Geodetic Datum
Introduction to Geodetic Datums
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Geodetic datums define the size and shape of the earth and the origin and
orientation of the coordinate systems used to map the earth. Hundreds of
different datums have been used to frame position descriptions since the
first estimates of the earth's size were made by Aristotle. Datums have
evolved from those describing a spherical earth to ellipsoidal models derived
from years of satellite measurements.
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Modern geodetic datums range from flat-earth models used for plane surveying
to complex models used for international applications which completely
describe the size, shape, orientation, gravity field, and angular velocity
of the earth. While cartography, surveying, navigation, and astronomy all
make use of geodetic datums, the science of geodesy is the central discipline
for the topic.
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Referencing geodetic coordinates to the wrong datum can result in position
errors of hundreds of meters. Different nations and agencies use different
datums as the basis for coordinate systems used to identify positions in
geographic information systems, precise positioning systems, and navigation
systems. The diversity of datums in use today and the technological advancements
that have made possible global positioning measurements with sub-meter
accuracies requires careful datum selection and careful conversion between
coordinates in different datums.
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The Figure of the Earth
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Geodetic datums and the coordinate reference systems based on them were
developed to describe geographic positions for surveying, mapping, and
navigation. Through a long history, the "figure of the earth" was refined
from flat-earth models to spherical models of sufficient accuracy to allow
global exploration, navigation and mapping. True geodetic datums were employed
only after the late 1700s when measurements showed that the earth was ellipsoidal
in shape.
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Geometric Earth Models
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Early ideas of the figure of the earth resulted in descriptions of the
earth as an oyster (The Babylonians before 3000 B.C.), a rectangular box,
a circular disk, a cylindrical column, a spherical ball, and a very round
pear (Columbus in the last years of his life).
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Flat earth models are still used for plane surveying, over distances short
enough so that earth curvature is insignificant (less than 10 kms).
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Spherical earth models represent the shape of the earth with a sphere of
a specified radius. Spherical earth models are often used for short range
navigation (VOR-DME) and for global distance approximations. Spherical
models fail to model the actual shape of the earth. The slight flattening
of the earth at the poles results in about a twenty kilometer difference
at the poles between an average spherical radius and the measured polar
radius of the earth.
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Ellipsoidal earth models are required for accurate range and bearing calculations
over long distances. Loran-C, and GPS navigation receivers use ellipsoidal
earth models to compute position and waypoint information. Ellipsoidal
models define an ellipsoid with an equatorial radius and a polar radius.
The best of these models can represent the shape of the earth over the
smoothed, averaged sea-surface to within about one-hundred meters.
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Reference Ellipsoids
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Reference ellipsoids are usually defined by semi-major (equatorial radius)
and flattening (the relationship between equatorial and polar radii).
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Other reference ellipsoid parameters such as semi-minor axis (polar radius)
and eccentricity can computed from these terms.
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Many reference ellipsoids are in use by different nations and agencies.
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Earth Surfaces
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The earth has a highly irregular and constantly changing surface. Models
of the surface of the earth are used in navigation, surveying, and mapping.
Topographic and sea-level models attempt to model the physical variations
of the surface, while gravity models and geoids are used to represent local
variations in gravity that change the local definition of a level surface.
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Earth
Surfaces
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The topographical surface of the earth is the actual surface of the land
and sea at some moment in time. Aircraft navigators have a special interest
in maintaining a positive height vector above this surface.
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Sea level is the average (methods and temporal spans vary) surface of the
oceans. Tidal forces and gravity differences from location to location
cause even this smoothed surface to vary over the globe by hundreds of
meters.
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Gravity models attempt to describe in detail the variations in the gravity
field. The importance of this effort is related to the idea of leveling.
Plane and geodetic surveying uses the idea of a plane perpendicular to
the gravity surface of the earth, the direction perpendicular to a plumb
bob pointing toward the center of mass of the earth. Local variations in
gravity, caused by variations in the earth's core and surface materials,
cause this gravity surface to be irregular.
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Geoid models attempt to represent the surface of the entire earth over
both land and ocean as though the surface resulted from gravity alone.
Bomford described this surface as the surface that would exist if the sea
was admitted under the land portion of the earth by small frictionless
channels.
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The WGS-84 Geoid defines geoid heights for the entire earth.
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The U. S. National Imagery and Mapping Agency (formerly the Defense Mapping
Agency) publishes a ten by ten degree grid of geoid heights for the WGS-84
geoid.
