Earth Models and Projections

1
Earth Models and Projections

• Models of the earth
• Datums
• How ArcGIS handles coordinates
• Projections
• Summary of projections

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Models of the Earth

• The earth can be modeled as a
• Sphere,
• Oblate ellipsoid
• Geoid

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The Map in Printed Form

• The earth model used.
• The coordinate/projection system.
• The datum.

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Earth Shape Sphere and Ellipsoid
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Measuring the Ellipsoid

• Oblate ellipsoid predicted by Sir Issac Newton
  (ca 1700).
• French academy of sciences sent expeditions to
  Lapland and Peru (now in Ecuador) to measure the
  length of a degree along a meridian.
• La Condamine (1735) sent to Mitad del Mundo.
• Moreau de Maupertuis (1736-37) sent to Tornio
  river valley.

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Measuring the Ellipsoid

• Maupertuis reported a meridian degree as 57,437.9
  toises (1 toise 1.949 m).
• Meridian degree at Paris was 57,060 toises.
• Concluded earth was flatter at poles.
• Measures were erroneous but conclusions were
  correct.
• Published as La figure de la Terre (1738).

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Earth as Ellipsoid
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The Spheroid and Ellipsoid

• The sphere is about 40 million meters in
  circumference.
• An ellipsoid is an ellipse rotated in three
  dimensions about its shorter axis.
• Many ellipsoids have been measured, and maps
  based on each.
• The terms spheroid and ellipsoid tend to be used
  interchangeably in the literature.

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Earth As an Oblate Ellipsoid
Flatter, latitude line space 69.41 mi
Because the earth is flatter at the poles, close
to poles tangent must move further to change by
1 degree, hence 1 degree of lat. is longer at
poles than at the equator.
d 1
d 89
Curved, lat. line space 68.7 mi
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Measuring Latitude

• Geodetic latitude is always used.
• Geodetic latitude (d) angle of vector
  perpendicular to ellipsoid surface compared to
  plane of equator.
• Geocentric latitude (c) angle of vector
  perpendicular to spheroid surface, all pass
  through the center.

11
Earth As an Oblate Ellipsoid

• Angle from center of earth subtends an arc at
  surface of earth.
• 1 degree arc 68.7 mi at equator.
• 1 arc-second 1/3600th degree or 30.86 m.
• Tongass DEM is 2 arc-second.

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Which Ellipsoid?

• Hundreds have been defined depending upon
• Available measurement technology
• Area of the globe (e.g., North America, Africa)
• Map extent (country, continent or global)
• Political issues (e.g., Warsaw pact versus NATO)
• ARC/INFO supports 26 different spheroids!
• Conversions via math formulae
• Earth measurements
• Equatorial radius 6,378 km 3,963 mi
• Polar radius 6,357 km 3,950 mi
• (Flattened about 13 miles at poles)

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Which Ellipsoid?

• Most commonly encountered are
• Everest (Sir George) 1830
• One of the earliest ellipsoids India
• a 6,377,276 m b 6,356,075 m f
  1/300.8
• Clarke 1886 for North America
• Basis for USGS 7.5 Quads
• a 6,378,206.4 m b 6,356,583.8 m f
  1/295

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Which Ellipsoid?

• GRS80 (Geodetic Ref. System, 1980)
• Current North America mapping
• a 6,378,137 m b 6,356,752.31 m f
  1/298
• WGS84 (World Geodetic Survey, 1984)
• Current global choice
• a 6,378,137 b 6,356,752.31m
  f 1/298

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The Datum

• An ellipsoid gives the base elevation for
  mapping, called a datum.
• Examples are NAD27 and NAD83.
• The geoid is an ellipsoid that is based on local
  gravity.
• It is the most accurate, and is used more in
  geodesy than GIS and cartography.

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Earth Models and Datums

• Elevations defined with reference to a sphere,
  ellipsoid, geoid or local sea level will all be
  different.
• When linking GPS with GIS, the user must know
  what base to use.

