Preface I was first exposed to geodesy while studying geomatics at Texas A&M University. I was taking a course in geographic information systems (GIS) and, during an exercise, came upon a dialog box with a field whose value indicated the datum to which my data referred. I had no idea what this meant, so I asked another student. I was told, “Who knows? Just accept the defaults.” Unsatisfied, I began to search for the answer to what was seemingly a very simple question. This book is a result of my journey along the fascinating and wonderful trails that question propelled me. I now teach geomatics at the graduate and undergraduate levels, and I find my question as pertinent today as it was when I asked it as a student – probably more so. The field of geomatics has blossomed into a multibillion-dollar-per-year industry being supported by university programs around the world. All geomatics practitioners need to know some geodesy in order to understand their spatial data and to use them properly. In searching for a text with which to teach my own classes, I felt that the other geodesy books were either at too high a level or too low. My goal for this book is to introduce the concepts of geometrical and physical geodesy at a scope and at a level that geomaticians are likely to encounter in their practice. My students study natural resource management in the Department of Natural Resources and the Environment at the University of Connecticut. Natural resource managers need to know how to use a GIS, remote sensing tools, and land surveying tools, so I wrote this book with that audience in mind. As a consequence, this book can be useful to a broad spectrum of readers: from those who are specializing in one field of geomatics to those, like my students, who need to know something about them all. Geodesy sits squarely at the intersection of GIS, remote sensing, and land surveying, so it was appropriate to create a broad-based book. My students receive training on how to solve environmental problems, and some of the problem- solving methods have an engineering flavor. Therefore, this book contains many useful formulæ. In general, geodesy is a mathematically oriented field. Geometry and physics are the foundations of geodesy, and a working, practical knowledge of the subject necessarily requires understanding a fair number of mathematical formulæ. Most of the topics depend only on algebra and trigonometry, but the explanations of gravity and of map projections rely on a little calculus. I have tried to make the mathematics as accessible as I can by providing discussions illustrated by solved examples for almost all of the formulæ in the book. Each chapter includes a Your Turn section at the end, offering the reader an opportunity to exercise their newly acquired skills and knowledge. A good working knowledge of algebra and trigonometry will be needed to solve the problems. This book’s treatment of both geometrical and physical geodesy is somewhat unusual; most geodesy books focus strongly on one topic or the other. However, heights play a central role in many modern mapping projects, and geomaticians need an understanding of them as much as they need an understanding of map projection coordinates. The geometric geodesy portion is heavily laced with mathematical formulæ, but these formulæ are essentially algebraic in nature ix x PREFACE and, therefore, not unduly demanding. However, gravity is the subject of physical geodesy, and the mathematics of gravity begin with differential equations and go deeper from there. In keeping with the level of mathematical accessibility intended for this book, the Your Turn problems in the physical geodesy chapters are more conceptual than computational. The book is divided into three major parts: (I) Basic Concepts and Tools, (II) Geometrical Geodesy, and (III) Physical Geodesy. The introductory discussion of why studying geodesy is rele- vant for geomaticians in general is followed by a chapter on units of measure, which are numerous, important, and surprisingly subtle. Chapter 3 introduces the concept of reductions. Distances, angles, and other quantities of interest can have various forms. Reductions algebraically transform these quantities among their forms. This chapter also introduces the important direct and inverse problems, which form the basis of positioning. Chapter 4 presents geographic coordinates, which are the coordinates of choice in geodesy, and reference ellipsoids, which are the mathematical sur- faces upon which geodetic computations are defined. Chapter 5 answers my question about datums and also presents the major coordinate systems of geometrical geodesy, which, in my view, are the core of geometrical geodesy. Chapter 6 presents many of the various types of distances that arise in various coordinate systems, and chapter 7 explains their various types of angles. Chapter 8 introduces map projections and the grid coordinate