DEMs and ellipsoids

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DEMs and ellipsoids
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How high is that mountain?

• What is elevation?
• Measured from what?
• Sea Level

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Sea level, the ellipsoid, and the geoid

• from sea level (high tide or low tide?)
• okay, mean sea level (an average)
• suppose the earth was completely covered in water
  -
• what shape would it be ?
• A. A sphere (rough approximation).
• B. An ellipsoid (the ellipsoid)(a better
  approximation).
• C. A very slightly lumpy ellipsoid (the
  geoid)(the best so far).
• So now we have three surfaces
• Geoid where the mean sea surface would be.
• Ellipsoid a geometric shape defined by an
  mathematical equation that approximates the
  geoid.
• Topography the actual surface of the earth.

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Reference ellipsoid

• Geometric shape that closely matches the shape of
  the earth.
• Several different ones have been developed and
  optimally fit only one area of the world.
• An offset (a datum) is often applied for specific
  areas.
• Some common ellipsoids and datums are listed
  below.

Name (date) Semi-major axisflattening datum W
orld Geodetic System 1984 (WGS84) 6378137
298.257223563 Geodetic Reference System 1980
(GRS80) 6378137 298.257222101 World Geodetic
System 1984 World Geodetic System 1972
(WGS72) 6378135 298.26 World Geodetic System
1972 Clark (1866) 6378206 294.98 North
American Datum 1927
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Geoid and gravity

• Problem we dont know this shape exactly.
• Pretty close to the ellipsoid and defined by
  deviation from the ellipsoid.
• Depends on gravity variations and hence density
  variations.
• Surface perpendicular to a vertical line (plumb
  line).

The equipotential surface of the Earth's gravity
field which best fits, in a least squares sense,
global mean sea level (U.S. National Geodetic
Survey definition).
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So what?

• If we want to compare measurements, we need to
  use the same system.
• Ignoring it will lead to mostly small errors.
• Most GIS and software programs ask for the
  reference ellipsoid or datum.
• Handhelp GPS units usually use WGS84.
• Many topo maps we use are based on NAD27 (and the
  Clarke ellipsoid).
• Can be off by tens of m.
• Most pre-GPS topo maps are referenced to local
  sea level so are essentially referenced to the
  geoid.

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Elevation (and other data) maps

• Want to represent 3D data on a 2D surface.
• Contour
• Shaded
• Elevation
• Slope (or gradient)
• 3D visualizations.

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Contours

• Lines that connect points of equal elevation.
• Do not split or cross other contour lines.
• Closely spaced lines indicate slope widely
  spaced lines indicate flatter areas.
• Topography maps generally have smooth curves
  reflecting smooth changes.
• Faults in subsurface maps often have
  discontinuities.

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Creating contour maps

• An interpretation of data.
• Does not require evenly spaced data.
• By hand
• Can be better easily to include a priori
  information
• Impractical for large datasets
• By computer
• Often grid the data first
• May create artifacts
• Edge effects
• Misses control points
• Bulls eye
• Need to understand data and method

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Hand contouring

• Various methods
• Place contours divided linearly between points
  (basically linear interpolation)
• Parallel draw contours parallel to each other.
• Equal-space assume uniform slope over all areas.
• Interpretative do anything you want as long as
  the data is honored.

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Computer contouring

• Delaunay triangulation
• Creates a regular grid (or surface) from
  irregularly spaced data.
• Various methods
• Linear interpolation
• Weighted interpolation
• Kriging and geostatistics (may be best for
  geology)
• Advantages of grids
• Even data distribution
• Allows easy application of mathematical
  operations
• Filtering
• Smoothing
• Display

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Example a set of numbers

• 0 0 0 2 10 22
• 0 0 0 3 15 26
• 0 0 3 10 21 32
• 0 0 4 13 22 29
• 0 0 0 7 17 26

hand
computer
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DEM

• Digital elevation models
• Gridded elevations
• Not always accurate (thats why it is a model)
• Getting more widely available.
• Useful for
• Shaded and perspective maps
• Hydrological measurements