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Original North American Datums

• 1900 US standard datum
• First nationwide datum
• Clark 1866 ellipsoid
• Origin Meades Ranch, Osborne County, KS
  (39-13-26.686 n 98-32-30.506 w)
• Determined by visual triangulation
• Approx. 2,500 points
• Renamed North American Datum (NAD) in 1913 when
  adopted by Mexico and Canada.

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Original North American Datums

• NAD27
• Clark 1866 ellipsoid
• Meades Ranch origin
• Visual triangulation
• 25,000 stations (250,000 by 1970)
• NAVD29 (North American Vertical Datum, 1929)
  provided elevation
• Basis for most USGS 7.5 minute quads

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Original North American Datums

• NAD83
• Satellite (since 1957) and laser distance data
  showed inaccuracy of NAD27.
• 1971 national academy of sciences report
  recommended new datum.
• Used GRS80 ellipsoid (functionally equivalent to
  WGS84, but not identical).
• Origin mass-center of earth.

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Original North American Datums

• NAVD88 provides vertical datum.
• Points can differ up to 160 m from NAD27, but
  seldom more than 30 m.
• Conversion from NAD27 see USGS Bulletin 1875
  for conversion tables (in ARC/INFO).

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Coordinate Systems

• Standardized method for assigning location codes
  to objects.
• Standardized coordinate systems use absolute
  locations (lat/long).
• A map captured in the units of the paper sheet is
  based on relative locations or map millimeters.
• In a coordinate system, the x-direction value is
  the easting and the y-direction value is the
  northing. Most systems make both values positive
  by introducing false values.

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Geographic Coordinates

• Earth's latitude and longitude system ranges from
  90 degrees south to 90 degrees north in latitude
  and 180 degrees west to 180 degrees east in
  longitude.
• A line with a constant latitude running east to
  west is called a parallel.
• A line with constant longitude running from the
  north pole to the south pole is called a
  meridian.
• The zero-longitude meridian is called the prime
  meridian and passes through Greenwich, England.
• A grid of parallels and meridians shown as lines
  on a map is called a graticule.

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Coordinate Systems for the US

• Some standard coordinate systems used in the
  united states are
• Geographic coordinates.
• Universal transverse Mercator system.
• Military grid.
• State plane.
• To compare or edge-match maps in a GIS, both maps
  MUST be in the same coordinate system.

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Latitude and Longitude Location on the Spheroid
90 N
Prime Meridian
Equator
0
90 S
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Latitude and Longitude Graticule
Lat and long measured in degrees minutes
seconds (60 1 60 1) Decimal
degrees, not minutes/seconds, best for
ArcView. dd d m/60
s/3600 Pixel size (m) pixel (dd)
cos(central latitude of the data)(earth
circumference (m)/360
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Decimal Degrees

• 121.135 degrees 121 degrees
• and 0.135 60 8.1 minutes
• and 0.1 60 6 seconds
• 121 deg., 8 min., 6 sec.

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Geographic Coordinates as Data

• Geographic coordinates in decimal degrees.

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Latitude and Longitude Location on the Spheroid

• LONGITUDE MERIDIANS
• Prime meridian is zero Greenwich, U.K.
• International date line is 180 EW.
• Circ. 40,008 km or 24,860 mi.
• (Equator to pole approx. 10,000,000 meters).
• 1 degree 69.17 mi at equator.
• 46.50 mi at 47.75 N (Pullman).
• 0 mi at 90 N/S.
• Changes as cos(latitude).

• LATITUDE PARALLELS
• Equator is zero.
• Circ. 40,076 km or 24, 902 mi.
• 1 degree 68.70 mi at equator.
• 69.41 mi at poles.
• (1 mile 1609.34 m 5280 feet).
• 1 nautical mile length of 1 minute of arc.
• 1852 m 6076 ft (international).
• 1843 m - 1862 m.

