Map Projections (1/2)

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Map Projections (1/2)

• Francisco Olivera, Ph.D., P.E.
• Center for Research in Water Resources
• University of Texas at Austin

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Overview

• Geodetic Datum
• Map Projections
• Coordinate systems
• Global Positioning System

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Definition

• A geodetic datum defines the size and shape of
  the earth, and the origin and orientation of the
  axis used to define the location of points.
• Over time, geodetic data have evolved from simple
  flat surfaces and spheres to complex ellipsoids.
• Flat earth models can be accurate over short
  distances (i.e., less than 10 Km), spherical
  earth models for approximate global distance
  calculations, and ellipsoidal earth models for
  accurate global distance calculations.

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Shape of the Earth
... when it is actually an ellipsoid, slightly
larger in radius at the equator than at the poles.
We think of the earth as a sphere ...
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Ellipse
Z

• An ellipse is defined by
• Focal length ?
• Flattening ratio f (a-b)/a
• Distance F1-P-F2 is constant for all points P on
  ellipse
• When ? 0 then ellipse circle

b
F2
O
a
F1
X
?
?

• For the earth
• Major axis a 6378 km
• Minor axis b 6357 km
• Flattening ratio f 1/300

P
P
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Ellipsoid or Spheroid
Rotate an ellipse around one of its axis.
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Standard Ellipsoids
Ref Snyder, Map Projections, A working manual,
USGS Professional Paper 1395, p.12
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Standard Horizontal Geodetic Data

• NAD27 (North American Datum of 1927) uses the
  Clarke (1866) ellipsoid.
• NAD83 (North American Datum of 1983) uses the
  GRS80 ellipsoid.
• WGS84 (World Geodetic System of 1984) uses GRS80.

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Earth Surfaces
Sea surface
Ellipsoid
Topographic surface
Geoid
Geoid is a surface of constant gravity.
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Earth Surfaces
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Elevation
P
z zp
Topographic Surface
z 0
Mean Sea level Geoid
Elevation is measured from the Geoid
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Standard Vertical Geodetic Datum

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

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Overview

• Geodetic Datum
• Map Projections
• Coordinate systems
• Global Positioning System

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Map Projections

• A map projection is a mathematical algorithm to
  transform locations defined on the curved surface
  of the earth into locations defined on the flat
  surface of a map.

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Map Projection
Projection
Scale
Scale Fraction Map distanceGlobe distance
Representative Fraction Globe distanceEarth
distance
(e.g. 124,000)
(e.g. 0.9996)
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Types of Projections

• Conic Screen is a conic surface. Lamp at the
  center of the earth. Examples Albers Equal Area,
  Lambert Conformal Conic. Good for East-West land
  areas.
• Cylindrical Screen is a cylindrical surface.
  Lamp at the center of the earth. Examples
  (Transverse Mercator). Good for North-South land
  areas.
• Azimuthal Screen is a flat surface tangent to
  the earth. Lamp at the center of the earth
  (gnomonic), at the other side of the earth
  (stereographic), or far from the earth
  (orthographic). Examples Lambert Azimuthal Equal
  Area. Good for global views.

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Conic Projections
Albers and Lambert
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Cylindrical Projections
Mercator
Transverse
Oblique
Tangent Secant
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Azimuthal
Lambert
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Albers Equal-Area Conic
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Lambert Conformal Conic
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Universal Transverse Mercator
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Lambert Azimuthal Equal-Area
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Distortion Projected Maps

• In the process of transforming a curved surface
  into a flat surface, some geometric properties
  are modified.
• The geometric properties that are modified are
• Area (important for mass balances)
• Shape
• Direction
• Length
• The difference between map projections has to do
  with which geometric properties are modified.
• Depending on the type of analysis, preserving one
  geometric property might be more important that
  preserving other.