87 3 Geodesy, Datums, Map Projec- tions, and Coordinate Systems Introduction Geographic information systems are ments on maps are affected by the distor- different from other information systems tion, we must use a map projection to because they include coordinates that reconcile the portrayal of the Earth’s define the location, shape, and extent of curved surface onto a flat surface. geographic objects. For effective GIS use, The second main problem in defining a we must clearly understand how coordinate coordinate system results from the irregular systems are established for the Earth, how shape of the Earth. We learn early on that coordinates are measured on the Earth’s the Earth is shaped as a sphere. This is a curving surface, and how these coordinates valid approximation for many uses, how- are converted for use in flat maps, either ever, it is only an approximation. Past and digital or paper. This chapter introduces present natural forces yield an irregularly geodesy, the science of measuring the shape shaped Earth. This shape affects how we of the Earth, and map projections, the best map the surface of the Earth, and how transformation of coordinate locations from we define flat coordinate systems. the Earth’s curved surface onto flat maps. Third, our measurements are rarely Defining coordinates for the Earth’s perfect, and this applies when measuring surface is complicated by four main fac- both the shape of the Earth and the exact tors. First, most people view geography on position of features on it. All locations a flat surface. We perceive a flat Earth depend on measurements that contain some because the curvature is barely perceptible error, and on analyses that require assump- at human scales. We’ve used flat maps for tions. Our measurements improve through more than 40 centuries, and although time, and so does the sophistication of our globes are helpful for visualization at analysis, so our positional estimates extremely small scales, they are impractical improve; this evolution means our esti- for most purposes. mates of positions change through time. A flat map must distort geometry in Finally, the physical locations of points some way because the Earth is curved. on the Earth change through time. Plate When we plot latitude and longitude coor- tectonics and vertical crustal movements dinates on a Cartesian system, “straight” mean the distance from San Francisco to lines will appear bent, and polygons will be Tokyo changed from 1950 to 2010, and distorted. This distortion may be difficult to continues to change today. Earth surface detect on detailed maps that cover a small rebound from the weight of past glaciers area, but the distortions become apparent as yields elevations in central Canada several the mapped area grows. Because measure- centimeters higher than they were a few 88 GIS Fundamentals decades ago. How do we specify positions choices, as long as we are clear in defining through time when the locations aren’t truly them. fixed? An example may help. Figure 3-1 shows Because of these four factors, we often the location of a U.S. survey mark, a pre- have several different sets of coordinates to cisely surveyed and monumented point. define the same location on the surface of Coordinates for this point are maintained by the Earth. Remember, coordinates are sets of federal and state government surveyors, and numbers that unambiguously define loca- resulting coordinates are shown at the top tions, and in a GIS data layer, we usually use right of the figure. Note that there are three an X (easting), Y (northing) and sometimes different versions of the latitude/longitude height value. But each of these values are location for this point. Here, the three ver- only “unique” to any given point for a speci- sions differ primarily due to differences in fied set of measurements, calculation the measurements, and how measurement assumptions, at a specified time. The coordi- errors were adjusted (the third factor, dis- nates depend on the reference system we use cussed above). The GIS practitioner may for measuring latitudes and longitudes well ask, which latitude/longitude pair (which depends on measurement and Earth’s should I use? This chapter contains the infor- shape), how we translate points from a mation that should allow you to choose curved Earth to a flat map surface (which wisely. depends on how we project), and to what set Note that there are also several versions of measurements we reference our coordi- of the x and y coordinates for the point in nates (our measurement methods and qual- Figure 3-1. The differences in the coordinate ity), and when (which depends on crustal values are too great to be due solely to mea- movement). We may, and often do, address