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State Plane Coordinate System

• Began in 1930s for public works projects Popular
  with interstate designers.
• States divided into 1 or more zones (130 total
  for US).
• Each zone designed to maintain scale distortion
  to less than 1 part per 10,000.
• Washington has 2 zones running E/W.
• http//data.geocomm.com/helpdesk/coordinates.html

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State Plane Coordinate System

• Different projections used
• Transverse mercator (conformal) for states with
  large N/S extent.
• Lambert conformal conic for rest.
• Some states use both projections (NY, FL, AK).
• Oblique mercator used for Alaska panhandle.

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State Plane Coordinate System

• Each zone also has
• Unique standard parallels (2 for Lambert) or
  central meridian (1 for mercator).
• False coordinate origins which differ between
  zones, use feet for NAD27, meters for NAD83.
• Scale reduction used to balance scale across
  entire zone resulting in accuracy variation of
  approx. 1 per 10,000, thus 4 times more accurate
  than UTM.
• See Snyder, 1982 USGS bulletin 1532, p. 56-63
  for details.

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State Plane Coordinate System Washington State
Example

• Two zones, N and S
• South zone Lambert conformal conic
• Standard Parallel 45.5
• Standard Parallel 47.5
• Longitude central meridian 120.5
• Latitude of projection origin 45.33
• False Easting 2000000
• False Northing 0
• Horizontal Datum NAD83
• Ellipsoid Name GRS80
• Semi-major axis 6378137
• Flattening ratio 1/298.257222101

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How ArcGIS Handles Coordinates and Projections

• The coordinate reference system of the display
  view is determined by the first layer opened in
  the view.
• Geographic (lat/long).
• Projected (by type and parameters).
• This may or may not be known.

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How ArcGIS Handles Coordinates and Projections

• As other layers are added, they are
• Re-projected on-the-fly to that of the view if
  their reference system and reference system of
  first layer is known.
• Displayed as is (and thus potentially
  incorrectly) if either is not known (warning
  issued).

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How ArcGIS Handles Coordinates and Projections

• If the projection of the first layer was not
  already recorded, you must select view/data frame
  properties and..
• Select Coordinate System tab to specify
  projection of view.
• Projection is for entire view, not any one layer.
• Once ArcMap has been informed of the original
  projection of the data, you can re-project the
  view.
• This applies to the display only. The underlying
  data files are not changed in any way.
• To change the underlying data files, use
  ArcCatalog.

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How ArcGIS Handles Coordinates and Projections

• For correct measurement only, you can select
  General tab.
• Map units may be unknown when data read in.
• User sets it based on actual units for raw data
  (e.g., decimal degrees).
• Display units for reporting measurements (e.g.,
  miles).
• Map units must be specified before display units
  can be set.
• If map units specified incorrectly, distance
  measures will be wrong!

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Map Projections

• A transformation of the spherical or ellipsoidal
  earth onto a flat map is called a map projection.
• The map projection can be onto a flat surface or
  a surface that can be made flat by cutting, such
  as a cylinder or a cone.
• If the globe, after scaling, cuts the surface,
  the projection is called secant. Lines where the
  cuts take place or where the surface touches the
  globe have no projection distortion.

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Standard parallels

• Secant conic projection.
• Lines of true scale, standard parallels.

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Map Projections
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Map Projections

• Projections can be based on axes parallel to the
  earth's rotation axis (equatorial).
• At 90 degrees to it (transverse).
• Or at any other angle (oblique).

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Secant map projections
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Map Projections

• A projection that preserves the shape of features
  across the map is called conformal.
• A projection that preserves the area of a feature
  across the map is called equal area or
  equivalent.
• No flat map can be both equivalent and conformal.
  Most fall between the two as compromises.
• To compare or edge-match maps in a GIS, both maps
  MUST be in the same projection.

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No flat map can be both equivalent and conformal.
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Universal Transverse Mercator (UTM)

• First adopted by US army in 1947 for large scale
  maps worldwide.
• Used from lat. 84N to 80S Universal polar
  stereographic (UPS) used for polar areas.
• Globe divided into 60 N/S zones, each 6 wide
  These are numbered from one to sixty going east
  from 180th meridian.
• Each zone divided into 20 E/W belts, each 8 high
  lettered from the south pole using C thru X (O
  and I omitted).