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Global Coordinate Systems
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Coordinate systems to specify locations on the surface of the earth have
been used for centuries. In western geodesy the equator, the tropics of
Cancer and Capricorn, and then lines of latitude and longitude were used
to locate positions on the earth. Eastern cartographers like Phei Hsiu
used other rectangular grid systems as early as 270 A. D.
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Various units of length and angular distance have been used over history.
The meter is related to both linear and angular distance, having been defined
in the late 18th century as one ten-millionth of the distance from the
pole to the equator.
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Latitude, Longitude, and Height
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The most commonly used coordinate system today is the latitude, longitude,
and height system.
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The Prime Meridian and the Equator are the reference planes used to define
latitude and longitude.
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The geodetic latitude (there are many other defined latitudes) of a point
is the angle from the equatorial plane to the vertical direction of a line
normal to the reference ellipsoid.
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The geodetic longitude of a point is the angle between a reference plane
and a plane passing through the point, both planes being perpendicular
to the equatorial plane.
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The geodetic height at a point is the distance from the reference ellipsoid
to the point in a direction normal to the ellipsoid.
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Earth Centered, Earth Fixed X, Y,
and Z
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Earth Centered, Earth Fixed Cartesian coordinates are also used to define
three dimensional positions.
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Earth centered, earth-fixed, X, Y, and Z, Cartesian coordinates (XYZ) define
three dimensional positions with respect to the center of mass of the reference
ellipsoid.
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The Z-axis points toward the North Pole.
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The X-axis is defined by the intersection of the plane define by the prime
meridian and the equatorial plane.
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The Y-axis completes a right handed orthogonal system by a plane 90°
east of the X-axis and its intersection with the equator.
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ECEF
X, Y, and Z
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Geodetic Datums
Datum Types
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Datum types include horizontal, vertical and complete datums.
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Datums in Use
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Hundreds of geodetic datums are in use around the world.
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The Global Positioning system is based on the World Geodetic System 1984
(WGS-84).
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Parameters for simple XYZ conversion between many datums and WGS-84 are
published by the Defense mapping Agency.
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Datum Shifts
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Coordinate values resulting from interpreting latitude, longitude, and
height values based on one datum as though they were based in another datum
can cause position errors in three dimensions of up to one kilometer.
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Datum Conversions
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Datum conversions are accomplished by various methods.
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Complete datum conversion is based on seven parameter transformations that
include three translation parameters, three rotation parameters and a scale
parameter.
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Simple three parameter conversion between latitude, longitude, and height
in different datums can be accomplished by conversion through Earth-Centered,
Earth Fixed XYZ Cartesian coordinates in one reference datum and three
origin offsets that approximate differences in rotation, translation and
scale.
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The Standard Molodensky formulas can be used to convert latitude, longitude,
and ellipsoid height in one datum to another datum if the Delta XYZ constants
for that conversion are available and ECEF XYZ coordinates are not required.
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References
Bomford, G. 1980. Geodesy. Oxford: Clarendon
Press.
Burkard, Richard K. 1983. Geodesy for the Layman.
Washington, DC: NOAA.
National Imagery and Mapping Agency. 1997.
Department of Defense World Geodetic System 1984: Its Definition and Relationships
with Local Geodetic Systems. NIMA TR8350.2 Third Edition 4 July 1997. Bethesda,
MD: National Imagery and Mapping Agency.
National Oceanic and Atmospheric Administration.
1986. Geodetic Glossary. Rockville, MD: National Geodetic Information Center.
Schwarz, Charles R. 1989. North American Datum
of 1893. Rockville, MD: National Geodetic Survey.
Torge, Wolfgang. 1991 Geodesy, 2nd Edition,
New York: deGruyter.
Earth
Surfaces
WGS-84
Geoid Heights
WGS-84 Geoid Height Model (240k .gif)
Shaded
Relief of NIMA 0.25° WGS-84 Geoid Height Model (140k .jpg)
ECEF
X, Y, and Z
Conversion
from ECEF XYZ to Latitude, Longitude, and Height
Conversion
from Latitude, Longitude, and Height to ECEF XYZ.
XYZ
Three Parameter Datum Conversion
Reference
Ellipsoid Parameters
Selected
Reference Ellipsoids
National
Geodetic Survey Geoid-96 (135 kbytes)
Equator
and Prime Meridian
Geodetic
Latitude, Longitude, and Height
Horizontal
Position Shifts from Datum Differences
Standard
Molodensky Datum Conversion