Determination of long-term changes in the Earth's gravity field from satellite laser ranging observations

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B10, PAGES 22,377-22,390, OCTOBER 10, 1997 Determination of long-term changes in the Earth's gravity field from satellite laser ranging observations M. K. Cheng, C. K. Shum and B. D. Tapley Center for Space Research, University of Texas at Austin Abstract. Temporal changes in the Earth's gravity field have been determined by analyzing satellite laser ranging (SLR) observations of eight geodetic satellites using data spanning an interval of over 20 years. The satellites used in the analysis include Starlette, LAGEOS 1 and 2, Ajisai, Etalon 1 and 2, Stella, and BE-C. Geophysical parameters, related to both secular and long-period variations in the Earth's gravity field, including the geopotential zonal rates (J2, J3, J4, is, and J6)and the 18.6-year tide parameter, werestimated. Th estimated values for these parameters are,)2 = -2.7+0.4(10-11/yr); J3 = - 1.3+0.5(10-11/yr); J4 =-1.4+1.0(10-11/yr); J5 = 2.1+0.6(10-11/Yr); - J6 = 0.3+0.7(10-11/yr); - p+2o cl 8.6 = 1.56 +20 +0.2 (centimeters) and S18.6 = -0.1+0.2(centimeters). The amplitude and phase for the 18.6-year tide are in general agreement with the effects predicted by the Earth's mantle anelasticity. The solution accuracy was evaluated by considering the effects of errors in various non-estimate dynamical model parameters and by varying the data span andata sets used in the solution. Estimates for J3 from individual LAGEOS 1 and Starlette SLR data sets are in good agreement. The lumped sum values for.)3 and J5 are very different for LAGEOS 1 and Starlette. The zonal rate determination is limited to degree 6 with the current SLR data sets. Analysis of the sensitivity of the solution for the zonal rates to the satellite tracking data span suggests that the temporal extension of the current SLR data sets will enhance the solution of zonal rates beyond degree 6. 1. Introduction Tidal and nontidal global mass redistribution, which occurs at all timescales, will cause variations in the Earth's gravity field. The variations are both secular and periodic, with periods ranging from 12 hours to 18.6 years. The secular changes in the Earth's gravity field, particularly, the variations in the zonal harmonic coefficients, are believed to be largely attributable to viscous change in the lower mantle associated with postglacial rebound (PGR) and the change in ice sheet mass. However, there are a large number of smaller contributing geophysical processes, including changes in water reservoirs, mountain glaciers, earthquakes, etc. The even zonal rates, Jt (l = 2, 4,...), are more sensitive to the lower mantle viscosity profile, while the odd zonal rates, Jt (l = 3, 5,...) are more dependent on present-day glacial mass balance [Ivins et al., 1993]. These variations, both secular and periodic, cause observable perturbations in the orbits of near-earth geodetic satellites. It has been demonstrated thathe accurate determination of the Jt provides critical constraints on the dynamic behavior of the Earth's system [Trupin et al., 1992; lvins et al., 1993; Mitrovica and Peltier, 1993]. The response of the solid Earth to tidal forces and fluid loading exhibits a frequency dependence due to the Earth's anelasticity and is subject to considerable dispersive effects [Wahr and Bergen, 1986]. The effects of anelasticity on the Copyright 1997 by the American Geophysical Union. Paper number 97JB01740. 