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Geodetic Reference System 1980
by H. Moritz
1-Definition
The Geodetic Reference System 1980 has been adopted at the XVII General Assembly of the IUGG in Canberra, December 1979, by means of the following:
"RESOLUTION N 7
The International Union of Geodesy and Geophysics,
recognizing that the Geodetic Reference System 1967 adopted at the XIV General Assembly of IUGG, Lucerne, 1967, no longer represents the size, shape, and gravity field of the Earth to an accuracy adequate for many geodetic, geophysical, astronomical and hydrographic applications and
considering that more appropriate values are now available,
recommends
a)that the Geodetic Reference System 1967 be replaced by a new Geodetic Reference System 1980, also based on the theory of the geocentric equipotential ellipsoid, defined by the following conventional constants :
.equatorial radius of the Earth :
a = 6378 137 m,
.geocentric gravitational constant of the Earth (including the atmosphere) :
GM = 3986 005 x 108 m3 s-2,
.dynamical form factor of the Earth, excluding the permanent tidal deformation :
J2 = 108 263 x 10-8,
.angular velocity of the Earth :
w = 7292 115 x 10-11 rad s-1,
b)that the same computational formulas, adopted at the XV General Assembly of IUGG in Moscow 1971 and published by IAG, be used as for Geodetic Reference System 1967, and
c)that the minor axis of the reference ellipsoid, defined above, be parallel to the direction defined by the Conventional International Origin, and that the primary meridian be parallel to the zero meridian of the BIH adopted longitudes".
For the background of this resolution see the report of IAG Special Study Group 5.39 (Moritz, 1979, sec.2).
Also relevant is the following IAG resolution :
"RESOLUTION N 1
The International Association of Geodesy,
recognizing that the IUGG, at its XVII General Assembly, has introduced a new Geodetic Reference System 1980,
recommends that this system be used as an official reference for geodetic work, and
encourages computations of the gravity field both on the Earth's surface and in outer space based on this system".
2-The Equipotential Ellipsoid
According to the first resolution, the Geodetic Reference System 1980 is based on the theory of the equipotential ellipsoid. This theory has already been the basis of the Geodetic Reference System 1967; we shall summarize (partly quoting literally) some principal facts from the relevant publication (IAG, 1971, Publ. Spc. n3).
An equipotential ellipsoid or level ellipsoid is an ellipsoid that is defined to be an equipotential surface. If an ellipsoid of revolution (semimajor axis a, semiminor axis b) is given, then it can be made an equipotential surface
U = U0 = const.
of a certain potential function U, called normal potential. This function U is uniquely determined by means of the ellipsoidal surface (semiaxes a, b), the enclosed mass M and the angular velocity w, according to a theorem of Stokes-Poincar, quite independently of the internal density distribution. Instead of the four constants a, b, M and w, any other system of four independent parameters may be used as defining constants.
The theory of the equipotential ellipsoid was first given by Pizzeti in 1894; it was further elaborated by Somigliana in 1929. This theory had already served as a base for the International Gravity Formula adopted at the General Assembly in Stockholm in 1930.
Normal gravity g = EQ \X\LE\RI(grad U) at the surface of the ellipsoid is given by the closed formula of Somilgiana,
g = EQ \F(age cos2 F + bgp sin2 F,\R(,a2 cos2 F + b2 sin2 F)) ,
where the constants ge and gp denote normal gravity at the equator and at the poles, and F denotes geographical latitude.
The equipotential ellipsoid furnishes a simple, consistent and uniform reference system for all purposes of geodesy: the ellipsoid as a reference surface for geometric use, and a normal gravity field at the earth's surface and in space, defined in terms of closed formulas, as a reference for gravimetry and satellite geodesy.
The standard theory of the equipotential ellipsoid regards the normal gravitational potential as a harmonic function outside the ellipsoid, which implies the absence of an atmosphere. (The consideration of the atmosphere in the reference system would require an ad-hoc modification of the theory, whereby it would lose its clarity and simplicity.)
Thus, in the same way as in the Geodetic Reference System 1967, the computation are based on the theory of the equipotential ellipsoid without an atmosphere. The reference ellipsoid is defined to enclose the whole mass of the earth, including the atmosphere; as a visualization, one might, for instance, imagine the atmosphere to be condensed as a surface layer on the ellipsoid. The normal gravity field at the earth's surface and in space can thus be computed without any need for considering the variation of atmospheric density.
If atmospheric effects must be considered, this can be done by applying corrections to the measured values of gravity; for this purpose, a table of corrections will be given later (sec.5).
