Numerical simulation of Earth s gravitational field recovery from SST based on the energy conservation principle

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1 CHINESE JOURNAL OF GEOPHYSICS Vol 49, No 3 May, 2006, -, 2006, 49 (3) Zheng W, Shao C G, Luo J, et al Numerical simulation of Earth s gravitational field recovery from SST based on the energy conservation principle Chinese J Geophys (in Chinese), 2006, 49 (3) , , 1, , ,, 120, EIGEN2GRACE02S ;, 2,GRACE, -,, (2006) P , Numerical simulation of Earth s gravitational field recovery from SST based on the energy conservation principle ZHENG Wei 1, SHAO Cheng- Gang 1, LUO Jun 1 3, HSU Houtse 2 1 Department of Physics, Huazhong University of Science and Technology, Wuhan , China 2 Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan , China Abstract Based on the measurement principle of Satellite-to- Satellite Tracking mission (SST), the new and effective observation equations of two satellites and three satellites are established, respectively using the energy conservation principle The high- accuracy Earthπs gravitational field up to degree and order 120 is recovered through numerical simulation by applying an improved pre- conditioned conjugate- gradient ( PCCG) iterative approach The simulated results show that the accuracy of the Earthπs gravitational field recovery using two satellites is close to the results of EIGEN- GRACE02S publicized by Jet Propulsion Laboratory (J PL ) in America, and the accuracy of the Earthπs gravitational field recovery using three satellites is about 2 times higher than that using two satellites Keywords Earthπs gravitational field, GRACE satellites, Satellite-to- satellite tracking mission, Energy conservation principle, Pre- conditioned conjugate- gradient approach ( ),,1977,, 3,,1956,,, hust edu cn

2 , E k 1 = D k n u n 1, (1) [ gr 12 - ( gr 12 e 12 ) e 12 ],gr 1,gr 2 9 t,gr 12, gr 12 = gr 2 - gr 1, (5) (3), e gr 2 = d V e 9 V e - d t + d t 9 t d V T 9 V T - d t d t 9 t, e 12 = r 12 Π r 12, g 12 e 12 ( gr 12 e 12 ) e 12 K g 12,, 120, K 2 211, E k 1, k O Keefe [1 ], D k n k n, n = L 2 max + 2L max - 3, L max, u n 1 [2 GPS( Global Positioning System) ], CHAMP ( Challenging r = F + f, (2) Minisatellite Payload) GRACE ( Gravity Recovery and, r, F Climate Experiment), GPS, F = F e ( r, t) + F T ( r, t), F e ( r, t), F T ( r, t) (,Jekeli [3 ], Han [4,5 ], ), r f Visser [6 ],Gerlach [7 ] [8 ], (2) GRACE , gr, [ ] gr r = gr ( F e + F T ) + gr f, (3), F e F T, GRACE K F e(t) = 9V e(t) Π9r, (4), V e, V e = V 0 + T e V 0, V 0 = GMΠr r, K,, r = x 2 + y 2 + z 2 x, y, z r [12 ] GM M G T e, V T (4) 1, 2 ( gr 1 + d V e(t) = 9 V e(t) d r + 9 V e(t) 1 gr 2 ) ( gr 12 e 12 ) e 12 2 ( gr d t 9 r d t 9 t 1 + gr 2 ) = F e(t) gr + 9 V e(t) (5) + gr fd t + E 0 9 ( V e + V T ) d t = V 0 + T e + V T - 9 t g 12 = 1 mπs, + gr fd t + E 0 (6) (6),, T e = E k - E f + V - V T - V 0 - E 0, (7), E k, E k = 1 2 gr 2 ; E f

