period as well as all the periods combined to accurately study the long-term historical pattern of change on the map.

Change in location over time The maps in figures 7.19, 7.20, and 7.21 are all examples of how change in attributes over time can be displayed. The same methods can be used to show change in location over time—the second type of qualitative-change map. Because point, line, and area features vary in the way they change location over time, we consider them separately. Point symbols Movements of point features are often mapped as events or stops along a route. An example is a map of the location of the eye of a hurricane over the course of a storm. The events or stopping points can be annotated with explanatory labels and other indicators of the change in location. For instance, figure 7.22 shows the westward route taken in 1805 by Lewis and Clark through Oregon and Washington on their historic journey to the Pacific Coast. Their westward movement is shown by a series of point symbols to indicate their campsites beside the Snake and Columbia Rivers, from October 10 to November 10. The map appears to give a full picture of their rate of travel, but only 24 campsite symbols are on the map. Because no information is given about sites in which they camped more than one night, your understanding of their actual westward rate of movement is incomplete.

Figure 7.22. This map shows the westward route taken by the Lewis and Clark expedition through Oregon and Washington, indicating their campsites from October 10 to November 10, 1805.

Line symbols There are several ways to show the movement of qualitative line features. The simplest method is to create a set of small multiple maps that show the positions of the features on different dates. You may have seen military maps that show the daily positions of battle fronts or troop lines during the course of battle. One example shows, with blue and red lines, the combined daily position of Union and Confederate troops, equipment, and control during the Civil War Battle of Gettysburg, on July 1–3, 1863 (figure 7.23). This map series shows more than the advance or retreat of the same battalions over the three days—new troops that arrived and reserves that were called into battle are also mapped. As you have seen in the introduction (figure I.8), modern animated and interactive story maps show these troop movements in a visually compelling manner that deepens your understanding of this famous battle.

Figure 7.23. The positions of troops during the Battle of Gettysburg are shown with blue and red lines on these three maps (based on a National Park Service brochure

for the Gettysburg National Military Park). Courtesy of the National Park Service.

Figure 7.24. This map shows superimposed qualitative line features (the jet stream on different days) to give you a better understanding of a weather forecast.

Another way to show change in the position of linear features over time is to superimpose the line features from different periods on a single map. For example, your TV weather channel may use a map that shows the current and forecast position of the jet stream (figure 7.24) to help explain changing weather conditions.

Figure 7.25. The change over time of an areal feature can be shown as an expanding or contracting front. This map depicts the spread of gypsy moths in the northeastern United States at various points in time, from 1890 to 1971.

Figure 7.26. This map shows the change in location over time of the ash plume from the 1980 Mount Saint Helens eruption. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Area symbols Change over time in the location of qualitative area features can also be mapped. Spatial diffusion is the term for changes over time and location—it is the transfer or movement of things, ideas, and people from place to place. Figure 7.25, for example, shows the diffusion of gypsy moths in the northeastern United States, from 1890 to 1971. The movement of qualitative areal features generally occurs at their edges along a linear front, the foremost line of advance. In figure 7.26, the ash plume from the 1980 Mount

Saint Helens eruption in Washington state is shown as a series of isochrones (lines of equal time difference). The area of ash in the upper atmosphere at a given time is the same as the area within the single isochrone for that time period. Most of the advance is at the front edge of the isochrones. It is also possible to map qualitative area change on a change map. The change map in figure 7.20 is a good example that shows the diffusion of urban land cover over a 16-year period.

Figure 7.27. Scan the QR code or go to http://invasionofamerica.ehistory.org/#0 to interact with this dynamic map, which shows how the United States took over an eighth of the world across two centuries. Courtesy of Claudio Saunt, ehistory.org at the University of Georgia.

DYNAMIC QUALITATIVE-CHANGE MAPS One disadvantage to all the time-related mapping approaches in the previous section is that the maps are static (lacking in movement, action, or change), while the phenomena they are designed to show are dynamic (characterized by change, activity, or progress). A better solution is to display the phenomena in a dynamic fashion so that you can intuitively understand that what you are seeing changes over time. Dynamic maps are animated to create the illusion of movement or change. Computers simplify the creation of dynamic maps that deal explicitly with change and time. For example, it is now common to show the continual movement of the jet stream and other weather elements in an animated sequence. You may find dynamic maps challenging to read if individual scenes are not displayed slowly enough or if they are not repeated often enough. The optimal situation is for you to have interactive control over the display of the animation, which is not always the case. These maps require a prominent indicator of the changing time period. This change in time can be shown as a time label, a timeline, a clock, a calendar, or through other clever methods. But if you do not understand the associated passage of time, the map can be

misinterpreted. For example, explore the interactive animated map, shown in figure 7.27 (go to the URL in the caption or scan the QR code), of “The Invasion of America,” between 1776 and 1887, when the United States seized over 1.5 billion acres from America’s indigenous people by treaty and executive order. The map shows, in one-year time steps, every treaty and executive order that gained territory for the United States. It also shows the creation of present-day federal Indian reservations. In the online map, you can click the play button to the left of the timeline at the bottom to start the animation, or you can move the slider along the timeline to see the status at the beginning of any year—the map in figure 17.27 currently shows January 1, 1858.

SELECTED READINGS Bertin, J. 1981. Graphics and Graphic Information-Processing. New York: Walter de Gruyter. Carnachan, R. 1993. Wisconsin Soil Mapping Vol. 4. Madison, WI: Wisconsin State Cartographer’s Office. Chaston, P. R. 1995. Weather Maps: How to Read and Interpret All Basic Weather Charts. Kearney, MO: Chaston Scientific. Dent, B. D. 1996. Cartography: Thematic Map Design, 4th ed. Englewood Cliffs, NJ: Prentice-Hall. Hole, F. D., and J. B. Campbell. 1985. Soil Landscape Analysis. Totowa, NJ: Rowman & Allanheld. Holmes, N. 1991. Pictorial Maps. New York: Watson-Guptill. Jackson, P. L., and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press. Loy, W. G., S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press. Monmonier, M., and G. A. Schnell. 1988. Map Appreciation. Englewood Cliffs, NJ: Prentice Hall. Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. Elements of Cartography, 6th ed. New York, NY: John Wiley & Sons. Robinson, V., ed. 1996. Geography and Migration. Brookfield, VT: Edward Elgar. Saint-Martin, F. 1990. Semiotics of Visual Language. Bloomington, IN: University Press. Stevens, S. S. 1946. “On the Theory of Scales of Measurement.” Science 103, no. 2684 (June 7): 677– 80. Tufte, E. R. 1997. Visual Explanations: Images and Quantities, Evidence and Narrative. Cheshire, CN: Graphics Press. Wrigley, N. 1985. Categorical Data Analysis for Geographers and Environmental Scientists. New York, NY: Longman.

chapter eight QUANTITATIVE THEMATIC MAPS QUANTITATIVE INFORMATION Quantitative-data collection Counts and amounts Spatial samples Census data Continuous surfaces Quantitative-data accuracy Data normalization Data transformation Data classification Class interval selection Number of classes SYMBOLIZATION OF QUANTITATIVE DATA Visual variables Levels of measurement SINGLE-THEME MAPS Point feature maps Proportional or graduated symbols Line feature maps Flow maps Area feature maps Choropleth maps Dasymetric maps Area feature point symbols Cartograms Prism maps Continuous-surface maps MULTIVARIATE MAPS Multivariate-symbol maps Composite variable maps Combined mapping methods MAPPING QUANTITATIVE CHANGE Small multiples Complementary-formats display Change maps Time composite maps Attribute change maps

INTERACTIVE DYNAMIC QUANTITATIVECHANGE MAPS SELECTED READINGS

8 Quantitative thematic maps In chapter 7, we discuss qualitative thematic maps, which emphasize the location of different kinds of environmental features. Sometimes you want to know not only what and where, but also how much of something exists at a location. In such cases, you turn to quantitative thematic maps—maps that show a single theme or a few related themes of quantitative information (information that carries a magnitude message). We explore these maps in this chapter.

QUANTITATIVE INFORMATION In chapter 7, you saw the methods that mapmakers use for thematic maps of qualitative information that has classes that vary in type but not quantity. In this chapter, we focus on thematic maps for quantitative information—numerical data that represents an amount, magnitude, or intensity. First, we look at how quantitative data is collected. Then, we examine the ways that quantitative data is normalized, used to derive new values, and classified. Next, we examine the ways that mapmakers show a single quantitative theme about point, line, and area features. We also show examples of how mapmakers create multivariate thematic maps that show the geographic relationships between two or more themes. Finally, we explore quantitative thematic maps that focus on changes in feature locations and attributes over time.

Quantitative-data collection Data collectors have several choices when gathering quantitative information. They can take physical measurements to describe a theme at different locations within a region. When they collect data for every feature in a theme, they obtain what is called a population count. The population count can then be mapped, such as in figure 8.1 in which point symbols are used to represent each of the ski areas in Oregon. The ski areas are symbolized to show the total number of day visits from 1999 to 2000. It is possible to collect the data for each point feature of interest when the number of features is fairly small or the data can be easily collected for the entire population. The map of ski areas in figure 8.1 is a good example of data collection for a manageable number of features. An alternative is to collect data for a representative portion of the region or population to identify patterns or trends in the larger dataset, which is called taking a sample. A map of household income in a set of cities can be made from surveys that are collected from each individual household. The entire population of households may have been surveyed, although more likely the map is made from a sample of households in each city. The key difference between a population count and a sample is that your goal with a population count is identification of the characteristics of the population, whereas the goal with a sample is to make inferences about the characteristics of the population from which the sample was drawn. We talk more about spatial samples later in the chapter. Quantitative thematic maps can be created from all or part of the measurements taken for population counts or sample data. This choice of what data to use is called cartographic selection, as you learned in chapter 6. The map in figure 8.2 of the types of high-technology companies in the Portland, Oregon, metropolitan area is an example of cartographic selection—it only shows companies with at least 300 employees, so some of the smaller high-tech companies are not included on the map.

Figure 8.1. This map shows the number of day visits for every ski area in Oregon.

Quantitative information can also be gathered though a census, which is also a method used to collect qualitative data, as you saw in chapter 7. Now we look at each of these data collection methods in turn. Counts and amounts Maps are made from a wide variety of counts and measurements. Counts are the total number of features, and measurements or amounts are the quantities associated with features. Counts and amounts can be collected for point, line, and area features. First, we look at data that describes individual point features. Figure 8.2 provides a good example of count data for point features. The features are the high-tech companies, and the counts are the number of employees. For the largest companies, the number of employees is shown right on the map. For smaller companies, the number in the point symbol refers to the table at the left, in which the number of employees is listed. As with point features, maps can also be made for counts or amounts that relate to line features. Counts or measurements within sections of a line feature are common. An example for counts is the highway traffic map in figure 8.3, in which the traffic volume is recorded for each section of the highway. Abrupt breaks in the highway symbol on the map, at the points where road sections meet, are your clue that each line segment is symbolized to show the number of vehicles that pass per section of road. The counts are taken from automatic traffic recorders (ATRs) at ATR stations permanently installed throughout the road network. They record the number of vehicles that pass each location and transmit

the data via telemetry (commonly through wireless data transfer, but also using radio, telephone, or wire communications) to computers at agency headquarters for immediate data processing and mapmaking. Traffic flow maps are created from counts taken at the ATR sampling stations for the entire network to show the traffic volumes for the region. Near continuous transmission of ATR data allows new maps to be created, in near real time (at the same rate, and sometimes at the same time, as things happen). An example of an amount collected within sections of a line feature is an estimate of the flow of water in streams and rivers. Data is collected in continuous-flow monitoring stations commonly called streamgages (or gaging stations). These locations are used by hydrologists and environmental scientists to monitor bodies of water on land. Most streamgages operate by measuring the height of the water in the river or stream, and then converting the water height (called stage) to a streamflow (or discharge) by using a formula that relates the height to a set of actual discharge measurements. Data is collected at USGS streamgages at 15- or 60-minute intervals, and then transmitted to USGS offices every one to four hours. When there are critical events, such as heavy rainfall or flooding, the data may be recorded and transmitted more frequently. As with traffic data, transfer is done by telemetry and can be analyzed and mapped within minutes of arrival at agency offices. Streamgage data is used to assign streamflow values to stream line features in a GIS. For example, the USGS compiled the National Hydrography Dataset (NHD), which represents the nation’s drainage network including features such as rivers, streams, canals, lakes, ponds, coastline, and dams. The US Environmental Protection Agency (EPA) enhanced the NHD dataset to create the NHDPlus dataset. NHDPlus integrates the line features in the NHD stream network with gridded land elevation data. This integration allows the catchment (contributing areas) for each stream segment to be computed. The catchment is then used to associate precipitation, temperature, and runoff data with each stream segment to estimate streamflow. Elevations along each stream segment are used to estimate velocities. In addition to streamflow and velocity, NHDPlus provides additional attributes, including stream order (see chapter 17) and a group of attributes that facilitate rapid stream network traversal and query.

Figure 8.2. Only high-tech companies with at least 300 employees are shown on this map of the Portland, Oregon, metropolitan area. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Figure 8.3. This traffic volume map shows the number of vehicles that were recorded for sections of roads in the Portland, Oregon, area. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

The map in figure 8.4 uses this data to display the mean streamflow for Oregon’s rivers. Area feature maps can also show counts or amounts. When you’re more interested in the spatial distribution of features than the location of individual features, a map that shows counts of features within areas will best suit your needs. For these types of maps, it’s important to understand the data collection units in which the counts or amounts are taken. Data collection units There are many kinds of data collection units—these natural or human-defined units are used to divide the entire study area for the purposes of data collection. We start by looking at human-defined data collection units. The data collection unit for a census, which, as you saw in chapter 7, is a count of individuals in a population, is called an enumeration district. This district is the geographic area assigned to a census taker, usually representing a specific portion of a city or county. Census data is discussed in the next section of this chapter. Other examples of human-defined areas that are commonly used for data collection are ZIP Codes, school districts, and health service areas (HSAs), which are single counties or clusters of contiguous counties that are relatively self-contained with respect to hospital care. Humandefined data collection units are usually irregular in shape and size, and maps made from count data inherently show the irregularities among areas.

Figure 8.4. Streamflow measured at selected sampling stations along the rivers was used to make this map of mean streamflow for rivers in the state of Oregon. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

A special case of a human-defined data collection unit is binning. In binning, population counts are based on a grid of identical square or hexagonal cells that cover a region. Data binning is often used to place point data into square or hexagonal cells so that the value of each cell represents the count of points that fall within it. If the mapmaker uses hexagonalcell data binning, they are called hexmaps. A variety of biological phenomena are counted within grid cells through binning. Maps of biomass, animal density, insect infestation intensity, and many other themes are made from this biological data. Maps can also be made for counts and amounts of natural data collection areas. These environmental area features include watersheds, lakes, stream reaches, and areas that have a uniform environmental characteristic, such as ecoregions, climate zones, and soil units. For example, limnologists (scientists who study lakes and streams) may take a large number of surface temperature measurements at different sample points on a lake, from which they compute an average temperature. Repeating this process for all the lakes in the region, they can build a database for the lake feature and attribute (temperature) data. The average temperature of the lakes in the region at a particular time can also be obtained from a thermal infrared image by finding the average value of all grid cells in the image that fall within each lake’s boundary. A cartographer can then create a map that shows the average temperatures of the lakes within the region from either of these datasets.

Spatial samples As mentioned earlier, quantitative thematic maps are often based on a spatial sample of features rather than a full population count. The idea behind sampling is to use a small part of the population to find out what you want to know about the entire population. Samples are used for several reasons to estimate what the entire population is like. The time and cost involved in obtaining a full population count may be prohibitive, so only a sample is feasible. Samples are also used for continuous phenomena—that is, phenomena that exist everywhere and have values that change gradually over space or from moment to moment. Examples include temperature and precipitation. It is impossible to measure these values constantly at every point on the earth, but a spatial sample of temperatures and rainfall obtained from weather stations can be used to estimate the temperature or precipitation at any location over a selected period. Maps are made from several types of spatial samples (figure 8.5). Data for a map that shows average temperature of this lake and others in the region may have been collected using a basic sampling method (the means of obtaining the sample from the total population). The simplest approach is to travel by boat to a number of geographic positions and measure the temperature at each position, which is called a spatial point sample. A second method is to have the boat slowly follow a transect line and obtain measurements along the line, either continuously or at a constant time interval. This method is a common way to collect bathymetric data for the depth of a lake. The third method is to navigate the boat to quadrat sampling areas predefined on a chart of the lake, using quadrats (rectangular sampling units) to do the sampling. An easy way to create a tool to define the quadrat area is to use plastic PVC pipe parts (figure 8.6). This area can then be subdivided into smaller quadrat areas using string or wire. With careful navigation, temperature readings can be taken within each quadrat as the quadrat tool is laid over the lake surface.

Figure 8.5. Different spatial sampling methods were used to find the average temperature of this lake. Sampling points, transect lines, and quadrats were randomly and systematically located across the lake, and a point sample stratified by water depth was also taken.

The arrangement of sample points, transect lines, and quadrats can be random or systematic, as shown in figure 8.5. In random samples, the locations are selected randomly so that all locations are given an equal probability of being included for data sampling. In contrast, systematic samples are arranged to collect data at regularly spaced distance intervals. In this approach, sampling is often done in a rectangular or triangular grid of sample points, along equally spaced parallel transect lines, or at equally spaced quadrats. In a stratified sample, known characteristics of a feature distribution are used to guide data collection. This type of sample is obtained by first subdividing the feature or region into zones, or strata, that possess distinctive traits. Random or systematic samples are then taken to reflect the importance of each stratum. In our lake example, the randomly selected sample points are stratified by water depth, because the shallow and deep portions of the lake are equally important to estimating the average temperature. The same number of sample points are randomly placed in the deep and shallow sections of the lake, reflecting their equal importance. Statisticians may argue about which form of sampling gives the best data for estimating a true average temperature, but one thing is certain—a higher density of randomly or systematically placed sample points, transect lines, or quadrats should provide a better estimate, particularly if the sample is stratified in an appropriate manner.

Figure 8.6. An easy way to create a tool to define a quadrat area is to use plastic PVC pipe parts that can be subdivided into smaller quadrat areas using string or wire. Courtesy of the US Geological Survey.

Figure 8.7. Population counts on this map are shown for each county in Oregon. The values shown are in thousands, and the height of each bar is proportional to the county population in the year 2010.

Census data Federal, state, and local government agencies conduct censuses to learn basic population characteristics. Private companies focus their census taking on product advertising and marketing. Census maps are produced to make it easier for you to visualize the spatial pattern of high and low counts. Such census maps show how many features were counted in each data collection area, but not where each feature was located at the time of the census. Figure 8.7 is an example of population counts within the counties of Oregon. Maps are made from a variety of census data. US Census Bureau data collectors enumerate (count) the number of people within households, from which the total population count within different data collection areas (census blocks, census block groups, census tracts, cities, counties, and states) can be determined by simple summation. Continuous surfaces Data can also be collected for continuous surfaces—surfaces that represent geographic phenomena that exist over the entire region and whose values are gradually

changing. These surfaces can be generated from remote-sensing data or from data that is collected through direct measurement at a number of sample points. You can think of average annual precipitation across your state, as in the map in figure 8.8, as a continuous surface, because annual rainfall can be computed for every location in the state, and it varies gradually from place to place. The data to make this map can be averaged from meteorological satellite imagery taken daily. But more often, an average precipitation map is made by interpolating between values for the weather stations across the state. Each weather station, of course, is a sample point from which the continuous surface is interpolated.

Quantitative-data accuracy It seems logical that a map that shows the quantitative attributes for a theme would be accurate. This assumption is true in most cases, but not always. A number of errors may reduce the quality of the data. The errors can come from instrumental, methodological, and human deficiencies during data gathering. Such errors are usually well disguised on maps, sometimes on purpose and sometimes inadvertently. Consider the US Census of Population and Housing, which is conducted for the United States each decade by the Census Bureau. Because every household in the country is supposed to be surveyed, maps produced from the counts are flawless . . . or are they? A full head count of over 300 million people is an immense job. So we look at how the data is gathered. Once every 10 years, the Census Bureau gathers a nationwide team from the ranks of the unemployed to track down nonresponders to mailed questionnaires. These census takers are asked to put their hearts into a low-paying job that lasts only a few weeks. They must brave strange neighborhoods, sometimes repeatedly. The people to be interviewed often aren’t home, and when they are home, they may be hostile or their dog might be. The people interviewed are supposed to respond truthfully, even though this candor means telling secrets that can get them into trouble with their landlords, the welfare office, or local authorities. Finally, after all the information is gathered, it still must be processed, analyzed, and mapped.

Figure 8.8. This map of annual precipitation was created using Oregon State University PRISM Climate Group data to produce a continuous surface from digital grid estimates of yearly averages. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

You can see the potential for all three types of errors: instrumental (questionnaire), methodological (interview procedure), and human (between the interviewer and the respondent, as well as the people doing the data processing). Yet the Census Bureau has maps and a website that, at any time, will give you the “actual” number of people in the United States. Be your own judge of how accurate this population count is. Physical measurements are subject to these same sources of error. Imagine determining the number of fish in a stream by netting at several locations. The net may not be fine enough to capture all species (instrumental), the sample locations may not be representative of the entire stream (methodological), and the person doing the netting may be too distracted or lazy to perform the job thoroughly (human deficiency).

Data normalization Physical measurements are an example of raw data—data in its most basic form that is not manipulated in any manner. Derived measures such as averages can then be computed from this raw data. For some maps, it is important that the raw data is normalized

by computing an appropriate derived measure. For example, on each map in figure 8.9, population density (people per square mile), and not total population, is being mapped. The mapmaker has normalized the raw data for the area of each county. Through normalization, the population appears uniform throughout the county—a constant number of people per square mile. You’ll find densities, percentages, rates (such as incidence of disease per 10,000 people), and similar derived measures on quantitative thematic maps. If you find counts such as total county population shown in a choropleth map, which we talk about later in this chapter, the cartographer has made a serious mistake in not normalizing the raw data for the differences in area among counties.

Data transformation If the quantitative information that is mapped ranges greatly in magnitude, mapmakers may have transformed the raw data values to a more convenient mathematical form. For example, if the population of the United States by state was mapped as raw data, the few very populous states are so much greater in magnitude than the majority of the lowpopulation states that the map would show little magnitude variation between most states. But by mapping the square root of the state population, differences among the lowpopulation states are exaggerated. Considering the impact of such data transformations on the appearance of a map, you should check the map legend and explanatory notes to see if the raw data values have been transformed by the mapmaker and, if so, how.

Data classification Quantitative information on maps is often classed. Classification is the ordering, scaling, or grouping of data into classes that simplify features and their attributes. For quantitative maps, there are several commonly used classification methods (or classing methods) —procedures used to assign class intervals to numerical distributions. The selection of class intervals, or ranges, and the number of classes both have a significant impact on the appearance of the quantitative information on a map. To explain and illustrate this concept, we use examples of quantitative information that are mapped as choropleth maps, so that the area features are symbolized using variations in color lightness. We talk more about choropleth maps later in this chapter. Class interval selection The impression of magnitude variation that you get from the map depends on the method used to define the class intervals (the ranges for each numeric class). Now we look at the types of class intervals you are likely to see on choropleth and other types of quantitative thematic maps discussed in this chapter. Quantile intervals The three maps in figure 8.9 have two, four, and eight quantile (or equal frequency) intervals, in which the number of observations (objects or values in the dataset)—in this case, counties—in each class are as close as possible to equal. With quantiles, the same number of observations are assigned to each class. If the number of classes is four, the classes are quartiles; five classes are quintiles; and so on. The three maps give the impression of high population density throughout northwest Oregon and low population density in the southeast quarter of the state. Are these maps a realistic portrayal of the distribution of people in Oregon? To answer this question, you must look carefully at the

range for each class. Look again at the eight-class map. Values that range from 150 to 1,735 people per square mile may seem high because the average county density for the nation is a little under 100 people per square mile. The maps make much of Oregon appear too high in population density. The quantile classification method gives the most faithful portrayal of the data when the range of values for each class is approximately the same. That is, to have similar class ranges, the number of low, medium, and high values in the data must be about the same. Use quantile intervals to focus attention on relative rankings—the highest values, the next highest values, and so on. With quantiles, outliers (extreme high or low values) are not as obvious because they are grouped with other high or low values. Using quantiles is also the best way to compare maps that have different data ranges and values, because you will always be comparing high values to high values and low values to low values. Additionally, the concept of a quantile (for example, the top quarter of counties in terms of average family income) is easy to comprehend by most map readers.

Figure 8.9. For each of these choropleth maps of Oregon population density made with the quantile classification method, each class has the same number of counties. Maps made with two, four, and eight data classes show how the number of classes can change the appearance of the map.

Equal (range) intervals A second way for the mapmaker to group quantitative data is by equal-range intervals, commonly called equal intervals (figure 8.10, top). The range of data values is merely divided by the desired number of classes to obtain equal intervals. For example, dividing the

1,734 (1,735 − 0.7) Oregon population density range by 4 gives a constant interval of 435, or lower class limits of 0.7, 435, 870, and 1,305 for the four classes on the map.

Figure 8.10. Choropleth maps of Oregon population density that use equal intervals (top), natural breaks (middle), and the unclassed method (bottom) for county population densities. The class interval selection method that is used changes the appearance of the map.

Equal intervals are intuitively meaningful and easy to understand. Numerically constant intervals appeal to the same basic human data-handling mechanism that makes percentage figures so attractive. Our minds are comfortable with the idea of segmenting numbers into equal fractional parts.

If each class contains an approximately equal number of observations, equal intervals produce the most meaningful map. For equal intervals, data values must be equally distributed (approximately the same number of low, medium, and high values) within their numeric range. Thus, when the data values are equally distributed, choropleth maps made with quantiles and equal intervals should look identical. Use equal intervals to highlight differences in the values among areas. Areas with similar values will have similar symbols (for example, color lightness), no matter how many areas are in the class. With equal intervals, the distribution of values, including outliers, is easier to see. For many themes, the data values are unevenly distributed across their class range. Using Oregon population density as a typical example, many counties have a low population density and a few have a high population density. In this situation, equal-range intervals produce a strange-looking map. Most counties fall into one class, while some classes are empty, with no counties at all. Oregon population density appears to be uniformly low throughout the state, except for the two counties that contain Portland and much of its large suburban area. Although this map (figure 8.10, top) may be useful in showing the vast differences in county populations, it poorly communicates the actual variation in population density. Natural-break intervals Another classification method is to establish class limits at natural breaks in the distribution of data values to create natural-break intervals. The idea is to minimize the variation within classes and maximize the variation between classes. One way that mapmakers find natural breaks in a set of data is to create a frequency diagram (or histogram)—a graphic display that shows what the set of values in a dataset are and the number of features with those values. It can be enhanced to also show the proportion of features that fall into each category of data. An example of a frequency diagram is shown for Oregon population density in figure 8.11.

Figure 8.11. In this frequency diagram of Oregon population density, natural clustering in the data and breaks between clusters can be seen.

One way to establish the breaks is visually. The cartographer looks for clusters of data values with natural breaks between clusters and places limits between classes in the middle of the break between each cluster. Using this approach for the Oregon population density data in figure 8.11, natural breaks are placed at 60, 175, and 1,250 persons per square mile. The choropleth map made with these class limits (figure 8.10, middle) has the county that contains Portland (on the northern border of the state) in the highest class, three suburban counties in the next highest class, most of the Willamette Valley south of Portland in the third class, and the rest of the state in the lowest class. Another way to establish the class breaks with this method is to compute them. First, minimize each class’s average deviation by its mean, and then maximize each class’s deviation from the means of the other classes. This computation method, described by American cartographer and geography professor George Jenks, is often called the Jenks optimization method or Jenks natural-breaks classification method. Use the natural-breaks classification method when you want to distribute the error uniformly across all areas on the map. For example, the middle map in figure 8.10 shows the nature of population variation in Oregon better than the previous maps discussed in this section, which are based on quantiles and equal intervals. Critical-value intervals Mapmakers may also use critical values to determine class limits when the critical value has special relevance to the map’s theme. It may be a physical aspect of the theme, such as the temperature below which a crop freezes. It may be a politically defined dividing point, such as the income level at which counties fall below an arbitrarily defined “poverty” line and

are thus eligible for government assistance. It may also be a statistic computed from the data, such as the median or mean. If you define median population density for Oregon as a critical value, the map at the top of figure 8.9 would be based on this critical value. Number of classes The choropleth maps in figure 8.9 have two, four, and eight population density classes. You will likely see choropleth and other quantitative thematic maps having from two to eight classes for a theme. Progressive subdivision of the data range into classes simply involves reducing the numeric intervals between class limits. The simplest map has only two classes —above and below the median in our first example (figure 8.9, top). More information is shown if more classes are used. A map shows the most information if the population density for each county is in its own class. However, as you learned earlier, it is harder to interpret maps that have continuous (unclassed) symbols. You can see that quantitative thematic maps made with different classification methods and numbers of classes vary greatly in appearance, as illustrated in figures 8.9 and 8.10. The choices made by the mapmaker should be based on the nature of the mapped information, not on an arbitrary design decision. You may wonder how meaningful the map classes you see really are, particularly when the range of data values is divided into a small number of classes. Look again at the twoclass population density map in figure 8.9. The “above median” and “below median” classes may tell you all you need to know, but the information content of the map is minimal. When only a few classes are used on the map, there’s likely to be significant within-class variation that you cannot see.

Figure 8.12. The visual variables that inherently impart a magnitude message are size, pattern texture, gray tone or color lightness, and color saturation.

Mapmakers increase the information content by using more classes, but this solution creates its own problems. Each additional class makes the graphic portrayal of the data more complex, as the eight-class map in figure 8.9 illustrates. When many classes are used, the within-class variability is less, but the visual complexity of the map makes it more difficult to read. Many cartographers suggest that a choropleth map should have five to seven classes. On maps of continuous surfaces, as in figure 8.8, more data classes can be used because similar colors appear next to each other in a visual progression, making them easier to distinguish.

SYMBOLIZATION OF QUANTITATIVE DATA Visual variables On quantitative thematic maps, cartographers use a variety of symbols to depict information that tells you how many, large, wide, fast, high, or deep things are. The visual variables that inherently connote differences in magnitude are size, pattern texture, gray tone or color lightness, and color saturation (figure 8.12). A well-designed series of symbols using

variations in one or more of these visual variables will appear to you as a progression of magnitudes, from small to large or low to high. Many types of quantitative thematic information can be portrayed using these visual variables. Data for a single-theme map, such as population density for different cities or the precipitation recorded at various weather stations (figure 8.13), relates to one variable or attribute for the features mapped. You can also find multivariate maps that are a composite of two or more related themes. Figure 8.14 shows two themes that relate to earthquake loss for Oregon counties. The two themes are displayed using different visual variables. One variable, or attribute, of the data —the total estimated earthquake loss for each county—is shown by the height of a narrow bar, and another attribute—each county’s earthquake loss ratio—is shown by the lightness of a purple hue.

Levels of measurement The qualitative data for the maps described in chapter 7 is at the nominal level of measurement. Nominal-level data (also called categorical data) consists of categories that distinguish different types of features within a theme—but have no information about the relative magnitude, size, or importance of each category. In contrast to qualitative data, the data for quantitative thematic maps is at the ordinal, interval, or ratio level of measurement. From an analytic perspective, these measurement distinctions are important. Ordinal-level data is ranked according to a “low to high” or “less than to greater than” system. How much more or less one class is than another isn’t specified because there are no numeric values. Examples of ordinal data include short, medium, and tall trees, minor and major highways, and low- to high-density housing. Figure 8.15 shows sightings of baleen whales in the Gulf of Maine. The number of whales sighted per 1,000 kilometers of survey track (transect line) was converted into a continuous ordinal level of low to high sightings per unit effort scale.

Figure 8.13. This quantitative thematic map shows precipitation regimes for a set of weather stations in the state of Oregon. The size of the bar and the color lightness are used together to show how much rain is recorded for each month at each location From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Figure 8.14. This map shows two related quantitative themes—color lightness is used to show the earthquake loss ratio, and the size (height) of the bars is used to show the earthquake loss estimates in dollars for each Oregon county. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Figure 8.15. Sightings of baleen whales in the Gulf of Maine per unit effort. From Esri Conservation Map Book (Esri, 2011).

Figure 8.16. This portion of a map of landforms for Oregon demonstrates the use of interval-level data to show elevation. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Interval-level data consists of numeric values on a magnitude scale that has an arbitrary zero value, or zero point. Land elevations (figure 8.16) are an excellent example, because the zero datum is arbitrarily defined as mean sea level (see chapter 1 for more on the definition of mean sea level). To see how arbitrary mean sea level is, you have only to think about how sea level rises and falls—the zero datum shifts over time and doesn’t denote the absence of elevation. When looking at interval-level data on a map, you should realize that only the numeric intervals between classes are valid mathematically. The elevation classes in figure 8.16 are in 500-foot intervals. You can correctly conclude that the difference in elevation for each class is the same because the elevation interval is the same. However, it is incorrect to say that an elevation of 2,000 feet is twice as high as an elevation of 1,000 feet. To further understand this concept, imagine sea level rising 999 feet so that the 1,000-foot elevation is now 1 foot and the 2,000-foot elevation is 1,001 feet. The ratio of the elevation difference is now 1,001:1, but this ratio does not mean that the higher elevation is 1,001 times as high as the lower elevation. Ratio-level data also consists of numeric values on a magnitude scale, but, in contrast to interval-level data, the zero point isn’t arbitrary. Instead, the zero point denotes absence of the phenomenon. Themes such as population density (or any other density), annual precipitation, crime rate, tree heights, and temperature in Kelvin units have zero points that

denote total absence. For example, the 0% class in figure 8.17 indicates that several Oregon counties have zero people working in foreign-owned companies. In ratio-level data, both the numeric intervals and the ratios between values are mathematically correct; therefore, mathematical operations, such as addition and subtraction as well as multiplication and division, are valid. When you look at a map such as in figure 8.17, with classes such as 0.25–0.49, 0.50–0.74, and 0.75–0.99 for the percentage of total employment, you can conclude that the density range is the same for all classes. You can also assume that a percentage of 0.50 is twice that of 0.25 and a percentage of 0.75 is three times higher.

Figure 8.17. This map uses ratio-level data—the 0% class means a total absence of jobs.

SINGLE-THEME MAPS The simplest quantitative thematic maps show a single theme at the ordinal, interval, or ratio level of measurement. It is convenient to subdivide map themes into point, line, and area features to explore how cartographers have devised sets of quantitative point feature, line feature, and area feature symbols.

Point feature maps

To show quantitative information at specific points, mapmakers vary one or more visual variables to portray variations in the magnitudes of the attributes for the features. If they vary the symbol size in proportion to the magnitude for each feature on the map, the symbols are called either proportional or graduated symbols. Proportional or graduated symbols Proportional symbols are used to represent the exact data values of an attribute because the visual variable (size, in this case) of each symbol is scaled in proportion to its data value (figure 8.18, center). Difficulty arises with proportional symbols when there are too many similar values, and the differences between symbols may become indistinguishable. An alternative is graduated symbols, also called range-graded symbols, in which the quantitative values are grouped into classes, and all the features within a class are shown with a symbol of the same size (figure 8.18, right). Although you can’t tell the value of an individual feature, you can tell that its value is within a certain range. Range grading can be either numeric (1–10, 11–20, 21–30, and so on) or at the ordinal level (low, medium, high). Quantitative point symbols are usually geometric, such as circles, squares, or triangles. Circles are the most commonly used geometric point symbol because they are the symbol that has the most compact geometric form, they are easy to scale, and they have more visual stability (that is, they cause less eye wandering). However, you will also see these types of maps made with mimetic symbols—miniature caricatures of the features they represent (see chapter 7 for more on these types of symbols). Most commonly, the point symbols on these maps vary only in terms of size (figure 8.18, left). For example, if a human figure of one size indicates a city population of 100,000 people, the same figure eight times as large indicates a city with 800,000 residents. Color lightness, color saturation, and pattern texture are also used to show different magnitudes of the same attribute that is shown with size. For example, in figure 8.18 (right), size, color lightness, and color saturation are used together to show county seat populations in the San Francisco Bay Area with circles that are larger, darker, and more saturated in color. Sometimes you will also see color hue used to show a second, qualitative variable. Reading geometric symbols isn’t as straightforward as it might seem. The difficulty arises because of the way the human eye and brain work. The brain doesn’t perceive the size of geometric symbols in proportion to their mathematically computed size. Instead, the size of geometric symbols is progressively underestimated as the area or volume of the symbols increases. This discrepancy between the apparent and absolute size of map symbols is minimal in respect to symbol height, is worse in respect to area, and becomes a major problem in the use of three-dimensional symbols. We judge the magnitude of three-dimensional symbols by their area rather than their volume. Thus, the area that a sphere or cube covers on the map, not its volume, is what you’re likely to notice (for example, see figure 8.19).

Figure 8.18. Proportional and graduated point symbols are commonly used to show quantitative point information. Mimetic symbols (left) may look interesting, but simple geometric symbols (center and right) are usually easier to read. Graduated symbols (right) aggregate the data into a small number of classes to make the symbols easier to read.

Despite the tendency of the human eye to not correctly judge the magnitude meaning of 3D symbols, using 3D points on proportional- or graduated-symbol maps still has its advantages. First, we tend to find things that are in 3D more visually appealing and interesting, likely because that is how we see the world around us rather than as a flat, twodimensional picture. Second, 3D symbols, when scaled for volume—for example, spheres scaled by their volume, in equation (8.1):

where r is the radius—overcome problems of not having enough space to show all the symbols clearly. This lack of space can occur when the range of data values is large (three or more orders of magnitude). With 3D symbols, there are fewer large symbols, and thus, less symbol overlap. It’s most difficult to read proportional point symbols when a sequence of symbols that vary only slightly in area is used. The human eye simply doesn’t function precisely enough to differentiate between such slight variations in symbol size. These slight variations among symbols may actually contribute to map reading error because of the effort required to estimate the magnitudes of symbols by visually interpolating them from the labeled symbols in the map legend.

Difficulties in reading proportional point symbols are largely avoided when symbols are also labeled, as in figures 8.1 and 8.7. Limiting the sizes to a small number of classes also helps, so graduated symbols are sometimes the better choice for point symbol maps that use size to show magnitude values. In this approach, the symbols are usually different enough in size that the eye can easily tell them apart (figure 8.18, right). Although information is lost by reducing the magnitude data to a few classes, graduated-symbol maps are generally the easiest to read of the single-variable quantitative point symbol maps. Bounded continuous size and bounded continuous color Bounded continuous size and bounded continuous color, which combine the best of both graduated and proportional approaches, are the default methods when making quantitative point symbol maps in ArcGIS Online and Portal for ArcGIS. In the map in figure 8.20 that shows the world’s largest cities, the upper bound, as shown by the slider handle on the histogram, is set to 20 million people so that all cities larger than that are shown using the same-size circle. The lower bound is set to 4.5 million people. By default, the upper and lower handles are placed at one standard deviation around the mean. This method tends to do well at showing variation across the data while not being overly influenced by outliers. The default bounds can be changed by manually dragging the histogram handles to fine-tune the message of your map. Subtle patterns can emerge through adjustments that can be made within just a few seconds, allowing users to explore their data and find interesting patterns quickly and easily.

Figure 8.19. Graduated sphere map shows consumption in European countries. Esri © 2006.

2004

energy

production

and

Figure 8.20. ArcGIS Online (and Portal for ArcGIS) offers a solution for displaying quantitative data with bounded continuous-size symbols, such as this map of the world’s largest urban areas shown with proportional point symbols. Courtesy of Mark Harrower and Esri.

Figure 8.21. Average annual discharge for the major streams in Georgia is shown by varying the width of the lines.

Figure 8.22. This section of the Seattle traffic flow map from the Internet shows four ordinal levels of congestion, from stop and go in black to wide open using three hues that connote different speeds. Courtesy of Washington State Department of Transportation.

Line feature maps For line features, mapmakers show quantitative information associated with the segments of the lines. The visual variables that can be used to show quantitative information for lines include size, color lightness, color saturation, and pattern texture. The best visual variable to use is size (line width), which varies relative to the value being shown. Figure 8.3 provides an excellent example. On this map, the width of the line and the associated value give you an exact understanding of differences in traffic volume. As with point symbols, line symbols can be either proportional or graduated and have the same associated problems. However, with line features, proportional scaling is sometimes the better solution because the resulting symbol has fewer obvious breaks and therefore more closely approximates the feature it represents. The rivers in figures 8.4 and 8.21 are good examples, because rivers tend to vary in width gradually, not abruptly. In figure 8.21, the thinnest line in the legend indicates 0–1 million gallons of water per minute, the next thinnest line 1 million–2 million gallons per minute, and so on. You can see that there are lines on the map with widths that are not shown in the legend (for example, look at the thinnest lines on the map). The lines are proportionally scaled, which requires you to interpolate the magnitude for lines that do not have the same widths as those in the legend.

On these types of maps, the magnitude at any point along a line is difficult to read, because the human eye isn’t sharp enough to discriminate between slight changes in width. One solution is to use graduated line widths and a small number of classes. Another technique that mapmakers use is to vary the line pattern texture, color lightness, or color saturation within cased line symbols (see chapter 7 for more on cased line symbols). Sometimes, you will also see color hue used for quantitative line symbology. For example, the degree of traffic congestion on central Seattle freeways and major highways is illustrated in figure 8.22, with the green, yellow, and red hues that we associate with stop lights, plus black for stop-and-go traffic. Although the primary visual variable here is color hue (which is more appropriate for qualitative data), you automatically associate red, yellow, and green colors with ordinal-level traffic speed categories (red is no speed, yellow is low speed, and green is higher speed). You must be careful to note when color hue is used to show quantitative information, because the tendency of your eye is to relate this visual variable to categorical, not numerical, data.

Figure 8.23. This distributive flow map required the cartographer to decide where to draw the flow lines and how to symbolize them. On this map, the size of the line is proportional to the number of jobs that foreign-owned companies provided to Oregon. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Flow maps Flow maps show direction or movement along a line feature using a symbol called a flow line. Distributive-flow maps show the distribution of commodities or some other flow that diffuses from one or only a few origins to multiple destinations. The flow line’s width is made proportional to some magnitude, such as where the jobs that foreign-owned companies provide to Oregon come from (figure 8.23). To show the direction of flow, arrows can be added to one end of the flow line. On many flow maps, the mapmaker must decide where to place the lines so that they do not overlap other flow lines.

Network flow maps, such as the Portland, Oregon, traffic volume map in figure 8.3, show interconnectivity between places and are usually based on transportation or communication linkages. Radial-flow maps have a spokelike pattern because the features and places are mapped as nodes (line endpoints), with one place as a common origin or destination node. The radial-flow map in figure 8.24 illustrates travel by Facebook users: lines are plotted between Facebook check-ins (nodes) that are at least a certain distance apart.

Figure 8.24. Radial-flow map illustrates travel by Facebook users. Courtesy of Esri.

Vector flow maps, such as the map of significant wave height and direction in the North Pacific Ocean (figure 8.25), use a regular grid of vectors that are identical in shape but with arrows that vary in orientation to show the direction of significantly high ocean waves at a particular time.

Figure 8.25. Vector flow map shows significant wave height and direction in the North Pacific Ocean on May 21, 2013. Vectors show wave direction, and isolines of equal wave height are enhanced by a blue-to-brown layer tint progression. Courtesy of US Navy.

Mapmakers may decide to add a visual variable other than size to flow lines that are constant in width. Changes in amount of flow can be shown by varying the color value, color saturation, or pattern texture of the lines. Say that a mapmaker wants to show how many artichokes are being freighted along a railway line from Southern California to Chicago. One segment of the line can be shown three times darker than another to show that three times more artichokes are being shipped along it. The problem is that the lines must be wide enough to make these differences in lightness visually evident. When you read a flow map, focus your attention on the magnitude information, not the flow line’s precise location. Mapmakers often distort geography to accommodate flow lines. A flow line that shows the volume of ship traffic through the Strait of Gibraltar, for example, might be too wide to fit into the small space on the map between Spain and Morocco. Thus, the strait must be widened on the map; otherwise, it will look as if the ships are traveling over land.

Area feature maps

Quantitative area features vary in magnitude from place to place, sometimes in patchwork fashion. For these maps, data collection areas have a constant magnitude within, and abrupt changes at, their boundaries. Some map themes related to human activity, such as tax rates by state, have a pronounced “stepped” character. More commonly, the quantity within each data collection area is conceived of as being distributed homogeneously so that it can be mapped, when in reality, it varies within each data collection area. The map in figure 8.26 is an example—the phenomenon being mapped (land capability classes) is more variable than the data collection units suggest because of local soil differences and other factors. Population distribution is another example because population patterns are a reflection of environmental and social factors, not of data collection areas. Population is not uniform within units and doesn’t change abruptly at their boundaries, although the uncritical map user might get this idea from population density maps, such as those in figures 8.9 and 8.10. You have seen that to show changes in magnitude on area feature maps, mapmakers vary the size, color lightness, color saturation, or pattern texture of the symbol for the areas. Even more eye-catching maps are made by varying the height of individual areas, or changing the map area of data collection units to be proportional to their magnitude. The five common mapping methods for varying the visual variables for area features include choropleth, dasymetric, area feature point symbols, cartograms, and prism maps. Choropleth maps Many quantitative thematic maps of ordinal-, interval-, or ratio-level area data are made using the choropleth mapping method. The term comes from the Greek terms “choro” for area or region and “plethos” for multitude. In this method, each data collection area is given a particular color lightness, color saturation, or pattern texture depending on its magnitude. Areas are assumed to be homogeneous within, and abrupt changes can occur at, the boundaries between areas. Choropleth maps that show ratio-level Oregon county population density data are shown in figures 8.9 and 8.10. Unclassed choropleth maps On unclassed choropleth maps, each data collection area is given a color lightness, color saturation, or pattern texture according to its magnitude. These maps are not subject to the same challenges as those with data that is classified. For example, classification can mask important details by overgeneralizing, and many mapmakers aren’t sure how many classes to use or which classification method is best for their data. However, the problem with unclassed choropleth maps is that they can be useless if there are one or two extreme outliers, because everything looks the same except for the one or two very large or very small outlier values. As with proportional point or line maps, with unclassed choropleth maps, it can also be hard to match symbols on the map with those in the legend. The unclassed choropleth map of Oregon population density (figure 8.10, bottom) has a continuum of brown tones that range from white for the lowest density county to dark brown for the highest density. The intermediate population densities are linearly scaled between these two extremes. This map gives you an unbiased picture of Oregon population because the mapmaker has not generalized the data by placing the values into a small number of classes.

Figure 8.26. On this map showing land capability classes in the Pacific Northwest, the large homogeneous areas shown on the map as suited for cultivation are more variable on the ground because of local soil differences and other local factors. From P. L. Jackson and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press.

The difficulty with reading unclassed choropleth maps is that unless the values are more or less evenly distributed through the range of all values, the map will show most areas to be either high or low in value, with only a few areas at the opposite extremes. The Oregon map is a classic example, with one county solid brown, one county medium brown, and all others a very light brown. The map looks similar to the equal-interval map (figure 8.10, top), except that the discerning eye can see small variations among the light-brown tones. Dasymetric maps On dasymetric maps, the mapped areas aren’t political data collection units such as counties but rather areas of inherent homogeneity in the data. In dasymetric mapping, the mapmaker starts with a map of the collection units. Ancillary data, such as land cover, is used to help the cartographer adjust the boundaries of the units to more realistically represent homogeneous areas. Then the values are recomputed to reflect the addition or subtraction of values within the units. As you might imagine, having access to different layers of information in a GIS greatly aids dasymetric mapping. In the Delaware County, Pennsylvania, example in figure 8.27, the rural part of the county population is reassigned to the developed areas, and the densities are recomputed. As a result, the population density in the western region decreases in the agricultural and forest

areas, and in the eastern region the density decreases in the water and wetland areas. You can see that a dasymetric map with the homogeneous areas modified to reflect the data should give a more faithful representation of population density than a choropleth map that shows the density class for each county. The challenge for the mapmaker is to find the right ancillary data to use, and then correctly recompute the area values. Area feature point symbols One of the more confusing types of quantitative thematic maps is one that uses point symbols to represent quantities in areal data collection units. The cartographer usually places the point symbol in the center of each data collection area (at the centroid) and designs the point symbol using the same cartographic principles that apply to symbolizing point features (see single-theme point feature maps earlier in this chapter). For quantitative data, proportional or graduated symbols are used to symbolize the points. The problem is that you are led to think that you’re looking at data for a point feature when the data is for an area. As you saw in chapter 7, one of the graphic functions of points is to represent twodimensional area features that are conceived of as points for mapping purposes.

Figure 8.27. Dasymetric mapping is used to determine a more realistic representation of the population density in Delaware County, Pennsylvania (upper left). Remotely sensed data (upper right) is used to redistribute the population (lower left) to areas in which the people are more likely to be living. Then the densities are recomputed, and the new areas are symbolized on the dasymetric map at the lower right. From Jeremy Mennis and Torrin Hultgren. 2005. “Dasymetric Mapping for Disaggregating Coarse-Resolution Population.” In Proceedings of the 22nd International Cartographic Conference, July 11–16, A Coruña, Spain.

Point symbols are easiest to read when data collection areas are approximately equal in size on the map and when the range of data values is small. A map of California county population (figure 8.28) illustrates map-reading difficulties when both the data range and range of county areas are large. The map shows total county population, with proportional squares placed at the center of each county. The large population data range (from 1,500 to 9,800,000) results in a set of squares that range from barely visible to those that cover several counties. Many of the squares extend past their county’s boundaries and overlap with the squares for neighboring counties. Reading the map is particularly difficult in the San Francisco Bay Area on the central West Coast and Los Angeles regions on the southern

West Coast, where small counties and large populations result in numerous symbol overlaps and county boundaries that are hidden from view.

Figure 8.28. California population data from Census 2010 is aggregated by county, with proportional squares placed at the center of each county to show the total county population.

Cartograms You may have seen maps that show an eye-catching portrayal of the relative magnitudes of area features rather than their exact spatial locations. At these times, you were most likely

looking at a sometimes odd-looking map called a cartogram or area cartogram. Mapmakers create cartograms by distorting the geographic size of data collection areas in proportion to their magnitudes. The size of each state or county, for example, might be made proportional to its population rather than its geographic area (figure 8.29). Although these maps look a little strange, they help overcome the problem of choropleth and area feature point symbol maps when the size of the data collection areas varies widely.

Figure 8.29. Noncontiguous (upper right), pseudocontiguous (lower left), and contiguous (lower right) cartograms for California county population, based on Census 2000 data. Courtesy of NCGIA Cartogram Central.

