1 Reference Earth Mdel - WGS84 (Cpyright, David T. Sandwell) ω spherid c θ θ g a parameter descriptin frmula value/unit GM e (WGS84) 3.9864418 x 1 14 m 3 s M e mass f earth - 5.98 x 1 4 kg G gravitatinal cnstant - 6.67x1 11 m 3 kg 1 s a equatrial radius (WGS84) - 6378137 m c plar radius (derived) - 635675.3 m ω rtatin rate - 7.9115 x 1-5 rad s -1 (WGS84) f J flattening (WGS84) dynamic frm factr f = (a - c)/a 1/98.57356-1.81874 x 1-3 (derived) θ g gegraphic latitude - - θ gecentric latitude - - Radius f spherid r θ θ θ = cs sin a( 1 f sin θ) (1) a c + -1/ Cnversin between gecentric θ and gegraphic θ g latitude c tanθ = tanθg r tanθ = ( 1 f ) tanθ g () a
Gravitatinal ptential in frame rtating with the Earth U GMe GMeJa = + ( 3sin 1 3 )- 1 θ ω r cs θ (3) r r Calculatin f the secnd degree harmnic, J frm WGS84 parameters J 3 a ω = f 3 3GM e (4) Calculatin f J frm the plar-c and equatrial-a mments f inertia J C A = Ma e (5) Kepler's third law relating rbit frequency-ω s, and radius-r, t M e ω s 3 r = GM (6) e Measurement f J frm rbit frequency-ω s, radius-r, inclinatin-i, and precessin rate-ω p ω ω p s a = 3 J i cs (7) r Hydrstatic flattening is less than bserved flattening f = 1 < H f 99 5 = 1. 98.57 (8)
3 Disturbing ptential and geid height T a first apprximatin, the reference ptential U is cnstant ver the surface f the earth. Nw we are cncerned with deviatins frm this reference ptential. This is called the disturbing ptential Φ and ver the ceans the anmalus ptential results in a deviatin in the surface away frm the spherid. The reference ptential is given in equatin (3). U = U + Φ (9) The geid is the equiptential surface f the earth that cincides with the sea surface when it is undisturbed by winds, tides, r currents. The geid height is the height f the geid abve the spherid and it is expressed in meters. Cnsider the fllwing mass anmaly in the earth and its effect n the cean surface. N U spherid(r ) excess mass Because f the excess mass, the ptential n the spherid is higher than the reference level U = U + Φ. Thus, the cean surface must mve further frm the center f the earth t remain at the reference level U. T determine hw far it mves, expand the ptential in a Taylr series abut the radius f the spherid at r. U U r U r r r r = + ( - ) +... (1) Ntice that g = -δu/δr s we arrive at Ur - U gr- r (11) Φ = gn This is Brun's frmula that relates the disturbing ptential t the geid height N.
4 Reference gravity and gravity anmaly The reference gravity is the value f ttal (scalar) acceleratin ne wuld measure n the spherid assuming n mass anmalies inside f the earth. U g = U = r r U U ˆ 1 r θ θ ˆ 1 ˆ r csθ φ φ (1) The ttal acceleratin n the spherid is U U g = + r 1 r θ 1/ (13) The secnd term n the right side f equatin (13) is negligible because the nrmal t the ellipsid departs frm the radial directin by a small amunt and the square f this value is usually unimprtant. The result is GMe Ja gr (, θ) = r ( sin θ ) r ω θ + 1 3-3 1 rcs (14) T calculate the value f gravity anmaly n the spherid, we substitute r( θ) = a 1 f sin θ (15) After substitutin, expand the gravity in a binmial series and keep terms f rder f but nt f. g( θ) = ge + 5 m f sin θ 1 (16) ω a m = GM e The parameter g e is the value f gravity n the equatr and m is apprximately equal t the rati f centrifugal frce at the equatr t the gravitatinal acceleratin at the equatr. In practice, gedesists get tgether at meetings f the Internatinal Unin f Gedesy and Gephysics (IUGG) and agree n such things as the parameters f WGS84. In additin, they define smething called the internatinal gravity frmula. g ( θ) = 987.3185 1+.57889sin θ +. 346sin 4 θ (17)
5 This versin was adpted in 1967 s fr a real applicatin, yu shuld use a mre up-t-date value r use a value that is cnsistent with all f the ther data in yur data base. Free-air gravity anmaly The free-air gravity anmaly is the negative radial derivative f the disturbing ptential but it is als evaluated in the geid. The frmula is Φ g θ g = - N (18) r r( θ) Summary f Anmalies nrmal t plumb spherid geid U line height spherid(r ) excess mass Disturbing ptential Φ U = U + Φ (19) ttal = reference disturbing ptential ptential ptential Geid height N N = g Φ ( θ) () Free-air gravity anmaly Φ g θ g = - N (1) r r( θ)
6 Deflectin f the vertical The final type f anmaly, nt yet discussed, is the deflectin f the vertical. This is the angle between the nrmal t the geid and the nrmal t the spherid. There are tw cmpnents nrth ξ and east, η. 1 N ξ = a θ () 1 N η = acsθ φ
3 4 - -6-4 6-8 9-6 9 6 4 1 1 15 64 18 18 - - -6-15 -4-15 -1-4 -1 - -9-9 -4 4 - -6-6 15-3 6 3 3 - -4 6-4 -4 - -3-3 6 - -6-3 3 6