Geodesy
Fundamentals Geodesy Geodynamics Geomatics History
Concepts Geographical distance Geoid Figure of the Earth (radius and circumference) Geodetic coordinates Geodetic datum Geodesic Horizontal position representation Latitude / Longitude Map projection Reference ellipsoid Satellite geodesy Spatial reference system Spatial relations Vertical positions
Technologies Global Nav. Sat. Systems (GNSSs) Global Pos. System (GPS) GLONASS (Russia) BeiDou (BDS) (China) Galileo (Europe) NAVIC (India) Quasi-Zenith Sat. Sys. (QZSS) (Japan) Discrete Global Grid and Geocoding
Standards (history) NGVD 29 Sea Level Datum 1929 OSGB36 Ordnance Survey Great Britain 1936 SK-42 Systema Koordinat 1942 goda ED50 European Datum 1950 SAD69 South American Datum 1969 GRS 80 Geodetic Reference System 1980 ISO 6709 Geographic point coord. 1983 NAD 83 North American Datum 1983 WGS 84 World Geodetic System 1984 NAVD 88 N. American Vertical Datum 1988 ETRS89 European Terrestrial Ref. Sys. 1989 GCJ-02 Chinese obfuscated datum 2002 Geo URI Internet link to a point 2010 International Terrestrial Reference System Spatial Reference System Identifier (SRID) Universal Transverse Mercator (UTM)

The Geodetic Reference System 1980 (GRS80) is a geodetic reference system consisting of a global reference ellipsoid and a normal gravity model.^[1][2][3]

Background

Geodesy is the scientific discipline that deals with the measurement and representation of the earth, its gravitational field and geodynamic phenomena (polar motion, earth tides, and crustal motion) in three-dimensional, time-varying space.

The geoid is essentially the figure of the Earth abstracted from its topographic features. It is an idealized equilibrium surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. and continued under the continental masses. The geoid, unlike the ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between it and the reference ellipsoid is called the geoidal undulation, or more usually the geoid-ellipsoid separation, N. It varies globally between ±110 m.

A reference ellipsoid, customarily chosen to be the same size (volume) as the geoid, is described by its semi-major axis (equatorial radius) a and flattening f. The quantity f = (a−b)/a, where b is the semi-minor axis (polar radius), is a purely geometrical one. The mechanical ellipticity of the earth (dynamical flattening, symbol J₂) is determined to high precision by observation of satellite orbit perturbations. Its relationship with the geometric flattening is indirect. The relationship depends on the internal density distribution.

The 1980 Geodetic Reference System (GRS 80) posited a 6378137 m semi-major axis and a 1⁄298.257222101 flattening. This system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics (IUGG) in Canberra, Australia, 1979.

The GRS 80 reference system was originally used by the World Geodetic System 1984 (WGS 84). The reference ellipsoid of WGS 84 now differs slightly due to later refinements.

The numerous other systems which have been used by diverse countries for their maps and charts are gradually dropping out of use as more and more countries move to global, geocentric reference systems using the GRS80 reference ellipsoid.

Definition

The reference ellipsoid is usually defined by its semi-major axis (equatorial radius) ${\displaystyle a}$ and either its semi-minor axis (polar radius) ${\displaystyle b}$, aspect ratio ${\displaystyle (b/a)}$ or flattening ${\displaystyle f}$, but GRS80 is an exception: four independent constants are required for a complete definition. GRS80 chooses as these ${\displaystyle a}$, ${\displaystyle GM}$, ${\displaystyle J_{2}}$ and ${\displaystyle \omega }$, making the geometrical constant ${\displaystyle f}$ a derived quantity.

Defining geometrical constants

Semi-major axis = Equatorial Radius = ${\displaystyle a=6\,378\,137\,\mathrm {m} }$;

Defining physical constants

Geocentric gravitational constant determined from the gravitational constant and the earth mass with atmosphere ${\displaystyle GM=3986005\times 10^{8}\,\mathrm {m^{3}/s^{2}} }$;

Dynamical form factor ${\displaystyle J_{2}=108\,263\times 10^{-8}}$;

Angular velocity of rotation ${\displaystyle \omega =7\,292\,115\times 10^{-11}\,\mathrm {s^{-1}} }$;

Derived quantities

Derived geometrical constants (all rounded)

Flattening = ${\displaystyle f}$ = 0.003 352 810 681 183 637 418;

Reciprocal of flattening = ${\displaystyle 1/f}$ = 298.257 222 100 882 711 243;

Semi-minor axis = Polar Radius = ${\displaystyle b}$ = 6 356 752.314 140 347 m;

Aspect ratio = ${\displaystyle b/a}$ = 0.996 647 189 318 816 363;

Mean radius as defined by the International Union of Geodesy and Geophysics (IUGG): ${\displaystyle R_{1}=(2a+b)/3}$ = 6 371 008.7714 m;

Authalic mean radius = ${\displaystyle R_{2}}$ = 6 371 007.1809 m;

Radius of a sphere of the same volume = ${\displaystyle R_{3}=(a^{2}b)^{1/3}}$ = 6 371 000.7900 m;

Linear eccentricity = ${\displaystyle c={\sqrt {a^{2}-b^{2}}}}$ = 521 854.0097 m;

Eccentricity of elliptical section through poles = ${\displaystyle e={\frac {\sqrt {a^{2}-b^{2}}}{a}}}$ = 0.081 819 191 0428;

Polar radius of curvature = ${\displaystyle a^{2}/b}$ = 6 399 593.6259 m;

Equatorial radius of curvature for a meridian = ${\displaystyle b^{2}/a}$ = 6 335 439.3271 m;

Meridian quadrant = 10 001 965.7292 m;

Derived physical constants (rounded)

Period of rotation (sidereal day) = ${\displaystyle 2\pi /\omega }$ = 86 164.100 637 s

The formula giving the eccentricity of the GRS80 spheroid is:^[1]

${\displaystyle e^{2}={\frac {a^{2}-b^{2}}{a^{2}}}=3J_{2}+{\frac {4}{15}}{\frac {\omega ^{2}a^{3}}{GM}}{\frac {e^{3}}{2q_{0}}},}$

where

${\displaystyle 2q_{0}=\left(1+{\frac {3}{e'^{2}}}\right)\arctan e'-{\frac {3}{e'}}}$

and ${\displaystyle e'={\frac {e}{\sqrt {1-e^{2}}}}}$ (so ${\displaystyle \arctan e'=\arcsin e}$). The equation is solved iteratively to give

${\displaystyle e^{2}=0.00669\,43800\,22903\,41574\,95749\,48586\,28930\,62124\,43890\,\ldots }$

which gives

${\displaystyle f=1/298.25722\,21008\,82711\,24316\,28366\,\ldots .}$

References

External links

• GRS 80 Specification