0148-0227/97/97JB-01740509.00 22,377 tidal response vary with frequency. Measuring the anelasticity of the Earth at the long-period tidal frequencies, such as the 18.6-year tidal period, provides an opportunity to improve our understanding of the frequency dependency of the anelastic processes and the dissipation of tidal energy within the Earth system. Accurate observations of geodetic satellites over long time intervals can isolate the effects of these phenomena. Since Yoder et al. [1983] and Rubincam [1984] reported solutions for J2 from analysis of the LAGEOS 1 orbit, the study of the temporal variations in the Earth's gravity field has become an important concern for space geodesy and geodynamics. The earlier results for J2 and the estimates of Jt (l = 2, 3, and 4) given by Cheng et al. [1989] were obtained from the analysis of a satellite laser ranging (SLR) data set covering a relatively short time span of 3-5.5 years. Furthermore, only data from a single satellite were used, along with the assumption that the 18.6-year ocean tide is in equilibrium. Consequently, the reported solutions for Jt contained the effects of the variations in the high-degree zonal harmonics and 18.6-year tides. The improvement in the solution for lower degree Jt inherently requires the use of multisatellite data with different orbital characteristics over an adequate time span to reduce the aliasing effects from higher degree Jr. The useful information about the anelasticity of the Earth can also be obtained from an accurate determination of the 18.6-year tide by analysis of long-term multisatellite SLR data [Eanes, 1995; Zhu et al., 1996]. As shown in Table 1, the existing SLR data sets include continuous SLR tracking data over a 20-year time span for two geodetic satellites, Starlette and LAGEOS 1, with the shorter observation records for other satellites such as Ajisai, Etalon

22,378 CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS Table 1. Summary of Satellite Data Sets and Nominal Orbit Fits Satellite Data Span Observations Stations Arcs Arc length years RMS Starlette March 17, 1975 to Dec. 31, 1996 639,775 93 21 1 Ajisai Jan 1, 1987 to Dec. 31, 1995 567,429 63 9 1 BE-C Jan. 1, 1980 to June 30, 1986 98,057 27 5 1 Stella Sept. 30, 1993 to Dec. 31, 1995 52,033 48 2 1 LAGEOS 1 May 7, 1976 to Dec 31, 1995 818,498 115 1 19.5 LAGEOS 2 Oct. 25, 1992 to Dec 31, 1995 17,597 54 1 3.2 Etalon 1 Feb. 8, 1989 to Dec 31, 1995 21,300 34 1 6.9 Etalon 2 Aug. 9, 1989 to Dec 31, 1995 18,372 25 1 6.6 175 61 119 288 * Shown in Figure 1. 1, Etalon 2, LAGEOS 2, Stella and BE-C. The analysis of the multi-satellite SLR data sets, with considerably improved force modeling, has allowed better separation of the secular changes in the even zonal harmonics from the 18.6-year tide effects and accurate estimates of the various tidal and non-tidal properties of the Earth's geopotential. A number of solutions for Jl have been reported previously. These solutions have varied due to the use of different models, computational techniques, and data sets. In addition, the estimated value for J3 from the satellite is considered in this section from the perspective of the orbital sensitivity of the individual geodetic satellites given in Table solutions differs from the value predicted from the global ice model ICE-3G [Ivins et al., 1993]. A reduction of uncertainties in the estimates of the zonal rates is a crucial issue for the geophysical application of the satellite results. As a first step, this paper focuses on the discussion of the methodology for determination of the Jl (l = 2, 3, 4, 5, 6) and 18.6-year tide parameters using a multisatellite SLR data set. This study uses an optimal weighting procedure to combine the inhomogeneous date sets [Yuan, 1991]. In section 2, the orbit sensitivity and separability of the zonal variations in the satellite orbit are studied using an analytic linear satellite orbit perturbation theory. The SLR data and the observed orbital perturbations for the eight geodetic satellites used in this study are analyzed in section 3. The solution methodology for the determination of the zonal variations from multiple SLR data sets is presented in section 4. Finally, the solution of Jl (l = 2, 3, 4, 5, 6) and the 18.6-year lunar tide, and the associated accuracy assessment, are discussed in section 5, with the conclusions given in section 6. 