Geography 4103 / 5103 Introduction to Geographic Information Science Map Projections and Coordinate Systems Updates/News Thursday s lecture Reading discussion 1 find the readings online open questions, material, essays Last Lecture We talked about map scale and resolution of spatial data Definitions and terms Learned about the differences and confusions that commonly exist Learned what generalization is and how it relates to scale What the consequences can (and always will) be 1
Today s Outline Measuring the Earth Ellipsoid (e.g. Clarke 1866, WGS84, GRS80) Geoid Datums (e.g. NAD27, NAD83) Displaying the Earth General coordinate systems (e.g. spherical & Cartesian) Projections & their distortion properties Specific coordinate systems (e.g. UTM & State Plane) What does this mean? Geodesy - science of measuring the size and shape of the Earth Datum - a reference surface (i.e. set of known points) e.g. a site datum - a reference height against which elevations are measured e.g. a site plan for a subdivision - establish datum as a fixed elevation at the lowest point on the property. All heights are measured relative to this site (or local) datum (Ferris State/ACSM) 2
Earth is slightly flattened roughly an Ellipsoid Mathematical approximation defined by two radii: r 1, along semi-major (through Equator) r 2, along semi-minor (through poles) Why are there so many different ellipsoids? How about today? The Earth s True Shape is Best Described as a Geoid +85m -106m Geoid A geoid is the measured surface (not math. defined) perpendicular to a plumb line determined by the pull of gravity (= a specified constant) Gravimeters; Satellites (GRACE) Ellipsoid Geoid Deviations from ellipsoid due to gravitational pull 3
Geoid Animation Source: Gravity Recovery & Climate Experiment http://www.csr.utexas.edu/grace/gallery/animations/world_gravity/ So we have three surfaces to keep track of at each point on Earth: 1. the ellipsoid 2. the geoid, and 3. the physical surface Mean Sea Level (geoid) Geoid Undulation Perpendicular to the ellipsoid Perpendicular to the geoid 4
Geoidal variation as the main cause for different ellipsoids employed in different parts of the world Horizontal Datum Horizontal datum: set of carefully surveyed reference points (surface) based on a given ellipsoid 1. Define the shape of the Earth (the ellipsoid) 2. Use triangulation surveys to define the location of a set of known control points for which the position on the ellipsoid is precisely known (geodetic datum) 3. This is the reference surface and network against which all other points will be measured 5
Survey Network for NAD83 Up to 200m difference between NAD27 and NAD83 NAD27: 26K and Clarke NAD83: 250K (2 mio distances) and GRS80 ellipsoid (from Schwartz, 1989) Ellipsoids and Datum Examples Ellipsoids: mathematically defined by 2 radii Clarke 1866 World Geodetic System of 1972 (WGS72) Geodetic Reference System 1980 (GRS80) World Geodetic System of 1984 (WGS84) (See also Bolstad, p73) Datum: reference points for establishing horizontal position NAD27 (uses Clarke 1866) NAD83 (uses GRS80) Geodetic Datum of Australia (uses GRS80) WGS84 reference ellipsoid (uses WGS84) (developed by US DoD; basis for GPS measurements) 6
Today s Outline Measuring the Earth Ellipsoid (e.g. Clarke 1866, WGS84, GRS80) Geoid Datums (e.g. NAD27, NAD83) Displaying the Earth General coordinate systems (e.g. spherical & Cartesian) Projections & their distortion properties Specific coordinate systems (e.g. UTM & State Plane) General Coordinate Systems Provide a spatial referencing system to locate points on the earth s surface using pairs/triplets of numbers. Spherical (= geographic/ geodetic) - Coordinates that describe locations on a sphere Planar (= Cartesian) - Coordinates that describe locations within a two dimensional Cartesian space (right angles) Spherical Coordinates Use angles of rotation to define a directional vector 7
Planar Coordinate Systems Rectangular grid with X and Y axes X: easting Y: northing Each point is defined by X,Y coordinate Coordinate Systems 1 degree long. = 111.3km at the equator How much at the poles? 1 degree lat. = 110.6km at the equator How much at the poles (more or less)? Meuhrcke & Meurhcke 1992 Difficult to use computationally/cartographically 8
Map Projection A means of fitting features from the 3-D globe to a 2-D medium (e.g., map, computer screen) Source: www.gma.org/surfing/imaging/mapproj.html Why Map Projections are Important Necessary to display a spherical object on a flat surface Data from different sources will often fail to display properly in ArcMap unless the projections match up Understand map distortions to minimize projection errors Map Projection A means to depict the spherical earth (in ref. to a datum) on a two dimensional medium (3D => 2D) All distortion properties cannot simultaneously be preserved in two dimensions: Conformality. Distance Direction Area 9
