Geodetic Reference System 1980

<h2 id="background">Background</h2>
<a href="/facts/Geodesy/EJH0Te0m">Geodesy</a> is the scientific discipline that deals with the measurement and representation of the <a href="/facts/Earth/HLLmDfj0">earth</a>, its <a href="/facts/Gravitation/bqAN7DdQ">gravitational</a> field and geodynamic phenomena (<a href="/facts/Polar_motion/79TCTF3k">polar motion</a>, earth <a href="/facts/Tide/zNstvWPv">tides</a>, and crustal motion) in three-dimensional, time-varying space.
The <a href="/facts/Geoid/6aR04HQl">geoid</a> 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 <a href="/facts/Reference_ellipsoid/uD1C2sTU">reference ellipsoid</a>, 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 (<a href="/facts/IUGG/jy6qBv8o">IUGG</a>) in Canberra, Australia, 1979.
The GRS 80 reference system was originally used by the <a href="/facts/WGS_84/V9wSJnjX">World Geodetic System 1984</a> (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.

<h2 id="definition">Definition</h2>
The reference ellipsoid is usually defined by its <a href="/facts/Semi-major_axis/l7JES2wo">semi-major axis</a> (equatorial
radius) 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and either its <a href="/facts/Semi-minor_axis/l7JES2wo">semi-minor axis</a> (polar radius) 
 
 
 
 b
 
 
 {\displaystyle b}
 
, <a href="/facts/Aspect_ratio/1LVBgtfT">aspect ratio</a> 
 
 
 
 (
 b
 
 /
 
 a
 )
 
 
 {\displaystyle (b/a)}
 
 or <a href="/facts/Flattening/32pP4JsR">flattening</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
, but GRS80 is an exception: four independent constants are required for a complete definition. GRS80 chooses as these 
 
 
 
 a
 
 
 {\displaystyle a}
 
, 
 
 
 
 G
 M
 
 
 {\displaystyle GM}
 
, 
 
 
 
 
 J
 
 2
 
 
 
 
 {\displaystyle J_{2}}
 
 and 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
, making the geometrical constant 
 
 
 
 f
 
 
 {\displaystyle f}
 
 a derived quantity.

Defining <a href="/facts/Geometrical/wTQf5ffQ">geometrical</a> <a href="/facts/Constant_(mathematics)/SxlxTWFT">constants</a>
Semi-major axis = Equatorial Radius = 
 
 
 
 a
 =
 6
 
 378
 
 137
 
 
 m
 
 
 
 {\displaystyle a=6\,378\,137\,\mathrm {m} }
 
;
Defining <a href="/facts/Nature/CRC3KQUg">physical</a> <a href="/facts/Constant_(mathematics)/SxlxTWFT">constants</a>
<a href="/facts/Geocentric_gravitational_constant/pzGlwadL">Geocentric gravitational constant</a> determined from the <a href="/facts/Gravitational_constant/GdQ8tcBD">gravitational constant</a> and the <a href="/facts/Earth_mass/G74DaKFv">earth mass</a> with atmosphere 
 
 
 
 G
 M
 =
 3986005
 ×
 
 10
 
 8
 
 
 
 
 
 m
 
 3
 
 
 
 /
 
 
 s
 
 2
 
 
 
 
 
 {\displaystyle GM=3986005\times 10^{8}\,\mathrm {m^{3}/s^{2}} }
 
;
Dynamical form factor 
 
 
 
 
 J
 
 2
 
 
 =
 108
 
 263
 ×
 
 10
 
 −
 8
 
 
 
 
 {\displaystyle J_{2}=108\,263\times 10^{-8}}
 
;
Angular velocity of rotation 
 
 
 
 ω
 =
 7
 
 292
 
 115
 ×
 
 10
 
 −
 11
 
 
 
 
 
 s
 
 −
 1
 
 
 
 
 
 {\displaystyle \omega =7\,292\,115\times 10^{-11}\,\mathrm {s^{-1}} }
 
;
<h2 id="derived-quantities">Derived quantities</h2>
Derived geometrical constants (all rounded)
Flattening = 
 
 
 
 f
 
 
 {\displaystyle f}
 
 = 0.003 352 810 681 183 637 418;
Reciprocal of flattening = 
 
 
 
 1
 
 /
 
 f
 
 
 {\displaystyle 1/f}
 
 = 298.257 222 100 882 711 243;
Semi-minor axis = Polar Radius = 
 
 
 
 b
 
 
 {\displaystyle b}
 
 = 6 356 752.314 140 347 m;
Aspect ratio = 
 
 
 
 b
 
 /
 
 a
 
 
 {\displaystyle b/a}
 
 = 0.996 647 189 318 816 363;
<a href="/facts/Earth_radius/PVsf0g2I">Mean radius</a> as defined by the <a href="/facts/International_Union_of_Geodesy_and_Geophysics/jy6qBv8o">International Union of Geodesy and Geophysics</a> (IUGG): 
 