To use a cartogram effectively, you must be able to compare the sizes of data collection areas on the cartogram with the same areas shown on a map with an equal-area map projection (figure 8.29, upper left). If the positions, shapes, and relative sizes of data collection areas on the map aren’t familiar to you, a cartogram can be difficult to use because you must compare the unfamiliar map with your mental image of the geographically correct map. For this reason, cartograms are most successful when they are used to map data collection areas that are familiar to the map reader, such as countries, states, and counties. Cartograms are also easier to read when there is a conventional map displayed along with the cartogram. One of the advantages of cartograms is that although the size of the data collection areas carries information about one attribute of the features, a different visual variable can be used as the area fill, thus displaying a second attribute. The second attribute can be either quantitative or qualitative. Of course, this additional information increases the complexity of the map and therefore the potential for misinterpretation. When creating cartograms, mapmakers often try to retain many spatial characteristics of conventional maps. These characteristics include shape, proximity, and contiguity. Preserving the shape of data collection areas makes it easier to relate a cartogram to a conventional map. Preserving the proximity (nearness) and contiguity (boundary connectedness) to neighboring areas also make the maps easier to read. But shapes cannot be preserved without altering the proximity and contiguity of neighboring areas, and vice versa, so either shape or proximity/contiguity is compromised in a cartogram. Now we look at these differences. Noncontiguous cartograms In noncontiguous cartograms, the shape of areas is maintained at the expense of proximity and contiguity of neighboring areas. These cartograms are created by increasing the size of the data collection unit relative to the magnitude of its attribute value, without regard to connectivity with adjacent areas. Noncontiguous cartograms are the most commonly produced because they are the easiest to make. To create a noncontiguous cartogram for the population of California counties (figure 8.29, upper right), the mapmaker enlarged or reduced each county in proportion to its 2000 population. The transformed data collection areas were then placed as closely as possible to their geographic positions on the map by positioning the centroid of the area on the cartogram in relation to the centroid of the corresponding area on the conventional map. Where areas overlap on the cartogram, the areas are shifted slightly to eliminate the overlap while retaining their correct position relative to the neighboring areas. Preserving the shapes of data collection areas makes it easy to recognize and compare them with their counterparts on a conventional map. But as figure 8.29 also shows, proximity relations are only roughly maintained, and contiguity is completely sacrificed. Pseudocontiguous cartograms Pseudocontiguous cartograms appear contiguous at first glance, but on closer inspection turn out to give only the illusion of contiguity (figure 8.29, lower left). The transformed data collection areas share common boundaries, but the boundaries aren’t the same as on a conventional map of the region. A pseudocontiguous cartogram is made by transforming each data collection area into a simple geometric shape that is proportional in size to the magnitude being shown. Rectangles are the shapes most commonly used. The new shapes are then arranged in

what resembles their relative geographic position. When rectangles are used, the result is called a rectangular cartogram. By sacrificing shape and proximity, these cartograms can maintain a considerable degree of contiguity. Pseudocontiguous cartograms may actually be no better than noncontiguous cartograms. They may even be worse, because some map readers may believe them to be truly contiguous when, in fact, they are not. The popularity of pseudocontiguous cartograms is probably better explained not by any map-reading advantage they might possess, but rather that they are easy to construct because only a simple drawing program is needed (and some examples are generated with spreadsheet applications, such as Microsoft Excel). Contiguous cartograms Contiguous cartograms are by far the most interesting to look at because they maintain the proximity and contiguity of neighboring areas, although sometimes at the expense of extreme shape distortion. If drawn by hand, the shapes of regions are distorted in a subjective, seemingly uncontrolled manner. Therefore, contiguous cartograms that are created in this manner by different mapmakers can look dissimilar even with the same theme. In the contiguous cartogram of California county populations in figure 8.29, lower right, although the shapes of some counties are fairly well preserved, other counties don’t look like themselves at all. Yet despite this shape distortion, the cartogram is effective—at only a glance, you can see precisely which counties have the greatest population. The variable quality of shape preservation from one region to the next isn’t always a major distraction. We appear able to accept a fair degree of shape distortion before areas become unrecognizable.

Figure 8.30. Three-dimensional prism map of California population by county. The height of each county area is proportional to its total 2010 population.

The biggest problem with contiguous cartograms is that, up until now, they have been difficult and time consuming to make. To draw even a simple cartogram well requires a vast amount of labor, not to mention artistic talent. As a result, cartograms have been regarded more as a novelty than as valuable tools for environmental understanding. But both problems—uncontrolled geometric distortion and difficulty of construction—are solved by computer mapping software. It now takes little effort or artistic talent to create contiguous cartograms such as the one for California population in figure 8.29. Although county shapes are distorted, the distortion is done in a recognizable way. With the increasing use of computer mapping software, you are likely to see more cartograms of all types. In deciding whether to use a cartogram or a conventional choropleth or point symbol map of a theme, you must decide whether you want to emphasize the region’s geography or the magnitude of the attribute. With conventional quantitative maps, you retain the recognizable geography of the data collection units but face the problem of decoding magnitude information from map symbols. With cartograms, physical space is distorted so you must rely on your own familiarity with the geography of the region, but the magnitudes are clear. The optimal solution, of course, is to use an equalarea basemap, such as in figure 8.29, upper left, as a reference for understanding the distorted geography seen on the cartogram. Prism maps A prism map, also called a stepped-surface map, shows the magnitude of an attribute by varying the heights of the data collection units. Such a map for California population by county is shown in figure 8.30. Each county area is raised vertically above the base level to a height that is proportional to its total population. The resulting map looks like a 3D stepped surface. Prism maps are visually impressive, but they have several problems. One drawback is that, although highs and lows are apparent, the exact height of a given area is difficult to determine. Another potential drawback is that they are drawn in oblique perspective. So the vantage point taken when the map is made is crucial to its appearance. Yet a poor choice of viewpoint may cause important data collection areas to be obscured from view. For instance, the relatively low-angle, east viewpoint used for the map in figure 8.30 causes two important counties, San Francisco and Santa Cruz (at the center of the upper edge of the map), to be obscured. Continuous-surface maps As you saw earlier in this chapter, another type of quantitative map shows a theme as a continuous surface. A continuous-surface map portrays the changing magnitude of some phenomenon from one place to another. On these maps, the changes are gradual rather than abrupt, as on a choropleth map. The annual-precipitation map in figure 8.8 is an example of a continuous-surface map that shows gradual change from place to place. The data for continuous-surface maps may be collected as points, lines, areas, or in raster format (grid cells or pixels in a row-and-column grid or matrix). As you saw for the precipitation map in figure 8.8, data collected at points can be interpolated to create the continuous surface. You learned in chapter 7 that data can be displayed as lines with equal values (isolines). Areas between isolines can also be used to create continuous-surface

maps. You will commonly find that GIS is used with raster data to make these maps, and it is easy to convert between many of these data types. For example, the map in figure 8.31 was created from point data of zebra mussel sightings in water bodies, from 1986 to 2011. Using spatial analysis, the data was converted to a density surface that shows where the sightings are concentrated. There are several ways to map a continuous surface. Many of the mapping methods were originally devised to portray the elevation surface—these methods are described in chapter 9. In this chapter, we focus on mapping continuous surfaces defined by other environmental or statistical quantitative data. The most common methods for mapping these continuous surfaces produce isoline, dot density, and three-dimensional perspective surface maps. We look at each of these methods in turn.

Figure 8.31. The density of zebra mussel sightings in water bodies is mapped as a continuous surface. Red areas have the highest concentration of sightings. Courtesy of the US Geological Survey and Esri.

Isoline maps The most common method used to map continuous-data surfaces is to connect points of equal value with isolines (the prefix “iso” means equal). Isolines are also known as isopleths or isarithms. Isopleths look identical to isarithms except that they show a conceptual surface in which the values can’t be measured physically at points. Different names are given to isolines according to the type of information they show—isotherms are lines of equal temperature, isobars are lines of equal atmospheric pressure, isohyets (as shown in figure 8.8) are lines of equal precipitation, isochrones are lines of equal time difference (see chapter 7), and so on.

In figure 8.32, constant-interval isoline mapping shows average annual hours of sunshine in the Pacific Northwest. Each isoline, called an isohel, is labeled with its value so that regional variations in hours of solar radiation can be studied. A strong regional pattern is evident. Hours of sunshine at first decrease as you move from west to east across the region, with a rapid increase to the east of the Cascade Range. In creating this map, the mapmaker had to make a series of choices. He had to decide how many isolines to draw and how close together to place them. In so doing, he had to decide the interval between successive isolines. Mapmakers can choose a constant or variable isoline interval (see chapter 9 for more on intervals). This choice has a major effect on the map’s appearance. For example, if more isolines are shown, you can see more detail than the seven isolines on the isohel map, although the regional pattern of sunshine remains the same. The isoline interval also affects your map reading, so be sure to note whether the interval is the same across the map (constant) or whether it is larger or smaller in some places (variable). For the map in figure 8.32, the cartographer decided to make 1,800 average annual hours of sunshine the lowest value shown, and he chose a constant isoline interval of 200 hours.

Figure 8.32. This isoline map shows average hours of sunshine received in the Pacific Northwest. From P. L. Jackson and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press.

Isolines can also be drawn for a density surface created from statistical data. This type of isoline is what is meant by the term isopleth. These isolines look identical to those for physical phenomena such as sunshine, but they differ in that they show a continuously varying density surface, on which the values can’t physically exist at points. A common way

to create this type of continuous surface is to collect data by census units, and then assign the data value to the centroid. Each data value is divided by the census unit area, creating a density value for the area (when the data value is a count) that is assigned to the centroid. Values between the centroids are interpolated to create the continuous surface. The map in figure 8.33 is an example of this type of isopleth map—the population density of California is mapped by creating a continuous surface using data values at the centroids of the counties. It’s impossible, for instance, that 1,000 people per square mile can exist at points along an isoline because they occupy an area. At a given location along the 1,000value isoline, you may find no people, you may find 1,000 people, or you may find 50. The map is only intended to give a general impression of varying density over space. It is not meant to be analyzed location by location.

Figure 8.33. This isoline map of California population density is created by drawing isopleths from a set of density values assigned to the center of each county.

Layer-tinted isoline maps The most common way to read an isoline map is to ignore individual isolines and focus on the overall isoline pattern so that you can visualize the surface. Where isolines are placed far apart, the surface varies only slightly; where isolines are dense, the surface slope is steeper. Some people find this hard to do, however. To assist map users who have trouble focusing on the pattern rather than separate isolines, some maps are made with a progression of quantitative area symbols (color lightness or gray tone, color saturation, or pattern texture) added between isolines (see figure 8.32). This is called layer tinting (see chapter 9 for more on this technique). If the symbols are selected properly, you can see a magnitude progression from low to high, with the isolines seen as outlining different magnitude zones. The isolines often aren’t labeled, which further encourages you to see the general pattern of highs and lows on the surface rather than concentrate on individual isolines. If the lines aren’t labeled, numeric range information for each area symbol is found solely in the map legend. One drawback of layer-tinted isoline maps is that when only a small number of isolines are drawn, only a few tints are used on the map, and thus it looks highly generalized. Furthermore, even though a continuous surface is being mapped with isolines, the progression of tints may leave you with the false impression of a stepped surface or distribution. Modern computer mapping applications make it possible for cartographers to create a larger number of isolines that bound layer tints in color progressions that appear continuous (see figure 8.8). Dot density maps Just as a series of isolines can be used to show a continuous surface, so can a set of point symbols. In fact, point symbols are the basis for one of the most effective (but not commonly seen) ways of showing variations in density across a surface. The method is called dot density mapping (sometimes called dot mapping), but be careful not to confuse this term with the point symbol mapping method in which each dot represents either the location of a truly one-dimensional feature or the location conceived of as a point for mapping purposes (see chapter 7 for more on the function of points on maps). What distinguishes dot density maps is that each dot represents more than one feature, and the density of dots gives the impression of a gradually varying continuous surface. Dot density maps usually include a legend that tells you the value of a dot.

Figure 8.34. Dot density map of California population. Each dot represents 10,000 people.

Although a circular dot is the most common symbol used in dot density mapping, any geometric figure, such as a square or triangle, can be used. The shape is irrelevant, because the symbol’s meaning lies solely in the changes in symbol density produced by its repetition. However, as you saw with proportional- and graduated-symbol mapping, using circles as the symbol has clear advantages. An understanding of the difference between a point symbol map and a dot density map is crucial to map reading. On point symbol maps, each symbol represents only one feature, such as a city. Changing a visual variable of the symbol—making it larger or darker, for instance—shows the change in magnitude from one feature to another.

Although point symbol maps show where individual features are located or where a particular feature is found with a given magnitude, the symbols on a dot density map can’t exist at the same locations as the features do in reality because each symbol represents more than one feature. The aim of dot density maps isn’t to give precise locational information about the features but to present an image of changing density of the population across the region. Thus, if the dot density map is well made, your eye won’t be attracted to individual dots but to a general impression of changing population density. However, not all dot density maps meet this criterion. On poorly designed maps, the dots may be too large or too small in size, or too few or too many in number. When incorrectly sized dots are used, you may receive a mistaken impression of the changes in density.

Figure 8.35. Dot density map of population in San Bernardino County, California. Courtesy of Daniel Smith and Alex Quintero.

To create a dot density map, mapmakers first choose a dot unit value (the amount that each dot represents) and a dot size (the size of the symbols they will put on the map) for each dot. For example, they may decide, as they did for the California population map in figure 8.34, that each dot represents 10,000 people. They then divide the number of features in the data collection unit (counties, in this example) by this dot unit value. The resulting number tells them how many dots to place within the data collection unit. When producing a dot density map, the mapmaker places the correct number of dots that represent the total number of features within each data collection unit in a randomappearing fashion to show the variations in density across the surface.

A well-made dot density map has variations in dot density that correspond to variations within the data collection areas. The map in figure 8.35 illustrates within-county variations in dot density. To make this map of San Bernardino County, California, city population statistics are used along with data layers for terrain, land-use, and transportation features within the county to determine zones of exclusion where people are unlikely to live. Dots placed according to this information are seen as smooth gradations, from low to high population density. Dot density maps look simple, but they can be difficult to read if you want to determine exact density values for particular locations. As we’ve already discussed with proportionaland graduated-symbol maps, the human eye tends to underperceive magnitudes. Psychological experiments have shown that as the density of dots increases, our estimates tend to fall below the actual density at an increasing rate. The result is that people reading dot density maps typically get the impression that the range in dot density—and therefore the contrast in density from one region to another—is less than it really is. For this reason, you may find it necessary to compensate mentally for underperception in your own density judgments to gain a true picture of the mapped distribution. Incorrectly made dot density maps can be confusing, if not downright misleading. The more clustered the distribution, the more pronounced the potential discrepancy between reality and the dot density map. For example, a poorly created dot density map of California population might give the impression that population is distributed evenly within each county, with sharp breaks in density between counties—this even distribution of dots is more like that of a choropleth map. In reality, most of California’s population is clustered within parts of the San Francisco Bay Area and the Southern California coastal counties. A classic mapping problem is illustrated by San Bernardino County, the biggest county in the conterminous United States—the county stretches from the eastern edge of the Los Angeles metropolitan area to the Arizona and Nevada borders. If dots were placed evenly within the county, you could conclude that the uninhabited Mojave Desert and other desert areas in the eastern part of the county have a medium population density. As you can see on the correctly made dot density map in figure 8.35, most of the desert areas are unpopulated, and the vast majority of people live in the far southwestern corner of the county that is part of the Los Angeles metropolitan area.

Figure 8.36. Three-dimensional perspective views of California population density from north (top) and south (bottom) viewpoints.

Three-dimensional perspective surface maps A continuous surface can also be shown as a three-dimensional perspective surface map. If the mapmaker constructs closely spaced line profiles in two directions and in perspective view, you gain an impression not of individual lines, but of a continuously varying 3D surface called a fishnet map (figure 8.36). (See chapter 9 for more on fishnet maps.) The fishnet map, also called a wireframe map, uses lines to depict the continuous surface. In a fishnet map, your attention is focused not on any one line or any one quadrilateral formed by the lines, but on apparent vertical undulations in the surface. It is actually the angle and the length of the sides of the quadrilaterals that depict the surface.

The fishnet map in figure 8.36 shows the variability in California population density differently from the prism map in figure 8.30.

Figure 8.37. The growth rates of Oregon counties and cities are shown using color lightness and intensity variations on the county choropleth map and on the graduated-circle map showing the population of cities and towns along with their growth rate. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

As with prism maps, your ability to see all locations on the fishnet map is determined by the viewpoint and viewing angle selected by the mapmaker. In figure 8.36, the California population density surface is shown at a 30-degree angle above the horizon from both a north (top) and south (bottom) viewpoint. Different peaks and valleys on the surface are hidden from view on each map. Two or more maps are often required to see all parts of the surface, depending on the data distribution. Animated three-dimensional perspective maps are ideal for viewing the details of the continuous surface when they are shown with perspective surface maps (see chapter 9 for more on these kinds of maps).

MULTIVARIATE MAPS Sometimes mapmakers show more than one quantitative variable or attribute of a feature on a single multivariate map (see chapter 7). As with multivariate qualitative maps, they can display multiple variables or attributes in two ways—using a different visual variable to show each attribute within a single symbol and showing a concept defined by a composite of attributes. The first method is multivariate-symbol mapping. The second is composite variable mapping. We look at each type of map in turn.

Multivariate-symbol maps One method to show multivariate quantitative data on a single map is through the use of multivariate symbols. Mapmakers can display two or more variables in a single symbol, or they can segment a symbol to show the relative magnitudes of subcategories of attributes for the features. Most often, multivariate-symbol mapping uses point symbols. Figure 8.37 illustrates the first approach of using a single point symbol to show two different types of quantitative information. Circle size represents population, whereas changes in color lightness and intensity within the symbol represent growth rate. Figure 8.38 illustrates the second approach of using segmented symbols, in which point symbols are segmented to show quantitative attributes for subcategories of the features. The total magnitude of each feature is shown using point symbols (cubes) that vary in size. Each cube is then segmented to show the relative magnitudes of two subcategories—income tax and property tax. Look carefully, and you will see that the cube divisions are related to the tax values in millions, and the two different categories are differentiated by color hues. The different hues give you a clue that the subcategories are qualitative, whereas the map theme is quantitative.

Figure 8.38. The data on this map is shown with segmented point symbols to convey the taxes due by county for both income and property tax in the State of Oregon for 1997. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

On the map in figure 8.38, you see categories of a theme shown by subdivisions of a regular geometric shape. You may also see subdivisions shown in bar graphs (figure 8.39), pie charts (figure 8.40), or other statistical graphics. In bar graphs, the graphs often extend outside the data collection areas or obscure other data collection areas because only the height of bars can vary. In pie charts, the symbols have the same scaling problems as proportional and graduated symbols. You should note on these maps whether the mapmaker uses the symbols to display raw data or derived values obtained by mathematically manipulating the data. On the map in figure 8.38, raw values for the income tax and property tax categories are shown. The symbols on the maps in figures 8.39 and 8.40 are segmented to show the proportions or percentages for the categories. As we mentioned, other statistical graphics can be used as segmented multivariate point symbols. In figure 8.41, wind roses show the percentage of time that prevailing surface winds come from different directions during the month of July in the Pacific Northwest. You can see from the few weather stations shown on the map that certain areas (for example,

the Columbia River Gorge along the border between Oregon and Washington) have a strong dominant wind direction relative to other parts of the region.

Figure 8.39. Bar graphs can be used to show subcategories of a theme such as major outdoor recreation activities. From P. L. Jackson and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press.

Figure 8.40. Pie charts segment circles so that you can estimate the proportion of the total magnitude—softwood-growing stock, in this example—of subcategories, such as landownership types. From P. L. Jackson and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press.

Your ability to read maps that have segmented point symbols requires you to be able to judge the relative length or area within each segment. For most people, adjacent vertical bars are somewhat easier to read and analyze with precision than sectors within pie charts, because bars require the estimation of only relative height. One advantage of multivariate point symbols is that they are good for the inspection of individual variables within a theme. It is best if the map reader is familiar with the form of point symbol used, such as the pie charts, bar graphs, or wind roses you have seen on the maps on the preceding pages. A disadvantage is that it may be difficult to estimate and compare proportions, especially if multiple visual variables are used. There are also the scaling issues mentioned earlier. Furthermore, the visual field effect—the modification of a symbol’s appearance by nearby symbols—can alter the perception of a symbol. And it may be difficult to compare parts of symbols that are widely separated on the map, especially if there are many intervening symbols. Nonetheless, you may find these maps compelling, so you may be more inclined to spend time reading and analyzing them.

Composite variable maps

Now we turn to the second way to show multivariate quantitative data on a single map— composite variable mapping. In this approach, data for multiple variables of the theme is combined to show a concept that represents a composite of the attributes. On composite variable maps, several attribute values are combined into a single numeric index called a composite index. These maps are sometimes called cartographic modeling or composite index maps.

Figure 8.41. Wind roses are used to show categories of prevailing surface wind direction. From P. L. Jackson and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press.

More complex composite indexes involve the weighting of several attributes. The categories of California city safety shown on the map in figure 8.42, for instance, involve creating a safety index by weighting the incidence of six basic crime categories. The data for the map is taken from the 2000 Morgan Quitno national awards for 322 cities. The six basic crime categories—murder, rape, robbery, aggravated assault, burglary, and motor vehicle theft—are used in a formula that measures how a particular city compares with the national average for each crime category. The outcomes for the six categories are weighted equally to obtain the safety index. The index values for the 80 cities on the map are sorted from low to high to determine quartiles. The quartiles are given the ordinal-level names “safest,” “safe,” “less safe,” and “least safe.” The circle symbols that represent the cities are mapped using a lightness progression of a brown hue.

At first glance, these circles look no different from circles that show a single quantitative variable. To completely understand what the circles show, you must look carefully at the legend and explanatory notes on the map. For example, only California cities that are greater than 75,000 in population and reported the crime rate in all six categories appear on the map. In addition, the variables used to define the index and the weighting of variables may not be clear. You may think that some variables, such as murder and rape, are weighted much higher than others, such as automobile theft, for example. You may also assume that additional city safety factors, such as earthquake or flooding frequency, are included. If no information is given about how the composite index is computed, you can easily misinterpret the map. Composite variables can also be computed and mapped for line and area features, as well as continuous surfaces. For example, the seismic-risk map for Alaska, in figure 8.43, considers several physical factors that contribute to earthquake damage. These factors are weighted and summed to give a seismic-risk index that is used to define the seven ordinallevel risk categories from lowest to highest, as shown on the map.

Figure 8.42. This map shows ordinal-level categories of California city safety as defined by a safety index on the basis of weighting the incidence of six basic crime categories. Data courtesy of Morgan Quitno Corp.

Figure 8.43. A seismic-risk map for Alaska shows areas of lowest to highest risk. The risk index considers several physical factors, which, when combined, cause earthquake damage. Courtesy of the US Geological Survey.

For these maps to be made correctly, the appropriate data for computation of the composite index must be available, the data must be combined appropriately (for example, should the values be added or multiplied?), and the appropriate weights must be used. Another limitation for the mapmaker, which may affect the quality of the map, is that composite variable mapping requires spatial data for each variable for the full extent of the mapped area. Often, mapmakers are reduced to using data that is available rather than data that is appropriate, so it behooves you to study any notes that describe how the map is made.

Combined mapping methods Another way to show multiple quantitative themes on maps is to combine mapping methods. In figure 8.44, for example, two different quantitative-mapping methods are combined to show related mining industry information. A choropleth map shows the number of people employed in the mining industry while proportional point symbols show mining production. Another good example is figure 8.45, which shows two important aspects of the spread of the West Nile virus. The height of each prism is proportional to the number of confirmed human cases in each state. The top surface of each prism is a choropleth map that is

colored according to its distance from the Mississippi River. The map shows the strong relationship between the incidence of the virus and the proximity of incidents to the Mississippi River.

Figure 8.44. By combining different mapping methods, two related variables— mining employment and production—can be shown on one map. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Figure 8.45. This multivariate map combines prism and choropleth mapping to show the strong relationship between human cases of West Nile virus in 2002 and distance from the Mississippi River. Courtesy of Golden Software Inc.

Figure 8.46. Small multiples are used to show this series of maps of the presidential election results in Oregon, from 1928 to 1996. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

The advantages for map users of combined mapping methods are that the resulting map is conceptually simple, it is good for displaying a few variables (two or three), and it is useful for inspecting individual distributions. Limitations are that readability decreases as the number of variables increases, and it is difficult to convey the relative importance of the various types of information on the map.

Figure 8.47. Small multiples of population density change in the United States are animated to show the change for each decade, from 1800 to 2000. You can scan the QR code at lower right or go to http://webhelp.esri.com/arcGISdesktop/9.2/index.cfm? TopicName=Population_change_over_time to view the animation. Courtesy of Esri.

MAPPING QUANTITATIVE CHANGE You saw in chapter 7 that portraying changes in the environment has become easier for mapmakers, and you are now able to find more maps of this type than in the past. Some of

the same methods that are used for qualitative-change maps can also be used for quantitative-change maps: those that show change in the ordinal-, interval-, or ratiolevel attributes of features over time. Others are unique to the display of quantitative data. We explore some of these methods here.

Small multiples Rather than mapping multiple variables or attributes on a single multivariate map, mapmakers can use multiple-display maps to show multivariate data. Multiple displays can be generated in either constant or complementary formats. Constant-format displays, commonly called small multiples, use multiple maps with the same graphic design and map scale to depict changes in magnitude over a given time period, as in figure 8.46. With this method, the consistency of map scale and design assures that your attention is directed toward changes in the data. Another example is the three maps that show US population density in 20-year increments, from 1820 to 1860 (figure 8.47). The choropleth method is used to create these visually striking small multiples. The three maps are actually snapshots taken from an animated map that shows the change in US population density in 10-year increments, from 1800 to 2000, which you can view at the website listed in the figure caption or scan the QR code in the figure. Another way to map small multiples is to give the impression that maps are being overlaid vertically while still allowing more than one map to be seen at a time. This method of superimposing maps is an effective way to show a set of georeferenced surfaces (see chapter 5 for more on georeferencing). The three-dimensional perspective views of the two precipitation surfaces for average December and June precipitation for western Oregon in figure 8.48 are shown from the same viewpoint and on the same vertical scale. Drawing a line (either mentally or physically) vertically downward from the December surface shows the corresponding point on the June surface. Even without drawing such a line, you can easily see that the amount of precipitation in western Oregon is much greater in December, and that the overall geographic pattern of precipitation is similar in December and June, although it varies in magnitude.

Figure 8.48. December and June precipitation surfaces for western Oregon are overlaid to show the large seasonal differences in rainfall.

Figure 8.49. Complementary-formats display is used to show the spatial (map) and temporal (graph) characteristics of streamflow in Oregon. From W. G. Loy, S. Allan, A. R. Buckley, and J. E. Meacham. 2001. Atlas of Oregon, 2nd ed. Eugene, OR: University of Oregon Press.

Figure 8.50. This temporal change map shows the percentage population change in the conterminous United States by state, between 2000 and 2010.

Complementary-formats display Using complementary-formats display, as shown in figure 8.49, the mapmaker combines maps with graphs, charts, plots, and other graphic data display formats to show changes in the data. For temporal data, the map-graph combination works well, especially when the graph is used to show variation over time. One advantage of displaying multiple formats is that they are good for comparing datasets (although not so good for distinguishing among datasets), especially if the displays are complex. If small multiples are used, they should be comparable, multivariate, small graphics that show shifts in relationships among variables. A limitation, especially for complementary-formats displays, is that the effectiveness is dependent, to some degree, on your ability and aptitude to understand each display format.

Change maps Quantitative-change maps show the increase or decrease in the magnitudes of mapped features over a specified period. The population change map in figure 8.50 is made by finding the percentage change in state populations between 2000 and 2010, as reported by the US Census Bureau. If a state had 10 million people in 2000 and 11 million 10 years later, the ratio is 11 million to 10 million, a 10 percent increase. You must be careful when reading this type of map, because an equal percentage change doesn’t usually mean that the same number of features—people, in this case—are added or lost. If a populous state, such as California, increases its population by 15 percent, it is gaining far more people than if a sparsely populated state, such as Nevada, adds 15 percent to its population. Quantitative-change maps can also show cyclical phenomena (phenomena that recur regularly). The climate of a region, for instance, can be illustrated on a map composed of annual graphs that show the yearly cycle of temperature and precipitation at different locations (figure 8.51). The six annual graphs for the state of Washington allow you to compare seasonal variations at the different weather stations, or study seasonal variations at a single station. For example, the eastern half of Washington shows low values of monthly precipitation and a greater range of monthly temperature.

Figure 8.51. Annual graphs show the yearly cycle of temperature and rainfall at selected Washington weather stations. From P. L. Jackson and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press.

Time composite maps Another way to show change over time is to superimpose data for several periods on a time composite map, which is also called a time series map. You can then see exactly what changes have taken place over the time span covered by the map. Time composite maps are often used to trace the path along which some feature moved and display the variation in a qualitative variable or variables along the path. One of the most famous quantitative time composite maps was made in 1869 by Charles Minard, a French pioneer in the use of graphics in statistics and engineering. The map (recolored from

the original for better visual clarity) uses flow lines to portray the disastrous losses suffered by Napoleon’s Grand Army in the Russian campaign of 1812 (figure 8.52). Beginning at the Polish–Russian border, the width of the brown flow line represents the size of the Grand Army at different locations and dates. The thick line at the beginning shows that about 420,000 soldiers began the march to Moscow. The progressively thinner lines show the losses along the route, as only about 100,000 troops made it to Moscow. The path of Napoleon’s devastating retreat from Moscow in the bitterly cold winter is depicted by the black flow line, below which is a graph of minimum temperatures endured during the retreat. The thin black line at the end of the retreat shows that only about 10,000 soldiers returned to Poland.

Figure 8.52. One of the most famous quantitative time composite maps, in which flow lines portray the disastrous losses suffered by Napoleon’s army in the Russian campaign of 1812, was made in 1869 by Charles Minard. From “Visualizing Time in GIS.” ArcNews Online (Winter) 2009–10.

Attribute change maps It is also possible to show the change in attributes over time. Say that you want to see the location of earthquakes of different magnitudes. You can review a set of maps on which earthquakes in different magnitude classes are shown sequentially, such as the animated map sequence in figure 8.53 which shows the locations of earthquakes and depths of magnitude 5 to 8 occurring throughout the world in 2010. This attribute change map allows you to see the locations of the earthquakes relative to magnitude rather than time. Although you don’t often see these types of maps, they can be useful and may prove to be interesting and appropriate for some types of data.

Figure 8.53. Animated map shows 2010 earthquakes of magnitude 5 to 8. You can scan the QR code at the lower left or go to http://modernsurvivalblog.com/wpcontent/uploads/2010/12/2010-earthquakes-animated-map-magnitude-5-6-7-8.gif to view the animation. Courtesy of modernsurvivalblog.com.

Figure 8.54. Interactive space-time cube of Napoleon’s Russian campaign of 1812. You can scan the QR code at the upper right or go to http://www.arcgis.com/apps/CEWebViewer/viewer.html? 3dWebScene=2b48caaabd0e44028724c5f109f3de97 to interact with the space-time cube. Courtesy of Esri.

INTERACTIVE DYNAMIC QUANTITATIVE-CHANGE MAPS As with qualitative-change maps, a disadvantage of quantitative-change maps is that the maps are discrete and static, even though what is being mapped is dynamic. Displaying the phenomena in a dynamic fashion instead allows you to intuitively understand what is changing on the maps. The same challenges apply as for dynamic maps that show qualitative data. Sometimes it’s difficult to gain a detailed understanding of the message on the map if you’re trying to read the information from one period that flashes briefly on the map and then moves to the next period. You may find these maps hard to read if you do not have display or playback control. With this type of control, you can give yourself enough time to see the maps in sequence and review them individually so you can more fully understand the maps and their animated display. Prominent legends and clear time indicators, such as timelines, are a must. In most cases, these displays are pleasing to the eye and, if presented correctly, can help you better understand the dynamic nature of what is being mapped. Mapmakers are now creating interactive dynamic quantitative-change maps that are not sequences of traditional 2D thematic maps, but true 3D displays, in which time is the third dimension. We revisit Minard’s map of Napoleon’s 1812–1813 Russian campaign shown in figure 8.52, but now treated as an interactive space-time cube in figure 8.54. The spacetime cube was introduced by Swedish geographer Torsten Hägerstrand in 1970 as a way of studying the social interaction and movement of individuals in space and time. The space-time cube includes a z-axis, which adds time as a vertically oriented axis to the familiar x,y map coordinates that mapmakers use. The original Minard map is a classic example of cartographic representation because of its simplicity, but transposing the temporal dimension on the z-axis and creating a spacetime cube gives you additional insights into the movements of Napoleon’s troops. The flow line is now a proportional “tube,” and the vertical columns illustrate when the army was stationary—in Wilma, Witebsk, and Moscow. Napoleon’s army remained several weeks in Moscow before retreating. If they had left Moscow sooner, they might have avoided the worst of the harsh winter, which contributed to the deaths of so many soldiers. This aspect of the campaign is not readily seen in the original 2D map. The space-time cube, published as a web scene in ArcGIS Online, is interactive, with pan, zoom, and rotate functions allowing you to see the march from different perspectives. You can retrieve data for each segment of the flow lines simply by clicking along the line. Vertical connectors give a visual link to the cities through which the march passed, and temperature is displayed along the path of the retreat. All components of the space-time cube can be clicked to explore the troop numbers or other data along the campaign route, and you can switch the different layers on and off as you explore the various parts of the cube.

SELECTED READINGS

Bertin, J. 1983. Semiology of Graphics. Madison, WI: University of Wisconsin Press. Dent, B. D. 1996. Cartography: Thematic Map Design, 4th ed. Dubuque, IA: Wm. C. Brown. Esri. 2011. Esri Conservation Map Book. Redlands, CA: Esri. http://www.esri.com/library/books/conservation_mapbook.pdf. Jackson, P. L., and A. J. Kimerling. 2003. Atlas of the Pacific Northwest, 9th ed. Corvallis, OR: Oregon State University Press. Mitchell, A. 1999. The Esri Guide to GIS Analysis, Volume 1: Geographic Patterns and Relationships. Redlands, CA: Esri Press. Petrov, A. 2012. “One Hundred Years of Dasymetric Mapping: Back to the Origin.” The Cartographic Journal 49 (3): 256–64. Robinson, A. H. 1967. “The Thematic Maps of Charles Joseph Minard.” Imago Mundi 21:95–108. Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. Elements of Cartography, 6th ed. New York: John Wiley & Sons. Slocum, T. A., and S. L. Egbert. 1991. “Cartographic Data Display.” Chap. 9 in Geographical Information Systems: The Microcomputer and Modern Cartography. Edited by D. R. F. Taylor New York: Pergamon. Tufte, E. R. 1990. Envisioning Information. Cheshire, CN: Graphics Press. ———. 2001. The Visual Display of Quantitative Information. Cheshire, CN: Graphics Press. Tyner, J. 1992. Introduction to Thematic Cartography. Englewood Cliffs, NJ: Prentice-Hall. Wesson, R. L., A. D. Frankel, C. S. Mueller, and S. C. Harmsen. 1999. Probabilistic Seismic Hazard Maps of Alaska. US Geological Survey Open-File Report, Series 99-36. Wright, J. K. 1944. “The Terminology of Certain Map Symbols.” The Geographical Review 34 (4): 654– 55.

chapter nine RELIEF PORTRAYAL ABSOLUTE-RELIEF MAPPING METHODS Spot elevations, benchmarks, and soundings Contours Types of contours Isobaths Hypsometric tinting RELATIVE-RELIEF MAPPING METHODS Planimetric-perspective maps Raised-relief globes Terrain models Raised-relief topographic maps Hachures Relief shading Bump mapping Oblique-perspective maps Fishnet maps Perspective-view regional maps Block diagrams Landscape drawings Hillsigns COMBINING METHODS ON MAPS Relief shading and contours Relief shading and hypsometric tinting STEREOSCOPIC VIEWS Stereopairs Stereoscopic polarization Anaglyphs ChromaDepth maps DYNAMIC RELIEF PORTRAYAL Animated methods Interactive methods DIGITAL ELEVATION MODEL DATA ETOPO5, ETOPO2, ETOPO1, and GLOBE Shuttle radar topography mission National Elevation Dataset Coastal Relief Model Lidar SUMMARY

SELECTED READINGS

9 Relief portrayal The earth’s terrain provides the foundation on which we play out our lives. Nothing in the environment is immune from the vertical differences on the earth’s surface, which is called relief. Our mobility, orientation, and environmental understanding are all affected by relief. Yet it’s easy to forget the significance of relief in our environment, because many of us live in an essentially flat world. The floors of our homes, schools, and offices are flat; yards and school yards are mostly flat; highways, railroads, and streets are primarily flat. Thus, we tend to think of geographic position in purely horizontal terms. This limited thinking is often a perfectly good way to simplify our world, but ignoring or misunderstanding the terrain surface can have tragic consequences. Ships run aground, airplanes crash into mountainsides, and hikers lose their way—all because the terrain wasn’t understood properly. In mapping, a terrain surface is a three-dimensional representation of data about the elevations of the physical environment, and cartographers use relief portrayal techniques to map this data. Maps treat relief in several ways. Some maps ignore it and give only horizontal information. These planimetric maps are useful when the mapped area is essentially flat or when facts about an area’s relief aren’t important to your needs. In such situations, relief information merely clutters the map with unnecessary detail. At other times, understanding the terrain surface is crucial to establishing your position and studying spatial associations with other things such as vegetation and patterns of rainfall. When relief information is important, it’s best to turn to topographic and other maps that show the three-dimensional nature of the terrain surface. There are two types of relief portrayal—absolute-relief mapping methods, for showing precise elevation information, and relative-relief mapping methods, for showing different landform features and giving a general impression of their relative heights.

ABSOLUTE-RELIEF MAPPING METHODS Engineers, scientists, surveyors, navigators, and other map users who work analytically with the terrain surface require maps that show more than the general form of terrain features. These map users need information about the absolute relief, or the actual elevation values at locations in the landscape. Absolute-relief methods provide the numeric elevation and water depth information they need. The most accurate elevation and depth data shown on maps is surveyed elevations at individual points on the earth.

Spot elevations, benchmarks, and soundings On aeronautical charts, topographic maps, engineering plans, and other large-scale maps, the elevation of the surface is given numerically at individual survey points. These elevation values, relative to the mean sea level (MSL) datum (see chapter 1 for more on mean sea level), are called spot elevations. Spot elevations are often located at positions or on features that are positively identifiable and are easily recoverable survey marks (that is, they can be easily found again). Searching for survey marks, or mark recovery (also sometimes improperly called benchmarking), at road intersections or forks is much easier than at changeable locations such as stream confluences or vegetation patch corners. On topographic maps, as shown in figure 9.1, spot elevations are symbolized with a small x, followed by the elevation value above MSL. When spot elevations are located at a road intersection, the symbol is omitted, because the location of the reference elevation is known.

Figure 9.1. Spot elevations and benchmarks as portrayed on a 1:24,000-scale topographic map. Courtesy of the US Geological Survey.

At some of these locations, the elevation is determined by precise leveling methods (see chapter 1 for more on leveling), and a permanently fixed brass plate, called a benchmark, is installed in the ground (figure 9.2). Benchmarks are symbolized on topographic quadrangles made by the USGS and other national mapping agencies by an x next to the elevation value (figure 9.1) and the identifier “BM” (for benchmark) preceding the elevation value. Soundings (water-depth readings) are used to determine the depth of lakes, rivers, oceans, and other water bodies. For thousands of years, mariners have measured shallow water depths using a calibrated lead line (a line with a lead weight tied to one end that is lowered into the water to determine depth by markings on the line) or a sounding pole (a pole marked with water-depth values). Today, soundings are obtained by electronic depth-measuring instruments (sometimes called depth sounders or depth gages) that determine the amount of time that acoustic pulses take to bounce off the water bottom and return to the instrument. This travel time is then converted to distance above the bottom because the velocity (speed of movement) of acoustic waves is a known quantity. Modern electronic instruments collect soundings more rapidly, accurately, and at greater depths than poles or lead lines allow. GPS receivers (see chapter 14 for more on GPS) integrated into the instruments also give the exact latitude, longitude, and grid coordinates at the instant each sounding is obtained.

Figure 9.2. Precisely surveyed elevation points are identified on the ground by circular brass benchmarks. A small triangle with a plus sign (+) in the middle marks the surveyed point. Courtesy of A2Zgorge.info/Thunderbolt Designs.

Figure 9.3. Soundings on US nautical charts are usually shown in fathoms for depths greater than 11 fathoms. Fathoms and feet are used together for shallower areas. Courtesy of the National Ocean Service.

Soundings are not relative to MSL, but rather to specific definitions of “low water.” The two datums (see chapter 1 for more on datums) used in North America to define low water are the arithmetic average of all the low-tide levels recorded over a 19-year period, called mean low water (MLW), and the arithmetic average of the lower of the two daily low tides recorded over the same 19-year period, called mean lower low water (MLLW). Canadian nautical charts use MLW, whereas, in the United States, MLLW is the official nautical chart datum of the US National Ocean Service (NOS), the agency responsible for preserving and enhancing the nation’s coastal resources and ecosystems. The datum used to define high water to support harbor and river navigation is mean high water (MHW). The MLLW datum is used on nautical charts for water depths because ship captains must decide whether there is enough clearance for their vessels between the changing water surface and a fixed submerged obstacle, such as the harbor bottom or a wreck at low tide. To do so, they must determine the minimum depth likely to be encountered at a given

position. If overhead clearance is the concern, as when moving under a bridge at high tide, the situation is reversed, and the MHW datum is more useful. Soundings have been printed on nautical charts since the early 1600s. Charts that cover the coastline of the United States (figure 9.3) traditionally use the fathom (six feet) as the unit of measurement for depth. In areas shallower than 11 fathoms (66 feet) in which more exact depths are required for safe navigation, fathoms and feet are used together. The value 91 in figure 9.3, for instance, is read as nine fathoms and one foot, or 55 feet. Nautical charts made in Canada and most other nations show depths in meters, and more recent US charts show meters as well.

Figure 9.4. A portion of a 1:24,000-scale USGS topographic quadrangle for Corvallis, Oregon, with a 20-foot contour interval. Index, intermediate, approximate or indefinite, supplementary, and depression contours are used to portray special landform features and to make the map easier to read and analyze. Courtesy of the US Geological Survey.

Contours

Contours are a special case of isolines (see chapter 8 for more on isolines). Contours are lines of equal elevation above a datum. If a contour was actually drawn on the earth, it would trace a horizontal path that is constant in elevation. It is common to use contours on topographic maps to show variations in relief and landform features such as hills and valleys. The portion of a 1:24,000-scale USGS topographic quadrangle shown in figure 9.4 is representative of this type of map. To understand the logic behind contours, imagine that you are on a small island in the ocean. It has a hill that is just over 700 feet (213 meters) high and another that is just over 500 feet (152 meters) high (figure 9.5). If you walked around the island at the shoreline when the tide is at mean sea level, you would trace the zero or datum contour and return precisely to your starting point. It is important to remember that all contours eventually close like this, although the closed curve can’t always be seen on a single map sheet.

Figure 9.5. The logic of the contour method of relief portrayal is illustrated here by the relationship of the contours that intersect the horizontal transect line on the map (top) to their corresponding locations on the elevation profile (bottom).

Now suppose that you start climbing up the highest hill until you reach an elevation of 100 feet above mean sea level. If you again followed a path around the island at this constant elevation, your path would trace a contour, this time the 100-foot contour, and again you would return to your starting point. The effect is the same as if you walked along the shoreline after the ocean level is raised 100 feet. If you did the same thing for elevations of 200 feet, 300 feet, and so on, and the paths you walked are projected vertically onto a flat

map, the result will be something like the map at the top of figure 9.5. If the island is viewed in profile, it will look as though it is sliced into layers by imaginary horizontal planes, as in the elevation profile, commonly called a terrain profile, at the bottom of figure 9.5 (the procedure to create a profile from a contour map is explained under “Profiles” in chapter 16). In our island example, the vertical distance between contours, called the contour interval, is 100 feet. The mapmaker, of course, may select a contour interval of 20 feet, 50 feet, or whatever seems appropriate, considering the extent of the area being mapped and the amount of relief in that area. The smaller the interval used, the more detailed the relief portrayal. Once you know the contour interval, it’s a simple matter of “reading between the lines.” If the spot you’re interested in lies halfway between the 500-foot and 600-foot contours, you can conclude that it’s around 550 feet high. On most maps that show contours, the MSL datum and a constant contour interval are used. You are usually safe in treating these factors as constants so only the size of the interval is a variable factor. In general, you will find that the greater the relative relief on a map, the larger the contour interval. Mapmakers use this relationship between relief and contour interval to keep contours from becoming too dense in areas of high relief and too sparse in areas of low relief. A special problem arises when the terrain changes markedly from one part of the map to another. When this change in elevation occurs, it is possible that several different but constant intervals are used on the same map. For example, this variable contour density might occur when the map sheets span high to low relief. Without variable density, closely spaced contours clutter the map and become difficult to differentiate, and widely spaced contours make it hard to determine slopes and predict elevations at points that fall between the lines. Types of contours On USGS and other maps, you are likely to see a variety of topographic contours to show the shape of the land surface, including index, intermediate, approximate or indefinite, supplementary, depression, and cut and fill contours (figure 9.6). We look at each of these contours in more detail. On topographic maps, you’ll find that not every contour is marked with its elevation value; instead, every fourth, fifth, or 10th line is labeled, depending on the scale of the map. To help map readers more easily determine elevations, these index contours are labeled and drawn with a thicker line (see figures 9.4 and 9.6). Intermediate contours are drawn with a thinner line between the index contours, at an equal spacing from each other and from the index contours. Index contour labels normally are spaced widely, meaning that you may have to trace the index contour for some distance before you see its value. On US maps, the labels are often laddered—that is, the laddered contour labels are placed in line with the label on the adjacent labeled contours so that you can easily read the elevation values from one index contour to the next. On US maps, the labels are also drawn relative to the bottom of the page rather than to the terrain. On maps used in Great Britain and other countries, you will find the contour labels laddered as well, but they are always oriented upslope, so in some places they appear upside down on the map. On Swiss topographic maps, the contour labels are randomly placed rather than laddered to give the impression that the symbols are more natural, like the terrain.

Several modifications to standard index and intermediate contours are used to portray special aspects of the land surface and simplify map reading. Supplementary contours are sometimes shown in areas in which elevation change is minimal (figure 9.4). They almost always represent half the contour interval for the rest of the map, so these lines are mostly used in areas in which the overall change in elevation is gradual—for example, floodplain areas in which a slight change in relief might have a major impact on the stream channel and flooding pattern. You will also sometimes see them in areas of rugged terrain to emphasize important or unique features. On USGS topographic maps, they are shown in a lighter tint than standard index and intermediate contours and are labeled so that you can clearly determine their elevation values (figure 9.6).

Figure 9.6. Types of contours are illustrated and their symbology is described in the Topographic Map Symbols document for 1:24,000-scale USGS topographic maps. The entire document is at http://pubs.usgs.gov/gip/TopographicMapSymbols/topomapsymbols.pdf. Courtesy of the US Geological Survey.

Sometimes you will come across dashed segments of standard contours or find dashed contours between the standard contours. These lines, called approximate or indefinite contours (figure 9.6), are used when conditions prevent the line from being confidently positioned according to accuracy standards for the map (see chapter 11 for more on map

accuracy standards). For example, on USGS topographic maps, horizontal positioning on the map must be within 0.02 inches (40 feet on the ground at 1:24,000 scale) of the true ground position, and vertical positioning must be within one-half a contour interval of the true ground position. When these standards cannot be met—for example, when the ground elevation is hard to determine under dense vegetation—the contours are shown as approximate or indefinite. In the map example in figure 9.4, they are used along the shoreline of the river, which is an area in which the elevation of the land surface is subject to frequent change. Although these lines contribute valuable terrain information, they should be viewed with caution because they are intended to give a general impression of the terrain, not precise elevations. To aid in identifying closed depressions (areas lower than the surrounding surface), small right-angle ticks may be added to the downslope (or inside) of contours (figure 9.6). These depression contours help focus the map user’s attention when the depression is small relative to the size of the area mapped (figure 9.4). Depression contours are especially useful in distinguishing depressions from small hills because there is seldom enough space for the mapmaker to label the contours of these features with their elevation values. Whether depression contours are labeled or not, the first (outside) depression contour is always the same elevation as the adjacent standard contour. Moreover, depression contours merely represent special cases of the standard contours on the map and thus share the same interval and elevation values. You will sometimes see a special type of depression contour called a cut contour, when a roadway has been cut through the landscape, drastically lowering the terrain (figure 9.6). Cut contours are usually accompanied by fill contours, which show where the terrain was raised to support a road or railway grade (figure 9.6). In areas of high relief where there is insufficient space to show all the contours, carrying contours may be used to show two or more contours merged into a single contour. These carrying contours are used to represent vertical or near-vertical topographic features such as cliffs and escarpments. If a contour shows the land that lies underwater, it is a bathymetric contour (also called a depth contour, depth curve, or isobath). Bathymetric contours are symbolized much like topographic contours for land but are shown instead in blue or black instead of brown (figure 9.6). Similarly, if a contour is on an ice mass, like a glacier or snowfield, the line for the glacial contour is shown in blue (figure 9.7). An uncommon type of contour is a submerged or underwater contour. It depicts the former river channel in an area that was inundated because of damming. It is a special case of the standard contours on the map and thus shares the same symbol standards, although it is usually shown in blue, and sometimes black or brown. The submerged contours in figure 9.8, for example, are also approximate or indefinite.

Isobaths Isobaths are lines of equal water depth below the MSL or MLLW datum. They are found on nautical charts and other bathymetric maps that show water depths. The first isobaths appeared on European charts of river estuaries in the late 1500s. By the end of the 18th century, enough soundings were taken that isobaths were drawn on a large number of nautical charts, especially in shallower areas. In this century, electronic depth sounders provide the detailed information required to draw isobaths on virtually all nautical charts

(see figure 9.3). In addition, information on deep-sea and ocean depths, collected by sonar methods and detailed analysis of satellite data, allows isobaths to be accurately drawn on bathymetric maps of the oceans and other deep-water bodies.

Figure 9.7. Types of glacial contours are illustrated and their symbology is described in the Topographic Map Symbols document for 1:24,000-scale USGS topographic maps. The entire document is at http://pubs.usgs.gov/gip/TopographicMapSymbols/topomapsymbols.pdf. Courtesy of the US Geological Survey.

Figure 9.8. Submerged contours, shown with blue dashed lines, indicate where the former river channel was and help explain why the boundary between Cowlitz and

Clark Counties in Washington state is located where it is. Courtesy of the US Geological Survey.

Isobaths on large-scale nautical charts differ from contours in that isobaths are based on the MLW or MLLW datum and also use uneven depth intervals. Charts produced by the NOS, for instance, show isobaths for 1, 2, 3, 5, and 10 fathoms, with greater depths shown in multiples of 10 fathoms, typically 20, 50, and 100. Shallow-water isobaths appear on charts for the same reason that shallow-water depths are shown in fathoms and feet— mariners need this information for safe coastal and harbor navigation.

Hypsometric tinting Hypsometric tinting (also called hypsometric coloring or layer tinting) is a method of “coloring between contour lines” that visually enhances the relative elevation cues for contours while maintaining the absolute portrayal of relief. On maps made with this technique, contours or isobaths can have a stepped appearance, much like a layer cake (figure 9.9). The elevation zones between contours are given distinct gray tones or colors, called discrete hypsometric or discrete layer tints. For the terrain surface, cartographers often try to use colors that relate to the type of land cover found at different elevation zones, such as green for low-lying, verdant valleys, brown for high-elevation treeless areas, and white for snow-capped peaks.

Figure 9.9. Green to brown discrete hypsometric tinting used to symbolize land and blue tinting used for water visually enhance land elevations and water depths for the contours and isobaths in the Crater Lake area in Oregon.

Figure 9.10. Continuous hypsometric tinting creates a smooth progression of tints for water depths and the land surface form.

When mapmakers use a series of blues to tint between isobaths, the usual rule is, “The darker the blue, the deeper the water.” If the map is well designed, discrete hypsometric tinting should give a visual impression that appropriately portrays the underwater surface being mapped. The simplest discrete hypsometric tinting is on US nautical charts, in which a light-blue tint is usually added to all water areas within the five-fathom isobath (see figure 9.3). Using discrete hypsometric tinting to visually enhance the elevation zones between contours helps you see differences in relief. It also allows you to more easily determine the elevation or depth range of any location. With newer mapping capabilities, the abrupt change between hypsometric tints can be minimized by gradually merging one tint into the next, giving a smooth appearance to tonal gradation. This continuous hypsometric tinting (figure 9.10) is readily achieved using computer mapping software and digital elevation model data, which we explore later in the chapter. This technique of continuous tinting uses a color ramp, which is a continuous series of colors between specified beginning and ending colors (figure 9.11). Color ramps can be strung together to make a progression across many colors. As with discrete hypsometric tinting, the colors selected are often assumed to relate to the ground cover typically found at various elevations, or to water depths.

Figure 9.11. A series of color ramps is strung together to create the hypsometric tinting for the Crater Lake maps in figures 9.9 and 9.10.

The key advantage of this method is that each elevation in the digital elevation model data is assigned a color that corresponds to the exact elevation rather than a range of elevations, as in discrete hypsometric tinting. However, the colors vary so gradually that you cannot see the individual “steps” of the elevation ranges. For this reason, it is harder to determine an exact elevation or elevation range at any location. To find this information, you must rely on contours if they are also shown (figure 9.12). Continuous hypsometric tinting is used more and more frequently; you can see it on poster-size wall maps that are more like art, on tourist maps, in recreational atlases and gazetteers, and on web maps. You will also find some maps, such as older wall maps, that show the highest elevation zone in dark brown or red, the intermediate zones in buff and light brown, and the lowest zone in dark green. The water in the valley bottoms is blue. These color progressions are based on the perceptual phenomenon that some colors, such as blue, appear to visually fall away from you, while others, such as red, visually rise toward you. Although this coloring scheme is widely used, its visual effectiveness has not been satisfactorily proven.