systems that are built using them. Chapter 9 is the first physical geodesy chapter, giving an introduction to gravitation, gravity, geopotential, and the geoid. Chapter 10 introduces how height systems are defined and how they are organized into vertical datums. Chapter 11 gives an introduction to how tides affect heights. An explanation of geometry-based concepts usually benefits from illustrations – much more so when the subject is three dimensional. I have included many figures in this book, which hopefully will be both engaging and illuminating. The creation of a work such as this is seldom the product of only one person. I am indebted to many people, including dozens of students who used these materials as course notes, and several reviewers who took the time and effort to help me clarify, organize, and present this material more effectively. I specifically want to thank my wife, Ann, for her love and support and her excellent organizational skills. Tom Meyer May 2010 Part I Basic Concepts and Tools 1 Chapter 1 Introduction to Geodesy According to the Geodetic Glossary (NGS 2009), geodesy can be defined as “The science concerned with determining the size and shape of the Earth” or “The science that locates positions on the Earth and determines the Earth’s gravity field.” It might be hard to see anything too interesting or exciting in these definitions. After all, generally speaking, the Earth is round, a fact known at least since the time of the ancient Greeks. The earliest known definitive work on this was done by Aristarchus of Samos (ca. 310-230 B.C.E.), an astronomer who worked out a geometric method to determine ratios of the distance between the Sun and the Earth with the distance between the Moon and the Earth, a method that required knowing that all three bodies were round (Maor 1998, p. 63). But what is not obvious from the definition of geodesy is that much modern technology is possible only due to geodesy. For example, without modern geodesy, navigation might still be done with ancient methods (e.g., don’t lose sight of the shore) and high-accuracy maps of moderate or larger regions could not exist. Perhaps surprisingly, geodesy plays a central role in a wide variety of cutting-edge sciences, such as geophysics, astronomy, and climatology. The Earth’s warming due to climate change is expected to produce rising sea levels in many places, and sea level rise cannot be observed and monitored without geodesy. The observable change in sea level is due to a combination of a change in the ocean’s mean surface level and any subsidence of the shore. Geodesy is necessary to tease these two apart. Global warming is also generally reducing the size of ice sheets and glaciers as well as dramatically increasing how quickly they move. Space-age positioning methods make it possible to determine not only the location of an ice sheet, but also its velocity, shape, and size. Geodesy is also a key component of geophysics by making it possible to observe tectonic plate motion, for example. Geodesy is usually subdivided into geometrical geodesy, physical geodesy, and satellite geodesy, although additional subdivisions are recognized as well. Geometrical geodesy is concerned with describing locations in terms of geometry. Consequently, coordinate systems are one of the primary products of geometrical geodesy. Physical geodesy is concerned with determining the Earth’s gravity field, which is necessary for establishing heights. Satellite geodesy is concerned with using orbiting satellites to obtain data for geodetic purposes. To understand the role of geodesy, it might be useful to look at how maps are often drawn. A map is an abstract, scaled, two-dimensional image of some region typically on the Earth’s surface, showing the features of interest in their correct relative locations, sizes, and orientations. According to this definition, a globe is not a map because it is not two-dimensional. Neither are directions to the neighborhood gas station drawn on a napkin for a friend. Although entirely adequate for its 3 4 CHAPTER 1. INTRODUCTION TO GEODESY purpose, such a rendering is a sketch, not a map, because, while schematically correct, it is not an accurate rendition of the features. Maps are drawn essentially the same way whether they are drawn on paper or on a computer screen. The process begins with someone determining positions for all the features of interest. A position is (usually) a pair or triplet of numbers, called coordinates. A location is a place in the real world, whose spatial placement is described by a position. A coordinate system specifies how coordinates are assigned to locations. In order to draw, say, a building on a map, we need to determine the positions of its defining points, like its corners. Linear coordinates, such as the familiar x, y, z, are descriptions of offsets from some point of reference, called an origin, each in the direction of its axis.