2. Satellite Orbit Dynamics The capability of using satellite measurements to observe the long-term changes in the zonal harmonics associated with the Earth's gravity potential is based on the fact that zonal mass variations produce detectable perturbations on orbits of the geodetic satellites tracked by a global SLR network. The observability of zonal variations from satellite measurements 1. Table 2 provides the orbit characteristics of these satellites, including the semimajor axis (a), eccentricity (e), inclination (i), mean anomaly (M) rate, argument of perigee (to) rate, period of the ascending node (12), and the period of the ascending node with respecto the Sun. The last parameter describes the resonance between the satellite orbital motion and the diurnal apparent motion of the Sun. In principle, a satellite at lower altitude can sense all changes in the Earth's gravity field. However, the variations in the nonzonal spherical harmonics are associated with relatively small scale or regional mass transport, which results in smaller-amplitude and higher-frequency oscillations in the satellite orbit and are difficult to detect. Only the effects of zonal terms, which produce long-period orbit perturbations, are considered in this theoretical discussion. On the basis of an analytic perturbation theory [Kaula, 19661, the linearized dynamic system for the zonal excitations on a satellite orbit can be expressed as Table 2. Summary of Satellite Orbital Characteristics Satellite a e km deg ø/d ø/d day day Starlette 7,340 0.0205 49.8 Ajisai 7,840 0.0009 50.1 BE-C 7,449 0.0243 41.2 Stella 7,183 0.0003 98.7 LAGEOS 1 12,271 0.0044 109.8 LAGEOS 2 12,161 0.0137 52.6 Etalon 1 25,500 0.0006 64.9 Etalon 2 25,500 0.0007 64.9 4973 3.297 91 73 4476 2.539 117 89 4811 5.177 85 68 5136-2.920 363 57,145 2298-0.214 1050 560 2329 0.437 569 223 767.6-0.004 10588 353 767.6-0.004 10588 353

CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS 22,379 Table 3. Sensitivity of Satellite Orbits to Even Zonal Variations (per unit 8J[)xlO - Starlett.e Ajisai Stella BE-C I col M[ col M[ m I e M[ m I e r.o[ M[ 12 LAGEOS 1 LAGEOS 2 2 4 6 8 10 40.3 9.3-47.9 30.7 6.8-37.2-35.7-37.5 12.0 61.2 0.0-1.8 41.9 0.0-1.8-44.2 0.0 16.9-8.5-2.9 26.3-3.4-2.2 15.5-36.1 12.9 17.9-67.6 31.5 7.1-34.3 15,8 4.0-18.3 10.4 16.5-19.2 16.8-13.6-10.1 8.3-5.8 2.5-0.3 13.6 62.9 24.0-51.7-2.6-4.0 4.2 5.3 0.7-7.7 22.6 0.0 22.5 0.5 0.0 1.5 3.9 0.0-0.6-61.1 15.1 21.0 0.9-0.3 0.3 0.3-0.2 0.5-20.3 16.3-16.4 0.5-0.2 0.0-0.5 0.2 0.1 52.0-27.7-7.6 0.1-0.1-0.0-0.1 0.1-0.0 where dafx -[{A[}, {A?}] x[ { 6Jr } 2sin( A[ :[IE[}, I rofl, IMfl, {1'-2;}] r Af :[{E/},{ mr!t, {M;,{..Q;}] r (2) and 6J represents the time variations of the unnormalized degree 1 zonal harmonics. The superscript "e" represents the even zonal excitation, while "o" represents the odd zonal excitations, and Ao is the vector consisting of the zonal variation induced changes in the satellite orbital elements: e,,m, and. Here a= /2 for e, and a=0 for,m, and. The changes in the orbital elements, a and i, are neglected due to their small magnitude. The orbital element excitation factors used to quantify the sensitivity of the satellite orbit to zonal variations are expressed as follows: E = FlFlop(1-1)Glp(.1) 1-e2/e E[ = 0,f2 =- FtF}opGl,(.1)/( 1-e2sin i) 1'2[ =- FlF'topGl, o/ ( I 1-e2sin i) (.of =-cos/1'-2 - FlFlopG p(_l)(1-e2)l/2/e (j.