What is Geodesy ?

1. What is Geodesy ?

2. 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 smoothlyshaped 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 requiresan accurate estimation of the earth’s center

3. 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

4. Shape of the Earth Œ flat l large scale mapping • l street plans • l engineering surveys spherel small scale mapping, low accuracy l geography l survey calculations (medium accuracy) Žellipsoidlaccurate (geodetic) mapping l geodetic & survey calculations geoidl accurate heighting l satellite orbits l high accuracy geodetic calculations

5. C • everyday heights are relative to geoid (MSL) • physically exists D • must be measured (ð errors) • difficult to describe mathematically • even more difficult to calculate with • reduction of survey observations • map projections Geoid Locus (surface) of points with equal gravity potential approximately at Mean Sea Level

6. Geoid covering the USA (NGS96) 7.2 m - 51.6 m Note: This image shows the height of the geoid above the US reference ellipsoid

8. Ellipsoid ( = spheroid) Approximate the geoid (not the earth's surface) • 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) • 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

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

10. Ellipsoids - examples name semi-major axis flattening Bessel 1841 6377397m1/299.15 WGS846378137m1/298.26 Clarke 18666378206m1/294.979 Bessel 1841: usage: • Europe (German influence sphere), Namibia, Indonesia, Japan, Korea • National control network and mapping. WGS84: usage: • the entire world • the GPS system in conjunction with a datum of the same name. Clarke 1866: usage: • USA except Michigan, Canada, Central America, Philippines, Mozambique • National control network and mapping.

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

12. Z-axis P Greenwich meridian H semi-minor axis semi-major axis f Y-axis l oblate at poles X-axis Geographic coordinates f= Geographic Latitude l= Geographic Longitude H= Ellipsoidal height

13. Due to different Geodetic Datums: f1 ¹ f2 f1 f2 Latitude is not unique ! nor is Longitude

14. 80 o N 75 o N Anchorage 60 o N Why not the shorter route? 45o N Washington 30o N Tokyo 15o N 0 o 15 o S 30o S 45o S 60o S 90 o E 120 o E 150o E 180 o 150 o W 120o W 90o W 60 o W 30o W 30o E 0 o President Ford’s secret Alaskan visit ?

15. Washington to Tokyo - Orthographic Projection Tokyo Washington Anchorage

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

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

18. Coordinate Reference System What errors can you expect? • Wrong geodetic datum: • several hundreds of metres • Incorrect ellipsoid: • horizontally: several tens of metres • 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

19. Types of Coordinate (Ref) System Earth curvature modelling

20. 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) coordinate systems for map data

21. Types of Coordinate Systems • (1) Global Cartesian coordinates (x,y,z) for the whole earth • (2) Geographic coordinates (f, l, 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

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

23. Geographic Coordinates (f, l, z) • Latitude (f) and Longitude (l) 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

24. Shape of the Earth It is actually a spheroid, slightly larger in radius at the equator than at the poles We think of the earth as a sphere

25. Ellipse An ellipse is defined by: Focal length =  Distance (F1, P, F2) is constant for all points on ellipse When  = 0, ellipse = circle Z b O a X   F1 F2 For the earth: Major axis, a = 6378 km Minor axis, b = 6357 km Flattening ratio, f = (a-b)/a ~ 1/300 P

26. Ellipsoid or SpheroidRotate an ellipse around an axis Z b a O Y a X Rotational axis

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

28. 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

29. Definition of Latitude, f m p S n O f 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 f, of point S

30. P Prime Meridian Equator Meridian plane Cutting Plane of a Meridian

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

32. =0-180°W =0-90°S Latitude and Longitude on a Sphere Meridian of longitude Z Greenwich meridian N Parallel of latitude =0° P • =0-90°N  - Geographic longitude  - Geographic latitude  E W O • Y R  R - Mean earth radius • =0° Equator  • O - Geocenter =0-180°E X

33. Length on Meridians and Parallels (Lat, Long) = (f, l) Length on a Meridian: AB = ReDf (same for all latitudes) R Dl D R 30 N C B Re Df 0 N Re Length on a Parallel: CD = R Dl = ReDl Cos f (varies with latitude) A

34. 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 • p radians = 180 º, so 1º = p/180 = 3.1416/180 = 0.0175 radians • For the meridian, DL = ReDf = 6370 * 0.0175 = 111 km • For the parallel, DL = ReDl Cos f • = 6370 * 0.0175 * Cos 30 • = 96.5 km • Parallels converge as poles are approached

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

36. Geoid and Ellipsoid Earth surface Ellipsoid 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)

37. Definition of Elevation Elevation Z P z = zp • Land Surface z = 0 Mean Sea level = Geoid Elevation is measured from the Geoid

38. 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

39. Converting Vertical Datums • Corps program Corpscon http://crunch.tec.army.mil/software/corpscon/corpscon.html Point file attributed with the elevation difference between NGVD 29 and NAVD 88 NGVD 29 terrain + adjustment = NAVD 88 terrain elevation

40. 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

42. Representative Fraction Globe distanceEarth distance = Earth to Globe to Map Map Projection: Map Scale: Scale Factor Map distanceGlobe distance = (e.g. 0.9996) (e.g. 1:24,000)

43. Geographic and Projected Coordinates (f, l) (x, y) Map Projection

44. Projection onto a Flat Surface

45. 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

46. Conic Projections(Albers, Lambert)

47. Cylindrical Projections(Mercator) Transverse Oblique

48. Azimuthal (Lambert)

49. Albers Equal Area Conic Projection

50. Lambert Conformal Conic Projection