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

Transcription

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, 1 3, 2 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 Satelliteto Satellite Tracking mission (SST), the new and effective observation equations of two satellites anhree 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, anhe 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, Satelliteto satellite tracking mission, Energy conservation principle, Pre conditioned conjugate gradient approach ( ).,,1977,,. 3,,1956,,,. hust. edu. cn

2 3 : 713 1,. OKeefe [1 ],,. GPS( Global Positioning System) [2 ] CHAMP (Challenging Minisatellite Payload ) GRACE ( Gravity Recovery and Climate Experiment), GPS,.,Jekeli [3 ], Han [4,5 ], Visser [6 ],Gerlach [7 ] [8 ]. GRACE , [911 ].,GRACE K., K, [12 ]., 1 2 ( gr 1 + gr 2 ) ( gr 12 e 12 ) e ( gr 1 + gr 2 ) [ gr 12 ( gr 12 e 12 ) e 12 ],gr 1,gr 2,gr 12, gr 12 = gr 2 gr 1, e 12, e 12 = r 12 Π r 12., g 12 e 12 ( gr 12 e 12 ) e 12. K g 12,, 120.,K g 12 = 1mΠs, : E k 1 = D kn u n 1, (1), E k 1, k. D k n k n, n = L 2 max + 2L max 3, L max. u n 1.,: r = F + f, (2), r, F, F = F e ( r, t) + F T ( r, t). F e ( r, t), F T ( r, t) ( ), r. f,. (2) gr, gr r = gr ( F e + F T ) + gr f, (3), F e F T F e(t) = 9V e(t) Π9r, (4), V e, V e = V 0 + T e. V 0, V 0 = GMΠr., r = r x 2 + y 2 + z 2. x, y, z r. GM M G. T e, V T. (4) d V e(t) = 9 V e(t) 9 r d r + 9 V e(t) = F e(t) gr + 9 V e(t). (5) (5) (3), 1 2 gr 2 = d V e 9 V e + gr f + E 0 + d V T = V 0 + T e + V T 9 V T 9 ( V e + V T ) + gr f + 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, E f [3] = gr f ; V, V= 9 ( V e + V T ) 9 e ( xgy t ygx), e ; E 0. (7),, : T e12 = E E f12 + V12 V T12 V 012 E 012,, 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, (8) gy l, m (, ) = gp l m ( cos) Q m (), Q m () = cos m m 0,= GM. sin m m < 0 r 1, r 2, 1, 2, 1, 2. R e. gp lm ( cos) Legendre, l, m. gc lm. (8) E, E = 1 2 ( gr 2 + gr 1 ) ( gr 2 gr 1 ) ; E f12, E f12 = ( gr 2 f 2 gr 1 f 1 ) ; V 12 [2 ], V 12 = e ( x 12 gy 2 y 2 gx 12 y 12 gx 1 + x 1 gy 12 ) ; V T12 ; V 012, V 012 = GM r 2 GM r 1 ; E 012,. (8), E, CHAMP,., GRACE K g12 = 1mΠs., Jekeli [3 ] Han [4 ], GRACE.. (8) E E = 1 2 ( gr 2 + gr 1 ) { ( gr 12 e 12 ) e 12 + [ gr 12 ( gr 12 e 12 ) e 12 ]}, (9), 1 2 ( gr 2 + gr 1 ),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, E,, E K, 2 ( A). 2 A B,., GRACE K g 12 e 12 ( gr 12 e 12 ) e 12. (9) 12 = 1 2 ( gr 2 + gr 1 ) { g E 12 e 12 + [ gr 12 ( gr 12 e 12 ) e 12 ]}. (10) 1 Table 1 Errors of kinetic energy difference (m 2 Πs 2 ) E = 1 2 ( gr 2 + gr 1 ) ( gr 2 gr 1 ) E = 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 ] 12 = 1 2 (gr 2 + gr 1 ) {g 12 e 12 + [ gr 12 (gr 12 e 12 ) e 12 ]} E = E E = 0101 E 12 = E 12 = 1 2 ( gr 2 + gr 1 ) g 12 e 12 E = E 12 = 1 2 ( gr 2 + gr 1 ) [ gr 12 ( gr 12 e 12 ) e 12 ] E = , K g12 = 1mΠs,

4 3 : Table 2 Commensurate relationship of accuracy A B ( m 2 Πs 2 ) ( m 2 Πs 2 ), g mπs mπs, f, r, r mπs mπs m m m m 3 Table 3 Numerical simulation parameters of satellite orbits EGM km 220 km days 10 s, gr, gr mπs mπs mπs mπs. (10) (8), : T e12 = 1 2 ( gr 2 + gr 1 ) { g 12 e 12 + [ gr 12 ( gr 12 e 12 ) e 12 ]} E f12 + V12 V T12 V 012 E 012. (11) (11),, : T e23 T e12 = ( E 23 E 12) ( E f23 E f12 ) + ( V 23 V12 ) ( V T23 V T12 ) ( V 023 V 012 ) ( E 023 E 012 ). (12), r gr., Runge Kutta 12 AdamsCowell. 3, 9h. gc lm,r, gr,k g 12 f.,(11),,. 2 A, Te12 = m 2 Πs 2,,, 10 4,,. 1, (11). 212 (1),,,. (1) D T k n 1, (11) (10 5 m 2 Πs 2 ) Fig. 1 Numerical computation errors of the observation Eq. (11) without random noise (10 5 m 2 Πs 2 ) D T kn E k 1 = D T kn D kn u n 1. (13) G n 1 = D T k n E k 1, S n n = D T k n D k n, (13) G n 1 = S nn u n 1. (14) [12 ]. P n n. P n n :, P 1 n n., P 1 n ns 1 n n. 2, S n n,,, l = 30,, 10. S n n,p n n 0,0.,S n n, P 1 n n S 1 n n., ( 1Π1000).

5 716 (Chinese J. Geophys. ) 49 3, 8, 8h. (14) P 1 n n P 1 nn G n 1 = P 1 nn S nn u n 1. (15) G n 1 = P 1 n ng n 1, S n n = P 1 n n S n n, (15) G n 1 = S nn u n 1. (16) 3 GRACE, Fig. 3 Comparison of cumulative geoid height errors among GRACE satellites, two satellites anhree satellites 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 3 intensity, anhe color bar values are denoted by denary logarithm. 3, EIGEN GRACE02S 120 GRACE. 120,20 cm. 120,. 120, 2. 4., g 12 e 12 gr 12 ( gr 12 e 12 ) e 12. K g 12,., K g12 = 1mΠs,K,,. :, EIGEN GRACE02S ;, 2.,,. (NASA) ( GFZ) GRACE. (References) [ 1 ] O Keefe J A. An application of Jacobi s integral to the motion of an Earth satellite. 266 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, 75 : [ 4 ] Han S C. Efficient determination of global gravity field from satellite tosatellite tracking mission. Celestial Mechanics and Dynamical

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