3-Computational Formulas
An equipotential ellipsoid of revolution is determined by four constants. The IUGG has chosen the following ones:
a equatorial radius,
GM geocentric gravitational constant,
J2 dynamical form factor,
w angular velocity.
The equatorial radius a is the semimajor axis of the meridian ellipse; the semiminor axis will be denoted byb. The geocentric gravitational constant GM is the product of the Newtonian gravitational constant, G, and the total mass of the earth, M. The constant J2 is given by :
J2 = EQ \F(C-A,Ma2) ,
where C and A are the principal moments of inertia of the level ellipsoid (C... polar, A... equatorial moment of inertia).
We shall also use the first excentricity e, defined by:
e2 = EQ \F(a2 - b2,a2) ,
and the second excentricity e', defined by :
e'2 = EQ \F(a2 - b2,b2)
Closed computational formulas are given in sec.3 of (IAG, 1971, Pub.Spc. n 3); we shall here reproduce this section practically unchanged.
The derivation of these formulas is found in the book (Heiskanen and Moritz, 1967) sections 2-7 to 2-10. Reference to this book is by page number and number of equation.
Computation of e2
The fundamental derived constant is the square of the first excentricity, e2, as defined above.
From p. 73, equations (2-90) and (2-92'), we find :
J2 = EQ \F(e2,3) \B( 1 - \F(2,15) \F(me',qo) )
This equation can be written as :
e2 = 3J2 + EQ \F(2me' e2,15 qo)
with :
m = EQ \F(w2 a2 b,GM)
(p. 69, eq. (2-70)) and with be' = ae it becomes :
e2 = 3J2 + EQ \F(4,15) \F(w2 a3,GM) \F(e3,2 q0)
This is the basic equation which relates e2 to the dataa, GM, J2 and w. It is to be solved iteratively for e2, taking into account :
2 q0 = EQ \B\BC(1 + \F(3,e'2) ) arctan e' - EQ \F(3,e')
= EQ \I\SU(n=1,(, \F(4(-1)n+1 n,(2n+1) (2n+3))) e'2n+1
with
e' = EQ \F(e,\R(1 - e2)) (second excentricity)
(p. 66, eq. (2-58), p. 72, second equation from top).
Geometric Constants
Now the other geometric constants of the reference ellipsoid can be computed by the well-known formulas:
b = a EQ \R(,1 - e2) (semiminor axis),
f = EQ \F(a - b,a) (flattening),
E = EQ \R(a2 - b2) (linear excentricity),
c = EQ \F(a2,b) (polar radius of curvature).
The arc of meridian from equator to pole (meridian quadrant) is given by :
Q = c EQ \I\IN(0,p/2,\F(dF,(1+e'2 cos2 F)3/2))
where F is the geographical latitude. This integral can be evaluated by a series expansion :
Q= EQ c \F(p,2) \B(1-\F(3,4) e'2 + \F(45,64) e'4 - \F(175,256) e'6 + \F(11025,16384) e'8)
Various mean radii of ellipsoid are defined by the following formulas :
arithmetic mean :
R1 = EQ \F(a + a + b,3) = a EQ \B(1 - \F(f,3)) ;
radius of sphere of the same surface :
R2 = c EQ \B(\I\IN(0,p/2,\F(cos F,(1+e'2 cos2 F)2)) dF)\s\up8()
= EQ c \B(1-\F(2,3) e'2 + \F(26,45) e'4 - \F(100,189) e'6 + \F(7034,14175) e'8)
radius of sphere of the same volume :
R3 = EQ \R(3,a2 b) .
Physical Constants
The reference ellipsoid is a surface of constant normal potential, U = U0. This constant U0, the normal potential of the reference ellipsoid, is given by :
U0 = EQ \F(GM,E) arctan e' + EQ \F(1,3) w2 a2
= EQ \F(GM,b) \B(1 + \I\SU(n=1,(, (-1)n \F(e'2n,2n+1) + \F(1,3) m))
(p. 67, eq. (2-61)).
The normal gravitational potential V (gravity potential U minus potential of centrifugal force) can be developed into a series of zonal spherical harmonics :
V = EQ \F(GM,r) \B(1 - \I\SU(n=1,(, J2n \B(\F(a,r)))2n P2n (cos ())) ;
where r (radius vector) and ( (polar distance) are spherical coordinates. The coefficient J2 is a defining constant; the other coefficients are expressed in terms ofJ2 by :
J2n = (-1)n+1 EQ \F(3e2n,(2n + 1) (2n + 3)) \B(1 - n + 5n \F(J2,e2))
(p.73, eqs. (2-92) and (2-92')).