Figure 9.12. In addition to the contours, elevation is shown with hypsometric tinting in a section of this map of the Crater Lake area in Oregon. The higher elevations are shown in white as snow-capped peaks. The continuous hypsometric tint progresses to brown for the rocky outcrop areas and then to green for the lower elevation valleys. Courtesy of the National Park Service.

Colors for hypsometric-tinted maps aren’t always properly chosen by mapmakers, nor are darker tones always used to signify lower water depths, especially when coupled with relief shading (described later in this chapter), as they often are. Thus, it’s especially important to check the legend before using maps that have hypsometric tinting.

RELATIVE-RELIEF MAPPING METHODS In our day-to-day lives, we’re usually concerned with the local elevation range between high and low heights, or the relative relief, rather than the absolute elevation values. We often think of relief in terms of terrain features such as hills and plains, mountains and valleys. Relative-relief methods are used on many maps to give a visual three-dimensional effect that makes terrain features easy to see. One way to achieve the perception of realism is to depict the landform surface on planimetric-perspective maps.

Planimetric-perspective maps Not to be confused with planimetric maps, which represent only the horizontal positions of features and not the vertical positions that topographic or bathymetric maps do, planimetric-perspective maps give an overhead view of the topography. Showing elevation variation on maps that have a viewpoint as if you are looking straight down from an airplane is challenging—it is easier to see surface variations from the side in perspective view. Nonetheless, mapmakers have developed a number of clever techniques to provide

an overhead perspective. We examine a few of them so that you know how to read these kinds of maps when you see them. Raised-relief globes Because globes present the truest picture of the earth as a whole, you might conclude that raised-relief globes provide the most realistic and useful portrayal of the vertical dimension. But there is a flaw in this reasoning. You saw in chapter 1 that if the earth was reduced to the size of a bowling ball (a common globe size), the earth would be “smoother” than the ball. To create a realistic impression of relief, mapmakers exaggerate the elevation. Vertical exaggeration (VE) occurs when the vertical scale is larger than the horizontal scale. This technique makes low-relief landforms visible, and it can also make the entire surface somewhat “realistic” in appearance (figure 9.13). In practice, a VE of about 20:1 is typical on relief globes of a 24-inch (60-centimeter) diameter. Therefore, in addition to the highly generalized landforms as a result of their small scale and the handling and storage inconvenience of globes, users of raised-relief globes also face the problem of large VE. Makers of these globes also face the problem that they are costly to produce, although emerging 3D printer technology may reduce their cost in the future.

Figure 9.13. Elevation variations on a raised-relief globe must be greatly exaggerated to give a reasonable impression of the earth’s landforms. Courtesy of NutriSystems Raised Relief Globes bathymetric 20-inch globe.

Terrain models You can minimize the problem of large VE by using physical terrain models rather than globes. You can think of terrain models as chunks of the surface of a giant raised-relief globe. They let you focus on a smaller portion of the earth at a larger scale than a raisedrelief globe, allowing relief to be shown with less VE than on globes. Some terrain models

are constructed with discrete elevation layers, much like discrete hypsometric tinting, but the areas between contours are flat planes at the same elevation (figure 9.14A). The stepped effect is almost always a result of the mapping technique, not the nature of the landform surface. Although layered landforms occur in regions of terraced agriculture, openpit mining, and horizontal sedimentary beds of varying resistance to erosion, these landscapes are relatively rare.

Figure 9.14. The layer-by-layer construction of some terrain models often leaves them with a stepped surface. Smoothing out the elevation differences between steps creates a more realistic portrayal (A), as does printing satellite images and reference map information on the model (B). You can see how the bottom model of the Salton Sea, California, region is made by scanning the QR code or visiting https://www.youtube.com/user/solidterrainmodel. (A) Courtesy of HowardModels.com. (B) Courtesy of Esri.

Terrain models are used to show a more realistic, smooth, continuous landform surface. Of all planimetric-perspective maps, these models provide perhaps the clearest picture of the terrain, although the relief is often exaggerated vertically to make the model appear more realistic. The Salton Sea, California, region terrain model (figure 9.14B) is an example of using digital elevation model data, described at the end of the chapter, to drive a large computer-controlled milling machine that cuts the terrain into a thick block of polyurethane foam. A custom 3D ink-jet printer then scans line by line just above the surface, printing a satellite image or traditional reference map on the model. As with all physical models, terrain models suffer the disadvantages of bulk, weight, and a high cost of production. For these reasons, they are normally used only in permanent or semipermanent displays. They are frequently found in national park or natural history museum displays; the lobbies of government and private agencies that deal with environmental problems; city-regional planning exhibits; and university geology, geography, and landscape architecture departments. People who are working on promotional schemes or research projects are especially fond of using terrain models. Urban renewal projects, malls, and dam sites are favorite subjects. For these purposes, the inconvenience of constructing models is offset by the true-to-life impression of the landscape they provide.

Raised-relief topographic maps The raised-relief topographic map is a type of terrain model that partially sidesteps the problems of weight and high cost. It is made by taking a flat topographic map, printing it on a sheet of plastic, and using heat to vacuum-form it into a 3D model that enhances the elevations depicted on the map through contours and possibly hypsometric tinting (figure 9.15). The resulting map isn’t a curved piece of the globe, as are many terrain models. It is rather a flat map sheet with a three-dimensional landform surface.

Figure 9.15. A raised-relief topographic map is printed on a sheet of plastic molded to form a three-dimensional surface. Data courtesy of the USGS National Elevation Dataset.

Raised-relief topographic maps are available from private mapping firms. They cost about 10 times more than a conventional map of the same size and area and are relatively fragile and difficult to store. Because the flat plastic sheet must be stretched during construction, some displacement of features on the raised surface is also common. But the realistic impression of relief often compensates for these drawbacks. A more serious concern is that the horizontal position of geographic features may not be portrayed accurately. This misrepresentation is inevitable because of the displacement caused by stretching the flat map into a three-dimensional plastic model. The inaccuracy is compounded by the fact that this type of relief map is usually produced at medium to small scales for wider sales appeal. And with small-scale maps, a small stretching of the plastic gives a greater distortion than the same amount of stretching for a large-scale model.

Hachures On large-scale flat maps, the terrain is sometimes rendered with hachures—short lines arranged so that they face downhill. Each hachure lies in the direction of the steepest slope, showing the amount of slope with some accuracy. The best-known hachuring method is called the Lehmann system, after its founder, Johann G. Lehmann, a Saxon military officer who introduced the method to Europe in 1799. In the Lehmann system, the thickness of hachures is varied—the steeper the slope, the wider the hachures (figure 9.16A). At times, the hachure technique is so rigorously applied that you can derive numeric slope values by comparing a zone of uniform slope with the map legend. More commonly, only a general impression of steepness is intended.

Figure 9.16. With the Lehmann hachuring method (A), relief is portrayed by increasing the width of hachures with increasing slope. The Dufour method (B) eliminates hachures on the northwest sides of hills to give a three-dimensional appearance to the terrain.

Hachuring is a poor relief portrayal method when applied to small-scale maps, as it was in the 19th century (figure 9.17). The hachure lines for mountain ranges are so simplified and stylized that they look like hairy caterpillars crawling across the map. In fact, cartographers refer to them as “hairy caterpillars” (or sometimes “woolly worms”). If you see these creatures on modern maps, realize that they’re intended to show only the general location of hills and mountain ranges.

Figure 9.17. Hachures on 19th-century small-scale maps, like this section of an 1887 county and township map of Oregon and Washington, look like hairy caterpillars crawling across the map.

In the mid-19th century, the Lehmann system was largely replaced by the partialhachuring system of the Swiss cartographer Guillaume H. Dufour. In Dufour’s method, hachures are eliminated on the northwest sides of hills on north-oriented maps (figure 9.16B). Eliminating hachures in this manner greatly improves the 3D impression of relief and makes terrain features far easier to identify. The problem with Lehmann and

Dufour hachures is that they are tedious to create and obscure other map features. Hence, hachures are rarely used in the United States today, but the Dufour system is the predecessor of the relief shading method so widely used throughout the world and can still be found on topographic maps made by the Swiss and other national mapping agencies. Relief shading Relief shading, also called hillshading or shaded relief, has been used on maps since the late 19th century to enhance the three-dimensional appearance of terrain features. You know from looking down at the earth from an airplane that patterns of light and shadow on hills give the strongest impression of relief. Relief shading attempts to re-create these tonal variations on maps to give the same 3D effect. Relief shading is not the same as the shadowing that appears on real earth features. For instance, compare the relief-shaded representation of Mount Saint Helens (left) in figure 9.18 to a midmorning vertical photograph of the area. The relief shading varies in darkness within shaded areas and stops at the base of valleys. In contrast, in the air photo, shadows cast by hills are equal in darkness throughout. These shadows also cross valleys to darken adjacent hillsides.

Figure 9.18. Relief shading (left) of Mount Saint Helens, Washington, using oblique illumination with the light source at the upper left. A vertical aerial photograph (right) of the same area illustrates the differences between relief shading and shadowed terrain. Courtesy of the US Geological Survey.

The principle underlying relief shading is that drawings of 3D objects appear correct to us when we use an imaginary light source from the northwest (upper-left) corner of the map to create the relief depiction (figure 9.19, left). The lightest shading occurs on northwest-facing slopes, which are at right angles to the imaginary light source rays, and the lightest tones are used for areas that have the highest slopes. The darkest shadows are cast over southeast-facing slopes, with darker tones assigned to steeper slopes. Lightness and darkness, then, are determined not only by slope steepness but also by terrain orientation with respect to the light source. If the opposite lower-right light source position is used, objects appear inverted. Relief reversal is exactly the opposite effect from what was intended—hills look like valleys and valley bottoms look like ridge tops (figure 9.19, right).

Another problem with relief-shaded maps is more serious than relief reversal. Relief features that are oriented roughly parallel to the imaginary light rays aren’t shaded enough to give proper relief cues. To circumvent this problem, mapmakers often use multidirectional relief shading, using multiple imaginary light sources to the map’s west and north sides so that terrain with north–south and east–west orientation also receives sufficient shading. This use of a multiple light source is probably the best way to show relative relief on a flat map. This effect can be achieved analytically with computer software (figure 9.20), or it can be generated through an artistic rendering (figure 9.21). The resulting map may look as real as a picture of a physical terrain model of the landform.

Figure 9.19. Relief-shaded maps of Mount Saint Helens, Washington, using oblique illumination with the light source from the northwest (left) and southeast (right). Relief reversal is evident in the map on the right because hills look like valleys and valleys look like hills.

Figure 9.20. Multidirectional hillshading (top) is an eye-catching alternative to the single light source direction method of hillshading (bottom) as illustrated on this map of Mount St. Helens, Washington. Courtesy of USGS, NGA, NASA, CGIAR.

Relief shading used to be a laborious manual chore that required a great deal of artistic skill and a thorough understanding of geomorphology (the study of landforms and terrain-forming processes). The expense involved in producing shaded relief maps by hand could rarely be justified. Thus, despite the dramatic visualization that relief shading affords, for practical reasons, not many such maps were made. The situation has changed dramatically, however, because relief shading by computer is now simple and the data

required is plentiful. Relief shading can be produced in seconds using computer mapping and GIS software coupled with digital elevation model data. Bump mapping Recently, maps are being produced with surface textures added to relief shading, such as the textured relief shading for Crater Lake National Park, shown in figure 9.22. The idea is to give a general indication of the “texture” of the terrain surface, with its land cover (trees, shrubs, rocks, and even buildings) adding to the visual realism of the relief shading. In this case, a method called bump mapping or texture shading is used to modify the original digital elevation model by adding elevation values around randomly scattered points of varied heights that represent different vegetation types in the landscape. The reliefshaded digital elevation model then casts shadows for the vegetation as well as the terrain.

Figure 9.21. The use of artistic shading enhances the impression of the landform surface on this Swiss 1:100,000-scale topographic map. Rock outcrop drawings also enhance visual realism. Reproduced by permission from swisstopo (BA071647).

Figure 9.22. A vegetation texture is added through bump mapping to this reliefshaded map of the Crater Lake area in Oregon. Courtesy of the National Park Service.

Oblique-perspective maps Several methods can be used to create maps that give us a “bird’s-eye view” of the landscape. These oblique-perspective maps have a more three-dimensional look than planimetric maps that portray the landscape from a vantage point directly above the mapped area. Oblique-perspective maps have strong intuitive appeal and a high level of readability—this ease of use accounts for their popularity. But there is a problem. On these maps, features are often displaced from their true horizontal locations. Positional displacement of a terrain feature occurs in direct proportion to the height of its symbol— the shift in horizontal position between the bottom and top of a tall mountain is much greater than for a small hill. The top, bottom, or middle of a hill can be placed in its correct horizontal position on the map, but not all three at once. Mapmakers create obliqueperspective views in several ways, so we start with terrain profiles and fishnet maps. Fishnet maps One type of oblique-perspective view resembles a fishnet draped over a terrain model. For this reason, they are called fishnet maps, or sometimes wireframe maps. The lines that make up the “fishnet” are created as profiles drawn horizontally and vertically across the terrain surface (figure 9.23). More closely spaced profiles produce a smoother fishnet surface. When computer software was first able to create fishnet maps, they increased in popularity. However, more realistic representations of the terrain, such as multidirectional hillshades with hypsometric tinting, have since supplanted their use. You will find that many of the concepts that relate to fishnet maps also apply to other oblique-perspective maps. We continue our discussion of oblique-perspective maps using fishnet diagrams as an illustration because they depict the terrain so clearly.

With fishnet and other oblique-perspective maps, the quality of relief depiction depends on the vantage point set by the mapmaker. An oblique viewpoint about 45 degrees above horizontal is generally the most useful. In figure 9.24, as the viewing angle of the vantage point decreases, the problem of terrain in the background of the map being obscured by terrain in the foreground of the map increases. To circumvent this problem, you can sometimes “look behind the hill” if you use two maps, each with a different viewing angle. Computers now allow the generation of dynamic displays that you can tilt and rotate as desired to see the landscape from multiple vantage points.

Figure 9.23. Parallel horizontal and vertical profiles can be combined to make a fishnet map. Courtesy of Golden Software Inc.

To create a realistic impression of relief, mapmakers exaggerate the vertical scale of most oblique-perspective maps. A VE between 2 and 3 is common with large-scale obliqueperspective maps (figure 9.25), while a 50:1 VE is often used for small-scale maps of larger states or countries. Regions of little relief are vertically exaggerated more than regions of substantial relief. Because different degrees of VE can produce different relief impressions, you should make it a practice to check the legend to see if the amount of VE is indicated.

Figure 9.24. Changing the vantage point drastically alters the terrain portrayal, as this sequence of 5-, 15-, 45-, and 75-degree viewing angles demonstrates. Courtesy of Golden Software Inc.

Figure 9.25. Cartographers normally use vertical exaggeration to increase the three-dimensional appearance of oblique-perspective maps and other threedimensional perspective views of the terrain. Courtesy of Golden Software Inc.

Perspective-view regional maps The oblique-perspective maps in the fishnet map examples show “bald” terrain, denuded of land-cover features. With the aid of computers, mapmakers overcome this limitation. It is now common to drape relief shading over the underlying fishnet surface to create a more realistic perspective-view regional map that approximates a block diagram but leaves out the subsurface geology characteristics (figure 9.26)—we talk more about block diagrams later in this chapter. Contours and other map symbols can also be draped over the map for even more useful terrain representations.

Figure 9.26. The fishnet map that underlies this perspective-view map of Mount Olympus, Washington, is draped with hillshading, contours, and other topographic map information. Mount Olympus, Washington, block terrain example from Map Illustrations website, www.mapillustrations.com.au.

Figure 9.27. Landsat Thematic Mapper imagery is draped over a digital elevation model on this oblique-perspective regional map of Los Angeles and the San Gabriel Mountains. Courtesy of NASA/JPL-Caltech.

Another approach is to drape aerial photography or satellite imagery over the terrain surface (figure 9.27). Mapmakers also drape other information—everything from land-use categories to surface geology data—over surfaces. These combinations are expanding as GIS and mapping applications allow three-dimensional tree symbols and other 3D map symbols to be displayed on the surface. These maps are usually constructed with an oblique orthographic map projection (see chapter 3 for more on orthographic projections). Because all data is in digital form, you can often examine perspective-view regional maps from any number of vantage points (see the “Dynamic Relief Portrayal” section later in this chapter). Block diagrams A block diagram looks like a terrain model—the difference is that terrain models are physical representations of the terrain, made of plastic or other solid material, and block diagrams are drawings. As with a terrain model, block diagrams portray a “chunk” of terrain as if it was cut out of the surface of the earth. In block diagrams, the vertical sides of the block allow the underlying rock formations or other subsurface geology information to be shown (figure 9.28). Block diagrams reach their highest degree of sophistication in the rendering of their smooth, continuous surface form to give a realistic picture of the actual land surface. The natural appearance of the land surface form can be achieved in two ways —by using sketch lines to accentuate the terrain features that give distinct form to the landscape or by relief shading the surface. Either technique makes the terrain easily understandable.

Figure 9.28. A line drawing block diagram shows terrain features and subsurface geology in the vicinity of the Uncompahgre uplift in Utah. Courtesy of the Utah Geological Survey.

Figure 9.29. Landscape drawings such as this ski map of Mount Bachelor, Oregon, are an artistic rendering of the terrain surface. Courtesy of Peter Powers.

Popular magazines and advertisements take advantage of the realistic terrain picture provided by block diagrams. The illustrations in geology, physical geography, and other environmental textbooks are also often block diagrams. Landscape drawings Artistically rendered, oblique-perspective maps, called landscape drawings, can provide nearly photorealistic terrain portrayals. Because of the immense amount of hand labor involved in their construction, the best examples are found for regions of special interest, such as national parks and popular tourist areas such as ski resorts. Landscape drawings are popular as ski-area posters, such as the Mount Bachelor, Oregon, poster in figure 9.29. Hillsigns The stylized drawing of hills from an oblique perspective is one of the oldest ways to show relief. From the medieval period until the mid-1800s, hills and mountain ranges on both large- and small-scale maps were represented by crude line drawings of highly stylized hills called hillsigns (figure 9.30). These drawings look like conical mole hills that may bear little resemblance to the actual features symbolized. Yet, even today, these simple drawings are an effective way to show the general nature of the terrain, particularly on medium- to small-scale maps. They are commonly found on recreational, advertising, and similar maps designed for the public, on which only a general impression of landform character is needed (figure 9.31).

Figure 9.30. Hillsigns on old maps, such as this section of a map of the California coastline, resemble mole hills.

Figure 9.31. A modern version of stylized hillsigns on a section of a landform map of the Willamette Valley, Oregon. Courtesy of Frederick N. Weston.

COMBINING METHODS ON MAPS Cartographers often combine different relief portrayal methods to enhance the threedimensional appearance of the terrain while also making it possible for map users to determine elevations or water depths. This work often involves a combination of absoluteand relative-relief portrayal methods, with a lot of possible combinations. We examine two here—relief shading and contours, and relief shading and hypsometric tinting.

Relief shading and contours A skillful combination of contours and relief shading is one of the most effective ways to portray absolute relief. The US Geological Survey uses this technique to produce topographic quadrangles, such as the one in figure 9.32. Producers of commercial travel and recreation atlases also create relief-shaded topographic maps for entire states, many available in digital form. Further enhancements are possible to show the terrain on maps that combine relief shading and contours. Large-scale Swiss topographic maps, as shown in figure 9.21, are noted for combining contours and relief shading with line drawings of rock outcrops and ridges. These manually produced maps are works of art, yet elevations can be determined easily from the contours, spot elevations, and benchmarks.

Figure 9.32. Relief-shaded topographic maps are produced by the US Geological Survey for certain topographic map sheets such as this 1:24,000-scale map section of the Sherman Lake, Maine, area. Courtesy of the US Geological Survey.

Figure 9.33. Both relief shading and hypsometric tinting are added to this section of a landform map of the United States. Courtesy of Tom Patterson, National Park Service.

Relief shading and hypsometric tinting It is also common to see maps that combine both relief shading and hypsometric tinting— indeed, it is unusual to find a map today that has hypsometric tinting without relief shading, primarily because both techniques can be easily achieved using computers and digital elevation models (figure 9.33). If the mapmaker can create one effect, it is easy to add the other. The challenge is to correctly assign the hypsometric tints to the elevation values. Unknowledgeable mapmakers sometimes apply a color ramp that does not mimic the typical land cover. This mis-characterization is especially problematic if the area being mapped is extremely large, such as a continent or the world. In these cases, the land cover varies greatly across elevation over the area. Although it is still possible to see the relative difference in elevation (and sometimes bathymetry), colors used for these large areas cannot possibly represent land cover accurately. Be careful to refer to the legends for maps such as these to see how the elevations are symbolized.

STEREOSCOPIC VIEWS Stereoscopy is any technique that is capable of recording three-dimensional visual information or creating the illusion of depth in a photograph, map, or other two-dimensional

image. The illusion of depth is created by presenting a slightly offset image to each eye. Many 3D displays use this method to convey depth to the reader, and maps are no exception.

Stereopairs One way that you can visualize the terrain in three dimensions is by viewing a stereopair —a set of two maps from slightly different vantage points that, when viewed together, give the impression of a three-dimensional surface (figure 9.34). This viewing process mimics the same thing our eyes do all the time. You can test it if you close first one eye and then the other—the image you see shifts slightly. Together, the two images allow you to see the world in 3D. The stereovision viewing device, called a stereoscope (or pocket stereoscope, for the portable version), is the same as that used to view photographic stereopairs, which are often used for air photo interpretation (see chapter 10 for more on air photos). For terrain, you look at a pair of relief-shaded maps or oblique-perspective views of the surface created from slightly different vantage points. When you view these two maps with a stereoscope, your mind merges them into a single 3D mental image of the terrain.

Stereoscopic polarization A more technically complex way to view the terrain stereoscopically is using a special computer monitor that alternates the two maps at least 30 times a second. With this method, called stereoscopic polarization or stereoscopic projection, the first map is displayed with horizontally polarized light and the second with vertically polarized light. Map viewers wear special goggles with polarizing filters that allow the right eye to see only the horizontally polarized map and the left eye only the vertically polarized map, so that the terrain is seen stereoscopically. Because this technique requires the maps to be seen on a computer, the viewer can also pan, zoom, and otherwise manipulate the map.

Anaglyphs A special form of stereopair is produced by printing the maps constructed from two vantage points in red and blue, and then superimposing one on the other with a slight offset. The resulting view is a special type of stereopair called an anaglyph. When you look at the anaglyph through special glasses equipped with red and blue lenses, you see the red map with one eye and the blue map with the other (figure 9.35). This viewing equipment allows you to see the map stereoscopically, so the terrain appears three-dimensional.

ChromaDepth maps Another method of stereoscopic view uses the colors in the rainbow (red, orange, yellow, green, blue, indigo, and violet) to create the impression of different heights in the display. You may have used inexpensive ChromaDepth 3D glasses to look at illustrations in comic books, children’s textbooks, and perhaps print or display maps, such as in figure 9.36. ChromaDepth 3D glasses, patented by Chromatek Inc., create a stereoscopic effect using holographic film lenses that combine refraction and diffraction of light to make each lens act like a thick glass prism. These glasses create the visual impression of certain colors in the visible spectrum being closer and others being farther from your eyes, with red in the foreground, blue in the background, and orange, yellow, and green in intermediate

positions. Terrain maps designed with a blue-to-red hypsometric-tint color ramp have the strongest 3D effect.

Figure 9.34. When you view a stereopair of relief-shaded maps stereoscopically, you see a truly three-dimensional image of the terrain. Try viewing this stereopair of Fiji with a pocket stereoscope. Courtesy of NASA/JPL-Caltech.

Figure 9.35. Anaglyph of the San Gabriel Mountains north of Los Angeles, California, created from Shuttle Radar Topography Mission data. Try viewing this map with the same kind of red-and-blue anaglyph glasses used to view 3D movies. Courtesy of NASA/JPL-Caltech.

Although stereopairs, stereoscopic polarization, anaglyphs, and ChromaDepth maps are effective visually and fun to look at, they are also somewhat impractical. They cost more to produce and require more viewing effort than standard relief portrayals. And because the relief impression is created in the brain, not on a sheet of paper or computer screen, the image is ephemeral and not subject to analytical map-use procedures. Despite these drawbacks, anaglyphs and other stereoscopic views are commonly used in certain earth sciences, such as geology. These stereoscopic views may become more prevalent in the future because they are easily produced by computer.

DYNAMIC RELIEF PORTRAYAL Computer technology makes it possible to put relief portrayal in motion. If relief portrayals from a sequence of vantage points are animated, you get the impression that you’re flying and that the terrain is passing under and around you—this phenomenon is called a flythrough. The visualization of space and movement can be so realistic that you may actually feel pangs of airsickness. This effect occurs because your mind finds it easier to accept your body moving than the terrain moving.

Figure 9.36. This hypsometric-tinted map of the Hawaiian Islands stands out vividly in 3D when viewed through ChromaDepth 3D glasses. Courtesy of Mike Bailey, Oregon State University.

Animated methods Technological advancements introduce the potential of creating animated maps that give you a sense that things are changing on them either over time or through space, or both. By viewing a motion picture that gives the impression of a camera flying over and around a region, you can gain a dramatic, dynamic impression of the landform. The effect is similar to that of walking around a physical model, such as the terrain models you saw earlier, from several positions. In both cases, you have the advantage of being able to change vantage points and, therefore, perspective (although in the cinema version, the sequence of movement is preprogrammed). In fact, animated terrain maps are commonly made by essentially rotating a movie camera over and around a physical model of the terrain using computers. Inexpensive video (television) cameras have brought new life to animated mapping in recent years. But the most important advancement has come with the advent of high-speed computers, and their high-quality video cards and high-resolution color monitors. With the

aid of computers and sufficient numeric data representing the terrain, there is no longer a need to photograph or videotape a physical model or the terrain surface itself. Instead, the images can be created from the digital database through a series of calculations that take into consideration the vantage point of the observer, the orientation of the surface, and the position of the illumination source. The images are then displayed onscreen and recorded digitally for subsequent playback. For example, scan the QR code or go to the website in figure 9.37 to fly through an animated three-dimensional perspective map of the Uracas, Makhanas, and Ahyi submarine volcanoes in the Mariana Volcanic Chain—something you could never physically do in real life.

Interactive methods Interactive maps take animated methods one step further. Although animation takes you over the terrain on the path chosen by the animator, interactive relief maps put you in the driver’s seat. You can call up any image on the screen, in any sequence. If you operate the controls, you can simulate movement realistically from one vantage point to any other. For example, simulators to train pilots and astronauts use this dynamic mapping method. You control the fly-through path, using either a mouse or a joystick with three axes of movement that are transmitted to a computer. You might start high above to get the overall view, and then fly in closer to get a better look at features of special interest. You can view the terrain from as many heights and directions as you want to get a feel for the nature of the surface. You get the sensation of flying over the terrain, with your continuously changing view, in the fly-through effect. Not only can you interactively control your movement over and around the terrain, you can also interact with data about the area that is mapped (figure 9.38). This type of interaction takes place in a series of steps. First, terrain data is displayed on an electronic screen using a relative-relief technique, such as relief shading. Second, you “point” at the location whose elevation you are seeking. The pointing may be done by tapping a touchsensitive screen or clicking a location with a mouse or similar electronic device. Next, you query the map about elevation at an indicated point or about a terrain profile along a given path. You may be able to do so using a keyboard, verbal command, or by pointing to or clicking an electronic menu of questions displayed at the side of the screen. Finally, GIS software such as ArcGIS notes the location or path you indicated, searches through the elevation data records used to make the map, performs the necessary computations, and provides you with the requested elevation data or a profile. Other datasets draped over the terrain can also be displayed and queried. Using these “point and ask” portrayals, the map merely provides a graphic version of the landform, and you direct analytical questions to the underlying digital terrain data or overlying draped data. Essentially, the map serves as a window into the data. By not burdening the map with the need to portray absolute-relief information, the quality of visualization can be improved. At the same time, elevation values can be determined with greater precision than is possible with a static graphic portrayal alone.

Figure 9.37. Still image of an animated fly-through of the Uracas, Makhanas, and Ahyi submarine volcanoes in the Mariana Volcanic Chain. Scan the QR code or go to http://oceanexplorer.noaa.gov/explorations/04fire/background/marianaarc/media/nw_ uracas2ahyi_video.html to view the animation. Courtesy of NOAA Ocean Explorer.

Figure 9.38. When a map displayed on an electronic screen is queried about the elevation at a point, the computer can search the database used to make the map to get the answer. Courtesy of MicroDEM.

DIGITAL ELEVATION MODEL DATA Most of the relief portrayal methods discussed in this chapter are now carried out in computer mapping software that relies on digital elevation model (DEM) data. DEM data is in raster format—a raster consists of a matrix of cells (or pixels) arrayed in rows and columns (a grid) in which each cell contains a value that represents something at the grid cell location. DEM data represents a continuous elevation surface, and all cells in the grid contain elevation values that are referenced to a common datum. People sometimes confuse DEMs with digital terrain models (DTMs) and digital surface models (DSMs). DTMs are a type of DEM in which the elevation surface is a representation of the bare earth. DSMs are data models that depict the elevation of the top of all surface features, such as buildings and vegetation. Some people think of DTMs as digital representations of attributes that relate to a topographic surface, such as slopes, gradients, aspects, and horizontal or vertical land surface curvature (see chapter 16 for more on these topographic attributes).

The level of detail represented by a raster is largely dependent on the cell size, or spatial resolution, of the data. Spatial resolution refers to the dimension of the cell size that represents the area covered on the ground. Therefore, if the area covered by a cell is 5 x 5 meters, the resolution is 5 meters. The higher the resolution of a raster, the smaller the cell size and, thus, the greater the detail. The cell must be small enough to capture the required detail but large enough so that computer storage and analysis can be performed efficiently. More features, smaller features, or greater detail in features can be represented by a raster with a smaller cell size. However, more is not often better. Smaller cell sizes result in larger raster datasets to represent an entire surface; therefore, there is a need for greater storage space, which often results in longer processing time. For example, for a given area, changing cells to one-half the current size requires as much as four times the storage space, depending on the type of data and data storage techniques used. These relationships are summarized in figure 9.39.

Figure 9.39. Differences between smaller and larger cells relationships to resolution, storage space, and processing time.

sizes

and

their

If they have done things correctly, cartographers create their relief portrayal maps from data at the same or higher spatial resolution than the ground resolution of the map display grid cells. The ground resolution of the map display grid cells depends on the scale of the map. An equation you can use that relates optimal map scale expressed as a representative fraction (1/x), DEM cell resolution, and map display resolution is in equation (9.1):

so that one centimeter on the map represents x centimeters on the ground. For example, the optimal map scale for a 30-meter (3,000-centimeter) DEM mapped at 100 pixels/cm is 1/200,000. DEM data is tremendously important in modern mapping and map use because computers can easily carry out computations on a matrix of DEM data to create maps of the terrain in seconds. As you saw earlier, relief-shaded, hypsometric-tinted, and perspectiveview maps can now be made easily using DEM data. To use these maps appropriately, it is useful to take a closer look at the nature and sources of some of the DEM data used to make the maps.

ETOPO5, ETOPO2, ETOPO1, and GLOBE Many government and private organizations are involved in creating DEM data. You can download Earth topographic ETOPO5, ETOPO2, or ETOPO1 global DEM data distributed by the US National Oceanic and Atmospheric Administration (NOAA) National Geophysical Data Center (NGDC) (http://www.ngdc.noaa.gov/mgg/global/global.html), as well as world maps of land and ocean topography made from this data. The ETOPO5 DEM is called an equal-angle grid because it is at a spatial resolution of five minutes (called arc minutes) in latitude and longitude, which means that each cell is given the average elevation of its 5´ × 5´ quadrilateral on the ground (four-ninths the area of a USGS 1:24,000-scale topographic map sheet). (An arc degree is the spherical distance along the earth's surface measured in angular units of latitude or longitude. One arc degree is divided into 60 arc minutes, and one arc minute is divided into 60 arc seconds.) The ETOPO2 equal-angle DEM has a higher spatial resolution of 2´ × 2´, and the ETOPO1 DEM, with 1´ × 1´ cells, is the dataset that is most used currently for global terrain mapping. The NGDC also distributes finer resolution GLOBE 30-arc-second DEM data for the world’s land areas (http://www.ngdc.noaa.gov/mgg/topo/gltiles.html). The USGS distributes a similar GTOPO30 grid. The GLOBE grid is global in extent, but grid cells in oceans are given an elevation of −500 meters, which means “no available data.” Because of the convergence of meridians (explained in chapter 1), these equal-angle grid cells vary in shape and area, from large, essentially square cells at the equator to tall, thin cells near the poles. Consequently, equal-angle grids normally are not displayed in their raw form, but are transformed using a map projection such as the equirectangular, Mollweide elliptical, or Lambert azimuthal equal-area map projections (see chapter 3 for descriptions of these projections). Near the equator, however, grid cells are close enough to square that each grid cell can be faithfully displayed as a square pixel. For example, in figure 9.40, grid cells from ETOPO5, ETOPO2, ETOPO1, and GLOBE DEMs covering a five-by-five-degree quadrilateral in west-central Ecuador are mapped at a display resolution of 200 pixels per inch. The terrain detail progressively increases as the spatial resolution of the grid increases. You would be correct to assume that accuracy of elevations at particular points increases with finer spatial resolution because elevations within smaller quadrilaterals on the ground are being averaged. You might also guess that ETOPO5 data is good to use for a world map, ETOPO2 and ETOPO1 data for continental maps, and GLOBE data for pageor screen-size national maps.

Shuttle radar topography mission

The highest resolution worldwide DEM is the three-arc-second dataset created from Shuttle Radar Topography Mission (SRTM) data. The SRTM space shuttle mission was an international project led by the US National Geospatial-Intelligence Agency (NGA) (formerly the US National Imagery and Mapping Agency [NIMA]) and the US National Aeronautics and Space Administration (NASA). The SRTM radar device, carried on the space shuttle Endeavor from February 11 to 22, 2000, acquired enough data during its 10 days of operation to create a near-global high-resolution database of the earth’s topography. The device had two radar antennae, one in the shuttle’s payload bay and the other on the end of a 200-foot (60-meter) mast that extended from the payload bay once the shuttle was in orbit. These antennae collected radar return data used to determine land elevations. The shuttle’s orbit allowed data to be collected between 60° N and 56° S latitudes. Elevations for land areas within these latitudes were determined within a spatial resolution of around 150 by 150 feet (50 by 50 meters). The horizontal accuracy of grid cells is within 60 feet (20 meters) of their true position, and the vertical accuracy of the elevations is plus or minus 32 feet (10 meters) of their true average elevation.

Figure 9.40. These continuous hypsometric-tinted maps of a five-by-five-degree quadrilateral covering west-central Ecuador (0° to 5° S and 77° W to 82° W) are created at a display resolution of 200 pixels per inch from the ETOPO5, ETOPO2, ETOPO1, and GLOBE DEMs. The beginning and ending columns and rows that span the quadrilateral are shown along the upper and left edges of the map margin. The area in the box on the GLOBE map is shown in figure 9.41. Courtesy of the National Geophysical Data Center.

You can download SRTM three-arc-second resolution data (90-by-90 meter cells at the equator) in 5° × 5° tiles for any land area within the latitude range of the shuttle’s orbit by going to http://earthexplorer.usgs.gov. The continuous hypsometric-tinted map in figure 9.41, for example, was created at a display resolution of 100 pixels per inch from a 0.5° ×

0.5° segment, 1 percent the area of an SRTM tile. Notice the tremendous increase in spatial detail compared with the GLOBE map segment of the same area. Elevation data at a onearc-second (approximately 30-by-30-meter) resolution can be downloaded for the continental United States. Maps of interesting terrain features are made from this higher resolution data. Most maps have oblique-perspective views, often draped with satellite images of similar spatial resolution such as the Landsat Thematic Mapper scene in figure 9.27.

Figure 9.41. This continuous hypsometric-tinted map of the 0.5° × 0.5° quadrilateral in west-central Ecuador, outlined in figure 9.40, is created at a display resolution of 100 pixels per inch from the SRTM global three-arc-second resolution DEM. Courtesy of the National Geophysical Data Center.

National Elevation Dataset In addition to high-resolution SRTM data, you can obtain DEM data for the United States from the USGS National Elevation Dataset (NED) website (http://ned.usgs.gov). The

NED was created by merging the highest resolution, best-quality elevation data available across the United States into equal-angle grids that cover the continental United States, Hawaii, Alaska, and the US island territories, such as Puerto Rico and Guam. The grid resolution is one arc second, except for Alaska’s two-arc-second dataset. The shaded relief maps in figure 9.42 illustrate the terrain detail that is visible at these two spatial resolutions. The NED is not a static product, but is updated bimonthly to incorporate the best-available DEM data. Currently, a 1/3-arc-second (approximately 10-by-10-meter) resolution grid is being assembled that eventually will seamlessly cover the continental United States.

Figure 9.42. These two shaded relief maps of Mount Saint Helens, Washington, illustrate the terrain detail inherent in the NED one- and two-arc-second DEMs. Courtesy of the USGS National Elevation Dataset.

Coastal Relief Model The Coastal Relief Model distributed by the NGDC over the Internet (http://www.ngdc.noaa.gov/mgg/coastal/crm.html) is the first DEM of the US coastal zone. The DEM extends from the coastal state boundaries to as far offshore as NOS hydrographic data supports a three-arc-second grid of the seafloor. This seaward limit reaches beyond the continental slope in many areas. The Coastal Relief Model contains data for the entire coastal zone of the conterminous United States, including Hawaii and Puerto Rico. A 24-arc-second (approximately 750-by-750-meter) grid for southern Alaska and the Aleutian Islands is also available (http://www.ngdc.noaa.gov/mgg/coastal/s_alaska.html), and eventually the Great Lakes and surrounding state coastal areas will be included as well. The bathymetric map shown in figure 9.43 is made from Coastal Relief Model data for the Central California coast, including the seafloor just west of Monterey Bay. You can easily see submarine features such as Monterey Canyon and marine avalanches.

Figure 9.43. This seafloor map made from NGDC Coastal Relief Model data shows submarine features such as Monterey Canyon and marine avalanches in a onedegree quadrilateral just west of Monterey Bay, California. Courtesy of the National Geophysical Data Center.

Lidar Light detection and ranging (lidar) is a remote-sensing system that uses rapid pulses of laser light striking the surface of the earth and features on its surface to determine their elevations. The lidar laser pulse emitter and detector is mounted in the bottom of a remote-sensing device, usually on an airplane, along with a GPS receiver used to determine the aircraft’s horizontal position and an altimeter to record the aircraft’s

altitude. Part of each laser light pulse is reflected or scattered back to the detector. The distance from the detector to the spot where the pulse was reflected is determined by measuring the time it takes for each light pulse to travel to the spot and be reflected back, because the pulse return time is proportional to its distance from the detector. Lidar systems can time and record the distances for more than one return per pulse—for example, if the pulse continues through a terrestrial feature, such as vegetation. First, second, and third pulse returns may be from the forest canopy, understory, and bare ground in a wooded area, respectively. In urban areas, the first pulse return is used to measure the elevations of the building rooftops and tree canopies, whereas the last return is assumed to be from the ground surface. Lidar datasets can be extremely large, often hundreds of thousands or even millions of points per square mile. The distance values for these points are processed mathematically to create high-resolution DSMs of small areas, such as the one-meter resolution DSM for the portion of West Linn, Oregon, displayed as a continuous layer-tinted and relief-shaded map in figure 9.44. The subtle differences in elevation allow you to see streets, buildings, and excavations on the map.

Figure 9.44. Streets, excavations, and building tops are visible in this continuous hypsometric-tinted and relief-shaded map made from one-meter resolution lidar data covering part of West Linn, Oregon. Courtesy of the City of West Linn, Oregon.

SUMMARY As you can see, you can learn about relief in the landscape in many ways from maps. These relief portrayals are important to map users. We not only find them intriguing, but they also help us move beyond the fundamental two-dimensional nature of a map. Because we experience relief in our environment every day, relief portrayal on maps is something that we are eager to pay attention to and try to learn from. You may view the portrayal of relief differently from the other symbols and elements on the map because it is something that you can directly relate to your own experience. For this reason, you may also be more apt to find fault with, or shortcomings in, the way that relief is portrayed on maps. Through airline travel and images we see from space, we understand better than previous generations the three-dimensional nature of our world. Cartographers have tried innovative ways to capture this third dimension, through such clever approaches as bump mapping, anaglyphs, and ChromaDepth maps. With the use of computers, fly-throughs allow an animated view of the landscape and, through interaction, users can now decide where and how high to fly over a digital terrain surface. More data at higher resolutions helps cartographers portray relief, and relief portrayal methods are being applied to places that have not been fully mapped before, such as ocean bottoms and other planets. Our understanding of the world around us, and even other worlds, is expanding through the techniques that mapmakers use to help us understand elevation on maps. With a greater understanding of the techniques they use, you have a better ability to read the relief they portray on maps.

SELECTED READINGS Baldock, E. D. 1971. “Cartographic Relief Portrayal.” International Yearbook of Cartography 11:75–78. Barnes, D. 2002. “Using ArcMap to Enhance Topographic Presentation.” Cartographic Perspectives 42 (Spring). Castner, H. W., and R. Wheate. 1979. “Reassessing the Role Played by Shaded Relief Methods in Topographic Scale Maps.” The Cartographic Journal 16:77–85. Curran, J. P. 1967. “Cartographic Relief Portrayal.” The Cartographer 4 (1) (June): 28–38. Eyton, R. J. 2005. “Unusual Display of DEMs.” Cartographic Perspectives 50:7–23. Grotch, S. L. 1983. “Three-Dimensional and Stereoscopic Graphics for Scientific Data Display and Analysis.” IEEE Computer Graphics and Applications 3 (11): 31–43. Hurni, L., B. Jenny, T. Dahinden, and E. Hutzler. 2001. “Interactive Analytical Shading and Cliff Drawing: Advances in Digital Relief Presentation for Topographic Mountain Maps.” In Proceedings of the 20th International Cartographic Conference. ICC, Beijing, China. Imhof, E. 2007. Cartographic Relief Presentation. Translated and edited by H. J. Steward. Redlands, CA: Esri Press. Irwin, D. 1976. “The Historical Development of Terrain Representation in American Cartography.” International Yearbook of Cartography 16:70–83. Jenny, B. 2001. “An Interactive Approach to Analytical Relief Shading.” Cartographica 38 (1&2) 67–75. Kennelly, P. 2002. “Hillshading with Oriented Halftones.” Cartographic Perspectives 43:25–42. Kennelly, P., and A. J. Stewart. 2006. “A Uniform Sky Illumination Model to Enhance Shading of Terrain and Urban Areas.” Cartography and Geographic Information Science 33 (1): 21–36.

Kraak, M. J. 1993. “Cartographic Terrain Modeling in a Three-Dimensional GIS Environment.” Cartography and Geographic Information Systems 20:13–18. Kumler, M. P. 1995. “An Intensive Comparison of Triangulated Irregular Networks (TINs) and Digital Elevation Models (DEMs).” Toronto: University of Toronto Press. Lobeck, A. K. 1958. Block Diagrams and Other Graphic Methods Used in Geology and Geography, 2nd ed. Amherst, MA: Emerson-Trussel. Patterson, T. 1997. “A Desktop Approach to Shaded Relief Production.” Cartographic Perspectives 28:38–40. Petrie, G., and T. J. M. Kennie, eds. 1991. Terrain Modelling in Surveying and Civil Engineering. New York: McGraw-Hill. Price, W. 2001. “Relief Presentation: Manual Airbrushing Combined with Computer Technology.” The Cartographic Journal 38 (1). Robinson, A. H., J. L. Morrison, P. C. Muehrcke, A. J. Kimerling, and S. C. Guptill. 1995. “Portraying the Land-Surface Form.” In Elements of Cartography, 6th ed., 527–48. New York: John Wiley & Sons. Ryerson, C. C. 1984. “Relief Model Symbolization.” The American Cartographer 11 (2): 160–64. Schou, A. 1962. The Construction and Drawing of Block Diagrams. London: Thomas Nelson & Sons. Tait, A. 2002. “Photoshop 6 Tutorial: How to Create Basic Colored Shaded Relief.” Cartographic Perspectives 42 (Spring). Watson, D. F. 1992. Contouring. New York: Pergamon. Yoeli, P. 1983. “Digital Terrain Models and Their Cartographic and Cartometric Utilization.” The Cartographic Journal 20 (1): 17–23.

chapter ten IMAGE MAPS AERIAL IMAGERY Black-and-white imagery True-color imagery Color-infrared imagery Low-altitude imagery High-altitude imagery Unmanned aerial systems Geometric distortions on aerial imagery ORTHOPHOTOS AND ORTHOPHOTOMAPS Orthophotos Digital orthophoto quadrangles Annotated orthophotomaps SATELLITE IMAGES Weather satellites Earth resources satellites Landsat ASTER MODIS SPOT Submeter-resolution systems Hyperspectral systems Hyperion Satellite image map examples DYNAMIC IMAGE MAPS ArcGIS Explorer ArcGIS Earth SELECTED READINGS

10 Image maps Although people primarily rely on their own eyes to learn about the environment, indirect experience is also important. Reports of distant places have traveled by word of mouth and other forms of communication since the beginning of time. But such methods of gathering data remotely provide information for only specific sites. In the last century, the significance of collecting data and images of the earth (and other planetary bodies) from a distance, called remote sensing, has grown enormously. We are now inundated with a vast array of remotely sensed images of our surroundings. Remote sensing lets you observe features in the environment by using cameras or other electronic imaging instruments (sensors) that are sensitive to the energy emitted or reflected from objects. A camera is an optical device in which light from a scene enters an enclosed chamber through a lens and is projected onto a light-sensitive medium that instantaneously creates an entire image. In the past, light-sensitive photographic film that was processed using traditional photochemical developers was the medium, and the processed image was called a photograph, or photo for short. Today, digital cameras have supplanted their film-based predecessors. In a digital camera, the imaging medium is most often a charge-coupled device (CCD)—a small twodimensional array of tiny light-sensitive detectors whose output is stored as digital numbers in an image file. The resulting digital image is termed a digital photograph, or digital photo for short. Images can also be created by noncamera digital sensors, which are generally mechanical scanners or electro-optical devices that build up an image line by line instead of instantaneously. The maps made from this digital image data are called image maps. Therefore, the more general terms “image” and “image map” are more appropriate to remote sensing today, although we will continue to use the familiar terms “photograph” and “photo” in this chapter for images created by traditional film or newer digital cameras. In addition to distinguishing images by the type of sensor used to collect the images, we can categorize images by the sensor platform (the vessel, craft, or instrument that the sensors are carried on). In this chapter, we talk primarily about two different types of sensor platforms—aircraft that operate within the earth’s atmosphere (up to about 10 miles or 16 kilometers above the earth) and spacecraft in orbits from 300 to 700 miles (500 to 1,000 kilometers) above the earth.

Some images come directly from the sensor. Others represent computer manipulations of the data recorded by the sensor. These images may contain artifacts that are characteristic of the photo-chemical or electronic processes involved. Many factors can influence the appearance of the resulting images, including the sensor’s vantage point, the sensor’s spectral sensitivity, the sensor’s technical quality, the image’s spatial resolution, and atmospheric conditions. Although remote-sensing images are excellent for showing many aspects of the environment, they may fail to depict others. Intangible features, such as political boundaries, aren’t picked up on photographic or scanner images unless physical features in the landscape happen to follow them or they happen to align with physical features. Such useful map aids as geographic names, reference grids, map scale indicators, and orientation indicators are absent from images. Features on images aren’t classified and identified in a key or legend. For all these reasons, remote-sensing images are often made more interpretable and useful by cartographic enhancement, with symbols for point, line, and area features, as well as text for labels. These symbols are laid over the image base, producing an image map. As with conventional maps, the image map may then be draped on a 3D terrain model, as you will see later in this chapter. This chapter examines the ways in which the images for these maps are obtained. We look first at aerial imagery (images of the ground taken with a camera from an aircraft, satellite, or other remote platform). We review how they can be digitally processed into geometrically corrected orthophotos (images in which distortion from the camera tilt and topography are removed). We then examine images and image maps that are produced by a variety of satellite sensing systems, and conclude by looking at dynamic image maps and animated image map display software.

AERIAL IMAGERY The electromagnetic spectrum is broad and rich in environmental information that can be sensed throughout its range. In the past several decades, we’ve learned to use images obtained simultaneously in different portions of the electromagnetic spectrum that are called spectral bands, each of which provides valuable insights into the nature of our surroundings. With the development of cameras and photographic films beginning in the early 1800s, the visible-light portion of the electromagnetic spectrum was imaged as shades of gray. Later, photographic films not only came close to duplicating the color-sensing capability of the human eye but also extended our imaging capability into the nearultraviolet and near-infrared portions of the spectrum at both ends of the eye’s sensitivity range (figure 10.1). In addition to broadening our view of the environment, these different films, when combined with filters for blocking unwanted wavelengths, made it possible to image specific bands of visible-light energy, such as blue, green, and red, as well as the near-visible light energy, such as ultraviolet (UV) and infrared (IR). Nonphotographic sensors, discussed later in this chapter, expand our imaging capability into the mid-infrared (mid-IR) (2–8 microns [mm]) and thermal-infrared (thermalIR) (8–14 mm) portions of the electromagnetic spectrum.

Figure 10.1. Wavelength range (in microns) for the near-ultraviolet to near-infrared portion of the electromagnetic spectrum.

These advancements in imaging capability are coupled with the development of platforms that can be used to remotely image the environment. Airplanes, towers, balloons, and even birds can be used as remote-sensing platforms. When the image is taken from above the ground with a camera, the result is traditionally called an aerial photograph, or air photo for short. What is imaged on an air photo depends on several factors. One is the type of camera used. To take air photos for mapping purposes, aerial photographers use the familiar kind of camera that produces individual pictures called frames. In the past, aerial mapping cameras (figure 10.2), specially designed to expose large, nine-by-nine-inch (23-by-23centimeter) frames on photographic film, were used. This larger film frame made it possible to image a larger area in detail on the photo. Complex, expensive camera lenses were used to minimize geometric distortion from lens defects. The mapping camera was placed in a gyrostabilized mount (one that stabilizes side-to-side motion), which allowed it to be pointed vertically downward at the ground.

Figure 10.2. A traditional aerial mapping camera placed vertically in an airplane. Courtesy of the Washington State Department of Transportation.

Digital mapping cameras that record photos electronically rather than on film have all but replaced traditional aerial mapping cameras. The digital mapping camera is large and uses the same high-quality lenses as aerial mapping cameras, but the imaging is done on a two-dimensional CCD array, not photographic film. The CCD is similar to that used in your own digital camera, except that the array has thousands more rows and columns. Rolls of exposed film are replaced by a digital image memory device that is hundreds of gigabytes in size and capable of storing more than a thousand black-and-white or color digital photos. Composed of pixels that have integer values that represent gray tones or colors, these digital photos are equal in quality to air photo frames on photographic film. Large-scale air photos (around 1:20,000) are obtained by flying the aircraft at around 10,000 feet (3,000 meters) above the ground along flight lines (paths that aircraft follow), typically in the north–south or east–west direction (figure 10.3). Photos are taken along the flight line so that there is a 60 percent to 80 percent overlap (duplicated image of the ground in two successive air photos). Typically, more than a single flight line is required to cover the area to be mapped, and adjacent flight lines are planned with a 20 percent to 30 percent sidelap to ensure that there are no gaps in the coverage.

