Earth ellipsoid
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Geodesy |
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An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the
It is a
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from
Types
There are two types of ellipsoid: mean and reference.
A data set which describes the global
While the mean Earth ellipsoid is the ideal basis of global geodesy, for
This is the reason for the "long life" of former reference ellipsoids like the
However, for international networks,
Reference ellipsoid
Geodesy |
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In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the geoid, which is the truer, imperfect figure of the Earth, or other planetary body, as opposed to a perfect, smooth, and unaltered sphere, which factors in the undulations of the bodies' gravity due to variations in the composition and density of the interior, as well as the subsequent flattening caused by the centrifugal force from the rotation of these massive objects (for planetary bodies that do rotate). Because of their relative simplicity, reference ellipsoids are used as a preferred surface on which
In the context of standardization and geographic applications, a geodesic reference ellipsoid is the mathematical model used as foundation by spatial reference system or geodetic datum definitions.
Ellipsoid parameters
In 1687
In geophysics,
The shape of an ellipsoid of revolution is determined by the shape parameters of that
In geodesy publications, however, it is common to specify the semi-major axis (equatorial radius) a and the flattening f, defined as:
That is, f is the amount of flattening at each pole, relative to the radius at the equator. This is often expressed as a fraction 1/m; m = 1/f then being the "inverse flattening". A great many other ellipse parameters are used in geodesy but they can all be related to one or two of the set a, b and f.
A great many ellipsoids have been used to model the Earth in the past, with different assumed values of a and b as well as different assumed positions of the center and different axis orientations relative to the solid Earth. Starting in the late twentieth century, improved measurements of satellite orbits and star positions have provided extremely accurate determinations of the Earth's center of mass and of its axis of revolution; and those parameters have been adopted also for all modern reference ellipsoids.
The ellipsoid
Determination
Arc measurement is the historical method of determining the ellipsoid. Two meridian arc measurements will allow the derivation of two parameters required to specify a
- .
For two arc measurements each at arbitrary average latitudes , , the solution starts from an initial approximation for the equatorial radius and for the flattening . The theoretical
where .[6] Then discrepancies between empirical and theoretical values of the radius of curvature can be formed as . Finally, corrections for the initial equatorial radius and the flattening can be solved by means of a system of linear equations formulated via linearization of :[7]
where the partial derivatives are:[7]
Longer arcs with multiple intermediate-latitude determinations can completely determine the ellipsoid that best fits the surveyed region. In practice, multiple arc measurements are used to determine the ellipsoid parameters by the method of
Regional-scale
Gravimetry is another technique for determining Earth's flattening, as per Clairaut's theorem.
Modern
Geodetic coordinates
Historical Earth ellipsoids
The reference ellipsoid models listed below have had utility in geodetic work and many are still in use. The older ellipsoids are named for the individual who derived them and the year of development is given. In 1887 the English surveyor Colonel Alexander Ross Clarke CB FRS RE was awarded the Gold Medal of the Royal Society for his work in determining the figure of the Earth. The international ellipsoid was developed by John Fillmore Hayford in 1910 and adopted by the International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use.
At the 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 (
The GRS-80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is based on the equatorial radius (semi-major axis of Earth ellipsoid) , total mass , dynamic form factor and angular velocity of rotation , making the inverse flattening a derived quantity. The minute difference in seen between GRS-80 and WGS-84 results from an unintentional truncation in the latter's defining constants: while the WGS-84 was designed to adhere closely to the GRS-80, incidentally the WGS-84 derived flattening turned out to differ slightly from the GRS-80 flattening because the normalized second degree zonal harmonic gravitational coefficient, that was derived from the GRS-80 value for , was truncated to eight significant digits in the normalization process.[10]
An ellipsoidal model describes only the ellipsoid's geometry and a
Note that the same ellipsoid may be known by different names. It is best to mention the defining constants for unambiguous identification.
