"mean" radius of the Earth

I usually consider the Earth as a sphere when thinking of map projection.
That's natural for meteorologist. The question is what's the radius we
should use in this context.

In the framework of international operational meteorology, i.e. World
Weather Watch programme of WMO, I think the Manual on Codes is the only
document that mention about the radius of spherical Earth.

: 6367.47 km:
GRIB Edition 1, code table 7,
GRIB Edition 2, code table 3.2, code 0 (which says 6367 470.0 m)
: 6 371 229.0 m:
GRIB Edition 2, code table 3.2, code 6,
Attachment for icosahedron-based grid
: 6 371 200 m:
GRIB Edition 2, code table 3.2, code 8

But I don't know which to choose, since these numbers do not have
definition.

From the viewpoint of cartography, there are several kinds of "spherical
radius" of Earth. Wikipedia http://en.wikipedia.org/wiki/Earth_radius gives
a convenient summary. Now I'm able to compute those values for well-known
spheroids. Interpretations:

: 6367.47 km:
Rectifying radius for (probably) GRS-67 or International 1967 spheroid.
Later GRIB2 added trailing zeroes without referring back to the
definition.
: 6 371 229.0 m:
Mean radius of Hayford 1910 (also known as International 1924) spheroid,
rounded to nearest metre.
Source of the last digit (of 0.1 m) is unclear, but probrably just an
inappropriate handling.
: 6 371 200 m:
Mean, volumetric, or authalic radius for Hayford 1910 spheroid, rounde to
nearest 100 m.
No better interpretation was found.

Now that the CBS Resolution says WGS 84 is the standard frame, I think
neither of above is appropriate any longer. It would be much more
reasonable to use 6 371 007.1 (authalic) or 6 371 008.7 (mean) for WGS-84,
depending on application. It might be meaningful to use rectifying radius
(6367449.1) if we are really interested in the length of meridian, but it's
unlikely.


::: program :::

#!/usr/bin/ruby
params = (<<EOF).split(/\n/)
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
Hayford (1910) 6,378,388 6,356,911.946 297 USA
International (1924) 6,378,388 6,356,911.946 297 Europe
NAD 27 (1927) 6,378,206.4 6,356,583.800 294.978698208 North America
Krassovsky (1940) 6,378,245 6,356,863.019 298.3 USSR
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
GRS-67 (1967) 6,378,160 6,356,774.516 298.247167427
South American (1969) 6,378,160 6,356,774.719 298.25 South America
WGS-72 (1972) 6,378,135 6,356,750.52 298.26 USA/DoD
GRS-80 (1979) 6,378,137 6,356,752.3141 298.257222101 Global ITRS
WGS-84 (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) 6,378,136.6 6,356,751.9 298.25642
EOF

for line in params
name, as, bs, finv = line.split(/\t/)
h = Hash.new
h[:equatorial] = a = as.gsub(/,/, '_').to_f
h[:polar] = b = bs.gsub(/,/, '_').to_f
f = 1.0 / finv.to_f
h[:mean] = (a + a + b) / 3.0
h[:volumetric] = (a * a * b) ** (1.0 / 3.0)
e = Math::sqrt(f * (2.0 - f))
halfq = 0.5 * (1.0 + ((1.0 - e * e) * 0.5 / e) * Math::log((1.0 + e) /
(1.0 - e)))
h[:authalic] = a * Math::sqrt(halfq)
h[:rectifying] = ((a ** 1.5 + b ** 1.5) * 0.5) ** (1.0 / 1.5)
for k in [:equatorial, :polar, :mean, :volumetric, :authalic, :rectifying]
fmt = sprintf("%%-11.%uf %%32s %%s\n", (k == :rectifying) ? 1 : 3)
printf(fmt, h[k], name, k.to_s)
end
end

::: RESULT :::