Black-and-white imagery Conventional (analog) black-and-white air photo negatives are based on a film emulsion that records electromagnetic radiation in the 0.3 to 0.7 mm visible spectrum. Because of its sensitivity to visible light, black-and-white film is often called panchromatic (meaning “all

colors”) film. The shorter (0.3 to 0.4 mm) near-ultraviolet (near-UV) wavelengths are scattered by the atmosphere, requiring a UV haze filter placed over the camera lens to increase the clarity of the photo. These filters correct for UV effects, which cause photos to look bluish and modify other colors, and eliminate haze, caused by dust particles in the air, from the photos. Positive photographic prints, as shown in figure 10.4, are made from the film negatives, or, more commonly today, from the digital image data for each frame. The greater the amount of visible light reflected from an object and gathered by the camera lens, the lighter its tone on the final photographic print or the higher its numeric pixel value in the digital photo. Soil moisture content, surface roughness, and the natural color of objects all influence a feature’s tone on a black-and-white film or digital photo. Moist soils, marshlands, and newly plowed fields tend to be darker than surrounding features, whereas human constructions such as roads and buildings tend to appear lighter. Historically, black-and-white air photos have been put to many mapping uses. Until recently, the US Forest Service used them to make timber inventories and map national forests. The US Geological Survey used them in the production of topographic maps in its 7.5 minute (1:24,000-scale) quadrangle series until the series was completed in 1992, but with map revisions continuing through the 1990s. The US Natural Resources Conservation Service, the primary federal agency that works with private landowners to help them conserve, maintain, and improve their natural resources, used these photos as the basis for mapping soils and agricultural activities. They also have been widely used for transportation, recreation, and land-use planning purposes. Indeed, by far the largest amount of remote sensing in past decades was done using black-and-white aerial photography.

Figure 10.3. Adjacent air photos are taken to ensure 60 percent to 80 percent overlap along the flight line and 20 percent to 30 percent sidelap between adjacent flight lines to ensure completely overlapping ground coverage.

Figure 10.4. Photographic print (reduced from the original) made from a nine-bynine-inch black-and-white negative air photo of downtown Chicago, Illinois. Courtesy of USGS NAPP.

True-color imagery For a long time after its development in the 1930s, true-color film was little used in mapping. Color film was more expensive than conventional black-and-white film, partly because color film contains three separate emulsion layers sensitive to the blue, green, and red portions of the visible spectrum. Special film-processing requirements added further to the cost. In addition, early color film had poor resolution relative to panchromatic film.

The resolution and clarity of color film has improved dramatically, so it is now comparable in quality to black-and-white film. Color photos are usually easier to read than black-andwhite photos because they capture the colors associated with landscape features, such as the terra-cotta color of the tile rooftops in figure 10.5. It is easier to distinguish subtle differences in color rather than shades of gray. Color is especially useful in revealing the condition of objects, such as the stage of a crop in its maturation (phenological) cycle. For such applications as vegetation and soils classification, geologic mapping, and surface water studies, using color photos is simpler and more accurate than working from the equivalent black-and-white images. True-color imagery (the color that a photograph would show) is rapidly growing in importance for understanding the environment, and the extra cost associated with color images is primarily related to printing on paper.

Figure 10.5. A true-color air photo print from color film, reduced in size from the nine-by-nine-inch original, of central Raleigh, North Carolina. Courtesy of the North Carolina Department of Transportation.

The US National Agriculture Imagery Program (NAIP) is the major source of digital true-color photos for federal and state agencies as well as the general public. It obtains imagery during the agricultural growing seasons in the conterminous United States. The default spectral bands for the color photos are red, green, and blue, or RGB, but some states are provided four bands of digital photo data (RGB and near-infrared). For information about the NAIP, go to http://www.fsa.usda.gov/programs-and-services/aerialphotography/imagery-programs/naip-imagery.

Color-infrared imagery During World War II (WWII), military researchers developed a special film called colorinfrared (CIR) film that was sensitive to near-infrared (near-IR) wavelengths (0.7 to 0.9 mm) as well as to visible light (figure 10.6). Some of the highest reflectance surfaces in the near-IR wavelength region are the leaves and needles of plants. Generally speaking, the healthier the vegetation, the higher the near-IR reflectance. This property turned out to be extremely useful for the military during WWII, because on a photograph produced with CIR film, artificial camouflage materials with low near-IR reflectance can be distinguished from live, healthy vegetation. Because CIR film was used initially for this purpose, it is sometimes called camouflage detection film. However, it is commonly known as CIR film because the spectral sensitivities of the dye layers in the film bear no relation to the natural colors of environmental features—the blue, green, and red emulsion layers record green, red, and near-IR energy, respectively.

Figure 10.6. A CIR air photo print from CIR film, reduced in size from the nine-bynine-inch original, of central San Diego, California. Courtesy of USGS NAPP.

An environmental feature that absorbs a lot of near-IR energy, such as clear water, appears black on CIR photos. Features such as buildings and unhealthy vegetation absorb less near-IR energy and appear blue or blue gray. The most obvious feature, aside from water, on CIR imagery is healthy vegetation, which appears bright magenta or red rather than green because of its high near-IR reflectance. The cut vegetation, green paint, and rope netting used to camouflage military installations are recorded in pinkish to bluish tones, in stark contrast to the background of bright reds produced by the surrounding healthy vegetation. Although the first important applications of CIR photos were in military reconnaissance, many other uses have been found. For vegetation studies, CIR imagery is now used for such mapping applications as crop inspection, tree growth inventories, and damage assessment of diseased flora. Plant diseases can often be detected on CIR photos well before they become visible to the unaided eye. Geologists find CIR photos useful in

mapping near-surface structural features such as faults, fractures, and joints. These features can be detected because they often collect water, encouraging lusher vegetation growth than in the surrounding area. CIR imagery also enhances boundaries between soil and vegetation and between land and water, making it useful in mapping these features. In addition, CIR imagery is valuable for urban mapping because it shows a sharp contrast between vegetation and cultural features and because near-IR wavelengths penetrate smog easily.

Low-altitude imagery Detailed low-altitude air photos are taken anywhere from just above the ground to around 1,500 feet (500 meters) above the surface. These low-altitude images (figure 10.7) usually cover a small ground area at a large map scale (see chapter 2 for more on map scale). Towers and balloons provided convenient sensing platforms in the 1800s, but these devices were superseded in the 20th century by light aircraft and helicopters, and more recently by unmanned aerial vehicles, discussed later in this chapter.

Figure 10.7. Low-altitude black-and-white air photo of the Memorial Union at Oregon State University, Corvallis, Oregon, taken from an altitude of around 1,000 feet (300 meters). You can see students on the grass in front of the building. Courtesy of the City of Corvallis, Oregon, Public Works Department.

Acquiring images at higher altitudes, from 1,500 to 10,000 feet (500 to 3,000 meters), provides less environmental detail because the image scale is much smaller. But this type of low-altitude aerial imagery has the advantage that a single photo can cover more ground area. Black-and-white aerial photography of the type used to obtain basic elevation data for topographic mapping is usually taken from an altitude of 10,000 feet, almost two miles (three kilometers) above the ground (figure 10.8, top). These photos are available in a standard nine-by-nine-inch frame format. Systematic coverage of the United States in this format began in the 1930s and has been repeated in many areas at intervals of 5 to 10 years. Other economically advanced countries have similar national air photo acquisition programs.

High-altitude imagery If there are advantages to taking pictures from two miles above the ground, then why not even higher? In 1987, the National Aerial Photography Program (NAPP) was established to develop a cloud-free air photo database of consistent scale and orientation and high image quality. The aim of this federal and state program, which ended in 2007, was to provide complete coverage of the United States, updated every five to seven years. For this high-altitude imagery, primarily black-and-white aerial photographs (but also CIR photos) were taken from 20,000 feet (6,000 meters) along north–south flight lines through the west and east halves of USGS 7.5-minute topographic maps. Ten nine-by-nineinch photos at a 1:40,000 scale, each covering a five-by-five-mile (eight-by-eight-kilometer) area on the ground, were needed for complete stereoscopic coverage of each topographic map (see chapter 9 for more on stereoscopic views). Many of these photos are scanned into digital image format that you can download. Go to https://lta.cr.usgs.gov/NAPP for more information on obtaining NAPP imagery, including index maps that show the location of available imagery. NASA ex-spy planes, such as the U-2 and SR-71 military reconnaissance aircraft, routinely took high-altitude photographs from altitudes as high as 10 miles (16 kilometers). The National High Altitude Photography (NHAP) program, which the US Geological Survey operated from 1980 to 1989, acquired black-and-white panchromatic and CIR photography of the conterminous United States taken from these aircraft. Each blackand-white aerial photograph, at a 1:80,000 scale, was taken from an altitude of 40,000 feet (13,000 meters) and centered on a USGS 7.5-minute topographic map. CIR air photos at a 1:58,000 scale were taken at the same time as the black-and-white photos (for more on obtaining NHAP imagery, go to https://lta.cr.usgs.gov/NHAP).

Figure 10.8. Portions of low-altitude (top) and high-altitude (bottom) black-and-white air photos of Corvallis, Oregon (both reduced in scale from the original), taken at altitudes of around 10,000 and 40,000 feet (3,000 and 13,000 meters), respectively. The black box drawn on the high-altitude photo outlines the area covered by the lowaltitude photo. Courtesy of USGS NAPP and NHAP.

An illustration of the quality of this high-altitude aerial photography is provided by the photo centered on the Corvallis, Oregon, topographic map, part of which is shown in figure 10.8, bottom. For most purposes, this photo is equivalent in geometry to the standard topographic quadrangle of the area, although the photo shows far more ground detail. The advantage of high-altitude photography is that a single photo can cover a large ground area. The high-quality photo coverage has many mapping applications. The damage

caused by earthquakes, floods, and droughts, for instance, can be quickly monitored and assessed. Forest resources, snow cover, crop yields, and many other environmental features can also be studied over seasonal and longer periods. Unmanned aerial systems Unmanned aerial vehicles (UAV), also known as uncrewed reconnaissance vehicles or, more commonly, drones, are capable of operating without a pilot onboard. They use a radio control link to a control center called a ground control station, and they can be preprogrammed for both flight and payload (the vehicle’s load—for example, cargo, sensors, and explosives, for military use) operations prior to launch. Because of the higher degree of sophistication now required to operate a UAV, these vehicles are now referred to as unmanned aerial systems (UAS). The electronics systems and sensors on a UAS consist of GPS and other instruments that constantly gather and transmit information about the UAS’s location and orientation. The UAS pilot uses this and other information to fly the UAS and control its imaging system, adjusting the camera to follow features of interest. The sensors instantly transmit image data, such as video streams, to the ground control station, where image interpreters can not only view the data but also create still images and mosaick them into image maps of larger areas. Mapping applications of drone imagery seem limitless, but we focus on two recent studies that illustrate this potential. The first is an agricultural application, in which large-scale truecolor images (figure 10.9, top) were collected from a 400-foot (120-meter) altitude over a vineyard in Texas in an effort to detect insect infestations on individual grapevines as well as assess the vines’ overall health. Plant-by-plant maps of plant health were made from the analysis of the drone imagery. The second study is a stream restoration project at a historic coal mining town in New Mexico (figure 10.9, bottom). A drone took a large number of overlapping digital photos covering the town site. The digital data for each photo was computer processed to obtain a point cloud of x,y,z values (the dots at the bottom of the image) similar to lidar data, discussed at the end of this chapter. This data was used to create a topographic model and oblique-perspective map of the stream restoration project area, as well as to map the heights of vegetation in the area.

Geometric distortions on aerial imagery On vertical aerial photographs, the image scale most likely varies radially away from its center. To provide a simple way to determine the photo center, aerial cameras create fiducial marks. Fiducial marks are small registration marks exposed on the film at the midpoint of each photo edge (figure 10.10). If you draw straight lines between opposite pairs of fiducial marks, these lines intersect at the center of the image, called the principal point. The point on the ground that is directly below the camera when the air photo is taken is called the nadir. The principal point on the image is the same as the nadir on the ground only when the aircraft is flying parallel to the ground so that the camera is truly taking a vertical photograph.

Figure 10.9. Digital imagery from fixed-wing drones used in an agricultural study of vineyards (top) and in a stream restoration project (bottom) involving creation of a topographic model of the site. Courtesy of Esri.

The primary reason for paying attention to radial scale variation on vertical photos is that objects of different heights are displaced radially away from the principal point of the image. The camera lens creates a central-perspective view of the ground, as you would see if you looked straight down at the ground from the sensor platform. Relief displacement is the apparent “leaning out” of the top of a higher object on a vertical photo. If the top of a feature is higher than the elevation of the nadir, it will be displaced outward and imaged at a slightly larger scale. For example, in figure 10.11, the sides and tops of tall buildings in downtown

Chicago, south of the river and along the lakeshore, are displaced outward and radially away from the principal point (PP) of the image, in the lower-left corner of the figure. The geometry underlying this radially outward pattern of relief displacement is illustrated in figure 10.12. Notice the relation between the horizontal position of the six buildings on the ground at different distances from the nadir and their associated appearance on the air photo. The greater the height of the buildings relative to the ground and the farther the features are from the principal point of the photo, the greater the relief displacement. This pattern of relief displacement also holds true for hills, valleys, and any other topographic feature that varies in elevation.

Figure 10.10. Format of a National Aerial Photography Program vertical air photo with the principal point and fiducial marks identified. Courtesy of USGS NAPP.

Figure 10.11. Northeast quarter of a high-altitude vertical air photo of Chicago, Illinois, showing relief displacement of buildings away from the principal point (shown in red by a plus mark (+) and the letters PP in the lower-left corner of the photo). Courtesy of USGS NAPP.

Figure 10.12. Relief displacement of features in central-perspective air photos (or digital images) can be seen in this side view of buildings shown horizontally on the ground relative to the nadir and their appearance on the photo.

The amount of scale variation and relief displacement on remote-sensing images is also influenced by the height of the camera above the ground and by the sensor platform tilting the camera from vertical when the image is taken, called camera tilt. The higher the flying height, the less the relief displacement and scale variation. Thus, vertical photos taken with cameras by space shuttle astronauts exhibit so little relief displacement and scale variation that they can be overlaid on topographic maps, with the only geometric discrepancies coming from the earth’s curvature. In contrast, scale change and relief displacement may be large on low-altitude vertical photos, in which features such as tall buildings and trees can appear to lean outward from the center of the photo. If the region is hilly or the camera is tilted when the image is taken, scale variation across the photo may

be so great that it is difficult or impossible to use it for mapping purposes. In such cases, these scale variations must first be eliminated (if possible) from the photos.

ORTHOPHOTOS AND ORTHOPHOTOMAPS Orthophotos Scale variation because of camera tilt in an aerial photograph—either taken on film or stored as a digital photo—can be removed by physically altering the geometry of the photo. The process of changing the perspective from tilted to central is called photo rectification and is relatively simple using optical equipment or computer software. Rectified air photos still contain relief displacement, however, because of differences in the heights of features. To remove relief displacement, you must turn the centralperspective photo into an orthophoto. An orthophoto is an air photo that has a uniform scale and is geometrically corrected to remove scale variation caused by camera tilt and differences in elevation. Rather than look radially outward from the principal point to each feature on an air photo, with an orthophoto, you look directly down on the landscape. Thus, all features on an orthophoto appear in their true planimetric position. Creating an orthophoto from aerial photography that covers an area is a mathematically demanding process carried out by digital computers. Before air photos of the area are acquired, ground control points, which are determined by traditional surveying (see chapter 1 for more on horizontal control points) or collected by high-accuracy GPS receivers, must be identified and marked on the ground so that they will be visible on the photos. These control points are used to rectify the photos for aircraft tilt, so that each photo is oriented vertically to the ground and to true or grid north (see chapter 13 for definitions of true and grid north). The mathematical corrections are made directly on digital photos, or on digital images of traditional photographic film air photos. In the latter case, the film photo is scanned to create a black-and-white or color digital image that comprises several thousand rows and columns of picture elements (pixels). Correction of scale difference and relief displacement in the center section of each photo is performed with the aid of DEM data covering the area (see chapter 9 for more on DEM data). Special computer software is used to geometrically match each digital image as best as possible to the DEM. The software then repositions each pixel on the digital image to remove both relief displacement and scale variation (figure 10.13). If in color, these planimetrically correct photo center sections are then color corrected to minimize the tonal and color differences between photos. Finally, the color-corrected photo center sections are pieced together into a planimetrically correct orthophoto that is constant in scale and placed on a map projection, such as the transverse Mercator (see chapter 3 for more on this map projection), used for the UTM grid coordinate system (see chapter 4 for more on this grid coordinate system) that covers the area. What cannot be corrected are the sides of the radially displaced buildings seen on the original photography—only the base of each building is in the correct geographic position.

Digital orthophoto quadrangles Orthophotos are such an important component of image mapping technology that federal agencies in the United States cooperated with state and local governments, as well as the private sector, to create them for the nation. Orthophotos are used to create digital

orthophotoquads (DOQs) that combine the image characteristics of the orthophoto with the geometric qualities of a USGS topographic quadrangle. The original orthophotos created by the US Geological Survey, beginning in 1987, are based on 1:40,000-scale black-and-white vertical aerial photographs processed, as described in the previous section, to remove scale variations and relief displacements. These orthophotos were then scanned to create digital images with pixels of approximately one-meter spatial resolution and used to produce USGS DOQs.

Figure 10.13. Relief displacement and scale variation in the air photo of Tenth Legion, Virginia (left), are removed in the orthophoto of the same area (right). On the orthophoto, the planimetrically corrected power line running over the hills appears crooked on the air photo because of relief displacement. Courtesy of the US Geological Survey.

Two types of DOQs are produced by the US Geological Survey. Full-quad DOQs cover the area of a 1:24,000-scale topographic quadrangle—7.5 minutes latitude by 7.5 minutes longitude. These DOQs are available for most of Oregon, Washington, and Alaska, with limited coverage available for other states. Quarter-quad DOQs, at a scale of 1:12,000, cover one quarter of a 1:24,000-scale topographic quadrangle, an area that covers 3.75 minutes latitude by 3.75 minutes longitude (figure 10.14). Quarter-quad DOQ coverage for the entire United States was completed in 2004. DOQs are available from a USGS online interactive map service called EarthExplorer (http://earthexplorer.usgs.gov) to use on your computer, tablet, or other mobile device. Also available are higher resolution black-and-white, true-color, and CIR orthophotos of major metropolitan areas, US state capitals, and the national capital, as shown in the segment of the Washington, DC, true-color orthophoto in figure 10.15. These orthophotomaps serve as

a uniform, high-quality base for a variety of mapping, geographic study, and planning activities.

Annotated orthophotomaps An orthophoto provides environmental detail that cannot be portrayed with conventional map symbols for point, line, and area features because you can see all the features in the landscape that are closest to the sensor (tops of trees, buildings, and so on). This level of detail contrasts with the advantages of a conventional map, on which the cartographer uses symbols, text, selection, classification, generalization, and other techniques of cartographic abstraction (see chapter 6 for more on these techniques) to highlight important features, patterns, and themes. However, mapmakers can overlay conventional map symbols on an orthophoto to create an annotated orthophotomap that emphasizes selected geographic information about the area. Annotated orthophotomaps are produced at different levels of graphic sophistication for a variety of map uses, as the four examples in figure 10.16 illustrate.

Figure 10.14. Section of a 1:12,000-scale USGS quarter-quad DOQ covering the northeastern part of San Francisco, California (reduced from original size for display). The straight streets running over hills are planimetrically correct. Courtesy of the US Geological Survey.

Figure 10.15. Segment of the USGS high-resolution true-color orthophoto of the Capitol Mall area in Washington, DC. Courtesy of the US Geological Survey.

Figure 10.16. Segments of annotated orthophotomaps used for a tourist map (upper left), street map (upper right), cadastral map (lower left), and forest management map (lower right). Historical feature boundary map from Ministry of Land, Infrastructure, and Transport of Japan; cadastral and forest management map courtesy of Esri.

On the tourist map (figure 10.16, upper left) that shows part of Okayama, Japan, including the Okayama Castle and Korakuen Garden, the mapmaker overlaid blue lines of different thicknesses and Kanji text on a color orthophoto to highlight the extent of the original castle and garden area. On the Monroe, Oregon, street map (figure 10.16, upper right), the mapmaker overlaid white lines of constant thickness over all streets on the orthophoto, and then placed black street names over the white lines. Gray lines are placed over rivers and streams, and a transparent gray tone is laid over the city park. School symbols and names are also added, as are railroads. The horizontal and vertical solid black lines are part of an arbitrary grid cell locator system, such as discussed in chapter 4.

The orthophotomap (figure 10.16, lower left) comes from Lithuania’s national cadastre system (see chapter 5 for more on the cadastre). Cadastral maps that are downloadable from its website are used for displaying land parcel ownership and taxation. Point, line, and area symbols as well as labels are overlaid on the orthophoto to identify buildings, property boundaries, and corresponding tax lot numbers. Major streets and their names, as well as water bodies, are also shown with conventional map symbols to give map users additional locational information. The fourth example (figure 10.16, lower right) is an annotated orthophotomap used for natural resource management by the State of Virginia Department of Forestry. Different characteristics of forest land parcels are shown by overlaying transparent colors on the parcels in the orthophoto and using white text to note each parcel identifier and acreage.

SATELLITE IMAGES We mentioned previously that the higher the altitude of the sensor platform, the less the relief displacement and scale variation on the images collected. Images obtained by satellites orbiting the earth are at high enough altitudes that relief displacement of tall features, such as mountains, is minimal, and the only scale variation on the image is because of the earth’s curvature. This scale variation can be corrected mathematically through the same computer processing software used to create orthophotos from aerial photography, and the resulting orthoimage (an image that is processed to remove geometric distortions) can be in any map projection (see chapter 3 for more on map projections). We use the term “orthoimage” to refer to a satellite image that is corrected geometrically. Conventional map symbols can then be overlaid on the orthoimage to create a satellite image map. Numerous satellites collect images of the earth and other planetary bodies, but we focus on two major types—weather satellites and earth resources satellites.

Figure 10.17. A satellite in a geostationary orbit moves easterly in the equatorial plane at an altitude of 22,236 miles (35,786 kilometers), with a 24-hour orbital period that matches the earth’s 24-hour rotation. A large portion of the earth’s surface can be imaged from this high altitude.

Weather satellites

If you are in the United States, your local television or newspaper weather report probably uses one or more weather satellite images created from Geostationary Operational Environmental Satellite (GOES) system imagery. Europeans have the Meteosat system, with Meteosat-8, -9, and -10 located over Africa and Meteosat-7 over the Indian Ocean. Russia operates the Geostationary Operational Meteorological Satellite (GOMS) (also known as Elektro) positioned over the equator south of Moscow, and China operates the Fēngyún geostationary satellites. India also operates geostationary satellites that carry instruments for meteorological purposes. Satellites can operate in several types of earth orbit. The most common orbits for environmental satellites are geostationary and polar. A satellite in geostationary orbit is always in the same position with respect to the rotating earth. This orbit is also called the Clarke orbit because science fiction author Arthur C. Clarke first proposed the idea of using a geostationary orbit for communications satellites in 1945. Because a geostationary orbit must be in the same plane as the earth’s rotation, it is always in the equatorial plane (figure 10.17). By orbiting in the same easterly direction as the earth, at an altitude of 22,236 miles (35,786 kilometers) with a 24-hour orbital period that matches the earth’s rotation, the satellite appears stationary (synchronous) to ground receiving stations. The most frequently obtained imagery from space is from weather satellites, such as GOES West and GOES East. Positioned in a geostationary orbit above fixed longitudes of 135° W and 75° W, respectively, the GOES West and GOES East satellites obtain the images of the western and eastern United States that are seen on television weather reports, in newspapers, and on mobile devices (figure 10.18). These two satellites record images of the western and eastern United States every 15 minutes and the full disk (nearly a hemisphere) every three hours. A single GOES image covers millions of square miles of ground surface, making it possible to see the cloud patterns over half the United States simultaneously. By studying images taken every 15 minutes, you can easily see the detailed movement of clouds. This information is especially useful for monitoring severe rain- and snowstorms, as well as hurricanes and typhoons. This imagery is used to produce weather system videos that are commonly seen on TV weather broadcasts. Video clips are also made of the atmospheric effects of natural disasters such as volcanic eruptions and large wildfires (figure 10.19). The resolution of ground detail on weather satellite imagery is particularly poor, however, because of its extremely high altitude. Consequently, the use of the imagery, often draped over a terrain model, is essentially restricted to broadscale atmospheric phenomena.

Figure 10.18. Examples of full NOAA GOES West (left) and GOES East (right) satellite images. Courtesy of the National Ocean Service.

Weather satellite image maps for the conterminous United States are made by mosaicking the GOES West and GOES East images collected in the visible and thermal-IR regions of the electromagnetic spectrum, and in a mid-IR spectral region that is tuned to water vapor. Mosaicking is the process of combining multiple images to produce a single image of the area covered by the multiple images. Minimal geographic information (graticule lines, state boundaries, coastlines, and large lakes) is added to help you orient the map. The GOES visible satellite image map in figure 10.20 (top) shows the amount of visible sunlight that is reflected back to the satellite sensor by clouds, the land surface, and water bodies. Thicker clouds reflect more light than thinner clouds, so they appear lighter on the image. However, you cannot distinguish among low-, middle-, and high-level clouds on GOES visible images, which is important for predicting the location and intensity of precipitation. The thermal-IR weather satellite image map (figure 10.20, middle) distinguishes low-, middle-, and high-level clouds by a white-to-blue color progression on the image. Warmer objects appear white and colder objects are blue on the image, so wispy cirrus clouds at higher elevations or more vertically developed cumulonimbus (thunderstorm) clouds with colder tops appear blue. On the GOES water vapor image map (figure 10.20, bottom), the darker brown the color, the less water vapor in the atmosphere. The yellow to blue cloud plumes indicate moistureladen thunderstorms, whereas the other clouds in the area, which are white in color, contain less moisture. One of the most striking differences between the visible and water vapor images is seen in the dark-brown band of dry air from Tennessee to New York on the right side of the water vapor image (bottom). The visible image (top) shows this area as cloudy, but the clouds must be the high-altitude, thin, cirrus type for the air to be low in water vapor.

Figure 10.19. GOES West still frame from a video clip showing the westerly movement of smoke from October 2007 wildfires in Southern California. To view the video clip, scan the QR code at the lower left or go to http://s3.amazonaws.com/akamai.netstorage/qt.nasa-global/ccvideos/jpl/goes20071024-large2.mov. Courtesy of NASA.

Earth resources satellites Multispectral remote-sensing devices are used to capture a ground scene in different spectral bands of the electromagnetic spectrum so that the resulting images are geometrically identical. Different bands can be combined to produce images that emphasize separate characteristics of the environment. Over the past three decades, earth-orbiting satellites have become the most important multispectral sensing platforms. If a coarseresolution sensor is used, images are acquired from several hundred miles above the earth’s surface and cover a relatively large ground area. If a high-resolution multispectral imaging device is used, a small ground area is recorded at a high spatial resolution (see chapter 9 for more on spatial resolution). Most images of ground scenes come from what are called earth resources satellites—satellites launched with the primary mission of

providing systematic, repetitive environmental image data, such as surface reflectance, wave height, surface temperature, or land elevation.

Figure 10.20. Composite GOES weather satellite visible (top), thermal-IR (middle), and water vapor (bottom) image maps of the conterminous United States allow you to see the clouds (visible and thermal-IR) and moisture conditions (water vapor) at various times of day or night. Courtesy of the National Ocean Service.

Most earth resources satellites are in near-polar orbits, circling the earth at a nearpolar inclination (the angle between the equatorial plane and the satellite orbital plane—a true polar orbit has an inclination of 90 degrees). A sun-synchronous orbit is a special type of near-polar orbit in which the satellite passes over the same part of the earth at

roughly the same local time each day. This regularity enables data collection at consistent times as well as long-term image comparisons. Near-polar orbiting satellites provide a nearly complete view of the earth built up over many orbits, because they orbit over all but the highest latitudes, which is not the case for geosynchronous satellites. We examine imagery obtained from remote-sensing devices on four earth resources satellites commonly used for image maps—Landsat, ASTER, MODIS, and SPOT. We then look at imagery from several newer submeter-resolution sensors, as well as from the Hyperion hyperspectral sensing device. Landsat In 1965, NASA began the Landsat project to explore the potential for monitoring the earth from space. The first satellite, Landsat 1, was placed in orbit in 1972. Subsequent successful launches in the Landsat series took place in 1975 (Landsat 2), 1978 (Landsat 3), 1982 (Landsat 4), 1984 (Landsat 5), 1999 (Landsat 7), and 2013 (Landsat 8). Landsat 7 and 8 are currently in operation (Landsat 6, launched in 1993, failed to reach orbit), with onboard remote-sensing instruments collecting hundreds of images of the earth every day. These images provide an excellent orthoimage base for satellite image maps that are made worldwide for applications in agriculture, geology, forestry, regional planning, global change research, nautical charting, and other fields. Landsat satellites are in sun-synchronous, near-polar orbits so that near-global coverage is possible because of the earth’s rotation beneath the satellite (figure 10.21). Landsat satellites 7 and 8 orbit the earth every 99 minutes from an altitude of 440 miles (709 kilometers), circling the earth roughly 14 times per day and passing over the same spot on earth at the same time of day every 16 days. On each descending orbit (the daylight pass from north to south), remote-sensing instruments record electromagnetic radiation from the ground over a strip that is 115 miles (185 kilometers) wide. This data is used to create Landsat images covering a ground area that is 115 miles wide by 105 miles high (185 by 170 kilometers).

Figure 10.21. The sun-synchronous, near-polar orbits of Landsat 7 and 8 make it possible to provide nearly complete coverage of the earth’s surface in 115-mile-wide (185-kilometer) swaths every 16 days.

Most image maps today are made from images acquired by the Enhanced Thematic Mapper Plus (ETM+) sensor on Landsat 7 or the Operational Land Imager (OLI) sensor on Landsat 8. The more recent and widely used OLI sensor provides 30-meter resolution images from the blue (bands 1 and 2), green (band 3), red (band 4), near-IR (band 5), and shortwave-IR (bands 6, 7, and 9) portions of the electromagnetic spectrum, as well as two 100-meter resolution thermal-IR bands (bands 10 and 11) and a 15-meter resolution panchromatic band (band 8). Pixel values for the more than 7,000 rows and

7,000 columns that make up an image in each band are transmitted to ground receiving facilities, at which the data is processed into geometrically corrected images. You can download these images from the USGS EarthExplorer website (http://earthexplorer.usgs.gov) in the UTM map projection (see chapter 3 for more on this projection). You can see on the right side of figure 10.22 how images of the same area appear differently in 10 of the 11 bands. Band 8 is not used in this image.

Figure 10.22. Images from bands 1 through 11 from the Landsat 8 OLI sensor provide different information about the environment, as seen in this portion of a Landsat 8 image showing Fort Collins, Colorado, and vicinity. Bands 3, 5, and 7 are displayed in blue, green and red, respectively, to create this true-color image map. Courtesy of NASA’s Goddard Space Flight Center.

Figure 10.23. Landsat OLI bands 2, 3, and 4 and bands 1, 5, and 7 are used to create 30-meter resolution true-color (top) and false-color (bottom) composite images centered on San Fernando Valley, California. Image data courtesy of the US Geological Survey.

Three of the 11 Landsat OLI bands are often combined into color-composite images. The combination you use depends on what you want to see in the image. For example, the color-composite image of Fort Collins, Colorado (figure 10.22), combines bands 3, 5, and 7 into a true-color image. The Landsat 8 composite image in figure 10.23 (top), centered on California’s San Fernando Valley, is a true-color combination of the blue, green, and red bands (2, 3, and 4), whereas the false-color composite image, in which the colors differ from those in a photograph, at the bottom is a combination of the blue, near-IR, and shortwave-IR (1, 5, and 7) bands.

ASTER ASTER (Advanced Spaceborne Thermal Emission and Reflection Radiometer) is a Japanese sensor on board the Terra satellite launched in 1999 into a 438-mile (705-kilometer) sun-synchronous orbit. ASTER provides 15- and 90-meter resolution, 37-mile-wide (60-kilometer) images of the earth in 14 different bands, ranging from visible to thermal-IR. The visible and near-IR bands are imaged using a backwardlooking sensor to create stereopairs, from which surface elevations can be determined (see chapter 9 for more on stereopairs). ASTER data is used to create detailed image maps of land surface reflectance, temperature, and elevation. These image maps are used in studies of vegetation and ecosystem dynamics, hazard monitoring, geology and soils, climatology, hydrology, and land-cover change, as well as in the creation of DEMs. The boundary between the Imperial Valley of California and Mexico is clearly seen in figure 10.24, a CIR 15-meter resolution image created from ASTER visible and near-IR bands.

Figure 10.24. The boundary between California’s Imperial Valley (top) and Mexico (bottom) is clearly seen in this CIR 15-meter resolution image created from ASTER visible and near-IR bands. Courtesy of NASA/GSFC/MET1/Japan Space Systems, and US/Japan ASTER Science Team.

MODIS Moderate Resolution Imaging Spectroradiometer (MODIS) sensors on the Terra (along with the ASTER sensor) and Aqua satellites, launched in 1999 and 2002 respectively, provide a complete view of the earth’s surface every one to two days from a 438-mile (705-kilometer) altitude. The two MODIS sensors collect data from 36 spectral bands ranging from 0.4 µm (blue) to 14.4 µm (thermal-IR). Bands 1 and 2 (red and near-IR) have a 250-meter spatial resolution at the nadir, bands 3 through 7 (blue through mid-IR)

have a 500-meter resolution, and bands 8 through 36 (blue through thermal-IR) have a 1kilometer resolution. The 2,330-kilometer swath allows a large portion of the earth to be imaged on each orbit at a coarser spatial resolution than other earth resources satellite sensors. Images such as that of the Hawaiian Islands in figure 10.25 give you a more general view of land and water features and earth phenomena. Scientists use this data and images as a means to quantify land surface characteristics such as land-cover type and extent, snowcover extent, surface temperature, and forest fire occurrence. The one- to two-day coverage period for the entire earth means that changes in these surface characteristics can be studied week to week or at monthly, seasonal, or annual time intervals. SPOT Landsat, ASTER, and MODIS aren’t the only earth resources satellite systems. Other countries have also contributed to the “commercialization of space.” The French space agency CNES, with the participation of Sweden and Belgium, has launched a series of land resources satellites known as the Systeme Probatoire d’Observation de la Terre (SPOT), operated by the SPOT Image Corp. SPOT 1 was launched in 1986, followed by SPOT 2 in 1990, SPOT 3 in 1993, SPOT 4 in 1998, SPOT 5 in 2002, SPOT 6 in 2012, and SPOT 7 in 2014. SPOT 5, 6, and 7 are still in operation. SPOT 6 and 7 are placed in sunsynchronous near-polar orbits at an altitude of 694 kilometers (431 miles) so that highresolution, 60-by-60-kilometer vertical-perspective images can be obtained for the same area on the earth every 26 days.

Figure 10.25. MODIS true-color images of the Hawaiian Islands and surrounding ocean created from blue, green, and red spectral bands at one-kilometer spatial resolution. Courtesy of Jacques Descloitres, MODIS Land Rapid Response Team at NASA GSFC.

It is possible to obtain SPOT 5 black-and-white, true-color, and CIR images at spatial resolutions of 2.5 or 5 meters. These SPOT images are created digitally from green, red, and near-IR data collected at 10-meter resolution and panchromatic data collected at 2.5- or 5-meter resolution. Pan-sharpened true-color and CIR images are created by using the panchromatic band data to define the shape and texture of features in the images, and then coloring the image with the coarser-resolution green, red, and near-IR band data. True-color and CIR SPOT 6 and 7 images at 1.5-meter resolution are made by combining 1.5-meter resolution panchromatic band data with 6-meter resolution green, red, and near-IR band data in a similar way. Sections of SPOT bands 5, 6, and 7 images from different parts of the earth are shown in figure 10.26 to illustrate the urban features distinguishable for Shaanxi, China (5-meter resolution, upper left); Canberra, Australia (2.5-meter resolution, upper right); Doha, Qatar (1.5-meter resolution, lower left); and Sydney, Australia (1.5-meter resolution, lower right). Satellite image maps can be made with SPOT images at progressively larger map scales, from 5-, 2.5-, and 1.5-meter resolution images like these, because all images are geometrically corrected to be constant in scale on a map projection surface.

Figure 10.26. True-color and CIR SPOT 5, 6, and 7 image examples at three spatial resolutions: Shaanxi, China, with SPOT 5 imagery at 5-meter resolution (upper left); Canberra, Australia, with SPOT 5 at 2.5-meter resolution (upper right); and Doha,

Qatar, with SPOT 6 imagery (lower left), and Sydney, Australia, with SPOT 7 imagery, (lower right), both at 1.5-meter resolution. Copyright CNES/SPOT Image Corp.

Submeter-resolution systems Since 1999, commercial satellite systems such as IKONOS, QuickBird, GeoEye-1, Pléiades, and WorldView-3 have provided digital images from space for paying customers at the same resolution as low-altitude aerial photography. Table 10.1 lists these five systems in order of launch dates, from 1999 to 2014, and shows their key characteristics. All are aboard satellites in sun-synchronous, near-polar orbits at altitudes ranging from 482 kilometers (300 miles) to 770 kilometers (479 miles). All systems have a blue-to-near-IR wavelength panchromatic sensor, as well as a blue, green, red, and near-IR band multispectral sensing device. The spatial resolution at nadir has improved over time for both panchromatic and multispectral sensors, so it is now possible to obtain 0.31-meter resolution panchromatic and 1.24-meter resolution multispectral image data from the WorldView-3 system. The five systems have pointable sensors, giving them the ability to collect images away from vertical (off-nadir). Pointable sensors allow virtually all locations on Earth to be imaged every few days, in swaths that vary from 11 kilometers (6.8 miles) for the IKONOS satellite to 20 kilometers (12.4 miles) for the Pleiades 1A and 1B satellites. Small portions of the earth can be imaged each day, ranging from 200,000 square kilometers (the area of Nebraska) for the QuickBird satellite to 1,000,000 square kilometers (slightly less than the combined area of Texas and California) for the Pleiades satellite constellation. The collage in figure 10.27, of IKONOS, QuickBird, GeoEye-1, and WorldView-3 pansharpened color images, shows how different locations on the earth appear as the spatial resolution progressively increases. As the spatial resolution increases, you can see buildings, roads, vehicles, athletic fields, and other features in greater detail (look for players on the tennis courts in the WorldView-3 image). Also look for the relief displacement of buildings in the IKONOS image—this displacement is your clue that the image is taken considerably off-nadir (the nadir is to the north of Sydney). Table 10.1 Submeter satellite remote-sensing systems

Hyperspectral systems Hyperspectral imaging systems are similar to multispectral systems in that digital image data related to energy reflected from or emitted by ground features is collected in different spectral bands. The difference is that hundreds of narrow bands are imaged so that the spectrum of visible through thermal-IR energy for each pixel in the scene is recorded.

Features have unique spectral signatures, which are the differences in their reflectance characteristics with respect to wavelengths in the visible through thermal-IR spectrum. Digital image processing software is used to identify features in the image by matching the values in different bands for each pixel with known spectral signatures for different types of features.

Figure 10.27. IKONOS 0.80-meter resolution image of downtown Sydney, Australia (upper left); QuickBird 0.65-meter image of central Los Angeles, California (upper right); GeoEye-1 0.46-meter image of Olympic Stadium in London, England (lower left); and WorldView-3 0.31-meter image of a Madrid, Spain, sports complex (lower right). All are pan-sharpened color images courtesy of Satellite Imaging Corp. You can view these and additional images from the satellite systems we have discussed so far at http://www.satimagingcorp.com/gallery. Courtesy of DigitalGlobe.

You can visualize the hundreds of bands that compose a hyperspectral image as a cube of images, with each sheet as an image of one band, stacked top to bottom from shortest to longest wavelength. The cube in figure 10.28 graphically represents 224 bands from an airborne hyperspectral sensor called AVIRIS. The top of the cube is actually not a spectral band, but rather a false-color image of the Moffett Field Airport and vicinity in the San Francisco Bay Area. Hyperion

In 2000, the Hyperion hyperspectral sensor was launched on the Earth Observing-1 (EO-1) satellite in the same orbit as Landsat 7, trailing it by about 50 kilometers (30 miles) so that near-simultaneous images can be acquired from the two sensors. The Hyperion sensor breaks the visible to mid-IR spectrum, from 0.4 to 2.5 µm, into 220 bands. Each 30meter resolution image is a long strip that covers a 7.5 × 100 kilometer (4.7 x 62 mile) ground swath. The two Hyperion images in figure 10.29 show how careful selection of spectral bands with narrow wavelength ranges can improve the imaging of a particular location or event such as an active wildfire. Hyperion images have a wide variety of uses in addition to wildfire response, such as monitoring volcanic activity, tracking the amount of carbon that plants take out of the atmosphere, and mapping surface minerals by their chemical composition.

Satellite image map examples Now we look at examples of satellite image maps that are produced from images collected by the earth resources satellites we have discussed. Like image maps made from aerial imagery, conventional map line work, area coloring, and lettering are commonly overlaid on geometrically corrected earth resources satellite images that serve as a locational basemap for the qualitative or quantitative information added to the image map. Although we can’t possibly inventory all the types of annotated satellite image maps, we can take a closer look at some representative samples. We focus on the three examples in figure 10.30, which give you an idea of the variety of annotated image maps produced today. The image map in figure 10.30 (top) is a segment of a Benton County, Oregon, satellite image road map. The mapmaker makes the county look green by printing the image from a single Landsat Thematic Mapper band in green, instead of black-and-white, ink. By using one color for the image, roads, railroads, and lettering can be printed in black, rivers and the wildlife refuge can be shown with translucent gray tones, and US Public Land Survey System section lines (see chapter 4) can be overlaid in white.

Figure 10.28. Hyperspectral cube shows 224 spectral bands from the AVIRIS sensor. Bands are shown with shortest (top) to longest (bottom) wavelength along the cube edges. Courtesy of NASA JPL.

Figure 10.29. Parts of two Hyperion image strips obtained October 23, 2007, show wildfires just south of Escondido, California. The true-color image on the left is a composite of red, green, and blue bands—the image on the right is a composite of shortwave IR bands selected to penetrate smoke plumes to provide a clear picture of the burning fires. Courtesy of NASA.

The section of the flood extent map for the Kirulo, Bulgaria, area (figure 10.30, middle) is an example of an annotated image map made by overlaying a transparent light-blue color for the flooded area on a SPOT CIR composite base image. You can see the flooded features on the ground because they appear to be covered by clear-blue water. Also, the normal course of the river is shown in dark blue, allowing you to see the degree to which the river overflowed its banks.

The map in figure 10.30 (bottom) shows an algal bloom that covered Lake Atitlán, Guatemala. The extent of the algal bloom in the lake, shown in yellow to red colors, was determined by digitally processing the same Hyperion data that was used to make the falsecolor background image of the lake and surrounding land shown on the map. A further enhancement to satellite image maps is achieved by draping the images over a DEM-created surface (see chapter 9 for more on relief portrayal) to create an obliqueperspective satellite image map (figure 10.31). Thematic data can also be overlaid on this base, as well as conventional map symbols. This example shows a satellite image map of Cuba and Florida overlaid with precipitation extent and height data for Hurricane Katrina. By showing the precipitation data as a 3D surface over the satellite image, meteorologists were able to better visualize the predicted path and three-dimensional structure of the storm as it neared land.

Figure 10.30. These three satellite image maps illustrate how mapmakers overlay conventional map symbols on satellite images to create annotated image maps. The top map is based on a Landsat Thematic Mapper band 2 image; the middle map has a SPOT CIR image as a base; and the bottom map is made from three Hyperion spectral bands. SPOT 5 CIR image is an extract from a rapid-mapping flood impact map that covers Kirulo, Bulgaria, and is derived from SPOT imagery for the International Charter Call 94, covering the flood on June 12, 2005. © Sertit 2005. The Hyperion algal bloom map is of an algal bloom that covered Lake Atitlán, Guatemala.

Figure 10.31. This oblique-perspective view of the satellite image map of Cuba and Florida is overlaid with Hurricane Katrina data that shows the horizontal distribution and vertical height of precipitation on August 25, 2005. Courtesy of NASA.

DYNAMIC IMAGE MAPS Dynamic image maps go beyond traditional static displays to include animated sequences of images that you can view using interactive navigation tools. In addition, these maps often allow you to interact with map symbols that you see in the animation, symbols linked to the text, pictures, or short video clips. You have probably used a dynamic image map system, such as ArcGIS Explorer or ArcGIS Earth, that gives you the feeling of flying over and around a part of the earth. You are likely to see fly-throughs (see chapter 9 for more on fly-throughs) in various visual presentations, such as news programs, training simulators, and video games. We begin by looking at a widely used program, ArcGIS Explorer (https://www.arcgis.com/explorer/), which illustrates how interaction and animation can be applied to image maps you can create instantly on your computer.

ArcGIS Explorer

ArcGIS Explorer allows you to view imagery, topographic maps, physical features, shaded relief, historical maps, street maps, and other maps and images for the entire world seamlessly at different levels of detail. You can combine these global maps and images with your own local cartographic data to create custom maps. You can also perform various map analysis tasks on the maps and images using ArcGIS functions such as determining what is visible from a particular point (see chapter 16 for more on visibility analysis). Incorporating these ArcGIS functions with a dynamic image map system allows you to answer geographic questions about the maps you generate and share the results with others.

ArcGIS Earth You may have taken virtual flights to exotic locales by viewing ArcGIS Earth animated satellite and orthophoto images. Developed by Esri, ArcGIS Earth is a 3D visualization application that allows you to view large amounts of GIS data on a virtual globe (a 3D software model or representation of the earth) (figure 10.32). Like the Microsoft Bing Maps image in figure 10.33, ArcGIS Earth works by continually superimposing digital images of the earth’s surface on an oblique-perspective map projection of a surface derived from DEM data (see chapter 3 for more on oblique-perspective projections). The DEM allows you to see the imagery of the earth in three-dimensional perspective at different verticalexaggeration levels that you can select. The three-dimensional perspective image map in figure 10.33 is one frame from a Bing Maps fly-through that you can take over Manhattan. Flight controls superimposed on the image map allow you to continuously change the viewing distance and horizontal view direction. When virtually flying over large cities, such as New York, you may also be able to turn on and off a 3D display of building outlines that gives you a general idea of how the buildings would look from your ever-changing viewpoint. In ArcGIS Earth, you can also add your own data to the image map and make it available to others on the web. The resulting virtual fly-through is superimposed with map symbols that show your own data. Many companies and organizations take advantage of this do-ityourself capability, greatly expanding the types of information you can look at and query using these apps. You can mimic this capability in your own web apps as well.

Figure 10.32. ArcGIS Earth initial image map. You can scan the QR code at the lower right or go to http://esri.com/software/arcgis-earth to learn more about what you can see and do with ArcGIS Earth. Courtesy of Esri.

Figure 10.33. This 3D bird’s-eye view map of Manhattan, New York City, from Bing Maps allows a virtual fly-through of the city, with buildings, streets and highways, and other features labeled. © 2015 Microsoft Corp.

The possibilities for dynamic image maps seem limitless, as data about the earth is being collected at an accelerating rate using devices such as GPS to quickly and accurately determine the ground positions of features. These geographically referenced points, lines, and areas are increasingly being stored and manipulated in GIS that, in turn, links the data to dynamic image map systems such as ArcGIS Explorer and ArcGIS Earth. These dynamic image map systems can be combined with spherical display systems such as OmniGlobe (figure 10.34) because any digital image map made on an equirectangular map projection (see chapter 3 for more on this projection) can be displayed on the globe. Such truly threedimensional map displays may revolutionize how you use and benefit from both traditional and image maps.

Figure 10.34. This 60-inch (1.5-meter) diameter OmniGlobe digital spherical display can project any global map or image, or a sequence of image maps, on a spherical screen so that the earth’s geometry is correctly shown everywhere. To see the kinds of conventional and image maps that can be displayed, scan the QR code at the lower right or go to http://www.arcscience.com. Courtesy of Arc Science Simulations.

The future of image maps looks bright and interesting. We have only discussed the aerial photography and satellite imagery that is most commonly used today to create image maps, but many other meteorological, earth resources, and military reconnaissance satellites are, and will continue to be, placed in orbit. Both static and dynamic image maps will continue to be made from imagery that is collected by the diverse and ever-expanding suite of remotesensing devices.

SELECTED READINGS Barrett, E. C., and L. F. Curtis. 1992. Introduction to Environmental Remote Sensing, 3rd ed. London: Chapman & Hall. Blom, J. D. 2010. “Unmanned Aerial Systems: A Historical Perspective.” Occasional Paper 37. Fort Leavenworth, KS: Combat Studies Institute Press. Campanella, R. 1996. “High-Resolution Satellite Imagery for Business.” Business Geographics (March): 36–39. Campbell, J. B., and R. H. Wynne. 2011. Introduction to Remote Sensing, 5th ed. New York: Guilford. Carleton, A. M. 1991. Satellite Remote Sensing in Climatology. Boca Raton, FL: CRC. Chuvieco, E., ed. 2008. Earth Observation of Global Change: The Role of Satellite Remote Sensing in Monitoring the Global Environment. New York: Springer Science. Ciciarelli, J. A. 1991. Practical Guide to Aerial Photography: With an Introduction to Surveying. New York: Van Nostrand Reinhold. Cook, W. J. 1996. “Ahead of the Weather: New Technologies Let Forecasters Make Faster, More Accurate Predictions.” US News & World Report (April 29): 55–57. Corbley, K. P. 1997. “Applications of High-Resolution Imagery.” Geo Info Systems (May): 36–40. Corbley, K. P. 1997. “Multispectral Imagery: Identifying More than Meets the Eye.” Geo Info Systems (June): 38–43. Dickinson, G. C. 1979. Maps and Air Photographs, 2nd ed. New York: John Wiley & Sons. Evans, D. L., E. R. Stofan, T. D. Jones and L. M. Godwin. 1994. “Earth from the Sky.” Scientific American (December): 70–75. Falkner, E. 1994. Aerial Mapping. Boca Raton, FL: CRC. Hamit, F. 1996. “Where GOES has Gone: NOAA’s Weather Satellite Imagery and GIS-Marketed.” Advanced Imaging (November): 60–64. Kramer, H. J. 2002. Observation of the Earth and Its Environment: Survey of Missions and Sensors. New York: Springer-Verlag. Lillesand, T. M., R. W. Kiefer, and J. W. Chipman. 2008. Remote Sensing and Image Interpretation, 6th ed. New York: John Wiley & Sons. Newcome, L. R. 2007. Unmanned Aviation: A Brief History of Unmanned Aerial Vehicles. Barnsley, UK: Pen and Sword. Office of Technology Assessment, US Congress. 1993. The Future of Remote Sensing from Space: Civilian Satellite Systems and Applications. Washington, DC: US Gov. Printing Office. Paine, D. P., and J. D. Kiser. 2012. Aerial Photography and Image Interpretation. Hoboken, NJ: John Wiley & Sons.

chapter eleven MAP ACCURACY UNCERTAINTY AND ERROR BIAS PRECISION AND ACCURACY TYPES OF MAP ACCURACY Positional accuracy Attribute accuracy Conceptual accuracy Logical consistency Temporal accuracy Currency Mapping period Elapsed time Temporal stability SOURCES OF ERROR Factual error Data source error Natural-variation error Processing error COMMUNICATING ACCURACY UNCERTAINTY Metadata Symbols and notations Legend notes Reliability diagram LIABILITY ISSUES SELECTED READINGS

AND

11 Map accuracy Maps are often regarded as being authoritative and accurate, with nary a second thought. The precise line work, careful selection of colors, exacting placement of text and symbols all lead to the impression of maps being an information source that you can trust as being error-free. This confidence in maps leads us to make sometimes life-or-death decisions on the basis of the map in front of us. Upon closer inspection, however, you will find that all maps have error and uncertainty—it’s simply a part of the mapmaking process. It is your responsibility as the map reader to understand how design decisions made by the mapmaker affect the reliability of the map you are using. In this chapter, we explore the factors that contribute to error and inaccuracy in maps so that you are better prepared to recognize and deal with them. The accuracy needed for a map can only be determined by knowing its intended use. What matters most is the relationship between the decisions that people try to make with the map and the quality of the information that is available to them. For example, in daylight, a pilot requires that a map be accurate only to within several miles, while in fog at night, its accuracy must be much higher. Map accuracy is a complex topic that involves several related concepts, including uncertainty, error, bias, precision, and quality. Although these sometimes confusing terms are often used interchangeably, understanding the subtle differences among them can help you use maps more judiciously and make maps more proficiently.