Reference ellipsoid name | Equatorial radius (m) | Polar radius (m) | Inverse flattening | Where used |
---|---|---|---|---|
Maupertuis (1738) | 6,397,300 | 6,363,806.283 | 191 | France |
Plessis (1817) | 6,376,523.0 | 6,355,862.9333 | 308.64 | France |
Everest (1830) | 6,377,299.365 | 6,356,098.359 | 300.80172554 | India |
Everest 1830 Modified (1967) | 6,377,304.063 | 6,356,103.0390 | 300.8017 | West Malaysia & Singapore |
Everest 1830 (1967 Definition) | 6,377,298.556 | 6,356,097.550 | 300.8017 | Brunei & East Malaysia |
Airy (1830) | 6,377,563.396 | 6,356,256.909 | 299.3249646 | Britain |
Bessel (1841) | 6,377,397.155 | 6,356,078.963 | 299.1528128 | Europe, Japan |
Clarke (1866) | 6,378,206.4 | 6,356,583.8 | 294.9786982 | North America |
Clarke (1878) | 6,378,190 | 6,356,456 | 293.4659980 | North America |
Clarke (1880) | 6,378,249.145 | 6,356,514.870 | 293.465 | France, Africa |
Helmert (1906) | 6,378,200 | 6,356,818.17 | 298.3 | Egypt |
Hayford (1910) | 6,378,388 | 6,356,911.946 | 297 | USA |
International (1924) | 6,378,388 | 6,356,911.946 | 297 | Europe |
Krassovsky (1940) | 6,378,245 | 6,356,863.019 | 298.3 | USSR, Russia, Romania |
WGS66 (1966) | 6,378,145 | 6,356,759.769 | 298.25 | USA/DoD |
Australian National (1966) | 6,378,160 | 6,356,774.719 | 298.25 | Australia |
New International (1967) | 6,378,157.5 | 6,356,772.2 | 298.24961539 | |
GRS-67 (1967) | 6,378,160 | 6,356,774.516 | 298.247167427 | |
South American (1969) | 6,378,160 | 6,356,774.719 | 298.25 | South America |
WGS-72 (1972) | 6,378,135 | 6,356,750.52 | 298.26 | USA/DoD |
GRS-80 (1979) |
6,378,137 | 6,356,752.3141 | 298.257222101 | Global ITRS[11]
|
WGS-84 (1984) | 6,378,137 | 6,356,752.3142 | 298.257223563 | Global GPS |
IERS (1989) |
6,378,136 | 6,356,751.302 | 298.257 | |
IERS (2003)[12] | 6,378,136.6 | 6,356,751.9 | 298.25642 | [11] |
See also
- Equatorial bulge
- Earth radius of curvature
- Geodetic datum
- Great ellipse
- Meridian arc
- Normal gravity
- Planetary coordinate system
- History of geodesy
- Planetary ellipsoid
References
- doi:10.1137/1027056.
- S2CID 126412032.
- ^ Choi, Charles Q. (12 April 2007). "Strange but True: Earth Is Not Round". Scientific American. Retrieved 4 May 2021.
- ISBN 3-11-017072-8
- ISBN 0-226-76747-7.
- ^ Snyder, John P. (1987). Map Projections — A Working Manual. USGS Professional Paper 1395. Washington, D.C.: Government Printing Office. p. 17.
- ^ a b Bomford, G. (1952). Geodesy.
- ^ National Geodetic Survey (U.S.).; National Geodetic Survey (U.S.) (1986). Geodetic Glossary. NOAA technical publications. U.S. Department of Commerce, National Oceanic and Atmospheric Administration, National Ocean Service, Charting and Geodetic Services. p. 107. Retrieved 2021-10-24.
- ISBN 978-3-642-12124-1. Retrieved 2021-10-24.
- ^ NIMA Technical Report TR8350.2, "Department of Defense World Geodetic System 1984, Its Definition and Relationships With Local Geodetic Systems", Third Edition, 4 July 1997 [1]
- ^ a b Note that the current best estimates, given by the IERS Conventions, "should not be mistaken for conventional values, such as those of the Geodetic Reference System GRS80 ... which are, for example, used to express geographic coordinates" (chap. 1); note further that "ITRF solutions are specified by Cartesian equatorial coordinates X, Y and Z. If needed, they can be transformed to geographical coordinates (λ, φ, h) referred to an ellipsoid. In this case the GRS80 ellipsoid is recommended." (chap. 4).
- ^ IERS Conventions (2003) Archived 2014-04-19 at the Wayback Machine (Chp. 1, page 12)
Bibliography
- P. K. Seidelmann (Chair), et al. (2005), “Report Of The IAU/IAG Working Group On Cartographic Coordinates And Rotational Elements: 2003,” Celestial Mechanics and Dynamical Astronomy, 91, pp. 203–215.
- Web address: https://astrogeology.usgs.gov/Projects/WGCCRE
- OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 1: Common architecture, Annex B.4. 2005-11-30
- Web address: http://www.opengeospatial.org