6355862.933 Plessis (1817) polar
6356078.963 Bessel (1841) polar
6356097.550 Everest 1830 (1967 Definition) polar
6356098.359 Everest (1830) polar
6356103.039 Everest 1830 Modified (1967) polar
6356256.909 Airy (1830) polar
6356456.000 Clarke (1878) polar
6356514.870 Clarke (1880) polar
6356583.800 Clarke (1866) polar
6356583.800 NAD 27 (1927) polar
6356750.520 WGS-72 (1972) polar
6356751.302 IERS (1989) polar
6356751.900 IERS (2003) polar
6356752.314 GRS-80 (1979) polar
6356752.314 WGS-84 (1984) polar
6356759.769 WGS66 (1966) polar
6356772.200 New International (1967) polar
6356774.516 GRS-67 (1967) polar
6356774.719 South American (1969) polar
6356774.719 Australian National (1966) polar
6356818.170 Helmert (1906) polar
6356863.019 Krassovsky (1940) polar
6356911.946 Hayford (1910) polar
6356911.946 International (1924) polar
6363806.283 Maupertuis (1738) polar
6366197.2 Plessis (1817) rectifying
6366702.5 Everest 1830 (1967 Definition) rectifying
6366703.3 Everest (1830) rectifying
6366708.0 Everest 1830 Modified (1967) rectifying
6366742.5 Bessel (1841) rectifying
6366914.6 Airy (1830) rectifying
6367327.6 Clarke (1878) rectifying
6367386.6 Clarke (1880) rectifying
6367399.7 Clarke (1866) rectifying
6367399.7 NAD 27 (1927) rectifying
6367447.2 WGS-72 (1972) rectifying
6367448.1 IERS (1989) rectifying
6367448.7 IERS (2003) rectifying
6367449.1 GRS-80 (1979) rectifying
6367449.1 WGS-84 (1984) rectifying
6367456.9 WGS66 (1966) rectifying
6367469.3 New International (1967) rectifying
6367471.7 GRS-67 (1967) rectifying
6367471.8 South American (1969) rectifying
6367471.8 Australian National (1966) rectifying
6367513.6 Helmert (1906) rectifying
6367558.5 Krassovsky (1940) rectifying
6367654.5 Hayford (1910) rectifying
6367654.5 International (1924) rectifying
6369628.860 Plessis (1817) volumetric
6369634.826 Plessis (1817) authalic
6369636.311 Plessis (1817) mean
6370223.708 Everest 1830 (1967 Definition) volumetric
6370224.517 Everest (1830) volumetric
6370229.209 Everest 1830 Modified (1967) volumetric
6370229.991 Everest 1830 (1967 Definition) authalic
6370230.799 Everest (1830) authalic
6370231.554 Everest 1830 (1967 Definition) mean
6370232.363 Everest (1830) mean
6370235.491 Everest 1830 Modified (1967) authalic
6370237.055 Everest 1830 Modified (1967) mean
6370283.158 Bessel (1841) volumetric
6370289.510 Bessel (1841) authalic
6370291.091 Bessel (1841) mean
6370453.310 Airy (1830) volumetric
6370459.655 Airy (1830) authalic
6370461.234 Airy (1830) mean
6370937.089 Clarke (1878) volumetric
6370943.691 Clarke (1878) authalic
6370945.333 Clarke (1878) mean
6370990.707 Clarke (1866) volumetric
6370990.707 NAD 27 (1927) volumetric
6370996.142 Clarke (1880) volumetric
6370997.241 Clarke (1866) authalic
6370997.241 NAD 27 (1927) authalic
6370998.859 WGS-72 (1972) volumetric
6370998.867 Clarke (1866) mean
6370998.867 NAD 27 (1927) mean
6370999.786 IERS (1989) volumetric
6371000.385 IERS (2003) volumetric
6371000.790 GRS-80 (1979) volumetric
6371000.790 WGS-84 (1984) volumetric
6371002.744 Clarke (1880) authalic
6371004.387 Clarke (1880) mean
6371005.250 WGS-72 (1972) authalic
6371006.177 IERS (1989) authalic
6371006.762 IERS (2003) authalic
6371006.840 WGS-72 (1972) mean
6371007.181 GRS-80 (1979) authalic
6371007.181 WGS-84 (1984) authalic
6371007.767 IERS (1989) mean
6371008.367 IERS (2003) mean
6371008.608 WGS66 (1966) volumetric
6371008.771 GRS-80 (1979) mean
6371008.771 WGS-84 (1984) mean
6371014.999 WGS66 (1966) authalic
6371016.590 WGS66 (1966) mean
6371021.085 New International (1967) volumetric
6371023.523 GRS-67 (1967) volumetric
6371023.591 South American (1969) volumetric
6371023.591 Australian National (1966) volumetric
6371027.476 New International (1967) authalic
6371029.067 New International (1967) mean
6371029.915 GRS-67 (1967) authalic
6371029.982 South American (1969) authalic
6371029.982 Australian National (1966) authalic
6371031.505 GRS-67 (1967) mean
6371031.573 South American (1969) mean
6371031.573 Australian National (1966) mean
6371064.744 Helmert (1906) volumetric
6371071.133 Helmert (1906) authalic
6371072.723 Helmert (1906) mean
6371109.694 Krassovsky (1940) volumetric
6371116.083 Krassovsky (1940) authalic
6371117.673 Krassovsky (1940) mean
6371221.266 Hayford (1910) volumetric
6371221.266 International (1924) volumetric
6371227.711 Hayford (1910) authalic
6371227.711 International (1924) authalic
6371229.315 Hayford (1910) mean
6371229.315 International (1924) mean
6376523.000 Plessis (1817) equatorial
6377298.556 Everest 1830 (1967 Definition) equatorial
6377299.365 Everest (1830) equatorial
6377304.063 Everest 1830 Modified (1967) equatorial
6377397.155 Bessel (1841) equatorial
6377563.396 Airy (1830) equatorial
6378135.000 WGS-72 (1972) equatorial
6378136.000 IERS (1989) equatorial
6378136.600 IERS (2003) equatorial
6378137.000 GRS-80 (1979) equatorial
6378137.000 WGS-84 (1984) equatorial
6378145.000 WGS66 (1966) equatorial
6378157.500 New International (1967) equatorial
6378160.000 GRS-67 (1967) equatorial
6378160.000 South American (1969) equatorial
6378160.000 Australian National (1966) equatorial
6378190.000 Clarke (1878) equatorial
6378200.000 Helmert (1906) equatorial
6378206.400 Clarke (1866) equatorial
6378206.400 NAD 27 (1927) equatorial
6378245.000 Krassovsky (1940) equatorial
6378249.145 Clarke (1880) equatorial
6378388.000 Hayford (1910) equatorial
6378388.000 International (1924) equatorial
6380564.1 Maupertuis (1738) rectifying
6386115.886 Maupertuis (1738) volumetric
6386131.541 Maupertuis (1738) authalic
6386135.428 Maupertuis (1738) mean
6397300.000 Maupertuis (1738) equatorial