UNCERTAINTY AND ERROR Uncertainty is the degree to which the mapped value for a feature is estimated to vary from its true value. Uncertainty can arise from a variety of sources, including limitations on the precision and accuracy of a measuring instrument; measurement error; the integration of data that describes features differently; the variable, unquantifiable, or indefinite nature of the features being measured; and the limits of human knowledge. Uncertainty in maps can be introduced in any of the steps in the mapmaking process, including collecting data about the features to include on the maps and encoding their properties as attributes; projecting and scaling the data; processing the data to draw out the essential characteristics of the geographic phenomena or distribution; and manipulating and displaying the results graphically. Generalization, selection, classification, and symbolization (described in chapter 6), which are required for making a map, can contribute to uncertainty in the final map. The more that errors are introduced into any of the steps in the mapmaking process, the greater the potential for a mapped value to be different from its true value. This difference between map values and real values is what we mean by uncertainty. Error is a value that we use to measure uncertainty. Attribute error is the misreporting of the characteristics of the feature. Attribute errors are incorrect quantities or descriptions associated with features, as well as missing or invalid values. Positional error is measured as the difference in distance between the coordinates of a feature on the map and its actual location. The more attribute and positional error, the more we cannot be sure where the feature is or how it should be described.

BIAS Once you understand what error is, we can examine its relation to bias. Bias, whether intentional or not, is a systematic distortion of the representation as opposed to a random error, which we can think of as random “noise.” For example, consider lines on a map that represent streams that are digitized from an air photo. The streams that pass under dense tree canopies likely have some positional error in their representations because of the difficulty or inability to see them on the image when they are digitized. This chance misrepresentation is random error. If, however, all the streams are offset in a particular direction because of misregistration of the air photo to control points on the ground (see chapter 10 for more on air photos and ground control points) or because a datum is used that is different from the rest of the information on the map (see chapter 1 for more on datums), the representation on the map has systematic distortion rather than random error. This systematic error is what we mean by bias.

PRECISION AND ACCURACY The terms “accuracy” and “precision” are often used interchangeably, but they should not be confused. We start by looking at accuracy. Accuracy refers to the closeness of the reported value to the true value—for example, how well the measured coordinates of a feature on a map conform to the true or accepted coordinates of the feature on the ground. This comparison sounds simple, but what criteria do we use to establish “true” values such as ground coordinates? “Truth,” in this case, is

defined relative to agreed-on ground coordinate system standards, such as the selection of an agreed-on horizontal and vertical datum. You will sometimes see the accuracy of a map described in terms of horizontal accuracy (position on the surface of the earth) and vertical accuracy (land elevation or depth below surface), but you may also see or need information about the mapped feature’s attribute accuracy or the map’s temporal accuracy. Although we discuss these types of accuracy in more detail later in the chapter, table 11.1 illustrates horizontal positional accuracy by showing the accuracy standards set by the US Geological Survey in 1941 for the maps it produces and uses. Table 11.1 Horizontal positional accuracy standards for USGS topographic maps at different scales

Map scale

Maximum ground error

1:1,200

3.3 feet

1:2,400

6.7 feet

1:4,800

13.3 feet

1:10,000

27.8 feet

1:12,000

33.3 feet

1:20,000

40.0 feet

1:24,000

40.0 feet

1:63,360

105.6 feet

1:100,000

166.7 feet

Note: See http://pubs.usgs.gov/fs/1999/0171/report.pdf for more details on USGS map accuracy standards.

To interpret these positional accuracy standards, a point or line on a map has a “probable” location somewhere within the maximum allowable ground error. So on a 1:24,000-scale USGS topographic map, for example, a feature on the map should be positioned within 40 feet of its true location on the ground. The level of accuracy that you require on a map will vary, depending on its purpose. For example, if you are trying to find the approximate extent of a national forest so that you can hike its trails, you can easily use a topographic map with confidence. However, if you want to know exactly where your property lines are so that you can put up a new fence, you need a more accurate map, so a better choice is a larger scale surveyor’s plat map (see chapter 5 for more on these types of maps). Related to accuracy is precision, which in map use has three meanings: the number of significant digits reported for a measurement, the repeatability of measurements or the agreement among measurements, and the rigor and sophistication of the measuring process. Achieving high precision for locational information means that positions of features are measured and recorded with more exactness. For example, a UTM grid coordinate of 253,217.62mE, 4,736,932.85mN is more precise than 253,217mE, 4,736,932mN, but it may or may not be a more accurate measurement of the true ground location. For example, if there is an undetected error in the map projection, the coordinate values will be inaccurate,

despite how many digits are used to record the values in meters. Beware of situations in which the position of mapped features is recorded with greater precision than warranted—a common artifact of data processed with GIS, which often records positions with six or more decimal places although the source from which they are derived is often much less precise. Highly precise attribute information means that the descriptive characteristics of features are captured in greater detail. For example, a vegetation polygon with an attribute of “deciduous forest + some shrub + bare ground” is more precise than “deciduous forest,” but it may not be a more accurate description of the actual types of vegetation on the ground within the polygon. For example, if an air photo is used to capture the attribute information and the photo is taken in the winter in the Northern Hemisphere, the area being mapped may be devoid of leaves. In this case, both the more and less precise attributes in this vegetation example will be inaccurate if “deciduous forest” is incorrectly interpreted. To understand precision as the rigor and sophistication of the measuring process, consider the types of instruments used to collect map information. A certain level of map accuracy at a certain map scale may be required to collect certain types of map data with the required precision. For example, surveying projects for road construction require highly accurate large-scale engineering plan maps so that map measurements made to a precision of hundredths of an inch or centimeter give highly accurate ground locations and distances between locations. In these cases, high-cost, survey-grade GPS-enabled total stations (see chapter 14 for more on total stations) are used to collect information about the features shown on the engineering plan maps. On the other hand, hikers who only need a ground location accuracy of 300 feet (100 meters) can make less precise measurements in tenths of inches on a 1:24,000-scale topographic map to achieve this lower level of accuracy. These maps are often compiled through image interpretation of high-altitude air photos or satellite imagery—a data collection method that is much less precise than using a total station surveying instrument. It is important to understand that precise data may be inaccurate, and accurate data may be imprecise. Maps made with data collected using instruments such as total stations may still be inaccurate because of problems such as collecting data for the wrong area or about the wrong features. In this case, the data used for mapping may be precise, but there are locational and other errors that greatly diminish the accuracy of the information used to make the map. Conversely, highly accurate data may be imprecise. For example, landcover data is often generalized into a few broad categories to increase the accuracy of the classification, but the broad categorization results in a map with lower precision than a less accurate map with many more land-cover categories. False accuracy means reporting your map use results at a higher level of accuracy than is possible to obtain from the map. False precision follows along the same lines—it is the reporting of results at a higher level of precision than is possible to obtain from the map. If locations on a map are measured to only within three feet of their true position, it makes no sense to report measured locations to a tenth of a foot. Knowing this, you should only use a map to the level of accuracy and precision for which it is made. Online map users have a built-in tendency to assume a greater level of accuracy and/or precision than the maps support. The ability to zoom to any scale gives users the impression that the underlying data increases or decreases in precision and accuracy. Although this more truthful representation is sometimes the case (more precise and accurate datasets are generally used to make maps at larger scales), the real levels of

accuracy and precision are tied to the source map scale—the accuracy and precision do not change if the user zooms in and out. Highly accurate and precise data for mapping can be time consuming and costly to obtain, so these data requirements are sometimes relaxed in the interest of time or cost when the map is made. You should always try to determine the accuracy of the data used to compile a map, especially if you are using the map to make precise measurements.

TYPES OF MAP ACCURACY Rather than treat map accuracy as a single issue, we examine five different but related types of map accuracy: positional accuracy, attribute accuracy, conceptual accuracy, logical consistency, and temporal accuracy.

Positional accuracy We depend on maps to show us a useful representation of where things are located. In many cases, location means the horizontal position of an object—for example, its latitude and longitude. But because we live in a three-dimensional world, location also has a vertical component. Thus, we speak of the elevation of the land surface or the depth of a rock stratum. Any discussion of positional accuracy, therefore, is in reference to horizontal position, vertical position, or both, as they relate to the horizontal or vertical datum used to make the map (see chapter 1 for more on datums). It is important to consider which type of positional accuracy is relevant to the map you are using. For example, a prism map of population, as in figure 8.30 (see chapter 8 for more on prism maps), requires only horizontal positional accuracy because the vertical aspect of the map is used to show population values, not land surface elevation. For a topographic map, both horizontal and vertical positional accuracy are important because distance and height measurements are often made using these maps. Topographic maps and many other maps produced by the US Geological Survey in the United States include the marginal notation, “This map complies with National Map Accuracy Standards.” To learn what this statement means, read the explanation in box 11.1, which follows. Here, you’ll find that, as far as US National Map Accuracy Standards (and Canadian Provincial Government standards) are concerned, both horizontal and vertical map accuracy are measured statistically in terms of how far off a set of tested points on the map are from their true position on Earth. For example, as noted in box 11.1, horizontal and vertical accuracy pertain only to “welldefined points.” In fact, only 20 or more well-defined points on a USGS topographic map are tested for the horizontal-accuracy standard—these points can include survey monuments, such as benchmarks or triangulation points; property boundary markers; road intersections; and the corners of large rectangular buildings. The horizontal positions on the ground of these test points are determined using sophisticated field surveying techniques and devices. If the tested well-defined points are within the maximum allowable ground variation tolerance, other mapped features are assumed to be as well. In practice, most features on the map fall within the stated horizontal accuracy standard, but there are exceptions, such as closely spaced linear features such as a road built next to a railroad. The mapmaker likely has displaced this part of the road or railroad line from its actual position to make both

line symbols distinguishable on the map. The result is that the road or railroad position at this particular locale does not meet the USGS map accuracy standard. Also in box 11.1, you see that the acceptable measure of map accuracy decreases progressively as the map scale decreases—this trend is generally the case with maps. For well-made maps, there is a positive correlation between the map scale and the map accuracy. It makes sense intuitively—larger scale maps show less ground in more detail so you expect the map accuracy to be higher. Smaller scale maps are more “zoomed out” so they show greater extents in less detail—you naturally expect these maps to be more generalized and therefore less accurate. Using the National Map Accuracy Standards for maps of scales larger than 1:20,000, not more than 10 percent of the points tested shall be in error on the map by more than 1/30 inch (0.08 centimeters), whereas for maps at scales of 1:20,000 or smaller, the error in position on the map can be no more than 1/50 inch (0.05 centimeters). Putting this maximum allowable error on the map in terms of ground error, 1/50 inch on a 1:20,000scale topographic map represents 40 feet (12.2 meters) on the ground. Maximum allowable horizontal ground error values for 1:24,000 and other commonly used map scales are shown in table 11.1. Table 11.2 ASPRS 1990 horizontal map accuracy standard (for class 1 maps)

Vertical map accuracy is assessed similarly to horizontal accuracy—by comparing the elevation measured on the map to the actual elevation on the ground. For example, as shown in box 11.1, not more than 10 percent of the elevations tested shall be in error by more than plus or minus one-half the contour interval used on the map. For example, on a

map with a contour interval of 20 feet (6 meters), the map must correctly show 90 percent of all points tested to within 10 feet (3 meters) of the actual elevation on the ground. In 1990, the American Society of Photogrammetry and Remote Sensing (ASPRS) developed an alternative spatial-accuracy standard for large-scale topographic maps, engineering plans, and other detailed maps of the ground. Two key differences between the ASPRS and the National Map Accuracy Standards used by the US Geological Survey are that for ASPRS: (1) the allowable horizontal and vertical error is in feet on the ground, and (2) accuracy is defined in terms of an allowable root mean square error that varies with map scale. The Ordnance Survey of Great Britain mapping agency and government topographic mapping organizations in other nations also use this type of map accuracy standard. Root mean square error (RMSE) is defined as the square root of the average of the squared discrepancies in position d of n well-defined points determined from the map and compared with higher accuracy surveyed locations of each point, in equation (11.1):

As with the National Map Accuracy Standards, the acceptable measure of map accuracy decreases progressively as map scale decreases. Maximum allowable horizontal RMSE values for several map scales are shown in table 11.2. The maximum allowable horizontal RMSE for class 1 maps of the highest accuracy is set in English units as 10 feet for 1:12,000-scale maps, 5 feet for 1:6,000-scale maps, and so on, down to 1 foot for 1:1,200scale maps. In metric units, the error tolerances are defined as 5 meters for 1:20,000-scale maps, 2.5 meters for 1:10,000-scale maps, and so on, down to 0.3 meters for 1:1,200-scale maps. Vertical accuracy for class 1 maps is similarly defined by an RMSE in the elevations of well-defined points of no more than one-third the contour interval used on the map. Maps of somewhat lower accuracy, called class 2 maps, have allowable RMSEs that are twice that of class 1, and still lower accuracy class 3 maps allow RMSEs that are three times greater than class 1. Because only a few points on a map are actually tested to see if the map meets standards such as the National or ASPRS Map Accuracy Standards, the accuracy of welldefined points on the map that are not tested is likely unknown. And because map accuracy for these standards is defined as a maximum allowable percentage of points exceeding an error tolerance, you cannot expect to determine the horizontal or vertical accuracy of a feature that is not one of the well-defined points used in the accuracy assessment. If you now feel map accuracy standards are vague about how close the mapped position of features are to their true ground location, you’re on even shakier ground with maps that don’t meet any accuracy standard. For these maps, there is simply no way to determine the positional accuracy of the features on the map. However, in general, if the mapmaker is conscientious, you can expect that the larger the map scale, the more reliable the positions of the features on the map. In contrast to concrete objects with clearly defined boundaries, some features, such as soils, wetlands, and forests, have indefinite boundaries that are transitional and difficult

to find in the field. To appreciate what we mean by features with indefinite boundaries, read the National Map Accuracy Standards material concerning vegetation mapping on standard topographic maps provided in boxes 11.2 and 11.3. In both, the definitions of vegetation type, density, vegetated areas, and clearings are all human conceptualizations. The intricacy of the phenomenon or pattern to be mapped, the map scale, and the importance assigned to features all enter into the cartographer’s judgment of how the feature should be mapped. Map accuracy is also dependent on the type of feature being mapped. Well-defined features, such as roads, buildings, property lines, or administrative boundaries, can be placed on a map with a high level of positional accuracy. Although roads may be under construction, buildings built or destroyed, and administrative boundaries modified, for the most part, these features can be easily located on the ground. The accuracy of features with indefinite boundaries, such as forested areas on topographic maps, is quite different from the accuracy of features with clearly identifiable locations. Mapping features related to such things as vegetation, soils, and climate is complicated by the fact that the ecotones (regions of transition) are indefinite, transient, or difficult to find on the ground. It is thus understandable that the map’s accuracy may be in question because of errors introduced by the cartographer’s interpretation of such fuzzy detail. Box 11.1 US National Map Accuracy Standards WITH A VIEW TO THE UTMOST ECONOMY and expedition in producing maps which fulfill not only the broad needs for standard or principal maps, but also the reasonable particular needs of individual agencies, standards of accuracy for published maps are defined as follows: Horizontal accuracy. For maps on publication scales larger than 1:20,000, not more than 10 percent of the points tested shall be in error by more than 1/30 inch, measured on the publication scale; for maps on publication scales of 1:20,000 or smaller, 1/50 inch. These limits of accuracy shall apply in all cases to positions of well-defined points only. Well-defined points are those that are easily visible or recoverable on the ground, such as the following: monuments or markers, such as bench marks, property boundary monuments; intersections of roads, railroads, etc.; corners of large buildings or structures (or center points of small buildings); etc. In general, what is well defined will also be determined by what is plottable on the scale of the map within 1/100 inch. Thus, while the intersection of two roads or property lines meeting at right angles would come within a sensible interpretation, identification of the intersection of such lines meeting at an acute angle would obviously not be practicable within 1/100 inch. Similarly, features not identifiable upon the ground within close limits are not to be considered as test points within the limits quoted, even though their positions may be scaled closely upon the map. In this class would come timber lines, soil boundaries, etc. Vertical accuracy, as applied to contour maps on all publication scales, shall be such that not more than 10 percent of the elevations tested shall be in error by more than one-half the contour interval. In checking elevations taken from the map, the apparent vertical error may be decreased by assuming a horizontal displacement within the permissible horizontal error for a map of that scale. The accuracy of any map may be tested by comparing the positions of points whose locations or elevations are shown upon it with corresponding positions as determined by surveys of a higher accuracy. Tests shall be made by the producing agency, which shall also determine which of its maps are to be tested, and the extent of such testing. Published maps meeting these accuracy requirements shall note this fact on their legends, as follows: “This map complies with National Map Accuracy Standards.” Published maps whose errors exceed those aforestated shall omit from their legends all mention of standard accuracy.

When a published map is a considerable enlargement of a map drawing (manuscript) or of a published map, that fact shall be stated in the legend. For example, “This map is an enlargement of a 1:20,000 scale map drawing,” or “This map is an enlargement of a 1:24,000 scale published map.” To facilitate ready interchange and use of basic information for map construction among all federal map making agencies, manuscript maps and published maps, wherever economically feasible and consistent with the uses to which the map is to be put, shall conform to latitude and longitude boundaries, being 15 minutes, 7.5 minutes, or 3.75 minutes of latitude and longitude in areal extent. Issued June 10, 1941, US Bureau of the Budget Revised April 26, 1943 Revised June 17, 1947 Source: Maps for America (Thompson 1988, 104) Box 11.2 Vegetation on standard topographic maps (FROM NATIONAL MAP ACCURACY STANDARDS) Many of the intricate vegetation patterns existing in nature cannot be depicted exactly by line drawings. It is therefore necessary in some places to omit less important scattered growth and to generalize complex outlines. Types The term “woodland” is generally used loosely to designate all vegetation represented on topographic maps. For mapping purposes, vegetation is divided into six types, symbolized as shown, and defined as follows: Woodland (woods-brushwood): An area of normally dry land containing tree cover or brush that is potential tree cover. The growth must be at least 6 feet (2 meters) tall and dense enough to afford cover for troops. Scrub: An area covered with low-growing or stunted perennial vegetation, such as cactus, mesquite, or sagebrush, common to arid regions and usually not mixed with trees. Orchard: A planting of evenly spaced trees or tall bushes that bear fruit or nuts. Plantings of citrus and nut trees, commonly called groves, are included in this type. Vineyard: A planting of grapevines, usually supported and arranged in evenly spaced rows. Other kinds of cultivated climbing plants, such as berry vines and hops, are typed as vineyards for mapping purposes. Mangrove: A dense, almost impenetrable growth of tropical maritime trees with aerial roots. Mangrove thrives where the movement of tidewater is minimal—in shallow bays and deltas, and along riverbanks. Wooded marsh: An area of normally wet land with tree cover or brush that is potential tree cover. Density Woods, brushwood, and scrub are mapped if the growth is thick enough to provide cover for troops or to impede foot travel. This condition is considered to exist if density of the vegetative cover is 20 percent or more. Growth that meets the minimum density requirement is estimated as follows: if the average open space distance between the crowns is equal to the average crown diameter, the density of the vegetative cover is 20 percent. This criterion is not a hard-and-fast rule, however, because 20 percent crown density cannot be determined accurately if there are irregularly scattered trees and gradual transitions from the wooded to the cleared areas. Therefore, where such growth occurs, the minimum density requirement varies between 20 and 35 percent, and the woodland boundary is drawn where there is a noticeable change in density. A crown density of 35 percent exists if the average open space between the crowns is equal to one-half the average crown diameter. Orchards and vineyards are shown regardless of crown density. Mangrove, by definition, is dense, almost impenetrable growth; crown density is not a factor in mapping mangrove

boundaries. Areas On 7.5- and 15-minute maps, woodland areas covering 1 acre (0.4 hectares) or more are shown regardless of shape. This area requirement applies both to individual tracts of vegetation and to areas of one type within or adjoining another type. Narrow strips of vegetation and isolated tracts covering areas smaller than the specified minimum are shown only if they are considered to be landmarks. Accordingly, shelterbelts and small patches of trees in arid or semiarid regions are shown, whereas single rows of trees or bushes along fences, roads, or perennial streams are not mapped. Clearings The minimum area specified for woodland cover on 7.5- and 15-minute maps—1 acre (0.4 hectares)—also applies to clearings within woodland. Isolated clearings smaller than the specified minimum are shown if they are considered to be landmarks. Clearings along mapped linear features, such as power transmission lines, telephone lines, pipelines, roads, and railroads, are shown if the break in woodland cover is 100 feet (30 meters) or more wide. The minimum symbol width for a clearing in which a linear feature is shown is 100 feet at map scale. Clearings wider than 100 feet are mapped to scale. Landmark linear clearings 40 feet (12 meters) or more wide, in which no feature is mapped, are shown to scale. Firebreaks are shown and labeled if they are 20 feet (6 meters) or more wide and do not adjoin or coincide with other cultural features. The minimum symbol width for a firebreak clearing is 40 feet at map scale; firebreaks wider than 40 feet are shown to scale. Source: Maps for America (Thompson 1988, 70–71) Box 11.3 Woodland boundary accuracy on standard topographic maps (FROM NATIONAL MAP ACCURACY STANDARDS) Clearly defined woodland boundaries are plotted with standard accuracy, the same as any other well-defined planimetric feature. If there are gradual changes from wooded to cleared areas, the outlines are plotted to indicate the limits of growth meeting the minimum density requirement. If the growth occurs in intricate patterns, the outlines show the general shapes of the wooded areas. Outlines representing these ill-defined or irregular limits of vegetative cover are considered to be approximate because they do not necessarily represent lines that can be accurately identified on the ground. The outline of a tract of tall, dense timber represents the centerline of the bounding row of trees rather than the outside limits of the branches or the shadow line. In large tracts of dense evergreen timber, sharp dividing lines between different tree heights may be shown with the fence- and field-line symbol. Published maps containing fence-line symbols that represent fences and other landmark lines in wooded areas bear the following statement in the tailored legend: “Fine red dashed lines indicate selected fence, field, or landmark lines where generally visible on aerial photographs. This information is unchecked.” Woodland is not shown in urban-tint areas, but it is shown where appropriate in areas surrounded by urban tint if such areas are equivalent to or larger than the average city block. Mangrove is shown on the published map with the standard mangrove pattern and the green woodland tint. Breaks in the mangrove cover usually indicate water channels that provide routes for penetrating the dense growth. Source: Maps for America (Thompson 1988, 71–72)

Attribute accuracy As we note in chapter 6, all maps are concerned with two primary elements—locations and attributes. So far, we have talked primarily about accuracy in relation to location, but we can also talk about accuracy relative to attributes. Attribute accuracy is faithfulness in the

description of the characteristics of geographic features. Attributes can vary greatly in accuracy and precision. Accurate attributes describe geographic features with little variation. Precise attributes describe geographic features in great detail. To prevent attribute accuracy problems, cartographers often try to define feature characteristics explicitly through quantifiable measures such as population density or tree height. As far as attribute precision (the amount of detail used to report the characteristics of a feature), the identification key (figure 11.1) used to record the attributes for vegetation data compiled from air photos provides successively greater precision at each level of description. If features cannot be described with the greatest amount of detail, the interpreter falls back to a lower precision description. The intricacy of the pattern to be mapped, the map scale, and the importance assigned to features all enter into the cartographer’s judgment of how the assignment of attributes should be carried out. As with positional accuracy, standards are applied to assure a minimum level of attribute accuracy. For example, a standard might state that in 90 percent of cases tested, the correct category from among those defined must be chosen. To determine whether a map meets these standards, ancillary source data with a higher confidence level is used. A statistically significant number of randomly chosen features are selected and checked for attribute accuracy.

Conceptual accuracy Conceptual accuracy means determining what amount of information is used and how it is classified into appropriate categories using appropriate mapping methods. It is the responsibility of the mapmaker to always try to present an accurate message. Sometimes error, lack of knowledge, or lack of attention lowers the conceptual accuracy. The map reader also shares in the responsibility of evaluating the map’s message. This need for correct interpretation is illustrated by the two maps in figure 11.2. The one on the left leads the map reader to the wrong conclusions for two reasons: (1) it shows the total number of deaths in each US county—because the data is not normalized (in this case, converted to a rate per 100,000 population—see chapter 8 for more on normalization), it is impossible to compare the values among counties; and (2) the areas in which there is no data are not readily apparent so they seem to be areas with low numbers of deaths. The map on the right corrects these mistakes so that map readers can easily see the number of deaths per county population, as well as counties in which there is no data.

Figure 11.1. Portion of a key used for air photo interpretation. The key provides more precise attribute information at successive levels of description.

Figure 11.2. In the map on the left, the representation is not conceptually accurate because the number of deaths is not normalized by the population in each county. In the map on the right, this problem is corrected, and counties with no data can be easily distinguished from counties with low data values. Data from University of Wisconsin Population Health Institute, County Health Rankings for 2013, www.countyhealthrankings.org.

Logical consistency Logical consistency, or the internal consistency of the representation, helps ensure that the proper spatial relationships are conveyed. The map at the top of figure 11.3 is difficult to read because the colors and widths of the road lines and the way that they connect do not make sense. The map at the bottom is corrected so that the roads and their connections are easy to understand. Corrections include making sure that overpasses cross both lanes of the freeway, on-ramps end at edges of freeway lanes, and freeway lanes are colored red to clearly set them apart from the other types of roads on the map.

Temporal accuracy Another important map accuracy concept is temporal accuracy, or how well the map reflects the temporal nature of the mapped features. An outdated map can lead to serious confusion, inaccurate measurements, and erroneous conclusions. For instance, if a new interchange is constructed after information is collected for your road map, you may miss the correct exit and become lost. Mapmakers sometimes try to extend a map’s practical life by mapping features such as interchanges that are planned but not yet built. The problem here is that if the proposed roads are never constructed or aren’t completed on schedule, the map may be more confusing than it is helpful. The question is, How dated can a map be and still be accurate enough to be useful? As we mentioned earlier, the acceptable accuracy of a map is always related to its intended use. You can probably tolerate different degrees of temporal inaccuracy with respect to different features and in different circumstances. The utility of a map depends on both the amount of change and your ability to imagine the change between the state of the environment when the map is made and the reality that now exists. Most of all, your tolerance for temporal inaccuracy depends on the consequences of arriving at a wrong conclusion about the environment through map use. Sometimes an outdated map may

merely be inconvenient—at other times, the outcome of its use may be more serious. Can you imagine the consequences of using an outdated map in marine or air navigation? It’s up to the user to determine that he or she has the latest information needed and is not fooled by a map that seems to be up to date but isn’t. Experience will teach you that you must always be ready to do a reality check with any map or chart. Sometimes the temporal accuracy of a map is reduced because maps of a certain region haven’t been mapped for a long time. Cartographers tend to give higher priority to mapping hitherto unmapped regions than to updating areas that are already mapped. Their logic is that it’s easier for the user to cope with an outdated map than no map at all, assuming that the reader will realize that the map is dated.

Figure 11.3. Whereas the road features in map A have logical inconsistencies; map B makes sense logically—the roads and their connections are easy to understand. Courtesy of Esri.

Sometimes all you have access to are outdated maps. At other times, although updated maps are available, you don’t happen to have one. Who hasn’t known the frustration of trying to use a 20-year-old road map unearthed from the depths of the glove compartment? These outdated maps can cause no end of confusion. If you are such a skilled map user that you can instantly spot changes in the environment and know what action to take, you may be able to manage with an old map. But most people caught in this situation might better spend their efforts in locating a new map. Your need for finding a new map must be carefully weighed against the variations you find between the map and the environment and any assumptions you can make about the validity of the differences. The temporal accuracy of maps can depend on several factors. We look at four: currency, mapping period, elapsed time, and the temporal stability of features. Currency One of the first things you should do when you pick up a map is to look for indicators of the map’s datedness, or currency. The “up-to-datedness” of the map tells you whether it reflects the present time. Or if you are interested in historical landscapes, the currency can tell you whether the map accurately represents your period of interest. Because the outcome of using an outdated nautical or aeronautical chart is potentially disastrous, dates on these types of maps are often easy to find. On other maps, finding the publication date may be more difficult. At times, the date is left off inadvertently. At other times, the mapmaker may deliberately avoid adding a date so that the map doesn’t appear outdated through long revision cycles. Even if there is a date, it may be difficult to find, or you’ll see that it’s written in a cryptic code (table 11.3). From the examples in the table, you can see that date definitions are also subject to generalization, as they can range from less to more specific. There are a couple of temporal accuracy indicators that will give you clues to the currency of the map. The first, more helpful indicator is the compilation date for the data used to make the map. The other, more often shown indicator is the date that the map was completed. The growing use of digital data makes the issue of currency in mapping more complicated and, potentially, more specific. Digital maps are usually compiled from data collected at different times. If the source date of all the digital data used to compile the map is recorded in its description or metadata (we talk about this subject more later in the chapter), it is possible to determine the currency of one of the most important aspects of the map—the date of compilation. In the absence of information about the data used to make the map, the date of map completion at least tells you the latest possible compilation date for any of the data the mapmaker uses. On many maps, the completion date is clearly shown. The date on remotesensing images and air photos used for image maps (see chapter 10 for more on image maps) tends to be the most specific. Dates printed on standard low-altitude photographs, for instance, are given to the nearest day. Dates on maps are usually less specific. As you might suspect, the features shown on most maps don’t represent data gathered at one instant in time, so the date is recorded less specifically to reflect this temporal inexactness. The completion date may be only one of several dates shown for the map. Dates for revisions or additions to the map may also be indicated. Notes on USGS topographic maps, for example, tell when photographs are taken and when field checking of the map compilation is completed. Maps that use census information

usually give the date of data collection as well as the date of map completion. Mapping period Another way to think about temporal accuracy is the time it takes to complete the map. Mapping period is the time between initial data collection and final map printing. If the mapping period is lengthy and if mapped phenomena change rapidly, the map can be out of date well before it is even completed. Although the completion date isn’t always given on a map, the length of the mapping period almost never is, so you usually have to infer it, if possible. Look at the mapping notes on the topographic map in figure 11.4. The publication and map editing date is 1992, but the map is compiled from 1985 aerial photography. The mapping period for this map is seven years. A new map may, in a sense, be old if the mapping period is long and the environment mapped changes rapidly. Table 11.3 International standard date and time notation

Date notation

Example

YYYY-MM-DD

1995-12-31

YYYYMMDD

19951231

YYYY-MM

1995-12

YYYY

1995

YYYY-Week#

1995-W52

YYYYWeek#

1995W52

YYYY-Week#-Day#

1995-W52-2 or 1997W522

YYYYWeek#Day#

1995W522

YYYY-Day of the Year

1995-365

YYYYDay of the Year

1995365

Time Notation

Example

hh:mm:ss

23:59:59

hhmmss

235959

hh:mm

23:59

hhmm

2359

hh

23

hh:mm:ss.fractions of a second

23:59:59.9942

hhmmss.fractions of a second

235959.9942

Date and Time YYYYMMDDT(ime)hhmmss

Example 19951231T235959

Confusion over mapping period sometimes arises where least expected, such as with image maps. It is true that individual air photos give you an instantaneous view of the environment taken at a precise instant in time. But image maps covering large ground areas at smaller map scales are often created by mosaicking several images to make an image map (see chapter 10 for more on orthophotomaps and chapter 14 for more on mosaicking). The images used to make the orthophotomap may be gathered over a period of hours or

days, or in the case of image maps made up of several satellite images, weeks or months. Usually, however, image maps have a shorter mapping period than other types of maps, making image maps a better data source for showing the current state of the environment at the date of publication. The mapping period of some maps is now being reduced to almost the same lag time as for data collection. Using computers and automated recording stations that continuously stream data to the mapping system, maps can be completed only minutes after information is collected. For example, the Seattle traffic flow map in figure 8.22, in chapter 8, shows current travel conditions. Computer maps of a short-lived or rapidly changing feature can be created while the feature still exists—hurricane tracking maps, such as the one in figure 11.5, are an example. Elapsed time Another way to consider the temporal accuracy of maps is the elapsed time, or the length of time between its date of completion and date of use. A weather map that was current when you viewed it last week is of little use to you today unless you are interested in historical conditions. The faster the environmental features on the map change, which we talk about in the next section of this chapter, the shorter the acceptable elapsed time since the map’s completion. With the advent of ever more powerful computer mapping systems and data collection methods, mapping period and elapsed time are shrinking—map data is collected, maps are produced, and maps are used within ever decreasing amounts of time. We are rapidly nearing a time when the whole mapping process will appear to be instantaneous. Temporal stability Another factor that contributes to the accuracy of maps is the temporal stability of environmental features—that is, how long something stays the same over time. Different features on the same map change at different rates. As a result, some features hardly change from the time the mapmaker begins compiling the map, while others are vastly different or may disappear altogether. Thus, we can say that some features are more sensitive than others to temporal instability because they are short term or intermittent. Roads are temporarily closed, for example, because of routine roadbed and bridge maintenance. Rivers swollen by spring runoff or storm-induced flooding may be unnavigable for short periods. Hurricane life cycles may run their course in as little as a day or last as long as a month.

Figure 11.4. It is good practice to pay attention to the dates printed on maps. The publication date on this USGS topographic map is 1992, but the date of the aerial photography used to compile the map is 1985, so the map may not have accurately reflected the environment even when it was first published. Courtesy of the US Geological Survey.

To read maps effectively, you must either understand the effects of these—and similar— short-term conditions not shown on maps, or hope that the cartographer embeds information about the temporal stability of the phenomena on the map through symbols or annotation. For example, the US Geological Survey defines streams in part by their temporal nature, as shown in this excerpt from Standards for USGS and USDA Forest Service Single Edition Quadrangle Maps. STREAM/RIVER—a body of flowing water. Characteristics Show the following STREAM/RIVERS based on the portion of the year they contain water: Intermittent—Contains water for only part of the year, but more than just after rainstorms and at snowmelt. Perennial—Contains water throughout the year, except for infrequent periods of severe drought.

Figure 11.5. This map of Hurricane Sandy from 2012 shows the path the cyclone followed over its eight-day life cycle. Courtesy of NOAA.

In figure 11.6, you can see that the map symbols used for these types of streams are different, so the way the features appear on the map gives you a clue as to their temporal stability.

Figure 11.6. This page out of the USGS document Standards for 1:24,000 and 1:25,000 Scale Quadrangle Maps shows the map symbols for various hydrographic features. The symbols used help map readers understand the temporal stability of some of the features. Courtesy of the US Geological Survey.

SOURCES OF ERROR Now that you have a better understanding of accuracy issues that relate to mapped features and their attributes, it will help to know how errors find their way onto maps. We now turn to

a discussion of sources of error that may affect the quality of a map. Some sources are obvious, but others can be more difficult to discern. Map error may be introduced deliberately. For example, the cartographer may purposely introduce what are called trap features—features added to or left off the map for copyright reasons. More often, errors are introduced incidentally, or even accidentally. For example, incomplete knowledge of the subject being mapped, lack of time, or personal bias on the part of the cartographer can result in mistakes. Even physiological characteristics of the mapmaker (for example, tiredness, poor eyesight, hand shakiness) can introduce map error in the form of physiological error. Generally, maps have four types of error: factual error, data source error, natural-variation error, and processing error. Generally, the first two types of error are easier to detect than the last two. We look at each type of error to see why.

Factual error Sometimes features are left off a map by mistake—a lake or town may be overlooked by the cartographer. In other cases, features found on the map may not actually exist in the environment. For example, sometimes a name or symbol is misplaced on the map, or the wrong symbol or name is used, as in the case of a railroad being shown as a road. At other times, a feature disappears from the environment, but its map symbol persists. These are all examples of factual error. Thorough map editing can minimize errors of this type, but because mapmakers rely on data from diverse sources, mistakes can be introduced at many points in the mapping process. If factual errors are known, they are corrected as a matter of professional pride. In the event that errors slip through, cartographers rely on feedback from map users to keep from repeating mistakes in subsequent map editions.

Data source error Another source of error relates to the data used to compile the map—if inappropriate data is used, the map is therefore less accurate. Data source error includes using data that is out of date, does not cover the entire area, or does not include all the required features or attributes. Often the desired data may not exist, and available data of a lesser quality is substituted. Although a valid relationship must exist between the replacement data and the phenomenon it is substituting for, error can still creep in, because data specifically about the phenomenon is not used. Even if the appropriate data does exist, lack of access or excessive cost can result in the need to substitute data.

Natural-variation error Sometimes error is introduced because it is “hidden” from the cartographer. Natural variation in the data being collected may not be detected during the collection process. Consider the process of mapping soils. Cartographic conventions force soil scientists to define soil units as crisply delineated, homogeneous areas. Information about change or differences within boundaries cannot be represented on conventional maps, even though we know that there is natural variation in the soil and that these variations are important for understanding pollution or optimizing soil fertilization in precision agriculture. To compensate, soils are characterized in terms of the impurities within units—impurities are defined as observations that do not fully match the requirements specified in the map legend. However, by varying the legend, the definition of what is a matching observation can be manipulated at will.

Today, increasing information on the variability of soil within units, and other natural phenomena such as water quality or species composition, can be found in the metadata report, described later in this chapter. Unseen natural variation of phenomena such as soil, lithology, or water quality can contribute greatly to the relative and absolute errors of the results produced when using maps. Geographers and other scientists have developed models to estimate how these errors propagate through analyses. In any of these cases, if the errors do not lead to unexpected results, finding this type of error may be extremely difficult. Although these sources of error may not be as obvious, careful checking reveals their influence on the map.

Processing error Processing error is the most difficult to detect by map users and must be specifically searched out. This search requires knowledge of the data and methods used to create the map. These errors are more subtle, and they can potentially happen in multiple ways. One type of processing error is numerical error, as when a computer program rounds off operation results. Numerical error affects other processing operations, such as conversion of scale, map projection, or conversion of data format (for example, from raster to vector format). Another type of processing error is topological error. An example is when multiple layers of data are overlaid in a GIS, and the result includes small line or polygon features that, in truth, should be part of larger adjacent features.

COMMUNICATING ACCURACY AND UNCERTAINTY Ideally, it should be possible for map users to determine the accuracy of any map they want to use because the quality of a map can be communicated to its users in several ways. You may find reports of the data and processes used to make a map in its metadata. Special symbols may be used to indicate the accuracy of features on the map. Legend disclaimers may include a statement about the map’s accuracy. Reliability diagrams are also sometimes included to show the data sources used to make the map. We look at each of these methods in turn so that you can recognize and use them to better assess the quality of your maps.

Metadata A report of the map’s pedigree recounts the steps in the map production process. Knowing the datasets and the processes used to compile the map, the savvy map user can tell whether unreliable data or unsound mapping processes have resulted in questionable or unknown map accuracy. Many agencies now provide reports about the data and processing steps used to compile their maps—this type of report is called metadata. Standards for these reports vary by agency and nation, although many contain similar components. We explore a metadata standard that is widely used in the United States—the Spatial Data Transfer Standard (SDTS) developed under the leadership of the US Geological Survey. Under this standard, all datasets used in mapping and GIS have a metadata file that contains a data quality report. Three elements of the data quality report are of particular interest to map users: positional and attribute accuracy, which we discussed earlier in the chapter, as well as lineage.

The SDTS defines lineage as the source material used to create the digital datasets for mapping, as well as any mathematical transformations of the source material. In this part of the report, you learn what person or agency is responsible for creating the source material, the date of the original source material collection and of information used in later updates, and the map scale and projection used. The dates of source material collection allow you to assess the currency of the map because you can easily determine the mapping period and elapsed time from the date of source material collection. The positional accuracy section of the report notes the type of positional accuracy assessment carried out on the map and the degree to which the map complies with a standard, such as the US National Map Accuracy Standard or ASPRS horizontal and vertical position standards we discussed earlier in this chapter. You may also find that instead of using these positional accuracy assessment standards, which are on the basis of an independent source of higher accuracy, a self-deductive estimate of accuracy on the basis of the mapmaker’s knowledge of errors occurring at each step in the map production process may be used in the report. The method of attribute accuracy assessment is also noted. Deductive estimates of misclassification errors on the basis of the mapmaker’s professional experience are acceptable, although tests of accuracy on the basis of comparison with ground sample sites of known categories are preferred. The idea is to determine the degree of agreement between the categories of ground sample sites and the categories shown on the map at the same location. You may also see the accuracy of area feature categories on the map estimated by digitally overlaying a more detailed larger scale map of higher accuracy on the map being tested. For example, the accuracy of land-cover categories on a 1:50,000-scale map can be assessed by overlaying a more detailed 1:24,000-scale map on it to determine the percentage correspondence between the two maps for each land-cover category. More detailed classes such as “single-family residential” and “high-density residential” shown on the 1:24,000-scale map may be combined into a general “residential category” on the 1:50,000-scale map being tested for attribute accuracy.

Symbols and notations The most direct way that cartographers communicate to users the varying degrees of map accuracy is with symbols and notes that indicate differences in accuracy. For example, a dashed, instead of solid, line symbol may be used for less accurately known features (figure 11.7A). On USGS topographic maps, you’ll find dotted and dashed lines used for less accurately positioned indefinite or approximate contours (see figure 9.6) or submerged contours (see figure 9.8) (see chapter 9 for more on different types of contours). In figure 11.6, you can see that the topographic map symbol for indefinite shorelines is a dashed blue line rather than a solid blue line. Another map symbol strategy is to decrease the clarity or sharpness of symbols for less accurate data. For example, a less accurate or transitional boundary, such as the extent of a wetland, animal range, or climate zone, may be drawn with a blurred or fuzzy symbol. A method almost as direct as using map symbols for inaccurate information is to add a notation describing areas of lower accuracy on the map. A note such as, “This region was not field checked” serves this purpose. On aeronautical and nautical charts, it’s common to find notes of this sort, especially to warn that the general magnetic declination information given on the map may be subject to local disturbance (see chapter 13 for more on magnetic declination).

Legend notes Sometimes a note in the map legend is the only indication of the accuracy of a map—for example, the statement, “This map meets National Map Accuracy Standards.” No rule states that the mapmaker must warn you of an inaccurate map, so this kind of legend statement is not always there to help you. The most common types of legend notes tell you the date of map production and source of the data (figure 11.7B). The date is especially useful if the mapped region undergoes rapid change. The source of data gives a hint of its reliability, because some data-gathering organizations have better reputations than others for doing professional-quality work. Data from promotional and private survey groups should be viewed with suspicion. A legend disclaimer that relieves the mapmaker of the responsibility for inappropriate map use is commonly found on commercial products that have government maps or charts as a base. For example, someone may enhance nautical charts or topographic maps with direction and distance information. For this value-added product (value-added means that the map or chart is enhanced with extra features), you’re likely to find the statement, “This map is for reference only and should not be used for navigational purposes.” The aim, of course, is to avoid lawsuits resulting from accidents ascribed to using the map. It’s a curious kind of message, however, since the map is clearly intended for navigational use.

Figure 11.7. The accuracy of mapped features is often indicated by (A) dotted or dashed symbols, (B) legend notes, or (C) reliability diagrams.

Reliability diagram The third way to communicate map accuracy is through a reliability diagram. A reliability diagram is a simple outline map that shows the sources of data used to produce the map (figure 11.7C). The diagram may appear in the legend or as an inset near the map margin. It can be included on a map, even when little to no accuracy information can be shown (figure 11.8, top). A simple diagram may be used when more specific information cannot be shown (figure 11.8, middle). A detailed diagram (figure 11.8, bottom) may show a representative

illustration of different reliability areas on the map if these areas differ in their source information and accuracy. In addition to the map illustration, information regarding the plotting accuracy, dates of information sources, datum/ projection information, and any other pertinent notes are sometimes included in the diagram.

Figure 11.8. Reliability diagrams give information about the accuracy of the features on the map. Courtesy of Esri.

LIABILITY ISSUES Map accuracy is a concern for both the mapmaker and map user in a society that is quick to sue. You compound your problems if you are both mapmaker and user, as is often the case when working with digital maps. The question is, To what degree do you place yourself at legal risk when making and using maps? We consider liability issues from the perspective of both mapmaker and map user. The interpretation of liability law makes the responsibility of the mapmaker quite clear. In essence, “every reasonable effort” must be made to ensure map or cartographic database quality. What is reasonable is judged by contemporary professional standards. Additionally, the mapmaker must inform or warn map users of potential problems and hazards in using the product. In the previous section on communicating map accuracy, we discuss how this can be done. The legal responsibility of the map user is less well defined than that of the mapmaker. But the same advice holds true—“every reasonable effort” must be made to ensure highquality map use. This advice is especially true if careless map use on your part has the potential to harm someone else. If your business is to market products based on using maps, your responsibility to conduct yourself professionally increases. From an ethical rather than legal standpoint, you probably have no business using maps if low skill on your part can harm yourself, others, or the environment. You especially have an obligation to understand that the map does not, and cannot, always show the whole truth. We discuss the many ways in which you can evaluate the truth of maps throughout this chapter. As we’ve noted, maps are graphic representations of the environment, and graphic representation, by its nature, involves distortion of reality. This enigma is the paradox of cartography. We use a distorted representation to make accurate decisions about spatial issues. Because accuracy problems can’t be eliminated—or we would have no maps—we must learn to live with them. Each time you use a map, you should evaluate its accuracy in light of the task at hand. You’ll want to ask, “Can I live with the accuracy of this map in this situation?” or “What special care should I take if I use this map for this purpose?” Many subjects that seem simple turn out, upon closer inspection, to be surprisingly complex. So it is with map accuracy. The more you think about the topic, the more facets you will discover. Knowing, as you do now, the various issues that relate to map accuracy will lead you to use maps more judiciously in the future.

SELECTED READINGS Bolstad, P. 2005. GIS Fundamentals: A First Text on Geographic Information Systems, 2d ed. White Bear Lake, MN: Eider. Branscomb, A. 2002. “Uncertainty and Error.” In Willamette River Basin Atlas, 2nd ed., 156–57. http://www.fsl.orst.edu/pnwerc/wrb/Atlas_web_compressed/PDFtoc.html. Buckner, B. 1997. “The Nature of Measurement, Part II: Mistakes and Errors.” Professional Surveyor (April): 19–22. Buckner, B. 1997. “The Nature of Measurement, Part IV: Precision and Accuracy.” Professional Surveyor (July–August): 49–52. Burrough, P. A. 1990. Principles of Geographical Information Systems for Land Resource Assessment. Oxford: Clarendon Press. http://landscape.forest.wisc.edu/courses/readings/Burrough_McDonnellCh9.pdf.

Burrough, P. A., and R. A. MacDonald. 1989. “Errors and Quality Control.” Chapter 9 in Principles of Geographical Information Systems. Oxford University Press. Chang, K-t. 2006. Chapter 8 in Introduction to Geographic Information Systems, 3d. ed. Boston, MA: McGraw Hill. Chrisman, N. R. 1991. “A Diagnostic Test for Error in Categorical Maps.” In Proceedings of AUTOCARTO 10, 330–48, Baltimore, MD: American Congress on Surveying and Mapping. Chrisman, N. R. 2002. Exploring Geographic Information Systems, 2nd ed. New York: John Wiley & Sons. Congalton, R. G., and K. Green. 1997. Assessing the Accuracy of Remotely Sensed Data: Principles and Practices. Boca Raton, FL: Lewis. Guptill, S. C., and J. L. Morrison, eds. 1995. Elements of Spatial Data Quality. London: Elsevier Science. Hopkins, L. D. 1977. “Methods of Generating Land Suitability Maps: A Comparative Evaluation.” Journal of American Institute of Planners 43 (4): 386–98. Lo, C. P., and A. K. W. Yeung. 2002. Chapter 4 in Concepts and Techniques of Geographic Information Systems. Upper Saddle River, NJ: Prentice Hall. Lodwick, W. A., W. Monson, and L. Svoboda. 1990. “Attribute Error and Sensitivity Analysis of Map Operations in Geographical Information Systems: Suitability Analysis.” International Journal of Geographical Information Systems 4 (4): 413–28. Longley, P. A., M. F. Goodchild, D. J. Maguire, and D. W. Rhind. 2005. Chapter 6 in Geographic Information Systems and Science, 2nd ed. Hoboken, NJ: Wiley. Merchant, D. C. 1987. “Spatial Accuracy Specification for Large Scale Topographic Maps.” Photogrammetric Engineering and Remote Sensing 53 (7): 958–61. Monmonier, M. 1996. How to Lie with Maps, 2nd ed. Chicago, IL: University of Chicago Press. Mowrer, H. T. 1999. “Accuracy (re)Assurance: Selling Uncertainty Assessment to the Uncertain.” In Spatial Accuracy Assessment: Land Information Uncertainty in Natural Resources, edited by Kim Lowell and Annick Jaton, 3–10. Chelsea, MI: Ann Arbor Press. Thompson, M. M. 1988. Maps for America, 3d ed. Washington, DC: US Geographical Survey. US Geological Survey. 2003. National Mapping Program Standards. http://nationalmap.gov/gio/standards/. (04/2003) Van der Wel, F. J. M., R.M. Hootsman, and F. Ormeling. 1994. “Visualization of Data Quality.” In Visualization in Modern Cartography, edited by A. MacEachren and D. R. F. Taylor, 313–31. New York: Pergamon.

