Earth ellipsoid
Geodesy  

Fundamentals
 
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Technologies
 
Standards (history)
 

An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations.
It is a spheroid (an ellipsoid of revolution) whose minor axis (shorter diameter), which connects the geographical North Pole and South Pole, is approximately aligned with the Earth's axis of rotation. The ellipsoid is defined by the equatorial axis (a) and the polar axis (b); their radial difference is slightly more than 21 km, or 0.335% of a (which is not quite 6,400 km).
Many methods exist for determination of the axes of an Earth ellipsoid, ranging from meridian arcs up to modern satellite geodesy or the analysis and interconnection of continental geodetic networks. Amongst the different set of data used in national surveys are several of special importance: the Bessel ellipsoid of 1841, the international Hayford ellipsoid of 1924, and (for GPS positioning) the WGS84 ellipsoid.
Types
There are two types of ellipsoid: mean and reference.
A data set which describes the global average of the Earth's surface curvature is called the mean Earth Ellipsoid. It refers to a theoretical coherence between the geographic latitude and the meridional curvature of the geoid. The latter is close to the mean sea level, and therefore an ideal Earth ellipsoid has the same volume as the geoid.
While the mean Earth ellipsoid is the ideal basis of global geodesy, for regional networks a socalled reference ellipsoid may be the better choice.^{[1]} When geodetic measurements have to be computed on a mathematical reference surface, this surface should have a similar curvature as the regional geoid; otherwise, reduction of the measurements will get small distortions.
This is the reason for the "long life" of former reference ellipsoids like the Hayford or the Bessel ellipsoid, despite the fact that their main axes deviate by several hundred meters from the modern values. Another reason is a judicial one: the coordinates of millions of boundary stones should remain fixed for a long period. If their reference surface changes, the coordinates themselves also change.
However, for international networks, GPS positioning, or astronautics, these regional reasons are less relevant. As knowledge of the Earth's figure is increasingly accurate, the International Geoscientific Union IUGG usually adapts the axes of the Earth ellipsoid to the best available data.
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 reference ellipsoid. For example, if the measurements were hypothetically performed exactly over the equator plane and either geographical pole, the radii of curvature so obtained would be related to the equatorial radius and the polar radius, respectively a and b (see: Earth polar and equatorial radius of curvature). Then, the flattening would readily follow from its definition:
 $f=(ab)/a$.
For two arc measurements each at arbitrary average latitudes $\varphi _{i}$, $i=1,\,2$, the solution starts from an initial approximation for the equatorial radius $a_{0}$ and for the flattening $f_{0}$. The theoretical Earth's meridional radius of curvature $M_{0}(\varphi _{i})$ can be calculated at the latitude of each arc measurement as:
 $M_{0}(\varphi _{i})={\frac {a}{\sqrt {1e_{0}^{2}\sin ^{2}\varphi _{i}}}}$
where $e_{0}^{2}=2f_{0}f_{0}^{2}$. Then discrepancies between empirical and theoretical values of the radius of curvature can be formed as $\delta M_{i}=M_{i}M_{0}(\varphi _{i})$. Finally, corrections for the initial equatorial radius $\delta a$ and the flattening $\delta f$ can be solved by means of a system of linear equations formulated via linearization of $M$:^{[2]}
 $\delta M_{i}\approx \delta a(\partial M/\partial a)+\delta f(\partial M/\partial f)$
where the partial derivatives are:^{[2]}
 $\partial M/\partial a\approx 1$
 $\partial M/\partial f\approx 2a_{0}(11.5\sin ^{2}\varphi _{i})$
Longer arcs with multiple intermediatelatitude 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 least squares adjustment. The parameters determined are usually the semimajor axis, $a$, and any of the semiminor axis, $b$, flattening, or eccentricity.
Regionalscale systematic effects observed in the radius of curvature measurements reflect the geoid undulation and the deflection of the vertical, as explored in astrogeodetic leveling.
Gravimetry is another technique for determining Earth's flattening, as per Clairaut's theorem.
Modern geodesy no longer uses simple meridian arcs or ground triangulation networks, but the methods of satellite geodesy, especially satellite gravimetry.
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 GRS67 (Geodetic Reference System 1967) in the listing was recommended for adoption. The new ellipsoid was not recommended to replace the International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy is required. It became a part of the GRS67 which was approved and adopted at the 1971 meeting of the IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in the South American Datum 1969.
The GRS80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is based on the equatorial radius (semimajor axis of Earth ellipsoid) $a$, total mass $GM$, dynamic form factor $J_{2}$ and angular velocity of rotation $\omega$, making the inverse flattening $1/f$ a derived quantity. The minute difference in $1/f$ seen between GRS80 and WGS84 results from an unintentional truncation in the latter's defining constants: while the WGS84 was designed to adhere closely to the GRS80, incidentally the WGS84 derived flattening turned out to be slightly different than the GRS80 flattening because the normalized second degree zonal harmonic gravitational coefficient, that was derived from the GRS80 value for $J_{2}$, was truncated to eight significant digits in the normalization process.^{[3]}
An ellipsoidal model describes only the ellipsoid's geometry and a normal gravity field formula to go with it. Commonly an ellipsoidal model is part of a more encompassing geodetic datum. For example, the older ED50 (European Datum 1950) is based on the Hayford or International Ellipsoid. WGS84 is peculiar in that the same name is used for both the complete geodetic reference system and its component ellipsoidal model. Nevertheless, the two concepts—ellipsoidal model and geodetic reference system—remain distinct.
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  
GRS67 (1967)  6,378,160  6,356,774.516  298.247167427  
South American (1969)  6,378,160  6,356,774.719  298.25  South America 
WGS72 (1972)  6,378,135  6,356,750.52  298.26  USA/DoD 
GRS80 (1979)  6,378,137  6,356,752.3141  298.257222101  Global ITRS^{[4]} 
WGS84 (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)^{[5]}  6,378,136.6  6,356,751.9  298.25642  ^{[4]} 
See also
References
 ^ Alexander, J. C. (1985). "The Numerics of Computing Geodetic Ellipsoids". SIAM Review. 27 (2): 241–247. Bibcode:1985SIAMR..27..241A. doi:10.1137/1027056.
 ^ ^{a} ^{b} Bomford, G. (1952). Geodesy. OCLC 489193198.
 ^ 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) (Chp. 1, page 12)