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UTM zones in the lower 48
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Universal Transverse Mercator (UTM)

• Coordinate origins are at the intersection of the
  equator and the zones central meridian
• This origin given a value of 0 meters north,
  500,000m east, thus no negative values.
• Military uses a different system for coordinate
  location dividing each UTM primary grid zone into
  10km by 10km squares and designates each by a
  double letter.

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Universal Transverse Mercator (UTM)

• The meridian halfway between the two boundary
  meridians for each zone is designated as the
  central meridian and a cylindrical projection is
  done for each zone.
• Scale of central meridian reduced by .9996 to
  minimize scale variation in zone resulting in
  accuracy variation of approx. 1 meter per 2,500
  meters.

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Review Map Projections

• GIS Capability
• Choosing a Projection
• Common Projections in GIS
• Classification of Projections

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Map Projections

• Location on the 3-D earth is measured by latitude
  and longitude.
• Location on the 2-D map is measured by X,Y
  Cartesian coordinates.
• Unlike choice of ellipsoid, choice of map
  projection does not change a locations lat/long
  coordinates, only its XY coordinates.

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Map Projections Classification

• Property Preserved
• Equal area projections preserve the area of
  features (popular in GIS).
• Conformal projections preserve the shape of small
  features (good for presentations) , and show
  local directions (bearings) correctly (useful
  for coastal navigation!)
• Equidistant projections preserve distances
  (scale) to places from one point, or along a one
  or more lines.
• Scale can never be correct everywhere on any map.
• True direction projections preserve bearings
  (azimuths) either locally (in which case they are
  also conformal) or from center of map.

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Map Projections Classification
Geometric Model Used

• Planar/Azimuthal/Zenithal image of spherical
  globe is projected onto a map plane which is
  tangent to (touches) globe at single point.
• Conical image of spherical globe is projected
  onto a cone which touches
• Along one line (tangent) or
• Cuts thru globe along two lines (secant)
  (usually parallels of latitude)
• Cone is then unfolded to create flat map

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Map Projections Classification
Geometric Model Used

• Cylindrical image of spherical globe is
  projected onto a cylinder which again may be
  tangent along one line or secant along two
  --again, cylinder is unfolded to create a flat
  map

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Choosing a Map Projection

• Rules of thumb.
• Choose a standard for your organization and keep
  all data that way.
• Retain lat/long coordinates as the GIS database
  if possible.
• For small areas, projection not too critical
  unless you will be combining layers from
  different sources (common in GIS) Then, each
  layer must be in same projection, with same
  parameters and same datum.

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Choosing a Map Projection

• Issues to consider
• Extent of area to map city, state, country,
  world?
• Location polar, mid-latitude, equatorial?
• Predominant extent of area to map E-W, N-S,
  oblique?

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Choosing a Map Projection

• Use equal-area projections for thematic or
  distribution maps, and as a general choice for
  GIS work.
• Use conformal projections in presentations.
• For navigational applications, need true distance
  or direction.

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Common Projections in GIS

• American Polyconic
• Early projection used by USGS Usually only
  encountered on older maps Being replaced by
  transverse mercator.
• Albers conic equal-area
• Often used for US base maps showing all of the
  lower 48 states
• Standard parallels set at 29 1/2N and 45 1/2N

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Common Projections in GIS

• Lambert conformal conic
• Often used for US base map of all 50 states
  (including Alaska and Hawaii), with standard
  parallels set at 37N and 65N
• Also for state base map series, with standard
  parallels at 33N and 45N
• Also used in state plane coordinate system (SPCS)

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Common Projections in GIS

• Transverse Mercator
• Used in SPCS for states with major N/S extent
• Universal transverse Mercator (UTM) used for
  world wide military (and other) large scale
  mapping

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GIS Capability

• A GIS package should be able to move between
• map projections
• coordinate systems
• datum
• ellipsoids