What is Geodesy ?

Satellite Observations of the Earth European Remote Sensing satellite, ERS-1 from 780Km ERS-1 depicts the earth’s shape without water and clouds. This image looks like a sloppily pealed potato, not a smoothly shaped ellipse. Satellite Geodesy has enabled earth scienetists to gain an accurate estimate (+/- 10cm) of the geocentric center of the of the earth. A worldwide horizontal datum requires an accurate estimation of the earth’s center

All right then – what is it ? "geo daisia" = "dividing the earth" Aim: determination of the figure of the earth or, more practically: determination of the relative positions on or close to the surface of the earth. oldest profession on earth but one geodesy, not geodetics

Shape of the Earth  flat  large scale mapping   street plans   engineering surveys   sphere  small scale mapping, low accuracy  geography  survey calculations (medium accuracy)   ellipsoid accurate (geodetic) mapping  geodetic & survey calculations   geoid  accurate heighting  satellite orbits  high accuracy geodetic calculations

Geoid equal gravity potential Locus (surface) of points with equal gravity potential approximately at Mean Sea Level  must be measured (  errors) l difficult to describe mathematically l even more difficult to calculate with o reduction of survey observations o map projections  l everyday heights are relative to geoid (MSL) l physically exists

Geoid covering the USA (NGS96) 7.2 m m Note: This image shows the height of the geoid above the US reference ellipsoid

Ellipsoid ( = spheroid) geoid Approximate the geoid ( not the earth's surface ) l good approximation possible  variations ~10 -5 (±60 m over earth radius ~ 6,400,000 m)  ellipsoidal calculus is feasible  can be defined exactly: semi-major axis (size) and flattening (shape) l computational aid only; no physical reality  ellipsoidal heights are not practical  many choices possible (~50)  optimum local fit with geoid (sometimes global fit)  rotation axis parallel to mean earth rotation axis  based on surface geodetic observations

The geoid and two ellipsoids N N Europe N. America S. America Africa typically several hundreds of metres

Ellipsoids - examples namesemi-major axisflattening Bessel m1/ WGS m1/ Clarke m1/ Bessel 1841: usage: o Europe (German influence sphere), Namibia, Indonesia, Japan, Korea o National control network and mapping. WGS84: usage: o the entire world o the GPS system in conjunction with a datum of the same name. Clarke 1866: usage: o USA except Michigan, Canada, Central America, Philippines, Mozambique o National control network and mapping.

Many ellipsoids …. many datums Approximate the local geoid with different ellipsoids...  different origins  different orientation of axes  different shapes and sizes different GEODETIC DATUMS  different GEODETIC DATUMS What is a ‘Geodetic Datum’? location (origin) orientation shape size of the ellipsoid in space

Geographic coordinates P  X-axis Y-axis Z-axis semi-major axis H Greenwich meridian semi-minor axis oblate at poles   = Geographic Latitude  = Geographic Longitude H = Ellipsoidal height

Latitude is not unique ! 11 22 nor is Longitude  1  2 Due to different Geodetic Datums:

President Ford’s secret Alaskan visit ?

Washington to Tokyo - Orthographic Projection Tokyo Anchorage Washington

Mercator projection Globular projection Orthographic projection Stereographic projection A familiarly shaped ‘continent’ in different map projections

Easting Northing Longitude East Longitude West equator Latitude North Latitude South A A Geographic and map coordinates (N,E) = F (Lat, Lon)  distortions

What errors can you expect?  Wrong geodetic datum: q several hundreds of metres  Incorrect ellipsoid: q horizontally: several tens of metres q height: not effected, or tens to several hundred metres  Wrong map projection:  entirely the wrong projection: hundreds, even thousands of kilometres (at least easy to spot!)  partly wrong (i.e. one or more parameters are wrong): several metres to many hundreds of kilometres  No geodetic metadata  coordinates cannot be interpreted  datum  ellipsoid  prime meridian  map projection Coordinate Reference System

Types of Coordinate (Ref) System Coordinate Ref System Coordinate System Characteristics GeocentricCartesian or spherical Proper 3D spatial modeling; spatial applications Geographic 3DellipsoidalLocations described relative to ellipsoidal surface Geographic 2DellipsoidalLocations described on ellipsoidal surface; for large national/continental geodetic control networks ProjectedCartesianFor national mapping; smaller area than Geographic 2D. Carefully controlled mapping distortions EngineeringvariousEarth curvature ignored; mostly flat-earth model ImageCartesian or oblique Cartesian Distortions due to earth curvature determined by data acquisition characteristics Verticalgravity-related, depth,barometric Gravity-related means relative to geoid (~MSL) Depth: complex reference surfaces (tidal) Earth curvature modelling

Geodesy, Map Projections and Coordinate Systems l Geodesy - the shape of the earth and definition of earth datums l Map Projection - the transformation of a curved earth to a flat map l Coordinate systems - (x,y) coordinate systems for map data

Types of Coordinate Systems l (1) Global Cartesian coordinates (x,y,z) for the whole earth (2) Geographic coordinates ( , z) l (3) Projected coordinates (x, y, z) on a local area of the earth’s surface l The z-coordinate in (1) and (3) is defined geometrically; in (2) the z-coordinate is defined gravitationally

