# Lecture 4. Map Projections & Coordinate System in GIS

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1 Lecture 4 Map Projections & Coordinate System in GIS GIS in Water Resources Spring 2015 Geodesy, Map Projections and Coordinate Systems Geodesy - the shape of the earth and definition of earth datums Map Projection - the transformation of a curved earth to a flat map Coordinate systems - (x,y,z) coordinate systems for map data 1

2 Objectives: By the end of this class you should be able to: describe the role of geodesy as a basis for earth datums Display data from GPS in ArcMap and Google Earth list the basic types of map projection identify the properties of common map projections properly use the terminology of common coordinate systems use spatial references in ArcMap so that geographic data is properly displayed determine the spatial reference system associated with a feature class or data frame use ArcGIS to convert between coordinate systems calculate distances on a spherical earth and in a projected coordinate system Readings: In ArcGIS Desktop 10 Help library/ Professional library/ Guide books/ Map projections 2

3 Spatial Reference = Datum + Projection + Coordinate system For consistent analysis the spatial reference of data sets should be the same. ArcGIS does projection on the fly so can display data with different spatial references properly if they are properly specified. ArcGIS terminology Define projection. Specify the projection for some data without changing the data. Project. Change the data from one projection to another. Types of Coordinate Systems (1) Global Cartesian coordinates (x,y,z) for the whole earth (2) Geographic coordinates (, z) (3) Projected coordinates (x, y, z) on a local area of the earth s surface The z-coordinate in (1) and (3) is defined geometrically; in (2) the z-coordinate is defined gravitationally 3

4 Global Cartesian Coordinates (x,y,z) Greenwich Meridian Z O Y X Equator Next 7 slides are from Dr Irmak at U of Nebraska olde Times The Sextant Sighting of a celestial object, measure angle between (sun at noon, or Polaris at night) and horizon. 4

6 How GPS works in five logical steps: 1. The basis of GPS is triangulation from satellites 2. GPS receiver measures distance from satellite using the travel time of radio signals 3. To measure travel time, GPS needs very accurate timing 4. Along with distance, you need to know exactly where the satellites are in space. Satellite location. High orbits and careful monitoring are the secret 5. You must correct for any delays the signal experiences as it travels through the atmosphere GPS Satellites 24 satellites 6 orbital planes 12 hour return interval for each satellite Satellites are distributed among six offset orbital planes 6

7 Distance from satellite Radio waves = speed of light Receivers have nanosecond accuracy ( second) All satellites transmit same signal string at same time Difference in time from satellite to time received gives distance from satellite Triangulation 7

8 Triangulation GPS location of Mabel Lee Hall 8

9 Geographic Coordinates (, z) Latitude ( ) and Longitude ( ) defined using an ellipsoid, an ellipse rotated about an axis Elevation (z) defined using geoid, a surface of constant gravitational potential Earth datums define standard values of the ellipsoid and geoid Shape of the Earth We think of the earth as a sphere It is actually a spheroid, slightly larger in radius at the equator than at the poles 9

10 Ellipse An ellipse is defined by: Focal length = Distance (F1, P, F2) is constant for all points on ellipse When = 0, ellipse = circle For the earth: Major axis, a = 6378 km Minor axis, b = 6357 km Flattening ratio, f = (a-b)/a ~ 1/300 F 1 P Z b O a F 2 X Ellipsoid or Spheroid Rotate an ellipse around an axis Z b a O a Y X Rotational axis 10

11 Standard Ellipsoids Ellipsoid Major axis, a (m) Minor axis, b (m) Flattening ratio, f Clarke (1866) 6,378,206 6,356,584 1/ GRS80 6,378,137 6,356,752 1/ Ref: Snyder, Map Projections, A working manual, USGS Professional Paper 1395, p.12 Horizontal Earth Datums An earth datum is defined by an ellipse and an axis of rotation NAD27 (North American Datum of 1927) uses the Clarke (1866) ellipsoid on a non geocentric axis of rotation NAD83 (NAD,1983) uses the GRS80 ellipsoid on a geocentric axis of rotation WGS84(World Geodetic System of 1984) uses GRS80, almost the same as NAD83 11