surement errors. The differences are due pri- these factors in a number of different ways, marily to how we choose to project from the and the coordinates for the same point will curved Earth to a flat map, and in part to the be different for these different choices. We Earth shape we adopt and the measurement can translate between these different system we use. Coordinates for a From Surveyor Data: Point Location Latitude (N) Longitude (W) NAD83(2007) 44 57 23.23074 093 05 58.28007 d Geo an de st ti NAD83(1986) 44 57 23.22405 093 05 58.27471 a c o S C u . r NAD83(1996) 44 57 23.23047 093 05 58.27944 v S . e y U X Y B k SPC MNS 317,778.887 871,048.844 MT ench mar SPC MNS 1,042,579.57 2,857,766.08 sFT UTM15 4,978,117.714 492,150.186 MT From Data Layers: X Y MN-Ramsey 573,475.592 160,414.122 sFT MN-Ramsey 174,195.315 48,893.966 MT SPC MNC 890,795.838 95,819.779 MT SPC MNC 2,922,552.206 314,365.207 sFT LCC 542,153.586 18,266.334 MT Figure 3-1: An example of different coordinate values for the same point. We may look up the coordinates for a well-surveyed point, and we may also obtain the coordinates for the same point from a number of dif- ferent data layers. We often find multiple latitude/longitude values (surveyor data, top), or x and y values for the same point (surveyor data, or from data layers, bottom). Chapter 3: Geodesy, Projections, and Coordinate Systems 89 Whenever we work with spatial data, we lected data are often converted to a different must choose how to address the first three datum, labeled as NAD83(yy) system, where factors: projection distortion, an irregularly yy is a version number. The other satellite shaped Earth, and measurement imprecision. positioning systems (GLONASS, BeiDou, If our data are of very high accuracy and pre- Galileo) typically report in a datum labeled cision and we wish to work across time peri- ITRF(zzzz), where zzzz is a version number, ods, we must address the fourth factor: usually the year of issue. We will describe vertical and horizontal movements of physi- WGS84, NAD83, and ITRF datums and how cal locations through time. they relate to each other in the first half of It is crucial to realize that different ways this chapter. of addressing 1) the Earth’s curvature, 2) the These various versions of X, Y, and Z Earth’s deviation from our idealized shape, Cartesian coordinates are then commonly 3) inevitable inaccuracies in measurement, converted to latitude, longitude, and height and 4) physical shifts, will result in different coordinates, and subsequently projected to coordinates. These differences are the root of “flat” coordinate values, suitable for layers many errors in spatial analysis. As a rule, in a GIS. This process applies for data you should know the coordinate system used directly collected with a GPS or other simi- for all of your data, and convert all data to lar satellite-based navigation system, or with the same coordinate system, for the same data that depend on satellite positioning, time epoch, prior to analysis. In some cases such as satellite or aerial images. Since these the differences when ignoring some of these coordinate systems differ, have changed four factors may be small in relation to the through time, and data are commonly con- spatial precision required by your analysis, verted one to another, it is easy to add error particularly for the fourth factor (time differ- to new data so that features don’t fall in their ences between coordinate measurements). true location. Knowledge of the history and As positioning technology improves, we can technology of datum development helps us make increasingly accurate and precise mea- understand how to best collect new data, and surements, so in many cases, the epoch of to integrate older data with newer measure- measurement becomes important. This chap- ments. ter describes how we define, measure, and convert among coordinate systems. Early Measurements Modern Coordinate Capture, In specifying a coordinate system, we must first define the size and shape of the Coordinate Systems, and Datums Earth. Humans have long speculated on this. Most GIS data collection relies directly Babylonians believed the Earth was a flat or indirectly on satellite-based positioning disk floating in an endless ocean, while the systems. These systems, described in detail Greek Pythagoras, and later Aristotle, rea- in Chapter 5, allow the rapid, accurate col- soned that the Earth must be a sphere. They lection of locations. Positions are referenced observed that ships disappeared over the to Earth-centered, three dimensional, Carte- horizon, the moon appeared to be a sphere, sian coordinate systems – the X, Y, and Z of and that the stars moved in circular patterns, 3D systems described in Chapter 2.