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Some useful DEM bathymetry

• National Elevation Data set (NED)
• US 30 m resolution (except Alaska)
• Seamless http//seamless.usgs.gov/
• Shuttle Radar Topography (SRTM)
• US 30 m http//seamless.usgs.gov/
• Worldwide 90m (between 60 N and 60S)
• Some gaps and problems in high slopes and lakes.
• World topo bathymetry etopo2http//www.ngdc.no
  aa.gov/mgg/

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Lidar laser ranging
meters
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Miscellaneous DEM DMM

• Mars (at 1 km resolution)
• High resolution offshore CA bathymetry.http//wrgi
  s.wr.usgs.gov/dds/dds-55/pacmaps/site.htm

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Projections

• A projection is a method of portraying the curved
  surface of the earth on a flay surface.
• Distortions of distance, direction, scale, and
  area always occur.
• Some projections preserve one property but
  distort the others (usually badly)
• Other projections distort all properties but less
  strongly.
• Selection of a given projection depends on the
  use.
• An airplane pilot might use an azimuthal
  preserving projection.
• Someone in the polar regions might use a polar
  projection.

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Types of projections http//mac.usgs.gov/mac/isb/p
ubs/MapProjections/projections.html

• Cylindrical projections
• Wrap a cylinder around the earth
• Mercator, UTM
• State plane
• Conic projections
• A cone rather than a cylinder
• Albers
• Azimuthal
• Project onto a plane
• Lambert
• Polar
• Others
• Robinson
• Sinusoidal

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Universal Transverse Mercator (UTM)

• Conformal cylindrical projections
• Scale correct only at central meridian
  distortion increases with distance from meridian
• 60 different zones 6 degrees wide for both north
  and south hemispheres
• Projection differs (different cylinder) for each
  zone
• Low distortion near equator
• Cannot combine zones
• A living fossil poorly suited for current uses
  but still widely used.
• California in in UTM zones 10 (S. Cal) and 11 (N.
  Cal).

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State Plane

• Usually similar to UTM but uses a different
  projection for each state/zone
• Used only in US
• A mix of NAD27 or NAD83
• Must be careful
• Errors of 10s feet possible
• Coordinates denoted in feet (or meters) as false
  northing and false easting from point of
  origin.

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DEM

• Often in a geographical (longitude,latitude)
  projections.
• Can be displayed as a variety of different
  projections.

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Remote sensing data

• Usually an image from an airplane or satellite.
• Original data is distorted in some way
  (especially for radar data).
• Most useful if registered (image-to-map) to some
  projection.

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GPS

• Started in 1978
• Now composed of at least 24 satellites
• Funded and controlled by US Dept. of Defense
• 12 hour orbit
• Sends signals at two wavelengths, L1 and L2 to
  compensate for ionospheric error.
• Sends location of satellite and satellite health
  information.
• Distance (range) from receiver to each satellite
  is measured.
• This can be used to solve (with least squares)
  the location of the receiver.
• Horizontal accuracy is much better than vertical
  accuracy.

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Sources of error

• Poor visibility of satellites due to trees,
  buildings or cliffs.
• Poor configuration of satellites.
• Multipath (reflected signals)
• Dithering by the US DOD.
• Typical error is 5 to 15 m horizontal for newer
  handheld GPS at least 10 m vertical.
• Check PDOP for estimate of error

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Other measurements

• Velocity (try a car)
• Bearing or azimuth
• Store waypoints along a route.
• Can be downloaded to a PC.
• Maps created in mapping program such as Arcview.
• Can get better accuracy with differential GPS.

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GPS and geology

• Field mapping
• Normal unit is not really quite good enough for
  detailed mapping.
• Differential GPS can solve this but requires
  extra effort afterwards.
• Use a total station or laser rangefinder for the
  most detailed mapping.
• Use GPS for absolute location
• Use total station for the relative locations.

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GPS and geology

• Differential and phase measurement GPS can be
  used to measure plate motions.
• Takes expensive equipment and considerable
  post-processing.

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S. California plate motions