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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
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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
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Global Cartesian Coordinates (x,y,z)
O
X
Z
Y
GreenwichMeridian
Equator
•
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Global Positioning System (GPS)• 24 satellites in orbit around the earth
• Each satellite is continuously radiating a signal at speed of light, c
• GPS receiver measures time lapse, t, since signal left the satellite, r = ct
• Position obtained by intersection of radial distances, r, from each satellite
• Differential correction improves accuracy
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Global Positioning using Satellites
r1
r3r2
r4Number
of Satellites
1234
Object DefinedSphereCircle
Two PointsSingle Point
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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
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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
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Ellipse
P
F2
O
F1
a
b
X
Z
An ellipse is defined by:Focal length = Distance (F1, P, F2) isconstant for all pointson ellipseWhen = 0, ellipse = circle
For the earth:Major axis, a = 6378 kmMinor axis, b = 6357 kmFlattening ratio, f = (a-b)/a ~ 1/300
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Ellipsoid or SpheroidRotate an ellipse around an axis
O
X
Z
Ya ab
Rotational axis
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Standard Ellipsoids
Ellipsoid Majoraxis, a (m)
Minoraxis, b (m)
Flatteningratio, f
Clarke(1866)
6,378,206 6,356,584 1/294.98
GRS80 6,378,137 6,356,752 1/298.57
Ref: Snyder, Map Projections, A working manual, USGSProfessional Paper 1395, p.12
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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
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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 thetangent plane(3) Angle pqr which this line makes with the equatorialplane is the latitude , of point S
O
Sm
nq
p
r
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Cutting Plane of a Meridian
P
Meridian
Equator
plane
Prime Meridian
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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 meridianand the cutting plane on the meridian through the point, P
P
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Latitude and Longitude on a Sphere
Meridian of longitude
Parallel of latitude
X
Y
ZN
EW
=0-90
°S
P
OR
=0-180°E
=0-90°N
•
Greenwichmeridian
=0°
•
Equator =0°
•
•=0-180°W
- Geographic longitude - Geographic latitude
R - Mean earth radius
O - Geocenter
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Length on Meridians and Parallels
0 N
30 N
Re
Re
RR
A
BC
(Lat, Long) = (, )
Length on a Meridian:AB = Re (same for all latitudes)
Length on a Parallel:CD = R Re Cos(varies with latitude)
D
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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 = 3.1416/180 = 0.0175 radians
• For the meridian, L = Re km
• For the parallel, L = Re CosCoskm• Parallels converge as poles are approached
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Representations of the Earth
Earth surface
EllipsoidSea surface
Geoid
Mean Sea Level is a surface of constant gravitational potential called the Geoid
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Geoid and Ellipsoid
Ocean
Geoid
Earth surface
Ellipsoid
Gravity Anomaly
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Definition of ElevationElevation Z
•
Pz = zp
z = 0
Mean Sea level = Geoid
Land Surface
Elevation is measured from the Geoid
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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
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Converting Vertical Datums• Corps program Corpscon (not in ArcInfo)
– 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
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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
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Earth to Globe to Map
Representative Fraction
Globe distanceEarth distance
=
Map Scale: Map Projection:
Scale Factor
Map distanceGlobe distance
=
(e.g. 1:24,000) (e.g. 0.9996)
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Geographic and Projected Coordinates
() (x, y)Map Projection
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Projection onto a Flat Surface
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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
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Conic Projections(Albers, Lambert)
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Cylindrical Projections(Mercator)
Transverse
Oblique
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Azimuthal (Lambert)
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Albers Equal Area Conic Projection
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Lambert Conformal Conic Projection
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Universal Transverse Mercator Projection
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Lambert Azimuthal Equal Area Projection
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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
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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
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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 State Mapping System - a statewide coordinate system for Texas
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Coordinate System
(o,o)(xo,yo)
X
Y
Origin
A planar coordinate system is defined by a pairof orthogonal (x,y) axes drawn through an origin
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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) = (xo,yo) = (500000, 0) in the Northern Hemisphere, units are meters
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UTM Zone 14
Equator-120° -90 ° -60 °
-102° -96°
-99°
Origin
6°
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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
• Texas has five zones (North, North Central, Central, South Central, South) to give accurate representation
• Greatest accuracy for local measurements
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Texas Centric Mapping System
• Designed to give State-wide coverage of Texas without gaps
• Lambert Conformal Conic projection with standard parallels 1/6 from the top and 1/6 from bottom of the State
• Adapted to Albers equal area projection for working in hydrology
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Standard Hydrologic Grid (SHG)
• Developed by Hydrologic Engineering Center, US Army Corps of Engineers
• Uses USGS National Albers Projection Parameters
• Used for defining a grid over the US with cells of equal area and correct earth surface area everywhere in the country
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ArcInfo 8 Reference Frames
• Defined for a feature dataset in ArcCatalog
• Coordinate System– Projected
– Geographic
• X/Y Domain• Z Domain• M Domain
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Coordinate Systems
• Geographic coordinates (decimal degrees)
• Projected coordinates (length units, ft or meters)
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X/Y Domain
(Min X, Min Y)
(Max X, Max Y)
Maximum resolution of a point = Map Units / Precisione.g. map units = meters, precision = 1000, thenmaximum resolution = 1 meter/1000 = 1 mm on the ground
Long integer max value of 231 = 2,147,483,645
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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)
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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
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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
• Reference Frame in ArcInfo 8 requires projection and map extent