Distortion Properties Conformality Conformal map projections preserve local shape; scale of the map at any point on the map is the same in any direction (conformal). Meridians (lines of longitude) and parallels (lines of latitude) intersect at right angles. Shape is preserved locally on conformal maps. Distance Distances are true only from the center of the projection (or a reference point) to any other place on the map (equidistant) but not between these places. Direction The shortest route between two points on a curved surface such as the earth is along the spherical equivalent of a straight line on a flat surface. That is the great circle on which the two points lie. True-direction projections maintain these directions rel. to center (true-direction). Area All mapped areas have the same proportional relationship to the areas on the Earth that they represent (equal-area). Preserves area Shape, angle and scale are distorted Area and shape distortions Why Distortion Varies Across Map Expansion Compression projection light source 10
Families of Map Projections Planar Cylindrical Conic Vary location of the light source Vary how to wrap the surface around it also Plane or Azimuthal Types of Planar Projections Gnomonic - center of globe Stereographic - at the antipode Orthographic - at infinity Source:http://www.fes.uwaterloo.ca/crs/geog165/mapproj.htm Developable Projection Surfaces Most common projections are based on developable surfaces (a geometric shape onto which Earth locations are projected) Also mathematical projections (from the ellipsoid onto flat surfaces) 11
Common Map Projections Common Map Projections Global UTM System 80 degr. South / 84 degr. North 12
Universal Transverse Mercator Coord. Sys. Each Zone is 6 degrees wide How many total zones? Zone location defined by a central meridian North origin at the equator (Northing = 0) South origin with False Northing of 10,000km False Easting: 500,000m west of the zone s central Meridian Coordinates always positive Coordinates discontinuous across zone boundaries Universal Transverse Mercator Coord. Sys. Each Zone is 6 degrees wide How many total zones? Zone location defined by a central meridian North origin at the equator (Northing = 0) South origin with False Northing of 10,000km False Easting: 500,000m west of the zone s central Meridian Coordinates always positive Coordinates discontinuous across zone boundaries UTM Zones for the U.S. 13
UTM Coordinates Examples (meters) Common Topo format 2 83 000m E Easting (East of the zone meridian; remember false easting concept) 39 04 000m N Northing (North of the Equator) Standard format UTM 10S 0545980E 4185742N So what is the difference between UTM and Transverse Mercator? US State Plane Coord. System Defined in the US by each state (starting in 1930a; NAD) Some states use multiple zones Different types of projections are used by the system Provides less distortion than UTM Preferred for applications needing very high accuracy, such as (property) surveying Projection distortions less than 1 part in 10,000 (4x better than UTM) Problem: each zone uses a different coord. system e.g., if metrop. area covers several counties and crosses state plane zones (e.g. Seattle & Tacoma fall in separate zones) 14
State Plane Coordinate System Zones US State Plane Coord. System - Units in feet - Projection is either - transverse Mercator (good for N-S zones) - Lambert s conformal conic (good for E-W zones) - Based on NAD83 datum - One or more zones for each state with a false origin southwest of the zone - 126 zones in total (so also 126 origins) Example of State Plane 15
Transformations/Conversions Which is harder : converting a road network between different datums (e.g. NAD27 to NAD83) or different projections (e.g. UTM 15N to Iowa State Plane North)? 16
What does this mean? Careful Most commonly used Tool: Reprojects/Transforms feature into a different datum or projection Use with Caution: For feature where the Datum/projection information is not attached. Most common use is for bringing in tabular coordinates from a table (e.g. GPS points). You must know the datum/projection they were gathered in. 17
Kennedy & Kopp 2001, A good, cheap, reference book on map projections Also, Bolstad Appendix C Summary A datum is a reference surface, a realization of the ellipsoid, against which locations are measured Horizontal positions defined relative to the datum. Projections specify a 2-D coordinate system from the 3-D globe All projections cause some distortion Errors are controlled by choosing the proper projection type and/or limiting the area applied A GIS project should rely on the same datum 18