12 Definition of Latitude, m S p n O q r (1) Take a point S on the surface of the ellipsoid and define there the tangent plane, mn (2) Define the line pq through S and normal to the tangent plane (3) Angle pqr which this line makes with the equatorial plane is the latitude, of point S Cutting Plane of a Meridian P Prime Meridian Equator Meridian 12

13 Definition of Longitude, = the angle between a cutting plane on the prime meridian and the cutting plane on the meridian through the point, P E, W W (-90 ) 90 E (+90 ) -60 P E, W 30 Latitude and Longitude on a Sphere Greenwich meridian =0 Z N Meridian of longitude Parallel of latitude P X W O Equator =0 R E - Geographic longitude - Geographic latitude Y R - Mean earth radius O - Geocenter 13

14 Length on Meridians and Parallels (Lat, Long) = (, ) Length on a Meridian: AB = R e (same for all latitudes) Length on a Parallel: CD = R R e Cos (varies with latitude) R e R R R e C B A D Example: What is the length of a 1º increment along on a meridian and on a parallel at 30N, 90W? Radius of the earth = 6370 km. Solution: A 1º angle has first to be converted to radians radians = 180 º, so 1º = /180 = /180 = radians For the meridian, L = R e km For the parallel, L = R e Cos Cos km Parallels converge as poles are approached 14

15 Curved Earth Distance (from A to B) Shortest distance is along a Great Circle Z A Great Circle is the intersection of a sphere with a plane going through its center. 1. Spherical coordinates converted to Cartesian coordinates. 2. Vector dot product used to calculate angle from latitude and longitude 3. Great circle distance is R, where R= km 2 X A B Y Dist R cos 1 [sin sin cos cos cos( )] A B Ref: Meyer, T.H. (2010), Introduction to Geometrical and Physical Geodesy, ESRI Press, Redlands, p. 108 A B A B Representations of the Earth Sea surface Mean Sea Level is a surface of constant gravitational potential called the Geoid Ellipsoid Earth surface Geoid 15

16 Three systems for measuring elevation Orthometric heights (land surveys, geoid) Ellipsoidal heights (lidar, GPS) Tidal heights (Sea water level) Conversion among these height systems has some uncertainty Trends in Tide Levels (coastal flood risk is changing) Charleston, SC ft/century Galveston, TX ft/century ft/century Juneau, AK

17 Geoid and Ellipsoid Earth surface Ocean Geoid Gravity Anomaly Gravity anomaly is the elevation difference between a standard shape of the earth (ellipsoid) and a surface of constant gravitational potential (geoid) Definition of Elevation Elevation Z P z = z p z = 0 Land Surface Mean Sea level = Geoid Elevation is measured from the Geoid 17

18 Vertical Earth Datums A vertical datum defines elevation, z NGVD29 (National Geodetic Vertical Datum of 1929) NAVD88 (North American Vertical Datum of 1988) takes into account a map of gravity anomalies between the ellipsoid and the geoid 18

19 Converting Vertical Datums Corps program Corpscon (not in ArcInfo) Point file attributed with the elevation difference between NGVD 29 and NAVD 88 NGVD 29 terrain + adjustment = NAVD 88 terrain elevation Importance of geodetic datums NAVD88 NGVD29 (cm) NGVD29 higher in East More than 1 meter difference NAVD88 higher in West Orthometric datum height shifts are significant relative to BFE accuracy, so standardization on NAVD88 is justified BFE: The computed elevation to which floodwater is anticipated to rise during the base flood. 19

20 Geodesy and Map Projections Geodesy - the shape of the earth and definition of earth datums Map Projection - the transformation of a curved earth to a flat map Coordinate systems - (x,y) coordinate systems for map data Earth to Globe to Map Map Scale: Representative Fraction Map Projection: Scale Factor = Globe distance Earth distance Map distance Globe distance (e.g. 1:24,000) (e.g ) = 20

21 Geographic and Projected Coordinates ( ) (x, y) Map Projection Types of Projections Conic (Albers Equal Area, Lambert Conformal Conic) - good for East-West land areas Cylindrical (Transverse Mercator) - good for North-South land areas Azimuthal (Lambert Azimuthal Equal Area) - good for global views 21

