GRS 80

GRS 80, or Geodetic Reference System 1980, is a geodetic reference system consisting of a global reference ellipsoid and a gravity field model.

Geodesy

Geodesy, also called geodetics, 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. 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, or, in simplest terms, the degree of central concentration of mass.

The 1980 Geodetic Reference System (GRS80) posited a 6,378,137 m semi-major axis and a 1:298.257 flattening. This system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics (IUGG). It is essentially the basis for geodetic positioning by the Global Positioning System and is thus also in extremely widespread use outside the geodetic community.

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.

Defining features of GRS 80

The reference ellipsoid is usually defined by its semi-major axis (equatorial radius) \(a\) and either its semi-minor axis (polar radius) \(b\), aspect ratio \((b/a)\) or flattening \(f\), but GRS80 is an exception.

Defining geometrical constants 

Semi-major axis = Equatorial Radius \(=a =\)6,378,137.00000 m;

Derived geometrical constants 

Flattening \(=f=\) 0.003352810681225;

Reciprocal of flattening \(=1/f=\)298.257222101;

Semi-minor axis = Polar Radius \(=b=\) 6,356,752.31414 m;

Aspect ratio \(=b/a=\) 0.996647189318816;

Mean radius as defined by the International Union of Geodesy and Geophysics (IUGG): R₁= (2a+b)/3 = 6,371,008.7714 m;

Authalic mean radius= 6,371,007.1810 m;

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

Linear eccentricity \(=(a^2-b^2)^{.5}=\) 521,854.0097 m;

Eccentricity of elliptical section through poles\(=((a^2-b^2)^{.5})/a=\) 0.0818191910435;

Polar radius of curvature \(=a^2/b=\) 6,399,593.6259 m;

Equatorial radius of curvature for a meridian\(=b^2/a=\) 6,335,439.3271 m;

Meridian quadrant = 10,001,965.7293 m;

Defining physical constants 

Geocentric gravitational constant, including mass of the atmosphere \(GM=\) 3986005·10⁸ m³/s²;

Dynamical form factor \(J_2\) = 108263· 10^-8;

Angular velocity of rotation \(\omega\) = 7292115·10^-11 s^-1;

For a complete definition, four independent constants are required. GRS80 chooses as these \(a\), \(GM\), \(J_2\) and \(\omega\), making the geometrical constant \(f\) a derived quantity. The formula is iterative; plugging in the defined constants gives

\(e^2 = \frac {(a^2 - b^2)}{a^2} = 0.00324789 + Z \left ( \frac {e^3}{(1 + \frac {3}{{e'}^2})(\arctan e') - \frac {3}{e'}} \right )\)

where  \(Z = \frac {(6.378137)^3(2.835996862572)}{797201}\)  exactly.

Replace \((e')^2\) with \(e^2/(1 - e^2)\) and \( \arctan (e')\) with \( \arcsin (e)\) and iterate to get \(e^2\); a rounded value comes out

\(e^2 =\) 0.00669 43800 22903 41574 95749 48586 28930 62124 43890

from which a rounded value of \(f\) calculates to be the reciprocal of

298.25722 21008 82711 24316 28366

The GRS80 reference system is used by the Global Positioning System, in a realization called WGS 84 (World Geodetic System 1984). (The WGS84 ellipsoid is very slightly different, with 1/f = 298.257223563 exactly).

References

• Additional derived physical constants and geodetic formulas are found in the following reference: Geodetic Reference System 1980, Bulletin Géodésique, Vol 54:3, 1980.

External links

• GRS 80 Specification

ca:GRS80