Part II Map analysis

In your study of map reading in part 1, you learn what you might expect to find on a map, and you gain an appreciation for the mapmaking process. This knowledge is necessary background for the second phase of map use—analysis. In map analysis, your goal is to analyze and describe the spatial structure of—and relationships among—features on maps. Map analysis involves making counts and measurements, and looking for evidence of spatial structure and association. Sometimes, you carry out this analysis directly in the environment. Often, however, it is easier and less expensive to analyze features on maps. A map cuts through the confusion of environmental detail and complexity, making spatial relationships easier to see. Yet even a map doesn’t make everything apparent at a glance. The purpose of map analysis, therefore, is to reduce what might appear to be a muddle of information on a map to some sort of order that you can understand and describe to other people. It is possible to do this analysis visually—to view mapped information and describe it by saying, for example, “This area looks hilly” or “That pattern is complex,” or “There seems to be a strong correlation between those variables.” Traditionally, map analysis has been performed in just such a way. There are problems, however, with a visual approach to map analysis. First, such terms as “hilly,” “complex,” and “correlated” are subjective and vague. They are merely verbal descriptions of the environment on the basis of personal experience. Two or more people looking at the same map will probably use different terms to describe it. You might also have a hard time conjuring up an image of the landscape on the basis of their nebulous descriptions. What picture comes to mind, for instance, if someone says that a hillside is “steep”? Your mental image of steepness may be different from that of the person sitting next to you, much less the person describing the landscape. How do you know that the judgment of the person who describes the hill as steep isn’t biased by his or her personal experience or poor physical condition and that what seems steep to him or her isn’t steep at all to you? Furthermore, problems with visual analysis are compounded as patterns grow more complex, as details on maps become more subtle, and as the number of mapped features increases. Obviously, then, if you’re going to extract information from a map so that someone else will understand what you have in mind, you need an objective way to describe mapped phenomena. By “objective,” we mean repeatable—two people looking at the same map pattern will describe it in the same way, so you can be sure that their descriptions are trustworthy. Such objectivity is provided by quantitative analysis, in which you use numbers rather than words and replace visual estimation with counting, measurement, and mathematical pattern comparison. These activities allow you to convert mapped information to numerical descriptions in a rigorous way. You may be satisfied simply with raw numbers —the area and depth of a lake, the length of its shoreline, or the number of houses along its shore. But it’s often more interesting and useful to combine and manipulate raw numbers to obtain more sophisticated information. We might, for instance, want to add up the number of

people in a city or state and divide it by the city or state area to get an understanding of the population density (the number of people per square mile or kilometer). You can then compare this figure with the population density for other cities or states. There are many ways to combine, compare, and manipulate quantitative spatial information, but some are used more often than others in map analysis. In theory, quantitative analysis is strictly repeatable. Using the same map and analytical procedures, each map user should arrive at the same conclusions. But there’s no limit to the number of mathematical methods to choose from. Moreover, the choice of best or most appropriate method is not always obvious, even for someone who’s knowledgeable about quantitative analysis. Even with rigorous analysis, there’s no guarantee that two or more people will come to identical conclusions when working from the same map. In general, however, variations in conclusions are far fewer for quantitative analysis than for visual. You must thus decide whether the added objectivity and precision gained by using quantitative methods is great enough to warrant the extra effort that is sometimes required. In many situations, verbal descriptions and estimates on the basis of simple observation are all you need. You can capitalize on the advantages of quantitative procedures more easily, of course, by letting computers and GIS do the mathematical work for you. This impromptu analysis is becoming common with the use of portable computing devices, such as smartphones and tablets, and you will often find both the general public and professionals engaging in map analysis, sometimes without even knowing it, when they use these devices. In theory, quantitative methods are absolute and precise. In practice, however, there are several potential sources of error. As you saw in our discussion of map reading in part 1, the map itself is not error-free—the tools, materials, and techniques of the data collector and the mapmaker lead to many distortions of mapped information, some more extreme than others. Even if it was possible to make a perfect map with no distortion, map analysts would still introduce their own errors through their use of the map. Some of these errors are systematic. For example, they might be caused by inaccuracies in the data collection methods or in the tools of analysis. Other errors are random and can be attributed to factors such as human bias and the fact that human vision has resolution limits. You can compensate for both types of error once you know they exist. Two other cautionary notes about map analysis are warranted. First, map analysis is based on the assumption that you’re working with Euclidean physical space, but the earth is basically spherical and thus non-Euclidean. This problem has plagued nearly everyone who tries to conduct analytical studies based on map information. The second caution is that map analysis gives you descriptions, not explanations or interpretations. Map analysis merely converts the complex pattern of symbols to a usable form. Analyzing a map is designed to facilitate map interpretation, not replace it. A fascinating thing about map analysis is that you can, in a sense, get more out of a map than was put into it. When mapmakers show a few features in proper spatial relationship, they allow you to determine all sorts of things—directions, distances, densities, and so on— that they may not have specifically had in mind. One of the beauties of map analysis is that it can make complex geographic relations more readily understandable. The following discussion of map analysis is divided into seven chapters. The first two describe how to determine distances (chapter 12) and directions (chapter 13) from maps. In chapter 14, we discuss how to use distance and direction information in position finding and route planning. Chapter 15 explores various ways to measure the surface area, volume,

and shape of features on maps. In chapter 16, we show how various aspects of the terrain and other continuous surfaces can be analyzed. In chapter 17, we look at spatial pattern analysis of features found on a map, and in chapter 18, we focus on ways to analyze spatial associations among patterns. As we move from map reading to map analysis, we shift our attention from the theory behind maps to their practical use. It is here that the real fun of map use begins. For, however beautiful a map may be in theory and design, it is at its most beautiful when it is being used.

chapter twelve DISTANCE FINDING PHYSICAL DISTANCE Determining distance Determining distance by physical measurement Determining distance by coordinates Manhattan distance Errors in determining distance Premeasured ground distances FUNCTIONAL DISTANCE Travel time maps Isochrones Functional distance by inference SELECTED READINGS

12 Distance finding In our fast-moving society, we sometimes overlook the importance of distance. However, knowing the distance between places can be important. Travel time and fuel or energy consumption, for instance, are closely tied to the distance traveled, so distance measurement is crucial when planning hikes, road trips, and voyages by sea or air. As we discuss this important aspect of map use, keep in mind that you can interpret “distance” in two ways. When working with maps, we usually think of the physical distance between places—this distance, also called ground distance, is measured in standard units such as feet or meters. Because we assume that the map shows locations correctly, you expect a close relationship between map and ground distance. We often convert distances on a map—usually along the shortest practical route for travel—to distances on the ground, in miles or kilometers. In our everyday lives, we often find ourselves thinking of, and talking about, not physical distance but functional distance—distance measured as an expenditure of cost, time, or energy. Many of us have little notion of how long a kilometer or mile is unless you habitually walk, run, bike, or drive this exact length. Instead, you more naturally think of distance in terms of the time or energy you must expend to get from here to there. You ask not “How many miles?” but “How long does it take?” or “How hard is it to get there?” Although some special maps provide information on functional distance, most are based on physical distance. This physical distance complicates your use of maps because the way you naturally think about distance is not how it is represented on maps. Our first concern, therefore, is for you to learn how to determine physical distance from maps. Then you can convert physical distance to a more personal way of thinking about distance. In this chapter, we look at a variety of ways that you can determine physical distance from maps. Then we examine some special maps that give you information about functional distance.

PHYSICAL DISTANCE We measure physical distance using systems of measurement (a set of units that can be used to specify anything that can be measured). Some of these units are based on the size of the earth, as you will see in this section, whereas others are purely arbitrary. One widely used system on the basis of arbitrary units is the system of English units, also known as Imperial units. This system is sometimes referred to as “foot-pound-second,” after the base units of length, mass, and time. To show how arbitrary this system is, the yard (three feet, or 36 inches) was decreed by King Henry I to be the distance from the tip of his nose to the end of his thumb, with his arm outstretched! When you use the English system to work with maps, you usually measure the ground distance in miles and the map distance in inches. This system is still the official system of weights and measures in the United States, Liberia, and Myanmar. The English system is still used in some countries, such as the United Kingdom, Canada, and other countries formerly part of the British Empire, even though they have officially adopted the metric system. You’ll still see English units of yards and miles used for road sign distances and miles per hour used for speed limits (and pints for beer) in these places, although it is becoming more common to see dual signage that shows both English and metric units. When people talk about a mile, they usually mean a statute mile (1,760 yards, or 5,280 feet), which is the commonly used distance of a mile on the ground, sometimes referred to as a land mile. When we use the term “mile” in this book, unless otherwise noted, we are referring to this familiar statute mile. But as you use maps, you’ll discover other types of miles. Unlike the arbitrary statute mile, these miles are based on the earth’s circumference and represent a specific part of a degree of latitude or longitude. One such measure is the geographical mile (6,087.0 feet or 1,855.3 meters), which is one minute of longitude along the equator. The widely used international nautical mile (6,076.1 feet or 1,852 meters) is one minute of latitude on a perfect sphere whose surface area is equal to the surface area of the ellipsoidal earth (see chapter 1 for more information on the earth’s shape and circumference). Although not as familiar to most of us, the international nautical mile is important because it serves as the standard unit of distance in water and air navigation. It also provides the basis for the knot, which is the velocity of one nautical mile per hour. Knots are used in maritime and air navigation and meteorology to compute the speed of ships, planes, and wind, respectively. In contrast to the English system, all distance units in the metric system were originally based on the earth’s circumference. The meter (39.37 inches, which is slightly longer than the yard) represents a physically meaningful distance on the earth’s surface. A meter was first defined as one ten-millionth of the distance from the equator to the North Pole along a meridian. Today, the meter is defined relative to the speed of light and is 1/299,792,458th of the distance that light travels in a vacuum in one second. Conveniently, each kilometer equals 1,000 meters, and each centimeter is one hundredth of a meter. Indeed, convenience is the metric system’s greatest advantage. The metric system is much easier to use than the English system because all units are powers of 10. The overwhelming convenience and worldwide use of the metric system is hard to ignore. When you use the metric system to work with maps, you usually measure ground distance in kilometers and map distance in centimeters. The United States is one of the few

nonmetric countries, yet it is drifting toward the metric standard for official government maps. You should be familiar with both the metric and English systems, because you will see both of them on the maps you want to use. In fact, you’ll often have to convert from one system to the other. Table B.1 in appendix B will help you make these conversions. All these distance units have little real meaning in themselves, of course. They take on significance only when you use them to specify the distance between geographic locations.

Determining distance When you need distance information, it’s usually impractical to measure distance between places in the environment directly. The solution is to turn to maps. The relationship between maps and the environment is so close that measurements made accurately on the right type of map can be nearly as good as ground measurements of distance, and certainly easier to determine. Using maps to measure distance, however, isn’t always straightforward. A map is always smaller in scale than the environment it depicts. Map distances of centimeters or inches represent ground distances of kilometers or miles. To convert a map to ground distance, you must know the map’s scale (see chapter 2 for more on map scale). An understanding of scale is vital in using maps to measure distances, and indeed, to find directions and measure area and volumes, as you will see in coming chapters. There are two ways to use a large-scale map to find distances between places. You can use the map’s scale to measure the distance, or you can compute the distance using simple algebra applied to the map’s grid coordinates. We look at each of these methods. Determining distance by physical measurement Recall that there are three ways that map scale is commonly expressed on maps—scale bars, representative fractions, or RF, and word statements (see chapter 2 for more on map scale). We explore how two of them, scale bars and representative fractions, can be used to determine distances on maps. We also look at how map distances can be determined using special map rulers and measurers. Scale bars The easiest way to find distance on a map is to use the scale bar—the rulerlike graphic found on many maps with markings that are proportional to the map’s scale. To use it, mark the distance between two map locations with ticks on a marker, such as a paper strip or a bit of string. Then place the marker with your distance on it along the scale bar (figure 12.1A). All you must do is use the scale bar like a ruler to measure the ground distance. For example, the paper strip with pencil tick marks a and b is placed on the scale bar so that tick a aligns with the 1,000-meter gradation. Tick mark b is then measured on the 100-meter gradations to the left of the zero mark on the scale bar as 520 meters. The total length is thus 1,000 + 520, or 1,520 meters. If the tick marks on the paper strip are farther apart than the width of the scale bar, mark one full scale bar length on the marker and then reposition it along the scale bar and measure again. The distance value you obtain is not likely as accurate as direct ground measurements, however, especially if you try to determine the length of a winding route (see figure 12.1B). Representative fractions

You can also accurately compute ground distance using the RF printed on most large-scale maps and, less accurately, on medium- to small-scale maps. First, measure the distance between two points with a ruler, and then multiply the number of inches or centimeters by the denominator of the RF. For instance, if the RF is 1:24,000 and the distance between points is 3.3 inches, then use equation (12.1):

Figure 12.1. You can use the scale bar on the map to compute the ground distance between two map features along (A) a straight line or (B) a curved line. Courtesy of the US Army.

You’ll want to convert this distance to more familiar ground units such as miles or kilometers. Thus, use equation (12.2): 79,200 inches ÷ 63,360 inches/mile = 1.25 miles, or

In chapter 2, you can find additional examples of how to make these kinds of conversions.

Figure 12.2. Map rulers like the MapTools 1:24,000-scale ruler (top) allow you to measure distances in different units of measurement. The roamer (bottom) is designed for common topographic map scales. Courtesy of MapTools.

Map rulers You can buy special map rulers that have scale bars for common map scales. MapTools rulers, in figure 12.2 (top), for instance, have rulers in meters, statute miles, minutes, and seconds for map scales ranging from 1:24,000 to 1:500,000. Simply place the ruler for your map scale on the map and directly read the ground distance between two points. Roamers,

like the one in figure 12.2 (bottom), are meant to be used with maps at common topographic map scales (see chapter 4 for an example of using a roamer). These rulers and roamers are metric and graded in tenths of kilometers. Map measurers The methods in the previous sections work well for straight-line measurements. Using them to measure curved lines requires repositioning the measurement tools and adding the consecutive measurements. Curvilinear measurement is not only more time consuming, but error can be introduced through the repositioning and calculations. You can speed your measurement of curved lines and obtain more accurate results with a map measurer, or opisometer. This device consists of a wheel and one or more circular distance dials. After setting the needle to zero, roll the wheel along the desired path. The dials will give you the map distance in inches or centimeters. To convert this value to ground distance, multiply it by the denominator of the map’s RF. Many map measurers have dials that directly give the ground distance at several common map scales. On the distance measurer in figure 12.3A, for example, there are kilometer dials for 1:200,000-, 1:400,000-, 1:500,000-, and 1:750,000-scale maps.

Figure 12.3. Mechanical measuring devices, such as the (A) Silva Map Measurer and (B) Scalex MapWheel, are helpful for making accurate distance measurements on maps in ground units, especially if the route is curvilinear. (A) Courtesy of Johnson Outdoors Inc., Birmingham, NY. (B) Courtesy of Scalex Corp.

More sophisticated (and expensive) map measurers (figure 12.3B) have digital displays and built-in functions that let you enter the map scale before you make measurements so that ground distances are automatically displayed. You can also connect the device to your personal computer to download measurements. Map measurers work best when the path between the two points is relatively smooth. Keeping the small wheel on a winding road or stream is a real exercise in finger control. If the wheel slips or binds on tight curves, errors result. Determining distance by coordinates Physical-distance measurements become tedious if you must make a lot of them. If you want to know the distance from each of 10 houses to each of the other nine, you must make

45 measurements—quite a chore. But there is an alternative. If you use grid coordinates to compute the distances, you must only figure out the coordinates of the 10 houses, and then use them over and over again in your distance computations (see chapter 4 for more on grid coordinates). These kinds of calculations are common in mapping applications and GIS, saving you time and effort and providing more accurate results, especially for complicated routes. The GIS or mapping software processes the grid coordinates of the locations or routes and computes the distances. Such digital procedures eliminate the need to know a great deal about mathematics or even, to some degree, maps—you get the distance you are looking for (often in both ground units and time), and you also often get directions on the basis of turn-by-turn navigation. You’ll feel more comfortable with the answers, however, if you understand the basic mathematical process used to get them. We work through some coordinate distance computations so that you know how they are done, although you may never have to perform them yourself. Grid coordinates Assume that the distance you want to measure is the distance “as the crow flies”—that is, the straight-line distance. Distance along a straight line is called the Euclidean distance, and it’s our most basic idea of physical distance. For example, consider the hypotenuse of a right triangle. The Pythagorean theorem tells us that if a and b represent the sides of a right triangle, and c is the hypotenuse, then c2 = a2 + b2 (figure 12.4A). Changing c to d, to represent distance, we can restate the Pythagorean theorem as equation (12.3):

We can then determine sides a and b from grid coordinates, where (x1,y1) and (x2,y2) are the easting (x-coordinates) and northing (y-coordinates) of the two locations between which the distance is to be determined (see chapter 4 for more on eastings and northings). Knowing the two grid coordinates, we can compute the distance using what is called the distance theorem, which is the grid coordinate version of the Pythagorean theorem. The distance theorem states that you can determine the straight-line distance between any two points on your map by taking the square root of the sum of the squares of the differences in the x and y grid coordinates of the two points, or in mathematical notation, using equation (12.4):

Figure 12.4. You can compute the distance between two points using a version of the (A) Pythagorean theorem called the (B) distance theorem, where grid coordinates are used to find sides a and b.

This calculation is illustrated in figure 12.4B. The usefulness of the distance theorem is demonstrated in figure 12.5, with data derived from the universal transverse Mercator, or UTM, and state plane coordinate, or SPC, grid coordinate systems found on the Madison West, Wisconsin, 1:24,000-scale topographic

map (UTM and SPC grid coordinate systems are discussed in chapter 4). Say, for instance, you want to find the distance between a building (P1) and a corner of the University of Wisconsin Arboretum (P2) to the northeast of it.

Figure 12.5. Grid coordinates are used to compute the distance between two points on the Madison West, Wisconsin, topographic map (see text for explanation). Courtesy of the US Geological Survey.

Here are the steps to find the straight-line distance: 1. Extend the marginal ticks on the map to form the boundaries of the UTM and SPC grid cells in which the two places are located. On the map in figure 12.5, we used solid black lines to extend the UTM ticks and dashed black lines to extend the SPC ticks. 2. Determine the UTM and SPC easting (x) and northing (y) for each point (how to do this is explained in chapter 4), in equation (12.5) for P1 and equation (12.6) for P2:

and

3.

Now transfer the numeric values determined in step 2 to the distance formula, in equation (12.7):

For this example, the distance using the SPC grid coordinates is 6,607 feet (2,014 meters). The UTM grid coordinates yield a distance of 2,013 meters (6,604 feet)—a discrepancy of only three feet between the two computations. These computed distances also compare favorably with the distance you can obtain by using the map scale method to compute distance. The measured map distance is 3.3 inches, which, at a map scale of 1:24,000, yields a ground distance as follows: 3.3 inches × 24,000 = 79,200 inches ÷ 12 inches/feet = 6,600 feet.

You can also find the length of a more complex line—for example, a route between locations—by breaking it into short, straight segments. You then either physically

measure or compute line segment lengths using the distance theorem. Finally, you add the segment lengths to obtain the total distance (figure 12.6A).

Figure 12.6. You can figure out the distance along a complex line if you (A) divide it into short, straight segments. You can also approximate curved lines using (B) straight-line segments.

You can use the same method to determine the approximate length of a smoothly curving line. All you do is approximate the curve with a series of straight-line segments, and then sum the segment lengths to get the total distance (figure 12.6B). You must decide how long to make the line segments, of course, and this decision affects the accuracy of your results. Although shorter line segments lead to less error, they also require more measurements or

computations if the segments are of different lengths. Clearly, you must reach a balance between accuracy and effort. Using grid coordinates to find distances is a simple and accurate method if you must determine the distance only between two points. Aside from its increased complexity when used for multipart lines, the other disadvantage of using grid coordinates is that you can use them only for relatively small regions because both locations must lie within a single grid zone. You might think that you can solve this problem by extending the grid coordinates over larger regions, but grids of greater extent decrease the accuracy of distance computations. Larger grids incorporate greater scale distortion because of the increasing scale distortion on the grid system map projection toward the edges of the grid zone (see chapter 3 for more on map projections for grid systems). So it is not a feasible solution. Your use of SPC grid coordinates is therefore limited to areas the size of a small state, and UTM coordinates are restricted to zones of six degrees longitude.

Figure 12.7. You can use the distance theorem to compute the Euclidean distance between two pixels in an image map if you know the column and row location of each pixel as well as the image spatial resolution. Landsat image courtesy of the US Geological Survey.

Of course, computers eliminate the need for tedious manual calculations. The popularity and widespread use of mapping applications to find distances is fast making the need to manually compute distances off maps a thing of the past. Knowing how it is done digitally, however, helps you better judge the accuracy of distances reported by your map apps. Image map pixel locations You can also use the distance theorem to compute the Euclidean distance between two pixels in an image map as long as you can determine the column and row locations of the pixels and you know the spatial resolution of the image. For example, we determined the column and row locations of points 1 and 2 along the river in figure 12.7 as (340, 714) and (457, 767), respectively—this is the column and row location from the (0,0) origin at the upper left of the entire image. The Landsat Thematic Mapper image map has a 30-meter ground resolution so that each pixel is a 30-by-30- meter square.

The form of the Pythagorean theorem used in the distance calculation is in equation (12.8):

where PR is the pixel resolution. The straight-line distance between pixels 1 and 2 in figure 12.7 is in equation (12.9):

You can think of this value as the ground distance between the centers of the two pixels, although it is also the distance between corresponding pixel corners (for example, from the lower-left corner of one pixel to the lower-left corner of the other). This computation works well for image maps that cover small ground areas, particularly if the image has a UTM or SPC zone map projection. As with grid coordinates, image maps of larger regions have progressively greater scale distortion because of increasing scale distortion on the projection used for the image map. Again, map apps and GIS can speed up and ease the calculation of distances on image maps. Familiarity with the computation helps you better understand what is happening in the background. Although others might be impressed by the seemingly magical results, you now know how the trick really works. Spherical coordinates You have seen that there are extent limitations when measuring distances using the distance theorem. Limitations arise because the distance theorem is based on the Pythagorean theorem, and the Pythagorean theorem is only appropriate for use when the coordinates lie in a flat, Cartesian plane with minimal distortion in scale. For smaller extents, the flat-earth solution works fine, but for longer distances, we need a different solution. You can find the approximate distance between any two places on Earth by assuming the earth to be a perfect sphere, with locations defined by spherical geographic coordinates. To find the distance between two locations, you must know only the geographic latitude and longitude of each location and use basic spherical trigonometry. This method is the same way that GIS calculates longer distances and provides you with geodesic distance

measurements (geodesic distance is the distance measured along the shortest route between two points on the earth’s surface). Suppose you want to know how far it is from Seattle to Miami along the great-circle route (the shortest path between two places on the globe) shown by line a in figure 12.8 (see chapter 1 for more on great circles). To find the geodesic distance, follow these steps: 1. Find the latitude and longitude of the two cities. You can use table B.4 in appendix B to help you find the latitudes and longitudes of major cities in the United States. This table shows you that Seattle is located at 47°36´ N, 122°20´ W, and Miami is located at 25°45´ N, 80°11´ W. 2. Imagine forming a spherical triangle, the sides of which are arcs of great circles. You can construct this triangle by connecting Seattle and Miami with the arc of a great circle, followed by extending arcs of great circles along meridians from each city to the North Pole. These arcs are called meridional arcs (shown with thicker lines in figure 12.8).

Figure 12.8. You can find the distance between widely separated places using their latitude and longitude coordinates and basic spherical trigonometry.

3. Consider what you know about this spherical triangle. You know that the distance from equator to pole is 90°. Thus, you can calculate side b by subtracting the latitude of Seattle from 90°, as follows: b = 90° − 47°36´ = 42°24´.

Similarly, side c for Miami is calculated as follows: c = 90° − 25°45´ = 64°15´.

Angle A is the difference in longitude between the two cities: A = 122°20´ (Seattle) − 80°11´ (Miami) = 42°09´.

4.

Convert the angles you found in step 3 to decimal degrees (see chapter 1 for conversion to decimal degrees): A = 42°09´ = 42.15°. b = 42°24´ = 42.40°. c = 64°15´ = 64.25°.

5.

Determine the sines and cosines for A, b, and c using equations (12.10) through (12.14):

6.

Transfer the numeric values from step 5 to the law of cosines from spherical trigonometry—used to calculate one side of a triangle when the angle opposite and the other two sides are known, as in equation (12.15):

7. Convert this angular distance to ground distance by finding the proportion of a circle of the earth’s circumference spanned by the angle, using equation (12.16) for miles: Distance = a°/360° × circumference = 39.547°/360° × 24,874 miles = 2,732 miles. Or, for kilometers:

So the great-circle (geodesic) distance from Seattle to Miami is 2,732 miles or 4,397 kilometers. If the two cities fall in different hemispheres, you must adjust the trigonometric equation. This situation occurs when the great-circle route crosses the 180° meridian, the prime meridian (0°), or the equator. If one city is north and the other is south of the equator, you must add, rather than subtract, the latitude of the Southern Hemisphere city to 90° to obtain its meridional arc. This calculation is illustrated for São Paulo, Brazil, and Rome, Italy, in figure 12.8. When both cities lie south of the equator, you can either add 90° to both latitudes or use the South rather than the North Pole as a meridional reference point. If one city is east and the other west of the prime meridian, like São Paulo and Rome, you add rather than subtract their respective longitudes to obtain angle A. Except for these slight changes, you compute spherical distance just as you did when both cities fall in the same hemispherical quadrant. You can also use your GIS or your GPS receiver as a convenient geodesic-distance calculator. To do so, merely enter the geodetic latitude and longitude coordinates of the two points, press a few buttons, and the app displays the distance between the two points. These distances are not based on spherical trigonometry but rather ellipsoidal geodesics, a complex mathematical method beyond the scope of this book that gives more accurate distance computations. Manhattan distance

The methods for determining distance that we have examined so far assume straight-line or great-circle distance. In some cases, the route between locations is restricted by an existing infrastructure. For example, to get across town, you can hardly travel “as the crow flies,” unless perhaps you happen to have a jet pack. In many cases, your route is a series of horizontal and vertical paths along an existing grid network, such as a road system. In these cases, determining the distance on the basis of straight-line distance will underestimate the result. Instead, you can use Manhattan distance—the distance between two points measured along axes at right angles. Manhattan distance, also called taxicab distance, alludes to the square, gridded layout of most streets on the island of Manhattan in New York City. The distance between two points is the sum of the absolute differences in their Cartesian grid coordinates, not the Euclidean distance computed by the Pythagorean theorem. Now we look at the hypothetical square, gridded street network in figure 12.9, with streets spaced at 100-meter intervals. The straight-line Euclidian distance, D, between points A and B, is computed by the Pythagorean theorem for grid coordinates in equation (12.17):

For the three different Manhattan taxicab routes, the total vertical and total horizontal distances are both 600 meters, so the total Manhattan distance is 1,200 meters—the square root of two, or 1.414, times longer than the Euclidean distance. Although the Manhattan distance is the same for the three routes, the time it takes the taxi to traverse each route probably varies. The stair-stepped blue route likely has the longest travel time, because a turn, and potentially a stop, is required at each intersection— the red route may be the fastest, particularly if there are no stops along the way. Our taxi is a good example of how functional distance (travel time) differs on the basis of alternate routes of the same physical distance—we discuss functional distance later in this chapter. Errors in determining distance No matter how you determine distance from maps, some error is bound to slip into your results. The methods you use, the judgments you make, or the calculations you make may be faulty. To minimize these types of errors, use only proven techniques and instruments and take care to avoid mistakes in your computation.

Figure 12.9. Euclidean (green line) and three Manhattan distance taxicab routes (red, yellow, and blue lines) between points A and B.

But even if you make no errors, your final distance figure is still likely to be inaccurate to some degree. The reason is in the nature of the map itself. A variety of distance distortions are built into the process of transforming reality into a map (see chapter 11 for more on map accuracy). You can compensate for these types of errors, however, if you know they exist. We look at some of these errors, and show how you can compensate for them. Slope error One problem is that we normally measure or calculate ground distance as if the surface is flat. This map distance is always shorter than true ground distance—except in the rare case of perfectly flat terrain. As figure 12.10A shows, the steeper the slope, the greater the slope error—the difference between the true ground and measured map distance caused by slope. To compensate for slope error, you can use a version of the Pythagorean theorem discussed earlier. Consider figure 12.10B. To compute ground distance c between points A and B, use the map distance (b) and the elevation distance (a), both of which you can determine from topographic and other types of maps. For the example, in figure 12.10, the discrepancy between horizontal and ground distance is five meters (25 meters versus 20 meters). Slope error increases with increasing map distance, as well as with steeper slopes. The two graphs in figure 12.11 show the amount of slope error in feet and meters for different map distances as the slope percentage (elevation distance/map distance × 100) increases

(see chapter 16 for more on slope analysis). On the graph on the right, for instance, you can determine that if the map distance you measure is one kilometer and the slope is 40 percent, the actual ground distance you’ll travel is 77 meters longer, or 1,077 meters. You can find the amount of slope error from the values on the left vertical axis of each graph by drawing a line horizontally to the left axis from the point at which the map distance and the percent slope value intersect. Add the left axis value to the map distance to get the ground distance. Alternatively, you can multiply the map distance by the ground distance multiplier coefficients—labeled on the percent slope lines, which are drawn at 10 percent increments on the graph. For example, a map distance of 750 meters (0.75 kilometers) along a slope of 60 percent gives a ground distance of 750 × 1.166, or 875 meters. Alternatively, using the left-axis value, the point at the intersection of 750 meters and 60 percent slope extends to the left axis at a value of about 125 meters. Added to the original map distance, this distance is 750 + 125 = 875 meters as well.

Figure 12.10. When measuring the distance between two points (A and C) on the map, it may be necessary to adjust for slope error.

GIS, such as ArcGIS for Desktop software and ArcGIS Online, can make distance measurements that take the surface of the ground into account. Beware, though—most map apps on mobile devices, such as smartphones, do not take slope into account when calculating distance. Smoothing error A second type of error you may encounter when determining distance on maps can be traced to the smoothing of line symbols on maps. When showing linear features such as roads and rivers, mapmakers often straighten curves and smooth irregularities to get the

symbols for the features to fit on the map (see chapter 6 for more on generalization). As a result, the measured route along a highway, hiking trail, or bike route may be shorter than the actual ground distance you drive, walk, or bike. Likewise, map distance down a stream or along a shore may mislead a map reader who is in a canoe, kayak, or boat moving downstream or along a shoreline.

Figure 12.11. Use the feet or meter slope error values on the left vertical axis or the ground distance multiplier to convert the map distance (horizontal axis) to ground distance.

This smoothing error has something in common with slope error—both result in underestimation of the ground distance. A ground distance route may be the same length or longer than the map distance estimate, but it can never be shorter. Unless the ground is completely flat and the route perfectly straight, your computed distance will always fall short of true ground distance. Although elevation information may be available to help you overcome slope error, you have no way of knowing how much the mapmaker may have smoothed linear features. There are, however, a few useful guidelines. One good rule to remember is that the smaller the map scale, the more smoothing mapmakers use. Thus, the discrepancy between map and ground distance increases as map scale decreases. Thus, theoretically, you can minimize the problem if you use large-scale maps. It’s not quite that simple, however. Even on large-scale maps, some roads, rivers, and boundaries are more smoothed than others. The more detailed and crowded the features on the map, the more that area will be simplified. The map legend is rarely of much help in sorting out the effects of this variable smoothing of line symbols—that is, the varying levels of line generalization applied to different features on the map. Intuition is your best guide. You naturally expect highways to curve more than railroads, country roads to have more bends than highways, and streams to be more sinuous than roads. In crowded city centers, roads may be smoothed to an extreme, and the line symbols may be thinned to allow room to show other features. Additionally, linear features in rugged, rocky regions, such as in figure 12.12, are more irregular (and thus more smoothed on the map) than the same features in flat, sandy

regions. For example, in the map of Alpes d’Huez (figure 12.12), the hairpin curves of the roads in the mountains are smoothed more than the other roads on the map so that the crowded symbols do not collapse in on themselves. As you use your common sense and anticipate these differences within the same map, you can learn to estimate ground distance better.

Figure 12.12. Variable smoothing on this map results in some lines, such as the hairpin curves in the mountains, being smoothed more than others, such as the roads in the valleys. Courtesy of Esri.

Even GIS and map apps are not completely immune to this problem, because the street data used to calculate route distances may also be generalized. One way to keep this error to a minimum is to use the largest scale maps you can find to make distance measurements in your areas of interest, as mentioned earlier. Scale variation The larger the mapped region, the greater the scale variation across the map. Because no map projection maintains correct scale throughout, it is important to determine the extent to which it varies on a map. If you are measuring distances on small-scale maps of continents or hemispheres, it is good advice to trust the scale only near the center point or standard parallels of the map, or along meridian lines because the ground distance between degrees of latitude along meridians is a constant 69.2 miles per degree on the sphere (see chapter 1 for more on the spacing of meridians and parallels).

Because distances are such an important aspect of our environment, aren’t there any small-scale maps designed to keep distances correct? The answer is yes and no. Spherical globes faithfully represent distances, except for small errors introduced by the earth being ellipsoidal in shape. For a small-scale map projection to preserve distance relationships perfectly, it must show the shortest spherical distance between any two points as a straight line—and do so at true scale. That’s a mathematical impossibility on a flat surface. We can, however, make measurements on map projections that have true distances along certain lines. Recall from chapter 3 that the azimuthal equidistant projection transforms great circles passing through the projection center into straight lines on the map. Distances from all points to the projection’s center are true to scale, but distances between all other points are incorrect. This limitation creates a problem—to use the azimuthal equidistant projection to find distances from different starting points, you will need a new map centered on each starting point. Disproportionate symbol error A fourth source of error that can affect distance determination arises because most map features aren’t symbolized at correct scale—that is, at their real ground size. If they were, the symbols wouldn’t be large enough to see. Take, for instance, a road that is 30 feet (9 meters) wide. It might be shown by a thin line that is 0.01 inches wide (0.025 centimeters) on a map at a scale of 1:125,000. This symbol size gives it an effective ground width of 105 feet or 32 meters. Disproportionate symbol error occurs when the size of the symbol used to map the feature is not proportional to its ground size. This error becomes more extreme when a feature such as a political boundary, which has no width, is shown by a symbol that does have width on the map. And the trouble becomes worse as map scale decreases. There’s simply not enough space to include everything. As a result, cartographers must overemphasize features they want to show on the map. By the very nature of symbolism, depicting one feature can lead to excluding, distorting, or displacing its neighbor (see chapter 6 for more on generalization and symbolization). Consider the case of a house on a 105-foot (32-meter) lot between a road and a river (figure 12.13A). If the river and road are both symbolized by lines that are 0.01 inches (0.025 centimeters) wide at a map scale of 1:63,360 (one inch equals one mile), the symbols alone take up 52.8 feet (16.1 meters), or half the space available. There isn’t enough room for the building symbol, which, at 0.1 by 0.1 inches (0.025 by 0.025 centimeters), itself covers 528 feet (161 meters)—five times the ground distance between the center of the river and the road. To make all three features visibly distinct, the mapmaker displaces them, or “pulls them apart.” Such positional displacement is essential to retain the relative spatial relations between features—we can’t have the road and river running through the house. But the result is that distance accuracy is compromised (see chapter 11 for more on map accuracy).

Figure 12.13. Disproportionate symbol error can have a major distorting effect on distance determinations between closely spaced features, as these (A) 1:630-scale and (B) 1:63,360-scale maps of the same three features demonstrate.

Be wary, then, of any distance measurements that involve closely placed map symbols. If, in figure 12.13B, you compute the distance between the centers of the river and the road on the map, you come up with a ground distance of 686 feet (209 meters), although the two are really 105 feet (32 meters) apart. If you don’t know about disproportionate symbol error and positional modification, you might even try to measure the distance across the river and find that the river is 52.8 feet (16.1 meters) wide when it is actually only 10 feet (3 meters) across. Map users who recognize the problems of disproportionate symbols on conventional maps sometimes overlook the fact that the same thing occurs on air photos and remotesensing images (see chapter 10 for more on these kinds of images). Contrary to what you might think, an aerial photograph doesn’t show all environmental features in correct proportion. Because photography is based on reflected light, a shiny, highly reflective object is imaged as disproportionately large on a photograph. Mapping teams have made use of this fact for years. Before taking digital air photos, they place highly reflective targets on the ground (figure 12.14). These targets stand out on air photos, helping mapmakers locate known positions and digitally correct any distance distortions when using the photos to compile topographic and other maps. High reflectivity of small objects may also be partly responsible for the fact that linear features on photos appear to be far larger than their natural size. Power lines are often visible on a photo, even though the large transmission towers that hold them up are not. Railroad tracks can be seen when a car can’t. On satellite images with a pixel size of 200 by 200 feet (61 by 61 meters), there is no logical way for a 60-foot-wide (18-meter-wide) road to show up—yet it does. This appearance may occur because highly reflective subpixel-size

objects often increase the digital values of adjacent pixels. These examples should remind us to keep disproportionate symbol error in mind when making measurements on image maps as well as conventional maps. Dimensional instability error Unless maps are printed on a dimensionally stable material such as plastic, glass, or metal, they may stretch or shrink with changes in temperature and humidity. A map the size of a USGS 1:24,000-scale topographic quadrangle, if printed on paper (22 inches wide by 27 inches high), can change as much as 1 percent in length or width when moved from a cool and dry to a hot and wet environment. This dimensional instability caused by the stretching and shrinking of the medium on which the map is produced introduces errors in your distance measurements, especially if you’re measuring long distances on the map. If the maps are not simply scans of paper maps, this instability is not an issue with digital maps displayed on screen.

Figure 12.14. Highly reflective survey targets are placed on the ground before acquiring aerial photography of the area for mapping purposes. Courtesy of the Ohio Department of Transportation.

As with slope, smoothing, scale, and symbol error, dimensional instability error can play havoc with distance calculations. No matter how precisely you compute distance from maps, no matter how carefully you physically measure with scale bars, representative fractions, map rulers, or map measurers, your final distance result will only be approximate. In the best of circumstances, however, you will at least have a good idea of just how approximate the distance is. Premeasured ground distances Taking the preceding errors into account and mitigating for them as much as possible, measurements made accurately on the right type of map can be nearly the same as ground measurements of distance. Even more accurate distances can be determined on maps if someone physically measures distances on the ground and puts the distance measures right on the map. That’s just what has been done on some maps—road maps, in particular. Premeasured ground distances are an invaluable aid in determining route length and choosing among routes. We look at some of the ways that ground distances are added to maps. Distance segments On some printed highway maps and atlases, such as Rand McNally Road Atlases, Michelin Road Atlases, and American Automobile Association (AAA) printed maps, premeasured ground distances are indicated between such places as towns or highway intersections with the use of distance segments (figure 12.15A). Routes are divided into segments, with the distance along each segment labeled, starting with the first-order route distance segment. To find the distance between two points, merely add up all the distances for the segments along the route. If your route is long, of course, you’ll soon tire of adding numbers. To help you out, mapmakers may provide a second level of distance numbers. Second-order route distance segments are longer than the first and have major cities and intersections as their endpoints. These special points are shown by such symbols as stars or arrowheads (figure 12.15B). The symbols and segment distances are usually printed larger or in a different color to reduce confusion with the first-order figures. Distance insets To show the distance between widely separated points, mapmakers often include distance insets. Insets are usually double-ended arrows with names of features, such as cities, and premeasured distances between them (figure 12.15C). Insets can be useful, although they may be hard to locate and are far from complete. All too often, you’ll find that the mapmaker did not include the length of the route of interest to you.

Figure 12.15. (A) Distance segments, (B) second-order distance segments, and (C) distance insets are often incorporated into the design of road maps to help determine route length.

You may also find these insets as single-ended arrows and annotation or simply text at the edges of maps (figure 12.16). Insets often indicate not only the distance to the nearest point of interest but also the travel time. You generally won’t find any indication of the route you need to take to the point of interest—unless the mapmaker included a mileage map in the map’s marginalia (see chapter 6 for more on map marginalia). Mileage maps In addition to segment distances and distance insets, a mileage map can be used to indicate the distances between places (figure 12.17). On such a map, routes between major locations are shown by straight lines that are labeled with the distance along each route. These lines aren’t meant to duplicate real routes but to show that one place is connected to another by a route. Consequently, only the premeasured mile or kilometer distance labels for each line are correct, not the relative lengths and locations of the straight lines that connect the places. Some of you may notice that many of the cities and physical features on the map in figure 12.17 are not in their actual geographic locations but are shifted to make the map more legible.

Figure 12.16. Distance insets are sometimes added at the edges of maps, but rarely do you also get an indication of the route itself.

Figure 12.17. A mileage map, such as this one for Oregon, shows premeasured ground distances between selected places in the state and to selected places in neighboring states. Courtesy of the Portland, Oregon, Visitors Association.

The main disadvantage of mileage maps is their size. They are usually relegated to a corner of a larger road map and are so small that most cities and other geographic features must be left out. Furthermore, you have the same problem you do with segment numbers— you must add a lot of numbers together for longer routes, so the more numbers you add, the more time consuming the task and the more room there is for error. To find the road distance from Portland to Ashland in figure 12.17, for instance, you must add six numbers. Distance tables The problem of having to add a lot of numbers together is avoided if the distances between pairs of locations are given to you. You have no doubt used a mapping application, such as ArcGIS Online or Google Maps, to input a starting and ending location. These apps give you the distance in metric and English units as well as the travel time. But for printed maps, this interactive capability is unavailable, so the next best thing is to give the map reader an indication of the distances between pairs of origins and destinations. This pairing is most commonly done with a distance table—a table that identifies the premeasured ground distance between pairs of key locations (generally major cities). There are two types of distance tables. On a rectangular distance table, key locations are listed alphabetically along the top and side to create a row-by-column table of

distances. To find the distance between two key locations, look up one name along the left (or right) side and the other along the top (or bottom). Then find the cell in which they intersect. On the rectangular distance table in figure 12.18A, to find the distance from Chicago, Illinois, to Fairbanks, Alaska, locate Chicago in the left margin and Fairbanks at the top, and then read the row-column intersection—3,804 miles. Or look up Fairbanks on the left and Chicago at the top—the result is the same. The rectangular distance table is 50 percent redundant. To allow you to look up either key location in the margins, the table’s upper-right half is the same as the lower-left half. To avoid this duplication and save space, mapmakers often use a triangular distance table instead. On this table, place-names appear only once. In figure 12.18B, you can find the distance between Chicago and Fairbanks by locating Chicago along the named edge of the triangular table and reading down the column until it intersects the row for Fairbanks. The distance is again 3,804 miles. As with distance segments or insets and mileage maps, distance tables aren’t allinclusive. You may find one of the places you’re looking for but not the other, or both may be missing from the table. In such cases, you can sometimes find an approximate distance by looking up the distance between nearby places.

FUNCTIONAL DISTANCE Physical distances, which are what we have primarily been discussing in this chapter, tell only part of the story. Consider the following: In Los Angeles, with everybody traveling by car on freeways, nobody talks about “miles” anymore, they just say “that’s four minutes from here,” “that’s 20 minutes from here,” and so on. The actual straight-line distance doesn’t matter. It may be faster to go by a curved route. All anybody cares about is the time. (Wolfe and McLuhan 1967, 38)

There’s more to distance than physical miles (or kilometers) on the ground or measured miles (or kilometers) on the map. Physical distance may not be as germane to our lives as functional distance—the distance measured as an expenditure of time, effort, or cost. As the preceding example illustrates, functional distance depends on many factors—mode of travel, driver circumstances, road conditions, and more—the list is endless. It’s sad to say, but mapmakers find it difficult to respond effectively to our concern with functional distance. In part, their own shortsightedness is at fault. This myopia is due, in part, to the data they are working with to make their maps—this data is perfectly suited to physical (and digital) measurement, but they have no concept of functionality—that is, map use. Cartographers concentrate on making maps ever more geographically and geometrically (or geodesically, as you have seen) accurate, thereby satisfying the demands of engineers, land surveyors, military strategists, and others who are concerned more with physical distance. In the process, they often overlook the day-to-day needs of everyone else.

Figure 12.18. Rectangular (A) and triangular (B) distance tables for Alaska (adapted from Alaska road map).

But some of the fault rests with the nature of functional distance as a measure of cost, time, or energy. Although physical distance is always the same—distance is distance, except when it isn’t because of slope and other factors. In contrast, functional distance measured in cost, time, or energy depends on the function, or use, and that depends on the user—you. Functional-distance maps are therefore human-centric maps and thus subject to the foibles of human nature. Maps based on physical distance aren’t only simpler and less costly to make, they’re also more general in purpose. They can be used by anyone at any time. Maps that show functional distance may be more meaningful for specific purposes and users, but to be effective they must be tailor-made for specific situations.

The most common functional distance added to maps is travel time, or the time it takes to travel by standard means between locations, because it’s the easiest functional distance to measure and show on maps. Mapmakers can determine travel times by monitoring speed limits along a route. Examples include highway traffic speeds, airline flight times, and maritime travel estimated times of arrival (ETAs) (see chapter 14 for more on navigation). All travel time figures are approximate, of course. They vary with the driver, weather, time of day, route, vehicle, and more. Maps provide average figures on the basis of real time (the actual time during which an event or process occurs) or historical time travel measures, so you must consider the circumstances for special cases or for your own needs. Mapmakers provide travel time information on maps in several ways. As with mileage maps, travel time maps are made by adding functional distance values as supplementary information for route segments on conventional maps. They can also make maps that show only functional distances. Another approach is to show travel time with isochrones. Of course, any of these methods can be used with either conventional printed maps or interactive online maps.

Travel time maps Travel time maps are essentially the same as premeasured ground distance maps, except that the ground distances are replaced with travel times. As with mileage maps, this approach can be used to produce travel time segments, travel time insets, and travel time tables. Figure 12.19A gives the travel time for each travel time segment. You find your total travel time by summing the travel times of the segments, just as you do to determine physical distance. As with mileage maps, the travel time between major locations can be shown by straight lines labeled with the time it takes to travel along each route (figure 12.19B). The lines show that places are connected, so only the travel times for each line are correct, not the lengths or locations of the lines and, sometimes, not even the places. Again, you find the total travel time by adding the numbers for each segment. GIS and mapping applications use a (usually proprietary) travel time database to show you the shortest route with the travel time between two places, as well as alternate routes that take slightly longer. Travel conditions are generally based on real-time travel times recorded by automatic travel recorders, so locations of current traffic congestion, road construction, and accidents can also be displayed on the route you select.

Figure 12.19. Functional distances may appear as (A) travel time segments or (B) on a travel time map.

Isochrones A second mapping method is to convey time as a continuous surface of travel time away from a point, line, or area feature. Travel times are usually shown by lines of equal time, called isochrones, as shown in the 1914 historic isochronic distance map in figure 12.20 (see chapter 7 for other examples of isochrones). Isochrones showing the travel time from London in days are drawn outward from the starting point (London), with the spacing between isochrones showing the relative rate of travel from day to day. This last way of showing functional distance on maps is more restrictive than the others, because only time from the starting point is meaningful. But all the methods are useful, especially if they are combined with physical distance measures on the same map. In these cases, you are given the best of both worlds—you can use physical distance measures to compute variables such as gas mileage and fuel costs and travel time figures to estimate arrival times, travel fatigue, and other such factors. If maps that show travel times are few, maps that give other types of functional distance information are even harder to find. Details such as travel comfort, physical energy consumption, and route safety are all of interest to map users. The core of the problem is that this kind of information is difficult to gather. And here, again, average figures aren’t always helpful to individual users. Advanced mapping applications are starting to meet the need. For example, you now expect your navigation device to tell you the functional distance in time travel as well as the physical distance on the ground. This information allows you to plan for your needs before you even start your journey—how much water and food to take hiking, how long between stopping for gas, and when you will want to stop at a rest area—all these factors can be

estimated in advance of your departure. Once you are en route, your navigation device also allows you to modify decisions about your timing, mode of travel, and the distances you travel.

Figure 12.20. Historic isochronic distance map from London with isochrones showing travel time from London in days.

As widespread computerized mapping becomes more practical, we are seeing maps being tailor-made for a particular person and trip. With these apps, you can give all the information related to your journey and in return be presented with your own personal and truly functional map. In addition, while en route, you can interact with the map app to get updates you need on the basis of real-time information. In the not too distant future, you’ll also be able to predict travel time on the basis of such things as weather, vehicle (for example, airplane or taxi) availability, and traffic congestion, combining a variety of travel modes, such as taxi, airplane, and car. Soon it will be commonplace to accurately measure physical distances and predict functional distances while incorporating a number of userdefined variables. For example, in the future, you might have a hiking map on your mobile device that uses the most up-to-date, highest resolution DEM data so that you can compute slope along the trail you are following. Walking speeds adjusted by slope angle along the trail, computed from Tobler’s hiking function (shown graphically in figure 12.21), will give you the information to estimate the time it takes for the average person to complete the hike on foot. You can modify this function with your own walking speed and the slope of the path you intend to traverse. Then you can augment this conversion with the traffic and weather conditions on a particular date at a particular time to achieve a highly accurate estimate of your travel conditions.

Figure 12.21. Tobler’s hiking function gives the general relationship between slope angle and walking speed upslope and downslope.

Functional distance by inference In the meantime, maps of functional distance are now commonplace–users of online map apps expect to see their travel routes recorded with travel time estimates. The burden no longer lies with you, the map user, to translate physical into functional distance. You can do this conversion by searching out the information elsewhere and adding it to your maps, but such a process is usually too involved to be worth the effort, so map apps do this translation for you. In instances in which you do not have a map app to guide you, you probably rely on the process of inference, in which you derive conclusions on the basis of what you already know, without even realizing it. When figuring the length of a trip, for instance, you take traffic conditions into account. You know that to drive across town takes longer during rush hour than at other times. Holiday and weekend traffic is equally predictable, so you leave earlier or stay longer to shorten your travel time. You can probably think of many other instances in which you must rely on inferential distance determination. Travel time is merely an example. By drawing on your experience and intuition, in combination with careful map study, you can make many inferences that

broaden your use of functional distance. Your imagination can add more to maps than cartographers can show.

SELECTED READINGS Atwill, L. 1997. “What’s Up (and Down) at the USGS.” Field & Stream (May), 54–55. Bovy, P. H. L., and E. Stern. 1990. Route Choice: Wayfinding in Transport Networks. Boston: Kluwer Academic. Buttenfield, B. P., and R. B. McMaster. 1991. Map Generalization: Making Rules for Knowledge Representation. Essex: Longman Scientific & Technical. Federal Interagency Coordinating Committee on Digital Cartography. 1988. “The Proposed Standard for Digital Cartographic Data.” The American Cartographer 15, no. 1. Krause, E. F. 1987. Taxicab Geometry: An Adventure in Non-Euclidean Geometry. New York: Dover. Maling, D. H. 1989. “The Methods of Measuring Distance.” Chapter 3 in Measurement from Maps, 30– 52. New York: Pergamon. McMaster, R. B., and K. S. Shea. 1992. Generalization in Digital Cartography. Washington, DC: Association of American Geographers. Monmonier, M. S. 1977. Maps, Distortion and Meaning. Washington, DC: Association of American Geographers. Muehrcke, P. C. 1978. “Functional Map Use.” Journal of Geography 77 (7): 254–62. Muller, J. C. 1978. “The Mapping of Travel Time in Edmonton, Alberta.” The Canadian Geographer 22:195–210. Muller, J. C., J. P. Lagrange, and R. Weibel, eds. 1995. GIS and Generalization. London: Taylor & Francis. Olsson, G. 1965. Distance and Human Interaction, Bibliography Series No. 2. Philadelphia, PA: Regional Science Research Institute. Peters, A. B. 1984. “Distance-Related Maps.” The American Cartographer 11 (2): 119–31. Tobler, W. 1993. Three Presentations on Geographical Analysis and Modeling: Non-isotropic Geographic Modeling Speculations on the Geometry of Geography Global Spatial Analysis. Technical Report (National Center for Geographic Information and Analysis) 93: 1. Watson, J. W. 1955. “Geography: A Discipline in Distance.” Scottish Geographical Magazine 71:1–13. Witthuhn, B. O. 1979. “Distance: An Extraordinary Spatial Concept.” Journal of Geography 78 (5): 177– 81. Vincenty, T. 1975. “Direct and Inverse Solutions of Geodesics on the Ellipsoid with Application of Nested Equations.” Survey Review 22 (176): 88–93. Wolfe, T., and M. McLuhan. 1967. McLuhan Hot & Cool. New York: Dial.

chapter DIRECTION FINDING AND thirteen COMPASSES GEOGRAPHIC DIRECTION SYSTEMS True north Grid north Magnetic north Magnetic declination Annual change Declination diagram Compass direction systems Compass points Azimuths Bearings Conversions MAGNETIC COMPASSES Types of compasses Rotating needle Rotating card Reversed card Electronic Local magnetic anomalies DIRECTION FINDING ON LARGE-SCALE MAPS Topographic maps Nautical charts Determining direction from grid coordinates DIRECTION FINDING ON SMALL-SCALE MAPS Great-circle routes Determining direction on a globe Determining direction in the gnomonic map projection Rhumb lines The navigator’s dilemma Determining direction from geographic coordinates SELECTED READINGS

13 Direction finding and compasses Why do some people navigate with ease, while others always seem to get lost? A 2014 Nobel Prize–winning discovery revealed that it’s partly because of our brains. Jane Fang (2014) wrote in IFL Science: The entorhinal region is the part of the brain that tells you which direction you’re facing and which direction you should be facing as you move towards your destination. It’s where our sense of direction comes from, [the researchers] found, and the quality of the signals from this brain region determines how good your navigational skills are. Your “internal compass” readjusts as you move through the environment. . . . For example, if you turn left, then your entorhinal region should process this to shift your facing direction and goal direction accordingly. If you get lost after taking too many turns, this may be because your brain could not keep up and failed to adjust your facing and goal directions. Although it might be convenient to blame people’s propensity to get lost on short-wiring in their brains, does it mean that they are doomed to live with this disability? Consider people who make a living on their ability to tell direction. Wilderness guides, spelunkers, and search-and-rescuers, for example, must know how to find their way, regardless of visibility. Whether they are in deep fog, dark caves, or blinding snowstorms, they need a compass in their mind that operates all the time under any conditions. Does it mean that they have a hyperdeveloped entorhinal region in their brain? More likely, it is because they have learned to be more observant about their environment than the rest of us and have developed their internal compass through necessity and practice.

If you learn how to think geographically, and, like wilderness guides, develop your internal compass through practice, you can improve your sense of orientation. This line of reasoning is good news, for it means that anyone can learn to tell direction and be able to share the distinction of having a good sense of direction. If we say that someone has a good “sense of direction,” what do we mean? That they always know where north is? Or that they can always get back to a starting point? Or that they can always find their way home? Direction, by definition, can only be determined in reference to something in the external environment. The reference point may be near at hand or far away, concrete or abstract. This reference point, whether it is an object or a known position, establishes a reference line, sometimes called a baseline, between you and it. Direction is measured relative to this reference line. In its simplest form, direction is determined egocentrically—that is, it is self-centered. Your reference line is established by the way that you are facing. You go left or right, “this way” or “that way,” straight ahead or back, up or down the road—all in relation to an imaginary line that points out from the front of your body. A common way to express egocentric direction (direction relative to your own body) is to use a symbolic clock face. You are assumed to be located at its center, facing 12 o’clock. Your reference line is the line that is projected from your position straight ahead to 12 o’clock. Now say that you want to find the direction to a distant object. The line from you to the object is called the direction line. Direction is given in hourly angular units as the difference between the reference line and the direction line. Something directly to your right is at three o’clock, behind you at six o’clock, and directly to your left at nine o’clock. If something is ahead of you to the right, you might say it is at two o’clock, as shown in figure 13.1A.

