EPSc 407 IP07

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EPSc 407 IP07
IP07Map Projections
Washington UniversityPath 201 Fall 2007
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IP07 - Map Projections

• General Concepts
• Characteristics
• Reference ellipsoids
• Latitude and longitude coordinates
• Datums
• Cylindrical projections
• Conic projections
• Azimuthal projections
• Pseudo-cylindrical projections
• ENVI map projection header information
• Image display tools
• Cursor location
• Grid lines

• Map coordinate converter
• ASCII coordinate converter
• Resampling and warp methods
• Nearest neighbor resampling and bilinear
  interpolation
• Changing map projections and datums
• Ground control points
• Georeferencing
• GCP collection
• Image warping

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Map projection general concepts

• Map projections attempt to display a portion of a
  planet's surface on a flat plane
• Distortion in area, shape, scale, or direction
  occurs in creating the projection
• There is no best map projection
• Each projection is designed to minimize
  distortion in area, shape, scale, or direction
• Projections that accurately portray area are
  known as equal-area projections
• A circle placed on anywhere on map represents
  the same amount of area
• Also known as equivalent projections
• Conformal projections show objects on a map
  without shape distortion
• All lines of latitude and longitude intersect at
  right angles
• Local scale is the same in all directions around
  any point
• Areas are generally distorted, except along
  certain lines

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Map projection general concepts

• Scale is the ratio of a distance portrayed on a
  map to the same distance on the planet
• No map projection correctly shows scale
  throughout the map
• Usually one or more lines on the map has a
  constant scale
• Equidistant projections show true scale between
  the center of map and other points
• Directions (azimuths) on a map are shown
  correctly relative to the center of the map on
  azimuthal projections
• Some azimuthal projections are also equal-area,
  conformal, or equidistant
• Some map projections have special characteristics
• Lines of constant direction shown as straight
  lines (Mercator), good for navigating over long
  distances
• Great circle arcs shown as straight lines
  (gnomonic) or as circles (stereographic)

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Map projection general concepts

• Reference ellipsoid used to approximate planets
  that are flattened at poles (polar axis is
  shorter than equatorial axis)
• Reference ellipsoids can be designed for local
  or global applications
• Earth ellipsoids for global use refined over
  past 200 years
• Geopotential surface is the surface of equal
  gravity potential
• Gravity vector is perpendicular to geopotential
  surface
• Geoid is geopotential surface at mean sea level
• Varies from ellipsoid by up to 100 m.
• Elevations on maps are usually relative to a
  geoid latitude, longitude, and planar
  coordinates are relative to reference ellipsoid

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Map projection general concepts

• Reference Ellipsoids
• a semi-major axis (ellipsoid equatorial radius)
• b semi-minor axis (ellipsoid polar radius)
• f flattening
• e eccentricity

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Selected reference ellipsoids
WGS World Geodetic System
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Latitude and longitude coordinate systems

• Planetocentric is relative to ellipsoid center
• Latitude is angle between equator and line from
  surface point to ellipsoid center
• Longitude uses right-hand rule (east positive)
• Planetographic (geodetic) is relative to
  ellipsoid normal
• Latitude is angle between equator and normal to
  ellipsoid at a surface point
• Longitude direction depends on planet rotation
  west positive for prograde planets Earth is east
  positive for global areas or west positive
  locally
• Planet Centered Cartesian have x, y, z coordinates

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Geodetic datums

• Provides framework for referencing planar
  coordinates
• Horizontal and/or vertical reference
• Requires reference ellipsoid and coordinate
  system origin
• Hundreds of datums in use for different regions
  of the Earth
• Datums can differ from each other by up to one
  kilometer in x, y planar coordinates
• Conversion from one datum to another will also
  change values of geodetic latitude and longitude.
• It is important to know what datum is being used
• Do not mix data that use different datums
• Common datums for maps and remote sensing of
  North America
• NAD27 Clark 1866 ellipsoid
• WGS84 WGS84 ellipsoid
• These differ by about 150 - 200 meters

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Datum conversion
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Map projections

• Projection Types
• Cylindrical
• Conic
• Azimuthal
• Pseudocylindrical

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Cylindrical projections

• Regular cylindrical projections partly formed by
  projecting points onto a cylinder wrapped around
  a globe at the equator
• Longitude lines are equidistant parallel straight
  lines on the projection
• Latitude lines cross longitude lines at right
  angles, but are not equally spaced
• Oblique or transverse projections result from
  rotating the cylinder relative to the globe