22 Conic Projections (Albers, Lambert) Cylindrical Projections (Mercator, Polar axis aligned with cylinder axis) Transverse Oblique 22

23 Azimuthal (Lambert) Albers Equal Area Conic Projection 23

24 Lambert Conformal Conic Projection Universal Transverse Mercator Projection (Transverse Mercator, Polar axis aligned equatorial plane) 24

25 Lambert Azimuthal Equal Area Projection Projections Preserve Some Earth Properties Area- correct earth surface area (Albers Equal Area) important for mass balances Shape - local angles are shown correctly (Lambert Conformal Conic) Direction - all directions are shown correctly relative to the center (Lambert Azimuthal Equal Area) Distance - preserved along particular lines Some projections preserve two properties 25

26 Projection and Datum Two datasets can differ in both the projection and the datum, so it is important to know both for every dataset. Geodesy and Map Projections Geodesy - the shape of the earth and definition of earth datums Map Projection - the transformation of a curved earth to a flat map Coordinate systems - (x,y) coordinate systems for map data 26

27 Coordinate Systems Universal Transverse Mercator (UTM) - a global system developed by the US Military Services State Plane Coordinate System - civilian system for defining legal boundaries Texas Centric Mapping System -a statewide coordinate system for Texas Coordinate System A planar coordinate system is defined by a pair of orthogonal (x,y) axes drawn through an origin Y Origin X ( o, o ) (x o,y o ) 27

28 Universal Transverse Mercator Uses the Transverse Mercator projection Each zone has a Central Meridian ( o ), zones are 6 wide, and go from pole to pole 60 zones cover the earth from East to West Reference Latitude ( o ), is the equator (Xshift, Yshift) = (x o,y o ) = (500000, 0) in the Northern Hemisphere, units are meters NY: UTM Zones 17&18 Good reading at: 28

29 State Plane Coordinate System Defined for each State in the United States East-West States (e.g. Texas) use Lambert Conformal Conic, North-South States (e.g. California) use Transverse Mercator New York has four zones (East, Central, West, Long Island) to give accurate representation Greatest accuracy for local measurements ArcGIS Spatial Reference Frames Defined for a feature dataset in ArcCatalog XY Coordinate System Projected Geographic Z Coordinate system Tolerance Resolution M Domain 29

30 Horizontal Coordinate Systems Geographic coordinates (decimal degrees) Projected coordinates (length units, ft or meters) Vertical Coordinate Systems None for 2D data Necessary for 3D data 30

31 Tolerance The default XY tolerance is the equivalent of 1mm (0.001 meters) in the linear unit of the data's XY (horizontal) coordinate system on the earth surface at the center of the coordinate system. For example, if your coordinate system is recorded in feet, the default value is feet ( inches). If coordinates are in latitude-longitude, the default XY tolerance is degrees. Resolution 31

32 Domain Extents Horizontal Vertical Distance along a line ArcGIS.prj files 32

33 Summary Concepts The spatial reference of a dataset comprises datum, projection and coordinate system. For consistent analysis the spatial reference of data sets should be the same. ArcGIS does projection on the fly so can display data with different spatial references properly if they are properly specified. ArcGIS terminology Define projection. Specify the projection for some data without changing the data. Project. Change the data from one projection to another. Summary Concepts (Cont.) Two basic locational systems: geometric or Cartesian (x, y, z) and geographic or gravitational (, z) Mean sea level surface or geoid is approximated by an ellipsoid to define an earth datum which gives ( and distance above geoid gives (z) 33

34 Summary Concepts (Cont.) To prepare a map, the earth is first reduced to a globe and then projected onto a flat surface Three basic types of map projections: conic, cylindrical and azimuthal A particular projection is defined by a datum, a projection type and a set of projection parameters Summary Concepts (Cont.) Standard coordinate systems use particular projections over zones of the earth s surface Types of standard coordinate systems: UTM, State Plane, Texas State Mapping System, Standard Hydrologic Grid Spatial Reference in ArcGIS 10 requires projection and map extent 34

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