Figure 13.1. Direction is defined as the angular distance from a reference line. Sometimes a clock face is used as (A) a directional reference system, but more commonly direction is measured in (B) degrees from north.

GEOGRAPHIC DIRECTION SYSTEMS Your direction-finding ability will improve if you learn to think geocentrically in terms of a geographic direction system. In this system, direction is measured in the angular units of a circle (figure 13.1B), with north at the top so that the north reference line points to 12 o’clock on the clock face. Orienting maps with north at the top has a long history, likely beginning in the second century AD with the Greek geographer Claudius Ptolemy of Alexandria making his maps in this orientation. This practice was adopted later by renaissance cartographers in Italy and elsewhere in Europe who marked north prominently on their maps. The north reference line is useful because it is no longer oriented to your own body, as it is with egocentric direction. No matter which way you turn, north remains the same on the map. To make this system valuable in direction finding, all you must do is find north and use that reference point to determine a reference line on the earth’s surface from where you are standing to north. The direction to a distant object can then be given as the angle between the north reference line and your direction line. No single north reference line is used on all maps, because there are actually three types of “north”—true north, magnetic north, and grid north. Each type has its advantages and disadvantages as the basis for a reference line, and each one is best suited for certain purposes. The reference line used on most maps is based on true north.

True north True north (also called geographic north, or sometimes geodetic north) is a fixed location on the earth—the north pole of the axis of earth rotation. A great-circle line from any point on Earth to the North Pole is known as a true-north reference line (see chapter 1 for more on great circles). In fact, a great-circle line that passes through the North Pole is also a meridian; thus, any meridian on a map can serve as your reference line to find true north. The advantage of using true north to determine direction is that you can find it in the field without any special instruments. You can determine direction simply by making reference to natural features. People have used the sun and other stars as directional reference points since they first began observing nature. North was probably chosen as the reference point on maps because there are such good celestial ways to find it. One of the oldest and most reliable celestial ways to find true north is to locate the North Star (Polaris). It is positioned in the Northern Hemisphere sky less than one degree from the north celestial pole (the spot in the sky directly above the North Pole). Finding Polaris is a simple matter because it is a relatively bright star and is conveniently located at the tip of the handle of the Little Dipper (Ursa Minor) constellation (figure 13.2). Because the earth rotates on its axis, all the stars seem to move in circular paths around Polaris (figure 13.3). If it is nighttime and the sky is clear, you can hardly ask for a better true-north reference point. But what if it is daytime?

Figure 13.2. True north can easily be determined by observing the stars in the Northern Hemisphere sky at night to find the North Star (Polaris).

Figure 13.3. A time-lapse photo of the night sky in the Northern Hemisphere captures the apparent circular movement of the stars around the north celestial pole and Polaris, the North Star. In fact, the earth, not the stars, is rotating. Courtesy of Jerry Pool.

During the day, you can use the position of the sun in the sky to determine a true-north reference line. For thousands of years, people have found the direction of true north by noting the position of the sun at noon (or at 1 p.m. daylight saving time). The direction to true north can be determined in a few seconds if you are wearing a watch that has hour and minute hands. Simply hold the watch horizontally at eye level, and then turn the watch until the hour hand points to the sun (figure 13.4). Now picture a spot on the dial halfway between the hour hand and noon (one o’clock for daylight saving time). In the Northern Hemisphere, a line from the center of the dial through that spot points to true south, so the opposite direction is true north. In the Southern Hemisphere, the line points to true north. This method works best if you are located at the center of a time zone in which the sun truly faces south at noon.

Figure 13.4. True north can be determined in the daytime with a watch that has hour and minute hands.

One problem with relying on celestial bodies as a reference point for true north is that the sun is visible only during the day and the stars, like Polaris, only at night. Another problem is that celestial reference points can be obscured from your view by clouds, dense fog, and other meteorological obstructions. To overcome these limitations, we can use technology based on a global navigation satellite system, or GNSS. A GNSS uses a constellation of satellites that constantly transmit their time and position. Most often, you will hear about the system referred to as GPS, which is the acronym for the American satellite navigation system—the NAVSTAR Global Positioning System. The only other operational GNSS is Russia’s GLONASS system, although some countries, such as China, the European Union, and India, are also developing global systems; and others, like France and Japan, are developing regional systems. GPS receivers use the satellite transmissions to calculate their positions on Earth on the basis of the time it takes for the signal to reach the receiver from the satellite. With an unobstructed line of sight between the receiver and at least four satellites, high-precision (within meters) positions can be determined. Because GPS satellites continuously transmit signals to the earth 24 hours a day, we now have an all-weather, day and night ability to determine “north.” But which “north” is recorded by the GPS receiver? GPS receivers can be set to use any of the three norths. Say that your GPS is set to true north, and you enter the latitude and longitude of a distant feature. Your GPS tells you to go in a particular direction—for example, 45°. So you take out your compass, find the direction to the location, and start walking. If the distance to the location is 750 feet (about 228 meters), you’ll miss your mark by about 40 feet because you didn’t take into account magnetic declination, which we discuss later in this chapter. This

inaccuracy may not be a problem because many recreational GPS receivers are accurate to about 10 meters (32 feet or so). Nevertheless, it pays to know which north your GPS is using so that you can take it into account, especially because the error increases with distance. Your smartphone compass app uses another technology to determine north—magnetic field sensors. Typically, three magnetic sensors are placed perpendicular to each other in the phone. Magnetic field strength measurements from the three sensors are used to calculate the direction of magnetic north (the direction to the north magnetic pole, discussed later in the chapter). The compass app in the phone also determines the magnetic declination (discussed later in the chapter) for your ground position. This angular difference between true and magnetic north is used to determine the direction of true north.

Grid north True north isn’t always the most useful direction reference line. On maps that have a grid coordinate system overprinted (see chapter 4), you will find a second north—grid north, which is the northerly direction indicated by the grid lines or ticks on the map. Unlike true north, grid north has no geographic reference point. It is purely artificial, established for the convenience of those who use maps to measure or compute directions. Grid north is useful because it allows you to use the grid lines on your map as your north reference line. The UTM, SPC, and OS grids (see chapter 4 for more on these grid coordinate systems) are standardized systems in which the lines of constant easting, which run from top to bottom of the map, are oriented to grid north. Grid north lines usually aren’t in the same orientation as true-north meridians on the map. Grid north lines are straight and parallel, whereas true-north meridians converge toward the North and South Poles. If a map is centered on a meridian shown with a vertical line, the grid line in the center of the map points vertically to true north only if the meridian is the central meridian for the grid coordinate system used on the map. For example, on maps made by the Ordnance Survey of Great Britain, grid north is the direction of a grid line that is parallel to the central meridian on the country’s national grid. The farther east or west the location is from the center of the grid zone, the greater the difference between grid and true north. The angular difference between these two norths is called grid declination. Grid declination can also be considerable at the edges of UTM and SPC grid zones (see figure 4.13).

Figure 13.5. Grid north for this map of the Madison, Wisconsin, area is indicated by the near-vertical UTM grid coordinate system lines, shown in light blue. The angles of these lines differ substantially from those for true north (indicated by the dashed line for the 89°25” W meridian) and for the SPC (black) grid coordinate system lines.

When two grid coordinate systems (for example, UTM and SPC) appear on a single map, you must determine which grid system the cartographer used for grid north. If grid north is not indicated along the edges or at the corners of the map, it may be difficult to determine which grid system is being used. Sometimes, there are conventions you can rely on. For example, USGS topographic maps always use the UTM grid coordinate system for grid north. Clearly, using a map with two grid coordinate systems can be confusing. For example, on a standard topographic map of the United States published by the US Geological Survey, the grid lines in the SPC system probably aren’t at the same angle as the grid lines in the UTM system. The difference between these two systems is illustrated in figure 13.5. The lines on the map represent the eastings and northings you would see for the UTM and SPC grid coordinate systems, as well as the graticule meridian at the center of the map, shown with a dashed-line symbol. The near-vertical UTM grid lines, shown in light blue, are at a different angle from the SPC grid lines, shown with a solid black-line symbol. And neither of the grid coordinate system lines are at the same angle as the meridians of the graticule.

Magnetic north True north is the standard map orientation and the north used for GPS, and grid north is a helpful aid to direction measurement and calculation. The third north—magnetic north—is convenient for the map user in the field who has a magnetic compass. Magnetic north is the direction to which your compass needle points. The earth’s magnetic North and South Poles are aligned with the earth’s magnetic field. A free-floating magnetic needle in a

compass aligns itself with the earth’s magnetic field, and the needle’s north pole points to the earth’s magnetic north pole, as shown in figure 13.6. If this end of the needle is marked distinctly—say with blue paint or with the letter N—it will show the magnetic north reference line at your location.

Figure 13.6. The earth’s magnetic field resembles that of a simple magnet.

Theoretically, the needle of a magnetic compass points to the magnetic north pole from any place on the earth’s surface. In practice, however, because of local magnetic disturbances, it rarely works this way. This is because the earth’s magnetic poles (positions on the earth’s surface at which the geomagnetic field is vertical—that is, perpendicular to the ellipsoid) are not the same as the earth’s geographic poles (ends of the earth’s axis of rotation). The difference between the earth’s axis of rotation and the axis of the earth’s magnetic field is shown in figure 13.6. This difference complicates things for compass-wielding map users and navigators because they cannot simply align the (magnetic) north direction line on their compass with the (true) north direction line on their map. To make things even more complicated, magnetic poles are not stationary—they move. As the earth’s magnetic field changes, the magnetic poles move, as you can see in figure 13.7. NOAA states that, “The magnetic north pole moves by a significant but variable amount from day to day and year to year (on the order of 40 kilometers or 25 miles).” Over the past century, the magnetic north pole has shifted more than 620 miles (1,000 kilometers) toward Siberia. Additionally, these magnetic north and south pole positions, called dip poles, are rarely antipodal. Magnetic declination This angular difference between true north and the magnetic north pole that a compass needle points to is called magnetic declination, also called compass variation on navigation charts. The actual declination at a location may vary from this angular difference

because of local disturbances in the magnetic field that can cause a compass needle to point away from the magnetic north pole. According to the US Geological Survey, at extremely high latitudes, a compass needle can even point south. Using declination charts, local calibrations, or online declination correction apps, however, compass users can compensate for the differences and point themselves in the right direction. Magnetic declination is shown on maps with isogonic lines, or lines of constant angular difference between true and magnetic north (see chapter 8 for more on isolines). A global isogonic map for the world using data from 2015 is shown in figure 13.8. The line of “no declination,” the green line labeled 0°, is called the agonic line. At any position along this line, the true and magnetic north poles are aligned and the compass needle points to true north. In contrast, magnetic declination exceeds 20° in the northwest of the North American continent and of northern Europe.

Figure 13.7. The location of the magnetic north pole shifts over time. In 2015, when this map was made, the magnetic north pole was located about 400 kilometers (250 miles) south of the geographic pole. To see both measured and predicted pole locations, view the web-based map app from NOAA, at http://maps.ngdc.noaa.gov/viewers/historical_declination. Courtesy of the National Ocean Service.

In figure 13.9, blue isogonic lines indicate negative declination. In these areas, the compass needle points to the east of true north and has easterly declination. In areas with red isogonic lines, there is positive declination where the needle points to the west of true north and has westerly declination. If you use a compass in areas in which the

positive or negative declination values are large, you can’t ignore the large difference between true and magnetic north; and if you try to use your compass, you can be in for a surprise. Later in this chapter, we explain how to adjust for magnetic declination when doing map and compass work. Declination is important to keep in mind, especially when you’re navigating a boat or an airplane. Of course, it is more critical to take note of declination for some uses rather than others, depending on how much accuracy is desired. Navigators and others who work with precise directional information must always be conscious of declination. You can undoubtedly think of other examples. For instance, someone building a solar house also wants a precise determination of true north so that the structure can be aligned as effectively as possible. The rest of us can sometimes disregard declination—at other times when accurate direction is important, we should not.

Figure 13.8. World isogonic map shows magnetic declination in 2015. Isogonic lines are in two-degree intervals. Red isogonic lines indicate positive declination, and blue lines indicate negative. Magnetic north and true north are aligned along green agonic lines (0°). Map developed by NOAA/NGDC & CIRES.

Figure 13.9. Map with a Mercator projection shows annual change in magnetic declination in 2015. Annual change lines are in two-minute intervals. Map developed by NOAA/NGDC & CIRES.

Annual change As with the locations of the magnetic north and south poles, the magnetic declination for a given location changes slightly from year to year, albeit in a somewhat predictable manner. For this reason, scientists can predict, or model, the change. Magnetic declination calculators online and on your navigation devices use these models to calculate the declination for a specified location. Some GPS receivers have built-in models of the earth’s magnetic declination and can be set to automatically compute magnetic declination for the receiver’s current position. The software in GPS devices can be updated to take account of these annual changes. Nautical charts tell how much the magnetic declination changes each year and in what direction. Charts for the United States give the annual increase or decrease in minutes of a degree per year. Canadian charts indicate the amount and direction of annual change (for example, 13´ W) but leave it to you to figure out whether it is an increase or a decrease in declination. It is important to note the date of the chart you are using to accurately calculate the magnetic declination. If the chart is used a decade or so after it is created, the declination may be significantly changed from that printed on the chart. The compass variation diagram on a 1998 nautical chart of Prince William Sound, Alaska, shown in figure 13.10, for instance, shows a compass declination of 24° E, with an annual decrease of 12´. By 2015, the 12´ annual decrease had accumulated over 17 years to 3°24 ´, so that the declination is computed as 24°00´ − 3°24´, or 20°36´. Looking carefully at the 2015 isogonic map in figure 13.8, you can see that the actual 2015 magnetic declination is about 18 degrees east, meaning that the annual decrease predicted in 1998 was too low, by about 2.5 degrees. In this case, using a 1998 nautical chart for sea navigation in 2015 was

a potentially dangerous decision if highly accurate compass directions were required for a successful voyage.

Figure 13.10. A typical compass rose from a US nautical chart for Prince William Sound, Alaska. This compass variation diagram is positioned on the chart so that 0° on the outer circle orients vertically with true north, while 0° on the inner circle shows magnetic north. Compass variation and its annual change are also indicated around the center of the diagram. Courtesy of the National Ocean Service.

Declination diagram Magnetic declination is shown on maps in a variety of, primarily diagrammatic, ways. On large-scale nautical and aeronautical charts designed to be used with a compass in a boat or aircraft, respectively, magnetic declination, or compass variation, is shown on a circular compass rose that is printed in one or more places (figure 13.10). The outer circle of the

compass rose is oriented to true north, whereas the inner ring is oriented to magnetic north. The angular difference between the zero-degree north points on the two rings indicates the magnetic declination at the time the chart is made. The annual change in magnetic declination is also shown near the center of the compass rose, along with the date the chart is made so that the navigator can determine the cumulative change in declination since the date indicated. Many maps provide us with declination information because they are designed for use with a compass in the field. On these maps, a declination diagram—a diagram that shows the angular relationships, represented by prongs, between grid, magnetic, and true norths—is printed in the margin of the map. An example is the diagram in figure 13.11, which shows the angular differences among the three north reference lines, taken from the Madison West, Wisconsin, USGS 1:24,000-scale topographic map. The grid, true, and magnetic north reference lines are identified at the top of the diagram with the letters GN, a star, and the letters MN and an arrow, respectively.

Figure 13.11. A declination diagram shows relationships between grid, true, and magnetic north reference lines on the Madison West, Wisconsin, USGS 1:24,000scale topographic map. Courtesy of the US Geological Survey.

Be sure to use the declination values printed on the diagram. Don’t measure the angles on the diagram with a protractor because small angular differences are often exaggerated on the declination diagram to make them more discernible. UTM grid declination and

magnetic north declination are given for the center of the map sheet. Declination in areas toward the left and right edges of the map can be slightly different. The larger the area covered by the map, the greater the change in declination from one side of the map to the other.

Compass direction systems Our systems for giving names or numeric values to directions come from the various ways we have for identifying directions on different kinds of compasses. Three compass direction systems—compass points, azimuths, and bearings—are important because of their widespread use in navigation, scientific work, and land surveying. We look at each of these direction systems in this section.

Figure 13.12. The standard compass card has 32 points, which consist of 8 points, 8 half points, and 16 quarter points.

Compass points

The oldest compass direction system is the use of compass points (figure 13.12), which are directions indicated by arrows or prongs on the compass card—the graphic face of the compass. Early mariners, who used the winds to make their way, devised the compass point system. The first mariner’s compass cards had eight points, representing the common directions of the principal winds. But eight directions, sailors found, weren’t exact enough. So they split the compass card further—first adding eight half points and later adding 16 quarter points. The mariner’s compass thus came to have 32 points. The card, which looks something like a 32-petaled flower, is also called a compass rose. It’s similar to the compass rose for navigational charts (see figure 13.10), but directions are indicated by compass points rather than degrees. Unlike the circular compass on a navigational chart, the points on a mariner’s compass are named using a standard lettering system for the common directions, as figure 13.12 shows. The sizes of the compass points establish the priority used to name direction. North, northeast, east, southeast, south, southwest, west, and northwest have first priority; the eight half points have second priority; and the 16 quarter points have third priority. A compass rose reading, therefore, is in the form of NE (verbally stated as “northeast”), ENE (stated as “east-northeast”), and NE by E (stated as “northeast by east”). Each of these terms, of course, has a direct numerical counterpart in degrees, because half points are 45 degrees apart and quarter points are 22.5 degrees apart. Today, the mariner’s compass point system is largely replaced by azimuth and bearing readings. Not only are azimuths and bearings less awkward and confusing to use, they are far more precise. Azimuths The most common system of defining compass directions is the use of azimuths. An azimuth is the horizontal angle measured in degrees clockwise from a north reference line to a direction line (figure 13.13). Azimuths range from 0 to 360 degrees and are written either in degrees, minutes, seconds (DMS), as 45°22´30˝, or in decimal degrees (DD), as 45.375° (see chapter 1 for more on converting between degrees, minutes, seconds and decimal degrees). A back azimuth is the opposite direction (180 degrees) from a given azimuth (in chapter 14, you see how azimuths and back azimuths are used in position finding). The rule for determining back azimuths is to add 180 degrees if the azimuth to the feature is less than 180 degrees, and subtract 180 degrees from the azimuth if it is greater than 180 degrees. Azimuths are named according to the north reference line that is used. Thus, there are true, grid, and magnetic azimuths. Compass roses for nautical (see figure 13.10) and aeronautical charts have both true and magnetic north azimuth circles on the azimuth card to simplify navigation by either true or magnetic compass directions, respectively.

Figure 13.13. Azimuths and back azimuths are measured in degrees clockwise from a 0° north reference line at the top of an azimuth card consisting of a circle graduated from 0 to 360 degrees.

Bearings The third compass direction system is the use of bearings. As with azimuths, bearings are angles that are given in degrees from a reference line. The difference is that whereas azimuth readings range from 0 to 360 degrees, bearings range only from 0 to 90 degrees. Bearings are measured clockwise (eastward) or counterclockwise (westward) from either a north or south reference line, whichever is closer to the direction line. To avoid ambiguity with this method, it’s essential to give both the reference line (north or south) and an orientation (east or west) in addition to the angular measure in degrees. When expressing bearings, keep in mind that they can be relative to a true, magnetic, or grid reference line and that the reference line is either north or south. Therefore, bearings are written as N45°E (meaning 45 degrees east of north), S45°W (meaning 45 degrees west of south), and so on, as illustrated in figure 13.14. A compass with a bearing card has a compass card similar to the one shown in figure 13.14. This type of compass is frequently referred to as a surveyor’s compass because it is long preferred as a surveying instrument. Surveyors like the bearing system because specifying the opposite direction (called a back bearing) is simply a matter of changing letters. For example, the back bearing of N45°E is S45°W. But because surveyors must often convert bearings to azimuths when recording the angles they measure, it’s common for bearings to be augmented with azimuths on the compass card (you will learn more about the use of bearings and back bearings in position finding in chapter 14).

Figure 13.14. Bearings and back bearings range from 0 to 90 degrees and are measured from north or south toward east or west.

Conversions With three systems for specifying direction, sooner or later you’ll want to convert from one type of compass direction system to another. For example, you might have to compute a grid azimuth from the UTM grid coordinates for two locations on a topographic map, and then convert it to a magnetic or true azimuth for use with your magnetic compass in the field. When you do this conversion, remember the declination diagram from figure 13.11. Also consider the mathematical relationships among the three systems of specifying direction. For instance, a true azimuth of 85 degrees converts to a true bearing of N85°E, and both are roughly equivalent to the “east” compass point. The declination information provided on large-scale topographic maps and nautical and aeronautical charts lets you convert true, magnetic, and grid azimuths from one form to another. To understand declination information more easily, it’s a good idea to make an enlarged sketch of the declination diagram and an approximately correct direction line. For example, in figure 13.15A we sketched a direction line for a 65-degree true azimuth at a locale with a 17° E magnetic declination and 3° W grid declination. By marking the two declinations and the true azimuth with arcs, you can see that the magnetic azimuth for this direction must be 65 degrees minus the 17-degree magnetic declination, or 48 degrees. Similarly, you can see that the grid declination must be added to the true azimuth when

converting to a grid azimuth. In this example, the grid azimuth is 65 degrees plus 3 degrees, or 68 degrees. Converting bearings to and from true, magnetic, and grid north reference lines is a little more complex, but a sketch of the angles involved again clarifies what must be added and subtracted. In figure 13.15B, we sketched a direction line for a 1° true azimuth at a locale with a magnetic declination of 22° W and grid declination of 2° E. We also marked the two declinations and the true azimuth (shown only in the bottom of the figure). The diagram includes both magnetic and true south and the 22° angular difference between the two. The magnetic bearing is 1° west of true south, so the true bearing is S1° W. Figure 13.15B also shows magnetic and grid south and the 2° difference between them. The direction line for the magnetic bearing is 1° to the east of grid south, so the grid bearing is S1°E.

Figure 13.15. It is sometimes necessary to make conversions from one compass direction system to another. Conversions of (A) azimuths and (B) bearings are described in the text and illustrated in these figures.

MAGNETIC COMPASSES One of the most useful aids to navigation and map work is the magnetic compass. This clever device has three main parts—a needle or floating disk, which is magnetized so that it aligns with the earth’s magnetic field (the north end of the compass needle is painted in red); a jewel pivot, which allows the needle or disk to move freely; and a dial, called a

compass card, marked with directions (figure 13.16, top). The transparent, sealed case that encloses all these parts is known as the compass housing or compass capsule. Orienteering compasses (figure 13.16, bottom) also have a base plate marked with a ruler and sometimes map scales, a degree dial to read the bearing, and an arrow that indicates the direction of travel. Orienteering is a sport that requires the use of a map and compass to navigate from point to point. Some compasses also have long or round levels for aligning the compass with horizontal (see figure 13.17). Sights and a viewing mirror are sometimes added to help you determine angles more accurately (see figure 13.18). A compass needle (or disk) that can rotate in any direction, not just horizontally, also indicates the intensity of the earth’s magnetic field at the locale. At points where the earth’s magnetic field is parallel with the earth’s surface, the compass needle is perfectly horizontal. In figure 13.6, the magnetic field lines are parallel with the earth’s surface at what is called the magnetic equator. The angle between the compass needle and the horizontal plane is the magnetic dip, also called the magnetic inclination. This dip is the angle of the compass needle when the compass is held in a horizontal orientation. Magnetic dip increases progressively from 0 degrees at the magnetic equator to 90 degrees at each of the magnetic poles. To get an accurate compass reading, the compass needle must be “balanced” on the pivot or in the fluid bowl so that it does not drag on the top or bottom of the housing. A typical feature of most magnetic compasses is that they function well only within a limited latitude range, because the changes in the magnetic field from equator to pole affect the vertical balancing of the needle. Because the magnetic dip varies considerably in different locations, a compass needle that “balances” perfectly in Australia, for example, will drag or stick on the compass card or housing in the United States or Canada. For this reason, the compass industry divides the earth into the seven compass balancing zones shown in figure 13.17. Commercially available compasses are balanced for the zone in which they are sold. Global compasses solve this compass needle balancing problem. The compass needle and magnet are built as separate units that function independently so the vertical dip of the earth’s magnetic field cannot tilt the needle downward. The compass magnet, separated from the needle, absorbs the downward force of the magnetic field and thus works well at any location. Because of the strong magnet, the globally balanced compass needle settles quickly and stops immediately at the correct azimuth, which gives an accurate reading.

Figure 13.16. Basic features of a rotating-needle compass. Orienteering compass courtesy of Brunton Inc.

Types of compasses Compass quality varies greatly, usually in direct relation to the price. You may be confused by the wide range in prices and designs, but it helps if you understand that there are only four basic types of compasses. For convenience, we refer to them as rotating-needle, rotating-card, reversed-card, and electronic compasses. Rotating needle Rotating-needle compasses have the direction system inscribed clockwise on the compass card (figure 13.18). The compass needle is balanced on the jewel pivot and

rotates independently of the compass card. This design has the advantage of being able to make either magnetic or true readings. To find magnetic north, rotate the compass card slowly until north on the dial lines up with the needle. With your compass set this way, you can take a sighting on (record the bearing or azimuth to) any object, and the reading on the compass is relative to magnetic north.

Figure 13.17. Compass zones divide the earth into regions in which compasses accurately indicate the magnetic dip. Data courtesy of NOAA. Compass zones courtesy of E. S. Ritchie & Sons Inc., after Chulliat et al.

Suppose, for instance, that you want to find the direction to a distant tree. With the needle stabilized on north, face the tree and project a visual direction line (or sight line) from the center of the compass to the tree. You can now read from the dial the number of degrees between magnetic north and the tree. Your reading, of course, is only approximate. A more precise reading is possible if a sighting aid is added to the compass. Sighting aids are of two forms. One type, resembling a standard raised gunsight, is mounted on opposite sides of the housing, as with the sights in figure 13.19. Instead of peering over your compass at the tree, simply rotate the sighting aid until you can see the tree through it. You can then read the angle between the sight line and magnetic north. Sighting aids make readings easier and more accurate. Most rotating-needle compasses don’t include these raised sights, however, because they

increase the cost and bulk of the compass, and approximate readings are usually good enough.

Figure 13.18. Basic components of a Brunton magnetic compass. Courtesy of Brunton Inc.

A second form of sighting aid is an orienting arrow along the direction line, as found on orienteering compasses (see figure 13.16, bottom). On these compasses, the housing is mounted on a transparent rectangular base plate. To use an orienteering compass, align the direction line, which is etched into the base, with your destination, and then rotate the compass housing until the orienting arrow aligns with the magnetic needle. Now read the direction on the degree dial that the needle points to. This magnetic reading is the direction to your destination. Median lines, which align with the north–south lines on a topographic map, help you orient the compass to the map. As we mentioned, an advantage of rotating-needle compasses is that they allow you to make true, as well as magnetic, readings. To find true rather than magnetic north, first look at a declination diagram or compass rose on the map or chart to find the local magnetic declination. Then rotate your compass east or west until the needle points to the amount of

declination for your area. For example, if the local declination is 5° E, line up the needle with 5° E on the dial. Now when you take a sighting to an object, you obtain a true direction reading, because you are compensating for magnetic declination. Therefore, a rotatingneedle compass is easy to use with maps when a true reading is most useful, as well as in the field when a magnetic reading is the most practical. Rotating card On a rotating-card compass, the magnetic needle and compass card are joined and work as a single unit floating in a fluid bowl. Consequently, you don’t have to bother with aligning the needle and compass card, as you do with the rotating-needle compass. As soon as the disk stabilizes and the needle points north, you’re set to take your sightings. Magnetic readings are thus simplified—an advantage when you are in the field without maps or charts. But the rotating-card compass points only to magnetic north, not true north. To obtain a true-direction reading, you must add (or subtract) the local declination to (or from) your reading. The mariner’s compass is probably the most widely used rotating-card compass. Used since medieval times, these compasses have a floating magnetized card in a fluid bowl of clear liquid (figure 13.20A). A vertical pin in the middle of the bowl centers and stabilizes the card while allowing it to turn freely to magnetic north with changes in the boat direction. The compass is suspended so that it remains horizontal as the ship pitches and rolls in the water. A lubber’s line parallel to the centerline of the ship is marked on the edge of the bowl to allow directions to be read relative to the direction of travel.

Figure 13.19. Reversed-card compass. Courtesy of Brunton Inc.

Figure 13.20. Rotating-card compasses include the (A) mariner’s compass, (B) lensatic compass, (C) dashboard compass, and (D) hand-bearing compass. Mariner’s and dashboard compasses courtesy of E. S. Ritchie & Sons Inc. Hand-bearing compass courtesy of Davis Instruments. Lensatic compass made for the US Army.

Lensatic compasses, available as military surplus items, also fall into the rotatingcard group (figure 13.20B), as do the common dashboard compasses that have floating dials (figure 13.20C). Another example is a floating-dial hand-bearing compass (figure 13.20D), which may be mounted on a pistol grip and have a vertical lubber’s line and gunsights, all to help you align the compass quickly and accurately. Reversed card Using either a rotating-needle or rotating-card compass, you must figure out the angle between true north and the object that you are sighting. A reversed-card compass, like the Brunton compass in figure 13.18, directly displays this angle for you. You must pay for such convenience, but for some people, like surveyors, hydrologists, and others who make much use of maps and compasses, the ease of reading direction is worth the extra cost. Reversed-card compasses, like rotating-needle compasses, have a freely rotating needle that is independent from the compass card. They are almost identical to rotating-needle compasses, except that they include a number of refinements that make them more useful. Many of these compasses have a mirror on the inside of their hinged lid. When the lid is open, the compass is reflected in the mirror. This reflection enables you to sight a direction or an object and see the compass heading at the same time (figure 13.21). To use this

compass, hold it level with the lid open approximately 45 degrees. Use the sighting aids to align the compass with the object. Then, looking into the mirror, turn the rotating housing to align the orienting arrow with the north needle.

Figure 13.21. A mirror in the hinged lid of a reversed-card compass allows you to sight an object and see the compass heading at the same time.

The first thing you’ll notice about the compass card on a reversed-card compass is that the direction system seems to be backward—west and east are reversed (see figure 13.19). The manufacturer hasn’t made a mistake—there’s a good reason for this design. To visualize the logic behind the reversed-direction system used with these compasses, imagine that you are facing north and the needle on your compass is pointing north. Now turn slowly and align the sight on the compass with the object whose direction you are seeking. Although the needle actually remains stationary, it looks as though it is moving around the compass card in the opposite direction from which you move. But because west and east are reversed on the compass card, the direction appears correct in the mirror that you look at to read the azimuth. The reversed-card compass is the most sophisticated, expensive, and versatile of the three types we’ve discussed so far. Like the rotating-needle compass, it can be set for local magnetic declination, thereby permitting true readings. Although rotating-needle compasses sometimes include a sighting aid, reversed-card compasses always do. More than one direction system is often incorporated on the compass card to make several different types of readings possible. Many professionals use a tripod with reversed-card compasses, just as photographers do with expensive and sensitive cameras, to increase the accuracy of the readings. All these features make for a useful compass. Reversed-card compasses are used almost exclusively by professionals. Forester compasses (so named because foresters use them to plan timber cutting lines) fall into this class. The best-known example of a forester compass is the Brunton Pocket Transit (see figures 13.18 and 13.19), which is actually a combination of surveyor’s compass, clinometer, hand level, and plumb, and can do engineer-level work. It can be used to measure azimuths, vertical angles, inclination of objects, percent grade, and slope.

Electronic Electronic compasses measure the relative strengths of magnetic fields passing through two magnetometers, which measure the strength and direction of the earth’s magnetic field. Sophisticated electronics convert these measurements into a continuous determination of the direction of the earth’s magnetic field, thus giving magnetic north. The instrument then computes the azimuth between magnetic north and your straight-ahead direction of travel. Electronic compasses use digital displays to show the direction that you, your boat, or your plane is pointing (figure 13.22). Most such compasses allow you to enter the magnetic declination so that you can determine and display true azimuths as well. Many smartphones can also be used as compasses. An internal magnetometer measures the earth’s magnetic field. This information is combined with data from an accelerator inside the phone to pinpoint the phone’s position in space and measure its tilt and movement. To determine directions, you need a compass app on the phone, and you must also calibrate the compass. The compass app displays cardinal directions, no matter which orientation the phone is in. It may also automatically turn on your camera so the compass card can be displayed on the view. Another form of electronic compass is the gyrocompass used on ships and aircraft. A gyrocompass finds true north using a motor-operated gyroscope whose rotating axis is held parallel to the axis of the earth’s rotation and thus points to true north, not the earth’s magnetic pole. Because gyrocompasses are not based on magnetism, they are not affected by iron or other ferrous metals, as in a ship or aircraft.

Local magnetic anomalies In certain areas on the world isogonic map (see figure 13.8), the isogonic lines appear to curve rather than form a straight line to the magnetic pole. This difference between true north and magnetic north (compass deviation) occurs when the compass direction is strongly affected by local anomalies. These anomalies may be caused by nonmagnetic forces, such as regional variation in earth density, or by additional local magnetism, such as that produced by magnetic-ore bodies. Localized magnetic anomalies that shift the compass needle 30 degrees in less than a mile exist on the earth but cannot be plotted on the small-scale isogonic map. On navigational charts, the exact location of these anomalies may be described in notes printed in the margins, or they may be labeled on the map, as shown on the segment of US Chart No. 1 (figure 13.23), which isn’t a chart at all but a book with all the symbols used on National Ocean Service nautical charts. Compass deviation isn’t always so obvious, though—it may even differ from what’s on your map. It may also be caused by local disturbances, such as the presence of power lines and iron objects, AM radio signals, and even thunderstorms. Because this second source of deviation doesn’t show up on isogonic maps, you must constantly watch out for it. Try to keep away from known disturbances, and be on the lookout for unknown ones, which may make your compass needle pull away from north or act erratically.

Figure 13.22. Electronic compasses with digital displays, including a watchcompass combination. Digital compass by PNI Corp.

Figure 13.23. This segment of US Chart No. 1 shows how magnetic anomalies are symbolized or noted on National Ocean Service nautical charts. Courtesy of the National Ocean Service.

DIRECTION FINDING ON LARGE-SCALE MAPS Determining the azimuths or bearings of direction lines is the basis for several positionfinding methods that we discuss in chapter 14. Even though computerized methods can be used to easily find directions on large-scale digital maps and charts, it’s good to know how

to determine directions in the traditional manual manner. You can accurately measure the azimuth or bearing of a direction line if you plot its endpoints on a large-scale map designed for navigation purposes. Topographic maps, and nautical and aeronautical charts are good examples, because all are made on conformal map projections that preserve the angles of the azimuths and bearings (see chapter 3 for more on conformal map projections).

Figure 13.24. A T square and protractor can be used to measure azimuths and bearings on paper maps and charts. Map courtesy of the National Ocean Service.

For paper maps and charts, a straightedge and protractor are all you need to measure these angles. A commonly used straightedge for map reading is a T square, which is a ruler with a short perpendicular crosspiece at one end. This tool is used together with a protractor, which divides a circle into equal angular intervals, usually degrees (figure 13.24). You’re probably already familiar with semicircular and circular protractors used for basic geometry. You can use these tools for direction finding, along with square, rectangular, and course-plotting protractors that are specially made for measuring directions on maps (figure 13.25). All protractors have the angle scale along their edge and an index mark at the center of the protractor circle.

Figure 13.25. Types of protractors used to measure azimuths and bearings on maps.

Topographic maps To determine the true azimuth for the direction line between points A and B on a topographic map, use the following procedure with a semicircular or rectangular protractor (figure 13.26): 1. Place the map on a rectangular table so that the bottom edge of the map is parallel with the bottom of the table. Align the bottom of the map with the top edge of a T square placed along the left edge of the table. The map is now oriented to true north. 2. Lightly draw the direction line on the map.

Figure 13.26. Finding the true azimuth of a direction line on a topographic map. See the text for the steps in the measurement procedure. Map courtesy of the US Geological Survey.

3.

Place the protractor along the top edge of the T square, and then move the T square vertically and the protractor horizontally so that the index mark is over the start of the direction line. 4. Read the true azimuth from the angle scale on the protractor. You can use the same procedure to determine a grid azimuth, except that you must first orient the map to grid north. To do so, slightly rotate the map on the table until either a northing line or edge ticks for the same northing align parallel to the bottom of the T square. If UTM easting lines are printed on the map, you can align the index mark and the 90° mark on the protractor with one of these vertical lines. Now move the index mark to the start of the direction line, and you can read the grid azimuth from the protractor. Magnetic azimuths are best measured by finding the true or grid azimuth for the direction line, and then converting to a magnetic azimuth using the procedure described earlier in this chapter under “Conversions.” You may be tempted to orient the map to the magnetic reference line on the declination diagram, and then measure the magnetic azimuth directly with your protractor. Don’t do this! Remember that the declination angles are usually so small that the angles between the reference lines are exaggerated to make them distinguishable. Only the declination values printed on the diagram are correct.

Nautical charts Accurately determining direction on nautical charts is a critical task in marine navigation. (The same is true for aeronautical charts and air navigation.) Centuries ago, chart makers simplified the task by putting one or more compass roses on each nautical chart, as shown in the examples from US Chart No. 1 in figure 13.27.

Figure 13.27. Examples of compass roses printed on NOS nautical charts. Courtesy of the National Ocean Service.

The compass rose on a nautical chart has an outer circular angle scale for true azimuths and an inner angle scale for magnetic azimuths (figure 13.27). Marine navigators use a special straightedge, called a parallel rule, together with a compass rose to determine directions on charts (figure 13.28). A parallel rule consists of two straightedges connected by metal bars that allow the edges to remain parallel as they are separated. To operate the parallel rule, follow these steps:

1. 2. 3. 4.

5.

Push the two straightedges together, and align the bottom edge with the direction line (figure 13.28A). Holding the top straightedge firmly in place, separate the straightedges so that the bottom straightedge is closer to the compass rose (figure 13.28B). Firmly hold the bottom straightedge, and move the top straightedge until it closes the gap between it and the top straightedge (figure 13.28C). If the compass rose is far from the direction line, “walk” the parallel rule to it by holding the bottom straightedge in place and bringing the top straightedge to it, and then hold the top straightedge in place and again move the bottom straightedge closer to the compass rose. Walk the parallel rule to the compass rose until you can place the bottom of the straightedge at the center of the compass rose (figure 13.28D). (Reverse this process if the compass rose is above the parallel rule.) The parallel rule should be parallel to the direction line so that you can read the true or magnetic azimuth from the outer or inner scale on the compass rose.

Determining direction from grid coordinates You can compute a grid azimuth if you know the grid coordinates (see chapter 4 for more on grid coordinates) for the endpoints of the direction line. Use simple trigonometry to do the computation, as illustrated in figure 13.29. First, you must determine in which quadrant the direction line lies. If the beginning point has easting and northing grid coordinates (E1,N1) and the ending point is at (E2,N2), the rule for quadrants is as follows: Quadrant I: E2 > E1 and N2 > N1 Quadrant II: E2 < E1 and N2 > N1 Quadrant III: E2 < E1 and N2 < N1 Quadrant IV: E2 > E1 and N2 < N1

Once you know the quadrant, you can figure out the azimuth of the direction line. First, form a right triangle. The lengths of the opposite and adjacent sides of the triangle are the differences of the eastings (E2 − E1 or E1 − E2) and northings (N2 − N1 or N1 − N2) for the endpoints of the direction line. The hypotenuse is the length of the line with endpoints (E2,N2) and (E1,N1). Angle α, which is adjacent to coordinate (E1,N1), is calculated by finding its trigonometric tangent. Remember that the tangent of an angle in a right triangle is the ratio of the opposite and adjacent sides to the angle. Slightly different equations are needed for each quadrant, because the difference in eastings defines the opposite side in quadrants I and III, and the adjacent side in quadrants II and IV. The beginning and ending northings and eastings for the direction line must also be reversed in some quadrants to always have positive distances for the opposite and adjacent sides. The appropriate equations to use are shown in figure 13.29.

Figure 13.28. Finding the true or magnetic azimuth of a direction line by using a parallel rule and a compass rose on a nautical chart. See the text for the steps in the measurement procedure. Map courtesy of the National Ocean Service.

After you find angle α, you must add 90, 180, or 270 degrees for quadrants IV, III, and II, respectively, to convert the angle to a grid azimuth. This conversion is also shown in figure 13.29. An example of this calculation is to find the grid azimuth for a direction line from a beginning UTM (easting, northing) coordinate (345,630mE, 4,335,480mN) to an ending coordinate (353,287mE, 4,308,592mN). Because the second easting is greater than the first and the second northing is less than the first, the direction line lies in quadrant IV.

Figure 13.29. Trigonometric coordinates.

basis

for

computing

grid

azimuths

from

grid

Knowing the quadrant, you can now solve the quadrant IV azimuth equation with a calculator by doing the following calculations, in equation (13.1):

GIS software such as ArcGIS uses calculations from grid coordinates to compute the azimuths between pairs of points, for line segments, and for perimeters of polygons that represent area features. Grid, true, and magnetic azimuths are not typically stored in the GIS database, but it is possible to find Internet services that compute them on the basis of the date the map is published and the location of magnetic north at that time.

DIRECTION FINDING ON SMALL-SCALE MAPS “Fly north to get east” was Charles Lindbergh’s philosophy when he flew from New York to Japan by flying across Canada, Alaska, and Siberia. Determining direction on a spherical earth, as every ship and airplane pilot knows, is far different from direction finding over short distances. Until now, we have defined direction as the angle between a direction line and reference line. But when dealing with directions between points that are thousands of miles or kilometers apart, the flat-earth methods we have discussed so far must be modified to account for the earth’s sphericity. On the spherical earth, there are two types of direction lines: great-circle routes, or the shortest path between two points on the globe (see chapter 1 for more on great circles); and rhumb lines, or lines of constant compass azimuth or bearing. We look at both types of direction lines.

Great-circle routes Directions along the great-circle route between two points on the earth’s surface are true azimuths because they are measured relative to true north. What makes great-circle routes most valuable to navigators is that they are the shortest possible routes between two locations on the spherical earth. To minimize travel time, navigators traveling long distances ideally follow a great-circle route.

Determining direction on a globe GIS software such as ArcGIS uses calculations from geographic (latitude, longitude) coordinates to compute the true azimuths between pairs of points along a great circle. However, a simple manual way to measure true azimuths along a great-circle route is to use a globe (figure 13.30) because all true azimuths along the path are correct. You can hold a string tightly against a globe to find the great-circle route between two locales. A friend can mark the angles from meridians to the great circle, and you can measure these angles on the globe with your protractor. In figure 13.30, the great-circle route between Seattle, Washington, and London, England, for example, has continuously increasing true azimuths, from around 20° at Seattle to about 160° as London is approached. Using a globe to determine the azimuths for a great-circle route is time consuming and difficult to do accurately, especially because the protractor and straightedge do not lie flat on the globe. Measurements can be made more precisely and easily using flat maps. Determining direction in the gnomonic map projection The only map projection in which all great circles project as straight lines is the gnomonic projection (see chapter 3 for more on this projection). You can use the gnomonic projection to find the true azimuth at a point along the great-circle path—for example, between two distant cities such as Seattle and London (figure 13.31). Here’s how: 1. Draw the great-circle route between Seattle and London in the gnomonic projection— this route is a straight line between the two points. 2. Find the westernmost meridian that the great-circle route intersects (120° W in this example). Mark the point of intersection with a dot (a red dot in our illustration). 3. Place your protractor so that the index mark is on the dot and the zero-degree mark on the angle scale aligns with the meridian.

Figure 13.30. True azimuths along a great-circle route are shown correctly on a globe.

4. Note the true azimuth, which is the clockwise angle from the meridian to the great-circle line (29° at the 120° W meridian.) Now repeat steps 1 through 4 for each major meridian. If you navigate using these measurements, you must change bearing each time you come to one of the points at which you determined the true azimuth. The projected ellipses on which the north–south and east–west directions are indicated at the intersection of meridians and the great-circle route resemble simplified “compass roses.” But they are not round—all are ellipses. Your measurement with the circular protractor is being done on ellipses that change in shape along the route. You could make accurate direction measurements if you had an elliptical protractor that let you constantly change its elongation, but no such device exists. Hence, your protractor allows only rough measurements of true azimuths.

Figure 13.31. The great-circle route from Seattle to London is a straight line when drawn on a map with the gnomonic projection. Simplified “compass roses” at points at which the route crosses major meridians are projected as ellipses, showing the distortion of directions that makes it possible to measure true azimuths only roughly along the route.

Rhumb lines When using a compass to navigate, navigators may soon grow weary of continually making directional measurements or computations and steering their craft to exactly follow a greatcircle route. They can alleviate these problems by following a rhumb line, a direction line that crosses each meridian at the same angle. Rhumb lines are routes of constant compass azimuths or bearings and are therefore extremely useful in navigation. If navigators follow one of these lines, they can maintain a course without constantly figuring out new azimuths or bearings and making turns. They merely check the compass to make sure that they are crossing each meridian at the angle of their rhumb line. Their path may curve, but their compass direction doesn’t change. Obviously, it takes longer to reach their destination, however, as this is not the shortest way to get there unless they are travelling along the equator or a meridian.

Rhumb lines that run east–west along the equator or north–south along a meridian eventually return to their points of origin and are great circles. Rhumb lines that run east– west along parallels other than the equator also close in on themselves but are small circles, so they are not great-circle routes (see chapter 1 for more on great and small circles). Rarely, however, are rhumb lines either great or small circles. More commonly, they cross meridians at an oblique angle. To cross each meridian at the same angle, the oblique rhumb line must keep curving on the spherical earth. Its path on the earth forms a spiral, known as a loxodromic curve. Figure 13.32 shows the loxodromic curve that results when a constant 70-degree direction line (one that intersects every meridian at an angle of 70 degrees) starts at the equator and converges on the North Pole. To show all rhumb lines on maps as straight lines, a special projection is required. Mercator solved this problem in 1569, as figure 13.33 shows, by creating a map projection that pulls the meridians and parallels apart at higher latitudes. The result is a conformal world projection that preserves directions at any point on the projection surface (see chapter 3 for more on the Mercator projection). In a Mercator projection, any straight line is a rhumb line, and loxodromic curves appear as straight lines. This valuable property makes navigation by compass and straightedge straightforward when using maps with the Mercator projection. No repeated computations are required, as there are with great-circle routes. The rhumb line route, however, is the shortest path only along a great circle, which, as we pointed out, is limited to meridians and the equator. The 87-degree rhumb line route from Seattle to London is much longer than the great-circle route, although it appears shorter in the Mercator projection. By replacing true azimuths with rhumb lines, navigators make their job easier but lengthen their route.

Figure 13.32. A rhumb line heading of 70 degrees starting at the equator traces out a loxodromic curve as it converges on the North Pole.

The navigator’s dilemma If you’re navigating a long distance, you face a dilemma. Should you use true azimuths along great-circle routes to save travel time, while doing a lot more work to determine each azimuth? Or should you use rhumb lines, simplifying navigation planning but lengthening the trip?

Figure 13.33. Portion of a world map with the Mercator projection showing both the rhumb line and great-circle route from Seattle to London. The great-circle route, obtained from plotting the line on a map with the gnomonic projection, is divided into 500-nautical-mile legs that together appear as a smooth curve. Because the Mercator projection is conformal, each simplified compass rose at the intersection of major meridians and the great-circle route is correctly projected as a circle.

There is a solution to the dilemma: use both. First, plot the great-circle route as a straight line between your origin and destination using a gnomonic projection (see figure 13.31). Next, transfer the great-circle route to a Mercator projection as a curve (see figure 13.33). Then approximate this curve, which is always concave toward the equator, using a series of straight line segments called legs or tracks. Using GIS, which now easily handles map projection variation, this geometric problem is a cinch. In the Seattle-to-London example, each leg starts at 20-degree longitude intervals—the interval you use will vary on the basis of how closely you want to follow the great-circle route. Now you can determine the true azimuth at the beginning of each leg with a protractor (or digitally) by finding the azimuth angle between the vertical meridian and the direction line for each leg.

Determining direction from geographic coordinates You can also compute the true azimuth (Az) at the starting point of the great-circle path. The azimuth computation uses a combination of the law of sines and the law of cosines from spherical trigonometry in an equation for the tangent of the true azimuth, where a and b are the latitudes of the starting and ending points and δλ is the positive difference in longitude between the two points. Here is an example of this computation to find the starting azimuth for a flight from a in Seattle (47.5° N, 122.33° W) to b in London (51.5° N, 0°), in equation (13.2):

The direction determination methods explained in this chapter are the foundation for the digital methods commonly used today in GIS and mobile devices. You can write functions for your GIS that use trigonometric calculations from grid or geographic coordinates to compute azimuths between pairs of closely spaced locales or distant points on the earth, and for line segments along the great-circle route between the points. In fact, researchers at the University of Missouri recently developed a prototype app that combines a smartphone’s GPS, compass, and imaging capabilities to calculate the exact location of distant objects and track their speed and direction. To find out more, go to http://www.gizmag.com/locationtracking-smartphone-app/23946.

SELECTED READINGS Blandford, P. 1984. Maps & Compasses: A User’s Handbook. Blue Ridge Summit, PN: TAB Books. Chulliat, A., S. Macmillan, P. Alken, C. Beggan, M. Nair, B. Hamilton, A. Woods, V. Ridley, S. Maus, and A. Thomson. 2014. The US/UK World Magnetic Model for 2015-2020. Boulder, Co: NOAA National Geophysical Data Center. Accessed February 9, 2016. doi: 10.7289/V5TH8JNW. Deutscher, G. 2011. “Does Your Language Shape How You Think?” New York Times Magazine. Accessed December 3, 2015. http://www.nytimes.com/2010/08/29/magazine/29language-t.html. Deutscher, Guy. 2010. Through the Language Glass: Why the World Looks Different in Other Languages. New York, NY: Metropolitan Books. Fang. Jane. 2014. “Why Some People Really Don’t Need to Ask for Directions.” IFL Science (December 19). Accessed December 3, 2015. http://www.iflscience.com/brain/why-some-people-really-dont-needask-directions. Jonkers, A. 2003. Earth’s Magnetism in the Age of Sail. Baltimore, MD: Johns Hopkins University Press. Kjellstrom, B. 2009. Be Expert with Map & Compass: The Complete Orienteering Handbook, 3rd ed. New York, NY: Macmillan General Reference. Makin. S. 2015. “The Brain Cells behind a Sense of Direction.” Scientific American (May 1). Accessed December 3, 2015. http://www.scientificamerican.com/article/the-brain-cells-behind-a-sense-ofdirection.

Maloney, E. 1988. Dutton’s Navigation & Piloting, 14th ed. Annapolis, MD: Naval Institute Press. Maloney, E. 2003. Chapman Piloting, 64th ed. New York: Hearst Marine Books. Selwyn, V. 1987. Plan Your Route: The New Approach to Map Reading. London: David & Charles. US Army. 1969. “Directions.” Chapter 5 in Map Reading. Field Manual (FM 21-26), 5-1 to 5-27. Washington, DC: Superintendent of Documents. ———. 2013. US Chart No. 1: Symbols, Abbreviations, and Terms Used on Paper and Electronic Navigation Charts. Washington, DC: National Oceanic and Atmospheric Administration and National Geospatial-Intelligence Agency.

chapter POSITION FINDING AND fourteen NAVIGATION ORIENTING THE MAP Inspection method Single linear feature Two linear features Prominent features Compass method DETERMINING YOUR GROUND POSITION Distance estimation Inspection method Compass method Resection method Inspection method Compass method Measurement instruments Stadia Laser range finders Altimeters GPS receivers LOCATING A DISTANT POINT Intersection method Inspection method Compass method Trilateration method Measurement instruments MAPS AND NAVIGATION Land navigation Printed maps and atlases Digital maps and apps Electronic vehicle navigational systems Marine navigation Nautical charts Hydrographic charts Marine navigation methods Electronic nautical navigational systems Air navigation Aeronautical charts Air navigation methods

Electronic aeronautical systems SELECTED READINGS

navigational

14 Position finding and navigation No feeling is as chilling as realizing that you don’t know where you are or how to get where you want to go. The feeling of being lost can be paralyzing, and the fear of becoming lost can be so overwhelming that you hesitate to leave your known environment. But you needn’t give in to fears of disorientation. If you know how to compare your surroundings with a map, you’ll always be able to find your way. Being oriented is knowing your position on the ground and where distant features are located in relation to you. This information lets you navigate (plan and follow a specific route) from place to place. For a long time, people relied solely on their powers of observation for position finding and route planning. Over the centuries, we have enhanced our orientation and navigation capabilities by inventing a variety of technical aids. Clocks, compasses, optical sighting devices, electronic direction and distance finders, inertial navigation systems (INS), and GPS (explained later in this chapter) all fall into this category. No matter how technically sophisticated your position and pathfinding aids, however, they share much in common with longstanding “eyeball” methods. They all use distance or direction information and rely on a few basic geometric concepts. When our modern technical gadgets fail or aren’t at hand, you must fall back on methods that rely on good, old-fashioned visual observation. We start this chapter by discussing traditional observation and compass techniques that have proven useful for centuries and that underlie modern satellite-based methods of position finding and route planning. We begin with orienting the map, then look at ways to find and map ground positions, and conclude with navigation planning and route-following methods.