3 714 (Chinese J Geophys ) 49 [3], E f = gr fd t ; V 9 ( V e + V T ) d t - 9 e ( xgy - t, V = ygx), e ; E 0 (7),, + [ gr 12 - ( gr 12 e 12 ) e 12 ]}, (9) T e12 = E - E f12 + V 12 - V T12 - V E 012,, 1 2 ( gr 2 + gr 1 ) (8), T e12, T e12 ( r 1, 1, 1, r 2, 2, 2 ) = T e2 ( r 2, 2, 2 ) - T e1 ( r 1, 1, 1 ) = L R e l = 2 m = - l - R e r 1 l l +1 R e r 2 l +1 gy lm ( 2, 2 ) gy lm ( 1, 1 ) gc lm, gy l, m (, ) = gp l m ( cos ) Q m ( ), Q m ( ) = cos m m 0, = GM sin m m < 0 r 1, 1, r 2 E, 1, 2, E K,, 1, 2 R e 2 ( gp lm ( cos ) A) 2 A B, Legendre, l, m gc lm (8) E,, E = 1 2 ( gr 2 + gr 1 ) ( gr 2 - GRACE K g 12 e 12 gr 1 ) ; E f12, E f12 = ( gr 12 e 12 ) e 12 (9) ( gr 2 f 2 - gr 1 f 1 ) d t ; V 12 E 12 = 1 2 ( gr 2 + gr 1 ) { g [2 ], V 12 = - e ( x 12 gy 2 - y 2 gx 12 - y 12 gx 1 + x 1 gy 12 ) ; V T12 ; V 012 1, V 012 = GM GM Table 1 Errors of kinetic energy difference - ; r 2 r 1 E 012, CHAMP, E 12 = 1 2 ( gr 2 + gr 1 ) g, GRACE K g 12 = 1 mπs, GRACE (8) E E = 1 2 ( gr 2 + gr 1 ) { ( gr 12 e 12 ) e 12,gr 12 = ( gr 12 e 12 ) e 12,gr 12 = gr 12 - ( gr 12 e 12 ) e 12 E = 1 2 ( gr 2 + gr 1 ) ( gr 12 e 12 ) e 12, E = 1 2 ( gr 2 + gr 1 ) [ gr 12 - ( gr 12 e 12 ) e 12 ] gr 12, gr 12, 1, K Jekeli [3 ] Han [4 ] g 12 = 1 mπs, 12 e 12 + [ gr 12 - ( gr 12 e 12 ) e 12 ]} (10) (m 2 Πs 2 ) E = 1 2 ( gr 2 + gr 1 ) ( gr 2 - gr 1 ) E = = 1 2 ( gr 2 + gr 1 ) [ ( gr 12 e 12 ) e 12 ] (8), E = 1 2 ( gr 2 + gr 1 ) [ gr 12 - ( gr 12 e 12 ) e 12 ] E, E 12 = 1 2 (gr 2 + gr 1 ) {g 12 e 12 + [ gr 12 - (gr 12 e 12 ) e 12 ]} E E = E E = 0101 E 12 = e 12 E = = 1 2 ( gr 2 + gr 1 ) [ gr 12 - ( gr 12 e 12 ) e 12 ] E =

4 Table 2 Commensurate relationship of accuracy A ( m 2 Πs 2 ) B ( m 2 Πs 2 ), g mπs mπs, f mπs mπs 2, r m m, r m m, gr mπs mπs, gr mπs mπs 3 Table 3 Numerical simulation parameters of satellite orbits EGM km 220 km days 10 s (10) (8), T e12 = 1 2 ( gr 2 + gr 1 ) { g 12 e 12 + [ gr 12 - ( gr 12 e 12 ) e 12 ]} - E f12 + V 12 - V T12 - V E 012 (11) (11),, T e23 - T e12 = ( E 23 - E 12) - ( E f23 - E f12 ) + ( V 23 - V 12) - ( V T23 - V T12 ) - ( V V 012 ) - ( E E 012 ) (12) 1, (11), (10-5 m 2 Πs 2 ) r gr Fig 1 Numerical computation errors of the observation Eq (11), Runge- Kutta 12 Adams-Cowell D T k n E k 1 = D T k n D k n u n 1 (13) 3, G 9h n 1 = D T k n E k 1, S n n = D T k n D k n, gc lm, r (13) gr,k g 12 f G n 1 = S n n u n 1 (14), (11), [12 ], P n n P n n 2 A, Te12, P - 1 n n, P - 1 n n S - 1 n n = m 2 Πs 2,, 2, S n n,, 10-4, l = 30,,, 10 S n n 1, (11), P n n 0, without random noise(10-5 m 2 Πs 2 ), S n n, P - 1 n n (1), S - 1 n n,, (1) D T k n ( 1Π1000)

5 716 (Chinese J Geophys ) 49 3, 8, 8h (14) P - 1 n n P - 1 n n G n 1 = P - 1 n n S n n u n 1 (15) G n 1 = P - 1 n n G n 1, S n n = P - 1 n n S n n, (15) G n 1 = S n n u n 1 (16) 2 S n n ( l = 30), 10 Fig 2 Block- diagonally dominant characteristics of matrix S n n ( l = 30) The value of the matrix elements are represented by color intensity, and the color bar values are denoted by denary logarithm 3, 3,, EIGEN- GRACE02S 120 GRACE (NASA) 120, 20 cm ( GFZ) GRACE 120 ( References), [ 1 ] O 120 Earth satellite, 2 4, g 12 e 12 3 GRACE, Fig 3 Comparison of cumulative geoid height errors among GRACE satellites, two satellites and three satellites gr 12 - ( gr 12 e 12 ) e 12 K g 12,, K g 12 = 1 mπs, K,,, EIGEN- GRACE02S ;, Keefe J A An application of Jacobi s integral to the motion of an The Astronomical Journal, 1957, 62 (1252) 265 [ 2 ],, GPSΠ ( ),2005,48(2) Li F, Yue J L, Zhang L M Determination of geoid by GPSΠGravity data Chinese J Geophys (in Chinese), 2005, 48 (2) [ 3 ] Jekeli C The determination of gravitational potential differences from SST tracking Celestial Mechanics and Dynamical Astronomy, 1999, [ 4 ] Han S C Efficient determination of global gravity field from satellite to-satellite tracking mission Celestial Mechanics and Dynamical

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