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Mercator projection

• Longitude lines are vertical, equally spaced, and
  parallel to each other
• Latitude lines are horizontal (cross longitude at
  right angles)
• Spacing increases toward the poles so that the
  projection is conformal
• Area is distorted, as is scale
• Used for marine navigation because straight lines
  are lines of constant azimuth
• Used to show large portions of globe, except for
  the poles

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Transverse Mercator projection

• Projection of the globe onto a cylinder tangent
  to a longitude line
• Latitude and longitude lines are no longer
  straight lines
• Distortion of scale, distance, direction, and
  area increase away from central longitude

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Universal Transverse Mercator (UTM) projection

• Special case of transverse Mercator
• Widely used for designating rectangular
  coordinates (in meters) on large-scale maps
• Earth divided into 60 zones (each 6 of longitude
  wide)
• Scale variation within a zone is 1 part in 1,000
• Zone origin is equator at central longitude, with
  x value of 500,000 m and y of 0 m for Northern
  Hemisphere
• X increases to east, y to the north

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Cylindrical Equidistant projection

• Latitude and longitude lines are parallel,
  equidistant, straight lines, intersecting at
  right angles
• Simple linear scaling of latitude and longitude
• Also known as simple cylindrical or geographic
  lat/lon projection (ENVI)
• Neither equal-area nor conformal

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Space Oblique Mercator Projection

• Modified cylindrical projection with map surface
  defined by satellite orbit
• Designed for displaying early Landsat images and
  other similar satellite data
• Central line of projection is satellite
  groundtrack
• Scale is true along groundtrack
• Used only for narrow band along the groundtrack

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Conic projections

• Surface projected onto a cone that intersects
  planet at one or two latitude lines (known as
  standard parallels)
• Scale is true along the standard parallels, but
  distorted elsewhere
• Can also be conformal, equal-area, or equidistant
  in limited portions of the map
• Used for areas of large east-west extent

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Lambert Conformal Conic projection

• Uses two standard parallels
• Is conformal
• Latitude lines are arcs of concentric circles
  with spacing decreasing toward center of map
• Longitude lines are equally spaced and intersect
  latitudes at right angles
• Scale is true along standard parallels

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Albers equal area projection

• Uses two standard parallels
• Is equal-area
• Latitude lines are arcs of concentric circles
  with spacing decreasing toward north and south
  edges of map
• Longitude lines are equally spaced and intersect
  latitudes at right angles
• Scale and shape are true along standard parallels

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Azimuthal projection

• Surface projected onto a plane, usually tangent
  to the planet
• Direction or azimuth from the center of the
  projection to every other point on the map is
  correctly shown
• For spherical form, great circles passing through
  the center of the map are shown as straight lines

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Orthographic projection

• Projection from a point infinitely far from the
  planet onto a plane tangent to the planet
• Makes the planet appear like a globe
• Latitude and longitude lines can be straight
  lines, ellipses, or circles
• Neither conformal or equal-area

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Stereographic Projection

• Projection from a point on a planet to a plane
  tangent to the planet and on the opposite side
  from the projection point
• A conformal projection
• Often used to show polar areas with North or
  South Pole at the center of the map

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General perspective projection

• Projections of a planet onto a plane through a
  single point
• Simulates the geometry of a framing camera
• Neither conformal or equal-area
• Other azimuthal projections are special cases of
  this projection

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Pseudocylindrical projections

• Resemble cylindrical projections
• Latitude lines are straight and parallel
• Longitude lines are curves

Sinusoidal
Mollweide

• Used for world maps
• Projection is equal-area
• Central longitude is a straight line
• Other longitudes are equally spaced sinusoidal
  curves
• Latitudes are parallel, equally-spaced, straight
  lines

• Used for world maps
• Projection is equal-area
• Central longitude is a straight line
• Remaining longitudes are elliptical arcs
• Latitudes are parallel, straight lines

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ENVI map projection header information

• Map projection information stored in ENVI ASCII
  header file
• Map info can be added by editing file
• No registration is performed by editing the
• Geographic Corners attribute
• EM File Edit ENVI Header

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Image display tools

• Cursor Location/Value
• IM Tools Cursor Location/Value
• Grid lines
• IM Overlay Grid Lines
• Add grids for pixel, map, or geographic
  coordinate systems
• Non-pixel coordinates require georeferenced image

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Grid line settings

• Save grid settings to file for later use

Set attributes for pixel, map, and geographic
grids
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Map coordinate converter