 
 
 
 R
 
 1
 
 
 =
 (
 2
 a
 +
 b
 )
 
 /
 
 3
 
 
 {\displaystyle R_{1}=(2a+b)/3}
 
 = 6 371 008.7714 m;
<a href="/facts/Earth_radius/PVsf0g2I">Authalic mean radius</a> = 
 
 
 
 
 R
 
 2
 
 
 
 
 {\displaystyle R_{2}}
 
 = 6 371 007.1809 m;
Radius of a sphere of the same volume = 
 
 
 
 
 R
 
 3
 
 
 =
 (
 
 a
 
 2
 
 
 b
 
 )
 
 1
 
 /
 
 3
 
 
 
 
 {\displaystyle R_{3}=(a^{2}b)^{1/3}}
 
 = 6 371 000.7900 m;
Linear eccentricity = 
 
 
 
 c
 =
 
 
 
 a
 
 2
 
 
 −
 
 b
 
 2
 
 
 
 
 
 
 {\displaystyle c={\sqrt {a^{2}-b^{2}}}}
 
 = 521 854.0097 m;
<a href="/facts/Eccentricity_(mathematics)/SzgpW5R7">Eccentricity</a> of elliptical section through poles = 
 
 
 
 e
 =
 
 
 
 
 a
 
 2
 
 
 −
 
 b
 
 2
 
 
 
 a
 
 
 
 
 {\displaystyle e={\frac {\sqrt {a^{2}-b^{2}}}{a}}}
 
 = 0.081 819 191 0428;
Polar radius of curvature = 
 
 
 
 
 a
 
 2
 
 
 
 /
 
 b
 
 
 {\displaystyle a^{2}/b}
 
 = 6 399 593.6259 m;
Equatorial radius of curvature for a meridian = 
 
 
 
 
 b
 
 2
 
 
 
 /
 
 a
 
 
 {\displaystyle b^{2}/a}
 
 = 6 335 439.3271 m;
Meridian quadrant = 10 001 965.7292 m;
Derived physical constants (rounded)
Period of rotation (<a href="/facts/Sidereal_day/2JAVf8dy">sidereal day</a>) = 
 
 
 
 2
 π
 
 /
 
 ω
 
 
 {\displaystyle 2\pi /\omega }
 
 = 86 164.100 637 s
The formula giving the eccentricity of the GRS80 spheroid is:<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

e
          
            2
          
        
        =
        
          
            
              
                a
                
                  2
                
              
              −
              
                b
                
                  2
                
              
            
            
              a
              
                2
              
            
          
        
        =
        3
        
          J
          
            2
          
        
        +
        
          
            4
            15
          
        
        
          
            
              
                ω
                
                  2
                
              
              
                a
                
                  3
                
              
            
            
              G
              M
            
          
        
        
          
            
              e
              
                3
              
            
            
              2
              
                q
                
                  0
                
              
            
          
        
        ,
      
    
    {\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

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

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

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

which gives

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

<h2 id="external-links">External links</h2>
<ul><li><a href="https://web.archive.org/web/20051225213554/http://www.gfy.ku.dk/~iag/HB2000/part4/grs80_corr.htm">GRS 80 Specification</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Moritz, Helmut (September 1980). "Geodetic Reference System 1980" (PDF). Bulletin Géodésique. 54 (3): 395–405. Bibcode:1980BGeod..54..395M. doi:10.1007/BF02521480. S2CID 198209711. <a href="/wiki/Helmut_Moritz" target="_blank">/wiki/Helmut_Moritz</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Moritz, H. (1992). "Geodetic Reference System 1980". Bulletin Géodésique. 66 (2). Springer Science and Business Media LLC: 187–192. Bibcode:1992BGeod..66..187M. doi:10.1007/bf00989270. ISSN 0007-4632. S2CID 122916060. <a href="/wiki/Helmut_Moritz" target="_blank">/wiki/Helmut_Moritz</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Moritz, Helmut (March 2000). "Geodetic Reference System 1980" (PDF). Journal of Geodesy. 74 (1): 128–162. doi:10.1007/S001900050278. S2CID 195290884. Archived from the original (PDF) on 2016-02-20. Retrieved 2018-12-15. <a href="/wiki/Helmut_Moritz" target="_blank">/wiki/Helmut_Moritz</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Moritz, Helmut (September 1980). "Geodetic Reference System 1980" (PDF). Bulletin Géodésique. 54 (3): 395–405. Bibcode:1980BGeod..54..395M. doi:10.1007/BF02521480. S2CID 198209711. <a href="/wiki/Helmut_Moritz" target="_blank">/wiki/Helmut_Moritz</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Geodetic Reference System 1980 open-in-new

Geodetic Reference System 1980