ORIENTING THE MAP To orient a map means to determine how directions on the map align to directions on the ground. The term comes from medieval Europe, where church scholars drew maps of the known world with the Orient (China) at the top, but today, most maps are drawn with true north at the top so that all meridians converge at the North Pole (see chapter 13 for more on true north). When traveling or navigating, people tend to orient maps in one of two ways. Many people keep the map topside up, no matter which way they’re facing. This approach has the advantage that place-names, symbols, and features are easy to read. But its disadvantage is that map directions aren’t usually aligned with ground directions. When you’re heading south, right on the map is left in reality. Such reverse thinking can make the mind reel. Imagine trying to make a split-second decision about which way to turn when time is of the essence. Because this reversal may not be much of a problem in familiar surroundings, it’s easy to forget that an alternative exists. If you’re in an unfamiliar setting or confusing surroundings, it’s usually easier to find your way if you first orient the map to your direction of travel. Turning the map until ground and map features are aligned with your heading (the direction in which you, your car, your boat, or your airplane is facing) has the advantage that you can always determine directions directly. Although you may have to read place-names and symbols upside down or sideways, it is easier than trying to unscramble skewed or backward directions. You can finesse the problem of symbols and labels being upside down or sideways by thoroughly familiarizing yourself with them before you orient the map. A disadvantage of this method is that when following a route that changes direction often, you must constantly turn the map to stay aligned with the direction of travel.

Inspection method One of the easiest ways to orient a map in the field is by the inspection method of map orientation. With this approach, you simply look at (or “inspect”) ground features— you don’t need to know which way north is, and you don’t need any special tools. Two conditions, however, must be met. First, you must be able to see one or more linear features or prominent objects in your vicinity. Second, you must be able to identify these same features on the map. Various landmarks satisfy both conditions, and many are shown on maps used for navigation precisely for this reason. Single linear feature You have three options when orienting the map with a single linear feature. The most straightforward method is to position yourself on a line shown on the map, such as the road in figure 14.1. Then you need only turn the map until the mapped feature lines up with the real one in front of you (figure 14.1A, middle map). A second method is to take a position that lies on a straight-line extension of the ground feature and turn the map to align with the extension line (figure 14.1A, left map). Your third option is to position yourself to either side of the linear feature, and then turn the map so that it is parallel with the ground feature you are standing next to (figure 14.1A, right map). The first and third options share a serious problem if you happen to orient yourself in a location where you can’t see the beginning or end of the linear feature. In this case, when you line up the map feature with the ground feature, you might not know which direction is

which, and you can end up going in the opposite direction from what you intend. Because of this potential reversal of orientation, you shouldn’t rely on a single linear feature if you can avoid it. Two linear features A better practice is to rely on two (or more) linear features when orienting the map by inspection. For example, you might orient the map by inspecting a road intersection. As with a single feature, you have three options. You can move to a position on either of the two linear features (figure 14.1B, middle map), move to a point that lies on a straight-line extension of one of the features (figure 14.1B, left map), or take a position to one side of one of the features (figure 14.1B, right map). Then you simply turn the map until the two features on the map are aligned with the same features on the ground. In the case of the third option (figure 14B, middle map), map reversal can lead you to orient yourself in the opposite direction. Prominent features Reversal in map orientation can be avoided by using two or more different types of features on the ground that can also be found on the map, such as a house or a tree. First, you move to the location of one of the features (figure 14.1C, middle map) or to a position on a line that is extended through both features (figure 14.1C, left map). Next, use the hypothetical line between the two features on the ground to turn the map until it aligns with the hypothetical (or physical, if you draw it) line between the two features on the map. You can also use the approach of positioning yourself to the side of the features (figure 14.1C, right map). In this case, you must estimate your position relative to the line between the two prominent features.

Figure 14.1. You can use a number of methods to orient your map, including by inspection of (A) a single linear feature, (B) two linear features, or (C) prominent features.

The success of map orientation by the inspection method depends primarily on your ability to identify ground features that can be found on the map and then appropriately position yourself relative to the features. When linear or prominent features are unfamiliar or obscured (by vegetation, fog, or terrain, for example) or when they are locally nonexistent (as on an open plain or ocean surface), orienting a map by inspection is of little value. In these situations, a better method of map orientation is to use a magnetic compass.

Compass method

To use the compass method of map orientation, first be sure that you have both a compass and a map that shows the direction of magnetic north with a declination diagram, such as that shown at the bottom of figure 14.2 (see chapter 13 for more on declination diagrams). Then find the magnetic north indicator on the map—usually a barbshaped arrow placed in the margin of the map (see chapter 13 for more on symbology for north indicators). Holding the map under the compass, turn the map until the compass needle lines up with magnetic north on the map. Because this approach accounts for magnetic declination, your map is now properly oriented with true north as well.

Figure 14.2. When orienting a map to (A) true north or (B) magnetic north using a compass, you need information about the magnetic declination for the map you are using so that you can properly align the map.

Figure 14.2 shows why it’s essential to take magnetic declination into account when orienting a map with the compass method. In this example, the magnetic declination is 20° E. When the compass needle (the red part points to magnetic north) is oriented with the magnetic north reference line (A), true north is located 20° west of the compass needle. Conversely, when the compass needle is oriented with true north (B), the map is oriented to magnetic north (20° east of true north). Magnetic north, however, isn’t always shown on maps. Often the margins are trimmed off digital copies of topographic map sheets to make the file size more manageable and to allow mosaicking of adjacent maps by combining multiple, individual images of adjacent areas into a single image without gaps. The result is that the declination diagram indicating magnetic north is lost. For these reasons, it makes sense to understand the basic pattern of

magnetic declination in the area in which you’ll be using your map. For the contiguous United States in 2015, magnetic declination (see figure 13.8, in chapter 13) varies from approximately 16° W (in the Northeast) to 16° E (in the Pacific Northwest). Other parts of the world have even greater variation. For example, magnetic declination in South America ranges from almost 24° W in eastern Brazil to about 16° E in southern Chile. You’ll find a similar range of declination values in Russia. If you keep in mind the magnetic declination for the area in which you’re traveling or navigating, you will never be far off in estimating the direction of magnetic north. For large portions of the American Midwest and in France, India, many countries in Africa, and much of Indonesia, the declination is close enough to 0° that true north (indicated by longitude lines) provides a reasonable substitute for magnetic north. Elsewhere in the world, if you don’t know the declination, the compass method won’t work for you, and you’ll have to use one of the inspection methods described in this section.

DETERMINING YOUR GROUND POSITION At times, you’ll orient your map on the basis of a known ground position, or you’ll establish your position in the process of map orientation, a technique for orienting the map. At other times, you won’t know your ground location even after you’ve oriented your map. But once your map is oriented, you can figure out where you are on the map.

Distance estimation The most common technique for determining your ground position is distance estimation (or range estimation), in which you use the distance and direction to other objects to locate your position on the map. As with orienting the map, there are two useful methods for estimating distance—by inspection and using a compass. Inspection method When you use the inspection method of distance estimation, the idea is the same as using inspection to orient your map. With the inspection method, you find ground features on a map you’ve oriented and then estimate their distance from you. To use this method, first, orient your map. Then, sight on (look straight ahead at) a feature on the ground that you can also identify on the map to establish your direction line (see chapter 13 for more on direction lines). Next, estimate the distance from your ground position to the distant feature using the scale indicators (scale bar, RF, word scale) on the map. When you mark on the map or mentally estimate the computed map distance along the direction line, you establish your position (figure 14.3A). You can double-check your work by repeating the procedure with several features that are along different direction lines, but don’t be too discouraged if your results don’t agree. Distances are hard to judge accurately, although there are several things you can do to improve your estimates.

Figure 14.3. One way to find your position is to use distance and direction information gained by (A) inspecting your surroundings. Another approach is to (B) use a magnetic compass.

One trick is to use multiples of familiar distance units, such as a football field in the United States (100 yards) or a football pitch in Britain (100 to 110 meters). Still, most people have trouble visualizing these units. A further complication is that farther distances are harder to estimate than nearer distances. As your vision approaches the vanishing point (the point on the horizon at which parallel lines appear to converge), it becomes harder to judge distance. Another trick is to memorize what familiar objects look like at different distances. Because you are so familiar with the size of some objects, such as your house, your car, or a tree in

your yard, you can use that knowledge to estimate distances. Simply compare the size of a similar-looking object on the ground with the “template” you hold in your mind’s eye, and you will have a good idea of the distance to the object. This approach has problems, however. The object on the ground may not be the same size as the one you are familiar with. Even when it is the correct height or width, troubles may still arise. Weather, atmospheric conditions, the relationship of the sun’s rays to the object, and the intervening terrain or other obstructions may all compound the difficulties of judging distance on the basis of the size of familiar objects. A feature on a hillside, for example, appears to be at a different distance when seen from a low position over a flat surface than when viewed from a position on another hillside across from it. If an object is seen from above or below, it appears smaller and farther away than it actually is. An object lit from behind seems farther away than one lit from the front. A brightly colored feature appears closer than a dull feature, and both objects look closer on a bright, dry day than a humid or foggy one. For these reasons, a better method of distance estimation is to use a map and magnetic compass. Compass method The second form of distance estimation involves the use of a map and a magnetic compass, which indicates directions magnetically by the alignment of a magnetic needle or floating disk to the earth’s magnetic field (see chapter 13 for more on compasses). With the compass method of distance estimation, you plot a line from a distant feature to your location using the reading on your compass, and then use inspection methods to determine the distance between you and the object. An advantage of the compass technique is that you don’t need to orient your map before finding your position. Use your compass to sight on a distant feature and note the reading (45° in the example in figure 14.3B). This angle is the azimuth, measured clockwise in degrees between the direction line (the line between you and the object) and the north reference line (the line that points to north on the compass) (see chapter 13 for more on azimuths). Using the azimuth, you can figure out the back azimuth, the angle that is 180 degrees opposite of the azimuth. Back azimuths are calculated by adding 180 degrees to azimuths that are less than 180 degrees and subtracting 180 degrees from azimuths that are greater than 180 degrees. In the example in figure 13.13, in chapter 13, the azimuth is 45 degrees, so the back azimuth is 45 + 180, or 225 degrees. Next, draw using a protractor, or visualize mentally, a line on the map for the back azimuth, from the distant feature back toward your unknown position. You can determine the ground distance from your ground position to that feature using one of the inspection methods discussed earlier. Using the map scale, convert the ground distance to map distance units. Finally, find the estimated distance on the line that you draw or visualize, and that distance is your estimated ground position.

Resection method If accurate position determination is crucial, the resection method is a better technique for locating your position than the two distance estimation techniques described earlier. Instead of estimating the distance from some object on the ground to your position, you can determine the distance accurately and plot it on the map. Plotting multiple lines (called resection lines) from ground features to your position allows you to determine a more exact ground position because the lines cross, or resect, at your position. The position found using resection is sometimes called a cross fix. You can determine this position

using only two lines (figure 14.4A), although it is more accurate to use three lines. In the examples in this section, we explain how resection is done using three lines. You can use either the inspection method or the compass method to determine your position using resection. Inspection method With the inspection method of resection, first, you must be able to find three prominent features on both the ground and the map. With the map properly oriented, determine the direction line between you and one of the features—for this method, this line is called a resection line. Then, find the same feature on the map. Use a straightedge to draw the resection line on the map. This line is really a backsight (similar to a back azimuth) because it’s drawn from the known position of a distant feature back toward your position. Next, draw backsight lines for the other two features. You’re located where the three lines cross on the map. If you look carefully at the intersection of the three resection lines in the example (figure 14.4A), you’ll see that they don’t exactly intersect at a point, but rather form a small triangle called an error triangle. Your location is somewhere within the error triangle. For convenience, you can assume that it is at the center of the error triangle. If this triangle is small, you can feel fairly confident in your results. If the error triangle is large, however, it’s advisable to carefully repeat the procedure.

Figure 14.4. Position finding can be carried out by resection using the (A) inspection method or (B) magnetic compass readings.

The resection lines may not intersect at a point on the map for several reasons. You may have made an error in either positioning the known locations correctly on the map or plotting the backsight lines. But there’s also the chance that you didn’t sight on the same features that are shown on the map or that the mapmaker incorrectly located the features on the map in the first place. More important is that the size and shape of the error triangle depends, in large part, on the geometric arrangement of the three known locations and the backsight lines drawn from these locations. The ideal arrangement is to use features whose locations form an equilateral triangle with your position at its center. The worst arrangement is when two, or all

three, points are close to each other and are in nearly a straight line with your unknown location. In this arrangement, the lines drawn from the distant features typically intersect at a small angle, which may significantly shift the points at which the lines intersect. You’ll get the most reliable position or cross fix when all three of your resection lines cross at about 60 degrees. If this is not possible, try to get at least two of your lines to cross at angles close to 90 degrees. If you can, avoid using lines that cross at angles smaller than 45 degrees. With small angles, even a slight error in drawing the lines may add considerable error to the location at which the lines should intersect. If only two prominent point features are identifiable on both the ground and the map, you can still use resection, proceeding as you do when three features are available. Because only two resection lines are constructed, they cross at a point rather than form a triangle. Regardless of drawing errors on your part, this line intersection gives the illusion of accuracy. Any error you make in either of your sightings won’t be evident. Whenever two rather than three lines are used, you must watch out for this potential hidden error. Compass method Like the compass method of distance estimation, the compass method of resection also makes use of back azimuths—in this case, as resection lines (figure 14.4B). You merely use a compass, rather than a visual sighting, to determine the resection lines, the same as you do when you use a compass for distance estimation (see figure 14.3). Again, you make your resection plot with, preferably, three lines. For the three-point resection method, select three prominent features on the ground that you can also identify on the map. Next, sight on the three distant points with a compass from your ground position and note the azimuths. Using a protractor, plot the back azimuths for the three sightings from their respective map points. The three back azimuth lines form an error triangle, just as your resection lines do when made with the inspection method.

Measurement instruments As you have seen, determining your ground position requires knowing your distance from some object or objects and translating that information to the map. Instruments other than compasses can also be used to help you determine your position. Many gadgets are on the market. We look at three of these devices. Stadia Sophisticated and accurate surveying instruments and hunting scopes that measure distance use the stadia principle. The stadia principle is based on using a constant angle to determine a distance between a telescopic sighting device and a marker, such as the prominent ground features we discussed in the previous section, or special markers designed especially for this purpose. The sighting devices have a built-in pair of horizontal wires, called stadia, which are spaced to subtend a small angular distance (that is, they delimit the extent of an angle). The angle subtended is usually on the order of minutes of angle (MOA), an angular measurement used as the standard for the accuracy of sighting devices. One minute of angle (1/60 of a degree) subtends approximately 1 inch (actually 1.023 inches or 2.6 centimeters) at 300 feet (91.5 meters), 2 inches at 600 feet, and so on. Surveyors use stadia-equipped telescopes to sight on a graduated stadia rod, which is like a large ruler, and then compute the intervening distance using simple trigonometry (figure 14.5).

Laser range finders Newer technology for distance estimation includes the laser range finder (figure 14.6), which uses a laser beam to determine the distance to an object. Although somewhat expensive, these “point and read” instruments are much more accurate and handy. Infrared laser beams built into binoculars or a monocular (an optical instrument for viewing distant objects with one eye) measure distance up to 1,000 yards (over 900 meters) with an accuracy of one yard (almost a meter). The cost and weight of laser range finders are directly proportional to their range and accuracy. Varying from 9 ounces (255 grams) to 3 pounds (1.4 kilos), they’re all portable enough to carry. Laser range finders are indispensable around construction sites and Realtors’ offices, where they are known as electronic rulers. They are also used widely in the military, forestry, and even sports (such as golf, hunting, and archery) to provide accurate distances to the object of interest, whether it’s a tree, a target, or a bull’s-eye. Altimeters In some regions, particularly those that are mountainous, heavily forested, or cloud shrouded much of the time, you may be unable to locate enough features on both the ground and the map to use any of the inspection, compass, or resection methods of position finding accurately. Or you may not have one of your distance-finding instruments with you. In these cases, you might think a GPS receiver, a radio processor capable of processing the signal broadcast by GPS satellites, is your savior—but you may be wrong. GPS receivers have limited capability in heavily forested or dissected terrain because it is not possible for the receiver to get a good reading from the satellites it relies on. We explain why later in this chapter.

Figure 14.5. Distance estimation instruments based on the stadia principle include telescopic sighting devices and stadia rods.

Figure 14.6. Laser range finders time laser beam pulses to find the distance to a distant object.

A pocket altimeter may be your best position-finding aid in such circumstances (figure 14.7). An altimeter is an instrument used to measure the height above a fixed reference, usually sea level. With a topographic map and an altimeter, you can fix your position with the help of only a single feature that is common to your map and your surroundings. An altimeter position fix is easiest to determine if you are located on or near a linear feature, such as a trail, stream, or ridge line (see figure 14.1C, middle map). But it’s also possible to determine your position using an altimeter if you can make a sighting on a distant feature such as a mountain peak, tower, or lake. Before you hike into the wilderness with an altimeter in your pack, there’s something you should know. An altimeter’s primary drawback in position finding is that this instrument of portable size and weight is difficult to keep calibrated. The calibration problems can be traced to the fact that an altimeter is a form of barometer that determines elevation by measuring air pressure. Yet the relation between altitude or elevation and air pressure isn’t a simple one. Under ideal conditions, air pressure decreases systematically with increasing altitude or elevation, so the altimeter’s scale can, in theory, be calibrated in meters or feet to reflect changes in elevation. In practice, though, air temperature and humidity also influence atmospheric pressure and can change rapidly and unpredictably. All things considered, it is best to view pocket altimeter readings as approximate. You can do several things, however, to increase the accuracy of an altimeter’s readings. One is to pay a little more and obtain a temperature-compensated altimeter, which means that the measuring device is designed to compensate for the effect of temperature changes. Also, before going anywhere, always calibrate your altimeter—a process that

requires knowing the correct elevation or the correct sea-level pressure at the location of the instrument. You can use your topographic map to find the accurate elevation for the point at which you are calibrating the instrument. Finally, recalibrate your altimeter as often as possible during your trip. You can do so whenever you come to a landmark at which you can clearly discern the elevation. A trailhead, a hilltop, a saddle between two peaks, and the point at which a trail branches or crosses a stream are good spots to calibrate the altimeter because they are often marked on a map with elevation values or you can easily determine the elevation at these points (see chapter 16 for more on determining elevation). The more often you recalibrate your altimeter, the less chance that changing weather conditions will cause incorrect elevation readings. Linear-feature method Establishing your position with an altimeter when you’re hiking on a path, biking on a trail, traveling on a road, or following some other prominent linear feature is a two-step process. First, read the elevation from your altimeter, gently tapping the device several times to make sure that the pointer isn’t stuck in a wrong position. Next, follow the linear feature on your map until you come to the elevation indicated by the altimeter. This is your approximate ground position. This method works if the linear feature has a steadily changing elevation. Otherwise, it might be hard to pinpoint your position on a meandering path or other lengthy linear feature that varies little in elevation.

Figure 14.7. In mountainous, heavily forested, or cloud-shrouded terrain, a pocket altimeter can be an invaluable position-finding aid.

An example of position finding with an altimeter and a linear feature is shown in figure 14.8. Assume that you park your car at the parking lot with the intent of hiking southwest along the trail. You note that the parking lot is bisected by the 5,800-foot contour. Before starting out from the parking lot, therefore, you calibrate your altimeter to this elevation. When you get to the first stream, you check the altimeter and note that it is reading about 5,850 feet. When you cross the second stream, you note a reading of about 5,900 feet. Later, after hiking for a while through dense fog, you stop and read your altimeter again. The device indicates that you have climbed to an elevation of 7,000 feet. By tracing your progress up the trail, you decide that you must be located near point B, at which the 7,000foot contour crosses the trail. Backsight-line method If you aren’t located on or near a linear feature, you can still use your altimeter to fix your position if you can sight on a distant feature that is also marked on the map. Three steps are required. First, plot the backsight line from the distant feature on your map. To do so, you can use one of the inspection methods discussed earlier. Next, read the elevation from your altimeter. Finally, visually follow the backsight line until you come to this elevation value. This spot should be your approximate position.

Figure 14.8. You can use an altimeter to find your position along a trail and to figure out where you are when you are lost. Courtesy of the US Geological Survey.

An example of this backsight-line method of position finding is shown in figure 14.8. Suppose that you lose the trail in the fog shortly after making the position fix for point B. After a while, the fog clears enough so that you can see a mountain peak (point D) southeast of your position. Say that you have your compass so that you can use the compass method of distance estimation. Using this method, you determine that the magnetic azimuth to the mountain peak from where you are standing is 140 degrees. At your position, your altimeter indicates an elevation of 6,200 feet. By plotting the backsight line from the mountain peak on your map, and then following along the line until you get to the point at which it crosses the 6,200-foot contour, you establish your current position at point C. From this point, you decide that the easiest way to reach the trail is by following a magnetic north azimuth of 95 degrees back to point B. GPS receivers Another instrument you can use to help determine your position and navigate is a GPS receiver. The receivers, along with computers and satellites, make up the global positioning system, or GPS, which allows land, sea, and airborne users to determine their exact location, velocity, and time 24 hours a day, in all weather conditions, anywhere in the world. You probably have heard stories of unfortunate incidents when people rely too heavily on their GPS receivers and forget to also pay attention to their surroundings and trust their instincts. The best way to make use of this often indispensable technology is to have a basic understanding of how it works, as well as its advantages and limitations. Using radio signals transmitted along a line of sight from different satellites, the receiver’s position is determined by calculating the time difference for signals to reach the receiver. Because of the highly precise atomic clocks on the satellites, a receiver at a position on the ground can be used to calculate the exact time it takes to receive the GPS signals. The signal travel times are used along with the velocity of the signals to determine the distance from the satellite to the ground position, and multiple distances are used to determine location through a process called space trilateration. Trilateration is a method of determining relative positions from the measurement of distances, using the geometry of circles or spheres. For GPS, trilateration uses the known locations of two or more satellites, and the measured distance between the GPS receiver and each satellite to find an absolute position or fix. We explain how space trilateration works through the example shown in figure 14.9. Say that our receiver finds our distance from one GPS satellite to be 11,000 miles. This distance narrows down all our possible locations to be on the surface of a sphere that is centered on the satellite and has a radius of 11,000 miles, but many of those locations are out in space rather than on the surface of the earth. Then, say at the same time that the receiver picks up a signal from a second GPS satellite that’s 12,000 miles away. Now we know we’re not only somewhere on the surface of the first sphere, but we’re also on the surface of a second sphere as well. So we’re somewhere on the circle where these two spheres intersect. If the receiver simultaneously gets a signal from a third GPS satellite at 13,000 miles, our position is narrowed down even further, to the two points at which the 13,000-mile sphere cuts through the circle at the intersection of the first two spheres. So by ranging (finding the distance from one location to another) from three satellites, we can narrow our position to only two points. One of the two points is at a position too far from earth, so we can immediately reject it, leaving us the one point that is our GPS fix. Using the distance from a fourth satellite, we can further increase the accuracy of our position, if we have accurate

receiver clocks synchronized to GPS time (the atomic time recorded by the GPS satellites and GPS stations on the ground). The accuracy of a GPS-determined position depends on the GPS grade (the quality of the receiver) and the field conditions. Affordable recreation-grade GPS receivers determine their horizontal and vertical location to within a few meters. More expensive professional-grade receivers can achieve submeter accuracy through use of a method called differential GPS (DGPS). DGPS requires one receiver fixed at a known ground location (called a GPS base station) and one or more mobile or roving receivers, or rovers, that are used in the field. The base station “knows” where it is so it can correct the relative position of the roving receiver, provided the signals received by both the base and roving receivers are from the same GPS satellites. Field conditions include geographic location characteristics, length of time at the location, arrangement of the satellites in the sky, and even sunspot activity. GPS receivers need a clear view of the sky because the radio signals from the satellites are transmitted along a line of sight. So GPS receivers are only useful outside and do not work well in mountainous areas, under forest canopies, next to tall buildings, or in other locations that block or deflect radio signals. These limitations can be overcome to some extent through GPS augmentation, which involves using external information to improve the accuracy, availability, and reliability of the satellite signal. Many GPS augmentation systems are in place, and they vary on the basis of how the GPS sensor receives the information. Some systems, such as differential GPS, transmit extra information about sources of error (such as clock bias or ionospheric delay), others provide direct measurements of how much the signal was off in the past, while others provide additional navigational or vehicle information that is integrated in the position calculation process. One widely used augmentation system is the Wide Area Augmentation System (WAAS), created by the US Federal Aviation Administration (FAA) to make GPS reliable enough for commercial aviation in all weather conditions without the need for other navigation equipment. This system provides extra signals from several ground-based facilities and geostationary satellites.

Figure 14.9. A signal at distance d away from a single GPS satellite can be at a location anywhere on the surface of a sphere of radius d. Signals from two satellites at distances d1 and d2 intersect and define a position anywhere along the edge of a circle at the intersection of the two spheres. With three satellites, the signals with distances d1, d2, and d3 intersect at two points in space. You can discard one of the points because it is not on the earth’s surface. You are left with the position of the GPS receiver, which has an x,y,z coordinate.

GPS is widely used in a variety of navigation, data collection, and mapping applications because of its many benefits—ease of use, relative low cost, coverage over large areas, relatively high-speed data collection, and ability to collect a variety of data in any weather nearly anywhere. Knowing its shortcomings, which include a requirement for power,

potential radio interference, and susceptibility of the satellite signals to the geography of the area, will help you use this aid to position finding and navigation wisely.

LOCATING A DISTANT POINT It isn’t always enough to know how to locate your position on the map. What if you want to determine the position of something else instead? A pond, house, or tree, for instance, may be clearly visible from your point of view on the landscape but not be shown on the map in front of you. You can plot its correct position on the map using either the intersection or trilateration method.

Intersection method You use the intersection method to find the position of something you want to plot on the map. This method is the opposite of resection. As you have seen, resection lines are formed by backsights. Intersection lines, however, are made by foresights—lines from your position to a distant point. Using the intersection method, you sight on the spot with the unknown position from two (or more) ground positions whose map locations are known or can be computed. The point at which these foresights intersect establishes the feature’s location. As with resection, the accuracy of the method of intersection is improved by using three foresight lines rather than the two sighting points in the illustration (figure 14.10) that are accessible to the hiker. Again, the error triangle created by using three sight lines gives you an indication of confidence in how accurately you determine and plot the foresight lines from the formerly unknown position. To establish the position of a distant point by intersection, you can use techniques similar to those you use to establish your own position by resection. The two main techniques are the inspection method and the compass method. Inspection method With the inspection method of intersection, you first orient the map and determine your position on the map. Next, sight from your map position to the distant feature, which should be identifiable as a visible feature on the ground. Use a straightedge to plot this foresight line on the map (figure 14.10A). Now move to a second position that you can find on your map, sight on the same distant feature, and plot this second foresight line. The intersecting foresight lines establish the location of the feature. Although it is better to use three fixed positions to determine the location of a distant point, two can also be used, as illustrated in figure 14.10B. Fire towers, for example, can serve this purpose for firefighters. In this case, one fire station communicates to another the azimuth of a foresight line to the apparent origination of the fire’s smoke (as measured from the map with a protractor). The foresight azimuth is also determined from the other station. Together, these two foresights are enough to find the location of the fire. If only a single identifiable ground point is available, you can still use the intersection method if the location of a second point can be calculated. Simply follow the preceding procedure to make a sighting from the known point, and then repeat the procedure from a second point, which is a measured distance and direction from the first (figure 14.10B). As before, the sight lines intersect on the map at the desired ground position.

Figure 14.10. The intersection method is commonly used to determine the location of a distant point.

Compass method If a map and compass are both available, you can determine the location of a distant point from the compass method of intersection. First, locate your map position, sight on the distant feature, and record the foresight azimuth as noted on the compass (figure 14.10C). Next, move to a second known position, sight on the feature, and record the reading of this second foresight. If you plot the two foresight azimuths from the known positions, the point at which they intersect is the map location of the distant feature. The compass method of intersection can also be used when you work from two fixed positions

or from a fixed position and a computed position. The procedures are similar to the inspection methods discussed in our fire tower example.

Trilateration method As mentioned earlier, trilateration is a method for finding the location of a distant feature using only the geometry of circles or spheres that relate to the distances from known locations to the feature. You probably first become familiar with trilateration when you use a compass drawing tool to draw intersecting arcs of circles from two points on a sheet of paper. We extend this idea to an example for a large-scale map. In this case, you draw arcs of circles with radii that are equal to the ground distance from each point to a distant feature (figure 14.11). Assume that the distance to the distant feature from a radio tower (point A) is 6,700 feet (4,165 meters), from a hilltop (point B) is 3,800 feet (2,360 meters), and from another hilltop (point C) is 7,000 feet (4,350 meters). If you use the map’s scale bar to draw circular arcs of radii 6,700 feet and 3,800 feet outward from points A and B, you see that the arcs intersect at two points, each a possible position of the distant feature. Drawing the 7,000-foot radius circle from point C eliminates the ambiguity—the three circles intersect at a point on the map next to the creek. As with resection, the location is, in reality, within an error triangle.

Figure 14.11. Using trilateration and distance information alone, you can determine the location of a distant feature from (A) a radio tower, (B) a hilltop, and (C) another hilltop. Courtesy of the US Geological Survey.

The size of the error triangle gives you an indication of errors you may make when using the trilateration method. As with resection, you may make an error by not positioning the known locations correctly, or the mapmaker may have misplaced one of more of the features on the map in the first place. You may sight on the wrong feature or calculate the wrong distance from one or more of the features. You may also measure one or more ground distances along a sloping line of sight, but then forget to correct the slope distance to the horizontal distance plotted on the map. As you saw in chapter 12, the difference between the slope and horizontal distance is negligible for the slightly sloping line-of-sight distances that you normally measure, and if it is more, you can easily correct the distance measurement.

Measurement instruments

The aforementioned methods work fine for locating the position of something large and clearly visible, such as a radio tower or hilltop, but for more precise location finding, an alidade is often used. An alidade is a device with a straightedge and a telescope used to sight a distant object. The sight is used to draw a foresight line in the direction of the object or to measure the angle to the object from some reference point (figure 14.12). The Osborne Fire Finder is a type of alidade used by fire lookouts to alert fire crews to the location of a wildland fire. An alidade (including the telescope, its supports, the level, compass, and spindle) makes up the upper movable part of a theodolite (a surveying instrument used to measure horizontal and vertical angles). Alidades were the primary mapping instrument in the United States from about 1865 to the 1980s, when they were replaced by the total station (an electronic theodolite integrated with an electronic distance meter that reads distances and slopes from your position to a distant point).

Figure 14.12. This alidade has a pendulum device for establishing a horizontal foresight from which vertical angles can be measured. A cylindrical bubble, left, and magnetic compass, right, attached to the base are used to level and orient the instrument. This alidade was manufactured by the Keuffel & Esser Co. Courtesy of the US Geological Survey.

MAPS AND NAVIGATION Once you learn how to determine your position and that of distant features, you’re ready to use maps to navigate from one known location to another. Navigation is a two-part activity—planning your route and then following the route that you plan. Planning a route is more than drawing lines on maps. You must find out what features or amenities are at different locations, so you know when you have reached landmarks along the way or the places you want to visit. To understand route planning and following, it is important to

consider the types of special maps and charts available to assist you in navigation (maps used for navigational purposes, especially by ship and plane, are called charts.) We focus on maps and charts used for land, water, and air travel and how you can use them for navigation.

Land navigation Land navigation is often restricted to specific routes such as trails, roads, subways, or railroads. As a result, travelers are primarily concerned with their route network—they don’t much care about the environment as a whole, or about exact compass direction. Maps for such travelers can be greatly simplified, especially if an obvious origin, route, or destination exists. We look first at some of the maps and atlases used for land travel. Printed maps and atlases Often, the most convenient navigation tool at your fingertips is a printed map or a collection of maps in a book called an atlas. The advantage of these maps is that they are familiar to many people, and they do not require battery power, telephone service, or Internet access, as with many digital maps. Simple, schematic maps that focus on road travel have been around for a long time. The Romans simplified their road maps, such as the Peutinger Table (figure 14.13, top), by ignoring exact directions and distances, concentrating instead on showing the road network and cities along the routes. (The Peutinger Table is thought to be a medieval descendant of a road map prepared a thousand years earlier under the direction of Marcus Agrippa, a friend and ally of Roman Emperor Augustus Caesar.) The distortion of geographic space in schematic mapping that began with the Romans has been carried a step further in recent times. Often there has been a concerted attempt to mislead the map user. For example, in the book’s introduction, you saw that the scale and direction relationships on 19th-century railroad maps were often carefully manipulated to create a favorable impression of the company’s service in a region (figure 14.13, bottom). The practice of deliberate map distortion continues today, especially with subway and airline route maps. On these maps, the routes are not shown in their correct ground locations— instead, color-coded lines (often straight on subway maps and curved on airline maps) show the possible connections between origins and destinations. The modern hiking or canoeing guide (figure 14.14) is a similar type of simplified route map. In an effort to make the maps as simple and uncluttered as possible, only the route network and basic geographic reference information, such as prominent landmarks, are depicted. On these maps, however, an attempt is made to retain some indication of correct directions and distances so the hiker or boater can find his or her way along the route. Such maps are well designed for the amateur outdoors enthusiast but are generally inadequate for sophisticated map users. Furthermore, there is a danger that some unplanned event such as a washed-out trail or damaged canoe will force the user to alter course or engage in another mode of transportation for which the map is totally unsuited. The main problem with these simplified route maps, then, is that they don’t often provide flexibility. They keep you tied to a limited, predetermined route.

Figure 14.13. A section of the medieval Peutinger Table, top, and a 19th-century railroad map, bottom, are examples of land travel maps that are more schematic than planimetrically accurate.

The standard road map is a popular navigation tool. In the United States, free road maps were in abundance between the 1920s and 1970s, when major highways were being built— the maps came to symbolize the possibilities of the open road. These maps provide a general picture for route finding, and their coverage is defined by the state, region, or country for which they are made (figure 14.15). A variety of information is available to the traveler, including possible alternate routes such as railroads and ferry lines. Landmarks, such as parks and airports, are displayed as well as physical features such as mountain peaks. Map information selected for the purposes of road travel doesn’t paint a complete

picture of the environment, of course. The traveler concerned with details falling much beyond the view from the road will likely find road maps inadequate.

Figure 14.14. Canoeing and rafting map for the Penobscot River, Maine. Courtesy of Raft Maine.

Figure 14.15. State road maps usually show a variety of information in addition to highways, making them useful for general reference purposes as well as navigational assistance. This Washington road map also offers options for taking ferries across Puget Sound. Courtesy of the Washington State Department of Transportation.

Private companies publish a variety of road and recreation atlases, which are books for travel and trip planning that contain a series of maps covering large regions. Many travelers carry national road atlases produced by Rand McNally, National Geographic Maps, Michelin, or other private organizations. These atlases generally consist of maps for states, provinces, or other regions, showing different classes of roads, mileage along road segments, cities and towns, administrative boundaries, rivers, and other features of interest. Larger scale state road and recreation atlases are produced by such companies as DeLorme and Benchmark Maps. Their maps show in greater detail the same types of

information as in national road atlases, plus provide relief-shaded or layer-tinted topography (see chapter 9 for more on these mapping techniques) and detailed landownership information (figure 14.16). Graticule lines at small increments are printed over these maps to assist GPS users. Recreation features and other potential points of interest (POIs) to travelers are also added to these maps. Large-scale topographic maps are also used for detailed route finding. A 1:24,000-scale USGS topographic map, for example, shows topography, roads, railroads, streams, buildings, forested areas, and other features that are often important to planning and following a route, either by car or on foot. On Ordnance Survey of Great Britain maps, features such as footpaths, bridleways, national trails, and youth hostels are included for ramblers or hikers. The graticule and grid ticks along the map’s margins and in the map’s interior are aids to position finding. Digital maps and apps The popularity of digital maps and apps has skyrocketed in recent years, allowing users to quickly and easily plan a journey or follow a route using online maps. Examples include Google Maps, ArcGIS Online, Microsoft Bing Maps, MapQuest, and OpenStreetMap. Most smartphones come with a map app, and many others can be installed, often for free. In addition to their widespread availability and low or no cost, web map apps are often so easy to use that users do not require documentation. The digital maps generally cover wide geographic extents and can be zoomed out to show larger areas or zoomed in to show detailed coverage. These apps provide useful information for travelers, including the option to receive detailed driving directions, the ability to display traffic conditions, the choice of map view or satellite image view, and the ability to search for locations by name, address, or ZIP Code. A good example is the TripTik Travel Planner, a web map app designed specifically for road travelers in the United States.

Figure 14.16. Section of a map from the Benchmark Road & Recreation Atlas for California. Courtesy of Benchmark Maps.

The TripTik Travel Planner, a widely used digital map developed by the American Automobile Association (AAA), is an online app that allows you to plan your trip and get driving directions and distances. You can also find places to lodge and dine—it includes a feature to make reservations at hotels and restaurants. You can find nearby gas stations and get updated fuel prices, and you can even locate electric-vehicle charging stations and request roadside assistance. On screen, the best route for you to follow between two locations is highlighted on a basemap (figure 14.17). To the left of the map is an itinerary box that shows the total distance of your trip and the estimated travel time for the route

highlighted on the map as well as detailed driving directions. A slider bar allows you to zoom in on segments of the route to see more detailed maps. Amenity buttons allow you to toggle on and off nearby hotels, campgrounds, restaurants, and other facilities. At the highest zoom level, named residential streets are shown along with other details of the local area. In the event of an emergency, it also includes a basemap of the region neighboring the route. It takes a major deviation from your planned route to thrust you into “unmapped territory.” Electronic vehicle navigational systems GPS-based vehicle navigational systems can be found in many passenger cars, trucks, buses, delivery vehicles, and rental cars. These systems can either be installed (interfaced with the vehicle’s own navigational system) or portable. Portable systems are either on handheld GPS receivers or apps on one of the smart devices you carry with you in your vehicle. Smartphones, tablets, and laptops are examples of smart devices, which are generally connected to other devices or networks via mobile networks or wireless systems, that can operate, to some extent, interactively and autonomously. The Mercedes-Benz COMAND navigational system is an example of an installed vehicle navigational system in which the hardware and software are integrated with the car’s dashboard controls. You can display your Mercedes’ route and current position on either vertical or three-dimensional perspective-view dynamic digital maps (figure 14.18). You can enter your destination either by keyboard, by uploading the address from your mobile device, or verbally through the system’s voice recognition unit. The route from your current position to the destination can then be determined in several ways—the shortest distance, the fastest time with or without taking traffic into account, or the most economical. Alternate routes can also be computed that avoid minor highways and unpaved roads, toll roads, tunnels, ferries, or road construction. You can also choose to display POIs, such as gas stations, lodging, rest areas, and other features in the system’s map database. Another option is to view the current traffic flow information for the area displayed on the map. When you are on the road, the map can be displayed as north up or heading up, in which the map is constantly rotated to face your current direction of travel. When approaching a road intersection, the map changes to a split-screen display that shows the intersection in 3D perspective. The system then displays turning and lane-change information on the map as well as verbally. The Apple Maps app is an example of a portable navigational system. Whether driving, walking, or taking public transit, iPhone owners can use the app for route planning and navigation. The map displays your current position and planned route on either traditional vertical- or 3D-perspective basemaps. These constantly updated digital maps show the length of your route in physical distance (miles or kilometers) and estimated travel time. The directions can be in spoken form, so that you can concentrate on the road ahead rather than the map display. Real-time traffic delay information is also displayed on the verticalperspective basemap, and you can get information on the cause of the traffic backup, along with a display of alternate routes to save time. The 3D map also gives instructions in advance for turns you must take.

Figure 14.17. The American Automobile Association has popularized a type of online digital route map called a TripTik Travel Planner. Reprinted courtesy of the American Automobile Association (AAA). All rights reserved.

Another example of a vehicle navigational system for smart devices is Waze, a community-based navigation and traffic app currently accessible in almost 60 countries worldwide. Drivers share real-time traffic and road information so that users can calculate how long they will be stuck in traffic and receive alerts before approaching accidents, road hazards, highway construction sites, or police speed traps. The app helps you find the lowest nearby gas prices, as well as stores, hotels, and restaurants. When you connect to Facebook, you can coordinate the arrival times for picking up or meeting with friends. To use the app, you type a destination and then drive with the app open on your phone. You can passively contribute traffic and other road data, but you can also take a more active role by sharing current road information to give other users in the area advance warning of what lies ahead. Active online map editors use the passive data you generate as you drive to ensure that the data in their areas is as up to date as possible.

Figure 14.18. The Mercedes-Benz COMAND is an onboard navigational system that shows your car’s route and position on vertical or 3D perspective-view dynamic digital maps. Courtesy of Mercedes-Benz USA, LLC.

Marine navigation To understand marine navigation, it is advantageous to know about the types of maps used by mariners and how they are used for position finding and navigation. Using nautical charts is at the heart of marine navigation. Nautical charts The primary tool for marine navigation is the nautical chart, a map specifically designed to meet the needs of maritime navigators. This type of map shows the shoreline and prominent topographic and landmark features, but more importantly it shows offshore features that help navigators avoid hazards and arrive safely at their destinations. These type of maps are designed to be drawn on—courses (navigational routes) can be plotted, directions or bearings can be marked, distances and times can be noted, and more. In contrast to maps and atlases used for land travel, which are often produced commercially, nautical charts are most often produced by government agencies. Different types of charts are produced, but all have in common information about the subsurface water region and areas along land that can be seen from water. Nautical charts are designed to give all available information that is necessary for safe marine navigation, including soundings (water depth measurements), isobaths showing bathymetric contours (lines that connect points of equal depth below the hydrographic datum), obstructions and dangers that are hazards to navigation, bottom features and types, currents, anchorages, and the location and characteristics of navigational aids such as buoys, beacons, and lighthouses (figure 14.19). The scales of nautical charts published for the coastal waters of the United States by the National Ocean and Atmospheric Administration, or NOAA, range from 1:2,500 to about 1:500,000. Coverage includes the Atlantic, Pacific, and Gulf coasts and coastal areas of Alaska, Hawaii, and the United States possessions (Virgin Islands, Guam, Samoa, and

Puerto Rico). The US National Geospatial-Intelligence Agency, or NGA, provides charts for non-US waters. Oceanic mapping agencies, typically referred to as hydrographic organizations or agencies, as well as commercial companies, produce charts for other areas of the world. For example, the United Kingdom Hydrographic Office (UKHO) produces charts for waters under UK national responsibility. These charts are used not only by the Royal Navy but, as with NOAA and NGA charts, also by the navies of other nations and by a large portion of commercial shipping. The International Hydrographic Organization (IHO), an intergovernmental consultative and technical organization established in 1921, helps coordinate the activities of national hydrographic offices. It also tries to ensure the greatest possible uniformity in nautical charts and related documents. Coordinated by the IHO, the International Chart Series (INT Chart Series) is being developed as a worldwide system of charts to unify as many national chart series as possible. Hydrographic charts Hydrographic charts, published for inland waters at various scales, are designed primarily for navigational use. Most of these charts show the hydrography of water areas, together with the topography of limited areas of adjacent shores and islands, including docks, structures, and populated places visible by boat from the lakes and channels (figure 14.20A). Hydrographic charts are also used for flood control, dam safety, hydroelectric power management, water supply administration, shore protection, recreation, and other practices. In the United States, the US Army Corps of Engineers (CoE or USACE), a federal agency that delivers vital public and military engineering services, collects and converts inland waterway data that is commonly used for river and channel maintenance into electronic charts for inland waterways. These charts make up the Inland Electronic Navigation Charts (IENCs) (figure 14.20B). In addition, state agencies such as the Department of Natural Resources produce hydrographic maps for inland lakes and rivers of significant size or recreational potential.

Figure 14.19. Nautical charts contain information specifically for the marine navigator. Courtesy of the National Ocean Service.

Figure 14.20. Portions of the (A) Isle Royale, Minnesota, hydrographic chart and (B) US Army Corps of Engineers navigation chart for the confluence of the Missouri and Mississippi Rivers. (A) Courtesy of the National Ocean Service. (B) Courtesy of the US Army Corps of Engineers.

If you plan to navigate on a water body, a good rule is to obtain the largest scale chart available. Larger scale helps ensure that you have the most comprehensive graphic summary of information pertinent to making navigational decisions. Before you use the chart, study the notes, scale, symbols, and abbreviations placed near the title or next to the legend because they are essential for effective use of the chart. Chart No. 1 shows the symbols and abbreviations used on US nautical charts and is recommended to the mariner for study.

The date of the chart is of such vital importance to marine navigators that it warrants special attention. When charted information becomes obsolete, further use of the chart for navigation may be extremely dangerous. Natural and man-made changes, many of them critical to safe navigation, occur constantly. It is essential, therefore, that navigators obtain up-to-date charts at regular intervals or manually correct their copies with updates. In the United States, changes are announced in a weekly publication titled Notice to Mariners. Marine navigation methods In addition to nautical charts, you also need two special skills for successful and safe marine navigation: piloting and dead reckoning. In piloting, you plan your intended route, and then find your position and direction of movement by paying attention to landmarks, aids to navigation (such as buoys and beacons), and water depth. When dead reckoning, you estimate your present position relative to your last accurately determined location by using direction, speed, distance, and time information. We discuss these two navigational skills separately, realizing that dead reckoning is a key part of piloting. Dead reckoning The term dead reckoning (DR) is a shortened form of the phrase “deduced (ded.) reckoning” used in the era of sailing ships. The basic idea is to deduce your current position from the vessel’s direction of travel and speed. Dead reckoning is used both when planning a route and following the planned route. To plan your route, you use large-scale nautical charts to create a dead reckoning plot. You map your route on the charts as a series of intended dead reckoning tracks, or the straight-line routes you intend to follow between selected points. Two track lines are plotted in figure 14.21 for the last part of a voyage to Flounder Bay on Fidalgo Island, Washington. Assume that your boat has a maximum cruising speed (the speed at which the boat travels) of 10 knots (nautical miles per hour) and a draft (depth from the water line to the bottom of the keel) of three meters. Before plotting intended dead reckoning tracks, navigators highlight hazards and aids to navigation along the route (the circled features in figure 14.21). Tracks must be plotted to avoid obstructions and minimize hazards. Track 2, for instance, is drawn perpendicular to the international shipping lane (shown by the purple dashed lines) to minimize the time spent crossing it. Track 3 is the shortest straight line to Flounder Bay that gives safepassage distance around Williamson Rocks, Allan Island, and Young Island while having water depths greater than the boat’s three-meter draft. Dead reckoning does not take into account the effects of currents, winds, vessel traffic, or steering errors on tracks, although the navigator must correct for all these things during the actual voyage.

Figure 14.21. Route planning by dead reckoning consists of plotting intended tracks on nautical charts. Track information includes the course heading (C), usually shown as a magnetic azimuth, distance (D) in nautical miles, and boat speed (S) over the water in knots. Estimated arrival times at points along the track are also noted. You may also want to circle on the chart nearby hazards and aids to navigation. Reproduced with permission of the Canadian Hydrographic Service.

The course heading (direction) and distance for each intended track are calculated on the chart with a parallel rule and dividers, using the methods described in chapters 12 and 13. The nautical convention is to write the course heading (C) above and the boat speed (S) below the beginning of each track line, and to write the distance (D) below the center of the line under the track number. Track 2, for example, has course C051M (magnetic azimuth of 51 degrees), distance D3.0 (three nautical miles), and speed S10.0 (10 knots over the water). In strict nautical terminology, the azimuth is called a bearing. This distance and speed information allows you to calculate the time T required to complete each track from the equation T = D × 60/S, where 60 is the number of minutes in an hour. From equation (14.1), completing Track 2 should take 18 minutes.

A similar computation gives a crossing time of 26 minutes for track 3. This crossing time means that if you plan to begin track 2 at 8:00 a.m., your ETA at the start of track 3 is 8:18 a.m. and at the final destination, Flounder Bay, it is 8:44 a.m. These times are written on the chart at the start and end of the two tracks as four-digit numbers: 0800, 0818, and 0844, using the notation for military time rather than regular time. The difference between regular and military time lies in how the hours are expressed. Regular time uses numbers to identify the 24 hours in a day as 1 to 12, from both midnight to noon and noon to midnight. In military time, the hours are numbered from 00 to 23, so that midnight is 00, 1 a.m. is 01, 1 p.m. is 13, and so on. Minutes and seconds are expressed the same in both regular and military time. Piloting As mentioned previously, piloting is following your route using landmarks, water depth, and aids to navigation. As with dead reckoning, the first step in piloting is to acquire the large-scale charts that cover the area in which you will travel. For piloting in marine navigation, you also need other navigational publications, specifically tide tables and water current information for the area. In addition to knowing the draft of your boat, you must also know its maximum cruising speed. Using the dead reckoning plot in figure 14.21 as an example, imagine that you’re on a boat trip trying to hold the courses on the plot. This plot, however, doesn’t take water currents into account. Currents can be strong in this locale, so it’s best to look at a map or atlas that shows predicted currents at the time of your voyage. You find that page 10 in the Current Atlas (a document that shows the currents that marine navigators need to navigate and produced by the Canadian Hydrographic Service) for the Strait of Juan de Fuca to the Strait of Georgia (figure 14.22) shows the currents predicted for this area at 0800 on the day of your voyage. The arrows on the map show that the current is predicted to be light ( 0 implies a clustered arrangement. For our building count example, we computed Moran’s I using the inverse distance and polygon contiguity weighting methods previously discussed to obtain the following index values, which indicate a random to slightly clustered arrangement depending on the distance weighting method used: Inverse distance (for all pairs of quadrats): 0.0065. Inverse distance squared (for all pairs of quadrats): 0.0549. Polygon contiguity (for king’s-case neighbors): 0.0736. Polygon contiguity (for rook’s-case neighbors): 0.1781. Examining the range in the values for the Moran’s I index we computed using the different distance weighting schemes, you can see that the values increase as the number of quadrats used in the weighting is increasingly restricted to contiguous neighbors. This difference in values is a drawback of using this method of spatial arrangement analysis— the index values depend on the distance weighting scheme that you choose. For the testing of statistical hypotheses, the Moran’s I value is transformed to a z-score, and a p-value is computed. The z-score (critical value) is calculated in standard deviations from the expected index value, indicating whether you can reject the null hypothesis of no spatial clustering. The p-value (significance level) is the probability of whether the

difference between the expected and observed values is statistically significant. Z-scores and corresponding p-values are shown graphically in figure 17.15. To compute the z-score, the Moran’s I index value (referred to as the observed value) is compared to an expected index value, which is the value you get if there is a random distribution of the values for the features. The expected index value of Moran’s I for a perfectly random arrangement of n data collection units is obtained from equation (17.9):

which is −1/23, or −0.0435, for the 24 quadrats in our building count example. The mathematics used in the computation of z-scores are complex and beyond the scope of this book. However, for the building count example, we used the ArcGIS Spatial Statistics Toolbox to compute the Moran’s I index, z-score, and corresponding p-value for the polygon contiguity, rook’s-case weighting method. We obtained a Moran’s I value of 0.178, a z-score of 1.49, and a p-value of 0.135, indicating a 13.5 percent probability that there is no spatial autocorrelation (that is, you are 86.5 percent confident that spatial autocorrelation exists). Statisticians would reject the null hypothesis of no spatial autocorrelation if the z-score is >1.96 (or 100 percent) or below (