Normal gravity at the equator, ge, and normal gravity at the poles, gP, are given by the expressions :
ge = EQ \F(GM,ab) \B(1 - m - \F(m,6) \F(e' q'0,q0) )
gp = EQ \F(GM,a2) \B(1 + \F(m,3) \F(e' q'0,q0))
with
q0' = 3 EQ \B(1 + \F(1,e'2)) EQ \B\BC(1 - \F(1,e') arctan e') -1
and
m = EQ \F(w2 a2 b,GM)
(p. 69, eqs. (2-73) and (2-74); p.68, eq. (2-67)).
The constant :
f* = EQ \F(gp - ge ,ge) (gravity flattening)
is also needed.
A check is provided by the closed form of Clairaut's theorem for the equipotential ellipsoid :
f + f* = EQ \F(w2b,ge) \B\BC(1 + \F(e' q'0,2q0))
(p. 69, eq. (2-75)).
The Gravity Formula
Somiglinana's closed formula for normal gravity is
g = EQ \F(age cos2 F + bgp sin2 F,\R(,a2 cos2 F + b2 sin2 F))
For numerical computations, the form
g = ge EQ \F(1 + k sin2 F,\R(,1 - e2 sin2 F))
with
k = EQ \F(bgp,age) -1
is more convenient.
The conventional abbreviated series expansion is :
g = ge (1 + f* sin2 F - EQ \F(1,4) f4 sin2 2F)
with
f4 = EQ \F(1,2) f2 + EQ \F(5,2) fm
(p.77, eqs. (2-115) and 2-116)).
More generally, the above closed formula for normal gravity may be expanded into the series
g = ge EQ \B\BC(1 + \I\SU(n=1,(, a2n sin2n F))
where
a2 = EQ \F(1,2) e2 + k, a6 = EMBED Equation.3 e6 + EQ \F(3,8) e4 k,
a4 = EQ \F(3,8) e4 + EQ \F(1,2) e2 k, a8 = EQ \F(35,128) e8 + EQ \F(5,16) e6 k,
The average value of gravity over the ellipsoid is
EQ \X\TO(g) = EQ \I\IN(0,p/2, \F(g cos F d F,(1 - e2 sin2 F)2)) : EQ \I\IN(0,p/2, \F(cos F d F,(1 - e2 sin2 F)2))
= 1 + EQ \F(1,6) e2 + EQ \F(1,3) k + EQ \F(59,360) e4 + EQ \F(5,18) e2 k +
+ EQ \F(2371,15120) e6 + EQ \F(259,1080) e4 k + EQ \F(270229,1814400) e8 + EQ \F(9623,45360) e6k.
4-Numerical values
The following derived constants are accurate to the number of decimal places given. In case of doubt or in those cases where a higher accuracy is required, these quantities are to be computed from the defining constants by means of the closed formulas given in the preceding section.
Defining Constants (exact)
a =6378 137 m semimajor axis
GM =3 986 005 x 108 m3 s-2 geocentric gravitatio-
nal constant
J2 =108 263 x 10-8 dynamic form factor
w =7 292 115 x 10-11 rad s-1 angular velocity
Derived Geometric Constants
b = 6 356 752.3141 m semiminor axis
E = 521 854.0097 m linear excentricity
c = 6 399 593.6259 m polar radius of curvature
e2 =0.006 694 380 022 90 first excentricity (e)
e'2 =0.006 739 496 775 48 secondexcentricity (e')
f =0.003 352 810 681 18 flattening
f-1 =298.257 222 101 reciprocal flattening
Q =10 001 965.7293 m meridian quadrant
R1 = 6 371 008.7714 m mean radius R1=(2a+b)/3
R2 = 6 371 007.1810 m radius of sphere of same
surface
R3 =6 371 000.7900 m radius of sphere of same
volume
Derived Physical Constants
U0 =6 263 686.0850 x 10 m2 s-2 normal potential at
ellipsoid
J4 =-0.000 002 370 912 22
J6 = 0.000 000 006 083 47 spherical-harmonic
J8 =-0.000 000 000 014 27 coefficents
m =0.003 449 786 003 08 m = w2 a2 b/GM
ge =9.780 326 7715 ms-2 normal gravity at equator
gp =9.832 186 3685 ms-2 normal gravity at pole
f* =0.005 302 440 112 f* = EQ \F((gp - ge),ge)
k =0.001 931 851 353 k = EQ \F((bgp - age),age)
Gravity Formula 1980
Normal gravity may be computed by means of the closed formula :
g = ge EQ \F(1 + k sin2 F,\R(,1 - e2 sin2 F)) ,
with the values of ge, k, and e2 shown above.