Global Cartesian Coordinates (x,y,z) O X Z Y Greenwich Meridian Equator

Geographic Coordinates ( , z) Latitude (  ) and Longitude ( ) defined using an ellipsoid, an ellipse rotated about an axis l Elevation (z) defined using geoid, a surface of constant gravitational potential l 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

Ellipse P F2F2 O F1F1 a b X Z   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

Ellipsoid or Spheroid Rotate an ellipse around an axis O X Z Y a a b Rotational axis

Standard Ellipsoids Ref: Snyder, Map Projections, A working manual, USGS Professional Paper 1395, p.12

Horizontal Earth Datums l An earth datum is defined by an ellipse and an axis of rotation l NAD27 (North American Datum of 1927) uses the Clarke (1866) ellipsoid on a non geocentric axis of rotation l NAD83 (NAD,1983) uses the GRS80 ellipsoid on a geocentric axis of rotation l WGS84 (World Geodetic System of 1984) uses GRS80, almost the same as NAD83

Definition of Latitude,  (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 O  S m n q p r

Cutting Plane of a Meridian P Meridian Equator plane Prime Meridian

Definition of Longitude, Definition of Longitude, 0°E, W 90°W (-90 °) 180°E, W 90°E (+90 °) -120° -30° -60° -150° 30° -60° 120° 150° = the angle between a cutting plane on the prime meridian and the cutting plane on the meridian through the point, P P

Latitude and Longitude on a Sphere Meridian of longitude Parallel of latitude  X Y Z N E W   =0-90°S P O R =0-180°E  =0-90°N Greenwich meridian =0° Equator =0° =0-180°W - Geographic longitude  - Geographic latitude R - Mean earth radius O - Geocenter

Length on Meridians and Parallels 0 N 30 N  ReRe ReRe R R A B C  (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) 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

Representations of the Earth Earth surface Ellipsoid Sea surface Geoid Mean Sea Level is a surface of constant gravitational potential called the Geoid

Geoid and Ellipsoid Ocean Geoid Earth surface Ellipsoid 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 Mean Sea level = Geoid Land Surface Elevation is measured from the Geoid

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

Converting Vertical Datums l Corps program Corpscon Point file attributed with the elevation difference between NGVD 29 and NAVD 88 NGVD 29 terrain + adjustment = NAVD 88 terrain elevation

Geodesy and Map Projections l Geodesy - the shape of the earth and definition of earth datums l Map Projection - the transformation of a curved earth to a flat map l Coordinate systems - (x,y) coordinate systems for map data

Earth to Globe to Map Representative Fraction Globe distance Earth distance = Map Scale: Map Projection: Scale Factor Map distance Globe distance = (e.g. 1:24,000) (e.g )

Geographic and Projected Coordinates (  ) (x, y) Map Projection

Projection onto a Flat Surface

Types of Projections l Conic (Albers Equal Area, Lambert Conformal Conic) - good for East-West land areas l Cylindrical (Transverse Mercator) - good for North-South land areas l Azimuthal (Lambert Azimuthal Equal Area) - good for global views

Conic Projections (Albers, Lambert)

Cylindrical Projections (Mercator) Transverse Oblique

Azimuthal (Lambert)

Albers Equal Area Conic Projection

Lambert Conformal Conic Projection

Universal Transverse Mercator Projection Universal Transverse Mercator Projection

Lambert Azimuthal Equal Area Projection

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

Geodesy and Map Projections l Geodesy - the shape of the earth and definition of earth datums l Map Projection - the transformation of a curved earth to a flat map l Coordinate systems - (x,y) coordinate systems for map data

Coordinate Systems l Universal Transverse Mercator (UTM) - a global system developed by the US Military Services l State Plane Coordinate System - civilian system for defining legal boundaries l California State Mapping System - a statewide coordinate system for California

Coordinate System (  o, o ) (x o,y o ) X Y Origin A planar coordinate system is defined by a pair of orthogonal (x,y) axes drawn through an origin

Universal Transverse Mercator l Uses the Transverse Mercator projection Each zone has a Central Meridian ( o ), zones are 6° wide, and go from pole to pole l 60 zones cover the earth from East to West Reference Latitude (  o ), is the equator l (Xshift, Yshift) = (x o,y o ) = (500000, 0) in the Northern Hemisphere, units are meters

UTM Zone 21 &22 Equator -120° -90 ° -60 ° -102°-96° -123° Origin 6°

State Plane Coordinate System l Defined for each State in the United States l East-West States (e.g. Texas) use Lambert Conformal Conic, North-South States (e.g. Indiana) use Transverse Mercator l Texas has five zones (North, North Central, Central, South Central, South) to give accurate representation l Greatest accuracy for local measurements

California Mapping System l Designed to give State-wide coverage of California without gaps l Lambert Conformal Conic projection with standard parallels l 1927 Seven Zones - feet l 1983 Six Zones - meters

Coordinate Systems l Geographic coordinates (decimal degrees) l Projected coordinates (length units, ft or meters)

Summary Concepts 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)

Summary Concepts (Cont.) l To prepare a map, the earth is first reduced to a spheroid and then projected onto a flat surface l Three basic types of map projections: conic, cylindrical and azimuthal l A particular projection is defined by a datum, a projection type and a set of projection parameters

Summary Concepts (Cont.) l Standard coordinate systems use particular projections over zones of the earth’s surface l Types of standard coordinate systems: UTM, State Plane, California Coordinate System l Reference Frame in ArcInfo 8,& Geomedia requires projection and map extent