GRS 80

Geodesy
Fundamentals
Geodesy · Geodynamics Geomatics · Cartography
Concepts
Datum · Distance · Geoid Fig. Earth · Geodetic sys. Geog. coord. system Hor. pos. represent. Lat./Long. · Map proj. Ref. ellipsoid · Sat. geodesy Spatial ref. sys.
Technologies
GNSS · GPS · GLONASS · IRNSS
Standards
ED50 · ETRS89 · GRS 80 NAD83 · NAVD88 · SAD69 SRID · UTM · WGS84
History
History of geodesy NAVD29

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 (GRS 80) posited a 6 378 137m semi-major axis and a 1/298.257 222 101 flattening. This system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics (IUGG).

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 its later refinements (see WGS84).

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: 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.

Defining geometrical constants 

Semi-major axis = Equatorial Radius $=a =$ 6 378 137 m;

Defining physical constants 

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

Dynamical form factor $J_2$ = 108 263· 10⁻⁸;

Angular velocity of rotation $\omega$ = 7 292 115·10⁻¹¹ s⁻¹;

Derived geometrical constants 

Flattening $=f=$ 0.003 352 810 681 225;

Reciprocal of flattening $=1/f=$298.257 222 101;

Semi-minor axis = Polar Radius $=b=$ 6 356 752.314 14 m;

Aspect ratio $=b/a=$ 0.996 647 189 318 816;

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$=((a^2-b^2)^{.5})/a=$ 0.081 819 191 0435;

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;

The formula is iterative; plugging in the defined constants gives

$e^2 = \frac {(a^2 - b^2)}{a^2} = 0.003 247 89 + Z \left ( \frac {e^3}{(1 + \frac {3}{{e'}^2})(\arctan e') - \frac {3}{e'}} \right )$

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

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