The series expansion, given at the end of sec. 3, becomes :
g = ge (1 + 0.005 279 0414 sin2 F
+ 0.000 023 2718 sin4 F
+ 0.000 000 1262 sin6 F
+ 0.000 000 0007 sin8 F) ;
it has a relative error of 10-10, corresponding to10-3mms-2 = 10-4 mgal.
The conventional series
g = ge (1 + f* sin2 F - EQ \F(1,4) f4 sin2 2 F)
= 9.780 327 (1 + 0.005 3024 sin2 F
- 0.000 0058 sin2 2 F) m s-2
has only an accuracy of 1 mms-2 = 0.1 mgal. It can, however, be used for converting gravity anomalies from the International Gravity Formula (1930) to the Gravity Formula 1980 :
g1980 - g1930 = (- 16.3 + 13.7 sin2 F) mgal,
where the main part comes from a change of the Postdam reference value by - 14 mgal; see also (IAG, 1971, Publ. Spc. n 3, p.74).
For the conversion from the Gravity Formula 1967 to the Gravity Formula 1980, a more accurate formula, corresponding to the precise expansion given above, is :
g1980 - g1967 = (0.8316 + 0.0782 sin2 F
- 0.0007 sin4 F) mgal,
Since former gravity values are expressed in the units "gal" and "mgal", we have, in the conversion formulas, used the unit 1 mgal = 10-5 m s-2.
Mean values of normal gravity are :
EQ \X\TO(g) = 9.797 644 656 m s-2 average over ellipsoid,
g45 = 9.806 199 203 m s-2 g
at latitude F = 45.
The numerical values given in this section have been computed independently by Mr. Chung-Yung Chen, using series developments up to f5, and by Dr.HansSnkel, using the formulas presented in sec.3.
5-Atmospheric Effects
The table given here is reproduced from (IAG, 1971, Publ. Spc. n 3, p.72). It shows atmospheric gravity correction dg as a function of elevation h above sea level. The values dg are to be added to measured gravity. The effect of this reduction is to remove, by computation, the atmosphere outside the Earth by shifting it vertically into the interior of the geoid.
Atmospheric Gravity Corrections dg
(to be added to measured gravity)
h dg h dg
[km] [mgal] [km] [mgal]
0 0.87 10 0.23
0.5 0.82 11 0.20
1.0 0.77 12 0.17
1.5 0.73 13 0.14
2.0 0.68 14 0.12
2.5 0.64 15 0.10
3.0 0.60 16 0.09
3.5 0.57 17 0.08
4.0 0.53 18 0.06
4.5 0.50 19 0.05
5.0 0.47 20 0.05
5.5 0.44 22 0.03
6.0 0.41 24 0.02
6.5 0.38 26 0.02
7.0 0.36 28 0.01
7.5 0.33 30 0.01
8.0 0.31 32 0.01
8.5 0.29 34 0.00
9.0 0.27 37 0.00
9.5 0.25 40 0.00
6-Origin and Orientation of the Reference System
IUGG Resolution n 7, quoted at the begining of this paper, specifies that the Geodetic Reference System 1980 be geocentric, that is, that its origin be the center of mass of the earth. Thus, the center of the eliipsoid coincides with the geocenter.
The orientation of the system is specified in the following way. The rotation axis of the reference ellipsoid is to have the direction of the Conventional International Origin for the Polar Motion (CIO), and the zero meridian as defined by the Bureau International de l'Heure (BIH) is used.
To this definition there corresponds a rectangular coordinate system XYZ whose origin is the geocenter, whose Z-axis is the rotation axis of the reference ellipsoid, defined by the direction of CIO, and whose X-axis passes through the zero meridian according to the BIH.
References
W.A. HEISKANEN, and H. MORITZ (1967) : Physical Geodesy. W.H. Freeman, San Francisco.
International Association of Geodesy (1971) : Geodetic Reference System 1967. Publi. Spc. n 3 du Bulletin Godsique, Paris.
H. MORITZ (1979) : Report of Special Study Group N539 of I.A.G., Fundamental Geodetic Constants, presented at XVII General Assembly og I.U.G.G., Canberra.
Editor's Note:
Additional useful constants can be obtained from :
"United States Naval Observatory, Circular N167, December 27, 1983, Project MERIT Standards", with updates of December 1985.
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