• EM Map Map Coordinate Converter
• Change projections and datums to desired
  settings
• Enter known coordinate
• Calculate in forward or reverse direction

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ASCII coordinate converter

• EM Map ASCII Coordinate Conversion
• Convert one or more files of coordinates or GCPs
  (ground control points)

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Resampling and warp methods

• Pixel resampling methods
• Nearest neighbor uses the nearest pixel without
  any interpolation
• Bilinear is a linear interpolation using 4
  neighboring pixels
• Cubic convolution uses 16 pixels to approximate
  the sine function using cubic polynomials
  significantly slower than other methods
• Warp methods
• RST (rotation, scaling and translation),
  requires at least four GCPs
• Polynomial, sometimes called rubbersheeting
• Degree of polynomial is dependent upon number of
  GCPs selected GCPs gt (degree 1)2
• Delaunay triangulation fits triangles to the
  irregularly spaced GCPs and interpolates values
  to the output grid.

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Projection conversion reverse mapping

• Projection conversion employs reverse mapping to
  derive output
• Example take an input grid and convert to a
  different projection

Output grid does not look square, but it will
be stored as a rectilinear grid of samples and
lines
Nearest neighbor resampling uses value of
reprojected input pixel nearest to center of
output pixel
Some output pixels may be 0
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Projection conversion bilinear interpolation

• Bilinear interpolation resampling is used to
  better approximate output
• Example

Start by changing projection
Use bilinear interpolation to apply weight of
distances from center of output pixel to centers
of four nearest reprojected input pixels
continued
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Projection converstion bilinear interpolation
Define positions x and y so that Find points x,0
and x,1 Find point x,y Also may be done using
points 0,y and 1,y
0 x 1 and 0 y 1 (x,0) (0,0) x
(1,0) (0,0) (x,1) (0,1) x (1,1)
(0,1) (x,y) (x,0) y (x,1) (x,0)

For this example, x 0.7 and y 0.25So f(x,0)
3 0.7 (4 3) 3.7 and (x,1) 4 0.7
(6 4) 5.4and f(x,y) 3.7 0.25 (5.4 3.7)
4.13
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Changing map projections and datums

• EM Map Convert Map Projection
• Select file and target projection
• Optionally save warp points to GCP file
• Set warping and resampling parameters

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Ground control points

• A set of image coordinates for an unregistered
  image corresponding to a known set of locations
• Sources of known locations may vary
• Registered images
• Maps
• DLGs (digital line graphs)
• GPS field readings
• Unregistered images (special case)
• GCPs saved in ASCII format

ENVI Registration GCP File image to map
ground control points warp image
D\work\hawaii\tm2\6347refl.img projection info
UTM, 5, North, North America 1983,
unitsMeters Map (x,y), Image (x,y)
260220.1560 2150032.0000 7767.3301
1345.3300 216666.2729
2117456.7980 6235.5000 2492.0000
259937.9468 2149536.2236
7757.5000 1363.5000 210845.5391
2119270.6655 6031.5000
2427.0000 194448.2449 2150410.6623
5449.5000 1335.0000
199282.3610 2120379.6392 5624.0000
2388.0000 193543.8578
2158687.8007 5417.5000 1041.5000
195590.4008 2163809.4374
5489.0000 865.0000 260206.2190
2151741.5000 7767.0000 1286.3300
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GeoreferencingGCP collection

• GCPs required to register image to a map
  projection
• Warp (unregistered) image must be displayed to
  collect GCPs
• Image to Map for registering to DLGs or field
  readings
• EM Map Registration Select GCPs Image to
  Map
• Destination projection, datum, and pixel size are
  specified
• Image to Image for registering to another
  image
• EM Map Registration
• Select GCPs Image to Image

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GeoreferencingGCP collection

• GCPs are entered and managed through GCP
  selection dialog

Location in unreferenced image
Location on surface
Degree of warp polynomial
RMS warping error
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GeoreferencingGCP collection

• Collected GCPs are displayed on warp image
• Map locations may be entered by hand, or
  automatically entered from vector window or
  existing registered image
• ENVI will predict warp image location given map
  location after four GCPs have been entered
• GCPs may be updated or deleted to minimize error
• For best results RMS error lt 1.5
• Save GCPs to file for later use

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Georeferencingimage warping

• Register image from GCP selection dialog
• Options Warp Displayed Band or
• Options Warp File
• or from ENVI menu
• EM Map Registration Warp from GCPs Image
  to Map or
• EM Map Registration Warp from GCPs Image
  to Image

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