}e =-cos/.,qi-fifiopgl (1-e2) /2/e e, Mf = Fi Flop[(1-e2)/eGip(4)-2(/+ 1)G/,(_o] m[ = FlFlop[(1-e 2)/eGjpo-2(l+ 1)Glpo] where R is the mean Earth radius, the index p = I/2 when l Gipq(e) are the eccentricity functions and their derivatives (1) [Kaula, 1966]. Since Glpq(e) =, O(e 14) and G pq(e)=, O(e26q0'qqt-1), where O(e m) represents the magnitude of e m, all of the sensitivity factors in Equation (3) are to zeroth order in eccentricity except, and depend on the property of the inclination function. The significant excitation factors (degree < 11) for Starlette, Ajisai, Stella, BE-C, LAGEOS 1, and LAGEOS 2 are shown in Tables 3 and 4. The excitation factors for EtaIon 1 and EtaIon 2 have very small magnitudes and are not included. Table 3 shows the sensitivity of the satellite orbits to even zonal variations. The factors which influence the sensitivity are the altitude and the inclination. The mean anomaly M is less sensitive to the even zonal variations (for degree < 6), with the M,[ negligible for all satellites. Since they are at a lower altitudes, Starlette, Stella, and BE-C are more sensitive to the zonal variations than other satellites. The ascending node for Starlette, Ajisai, and Stella are less sensitive to 6J4 and 6J8. The excitation factors M[ + co[ +cos/.o[ for these satellites contain important information related to 8J4 and 8J8. The 8J6 significantly affects the arguments of perigee of BE-C and Stella. Table 4 shows the sensitivity of the satellite orbits to odd zonal variations. The ascending node is insensitive to the odd zonal variations. Although the excitation factor for the argument of perigee is largest, it will be canceled by Mf or ee[ in the eccentricity vector. To zeroth order in e, the (3) along-track component T = M+ r.o+ cos/ becomes almost insensitive to the odd zonal variations. The primary contributions to the solution for the odd zonal variations are expected from the eccentricity excitation. Most of the is even, and p = (/-1)/2 when 1 is odd. Flop and F op represent the inclination functions and their derivatives, and Go,q(e)and satellites, except for BE-C, are less sensitive to J? than to J3 Table 4. Sensitivity of Satellite Orbits to Odd Zonal Variations (per unit 8J/)x10 ' Starlette Ajisai Stella BE-C LAGEOS 1 LAGEOS 2 1 E/ o ø E/ r.0/ E/ r.0/ E/ o EJ' r.o/ E/ co/ 3 5 7 9 11-13.4 327.2-9.6 494.3 15.7-7855 -13.8 337.5-8.9 460.9 12.2-6105 7.1-175.5 3.6-187.6 6.7 3371 10.7-265.7 5.2-270.7-1.9-962 -1.7 44.1-0.5 25.8-1.6-784 -17.6 364.7 0.63-72.2-1.11 47.0-1.7 34.8-0.17 19.2-0.59 25.5 12.9-270.3-0.14 15.8 0.03-1.2-1.8 40.2-0.05 5.2 0.06-2.4-6.9 147.1-0.01 1.0 0.01-0.3

22,380 CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS and is. BE-C could make significant contributions to separate 6J7 from the effects of the lower odd zonal variations. The separability of the effects of the zonal variations is studied by examining the determinant or eigenvalues of the normal matrix A t e and At. o Previous analyses have relied on the analysis of f2 only for even zonal rate determination [Yoder et al., 1983' Rubincam, 1984]. A nonzero determinant for the even zonal normal matrix, A[ (< 8), indicates that the number of degrees of freedom of the linear dynamic system, described by Equation (1), is more than one for all seven geodetic satellites, except Etalon. Consequently, in principle, there is at least one additional observation equation for the even zonal variations from a single satellite orbit. However, the ascending node provides the dominant contribution because drag has a smaller effect on this element. On the other hand, a very small determinant (<10-4) associated with the odd zonal normal matrix, Af (< 9), for the satellites LAGEOS 1, LAGEOS 2, Etalon 1, Etalon 2, and Stella, as well as small values for Ajisai (= 0.06) and BE-C (= 0.8), indicates a weak separability of the odd zonal variations with those satellites. Starlette, whose determinant is 4.7, is the only exception due to its larger eccentricity. In the case of zeroth order in eccentricity, the number of degrees of freedom, for the linear dynamic system, described by Equation (1), is reduced to one equation for the odd zonal variations for any geodetic satellite. Consequently, owing to the orbit geometry, current geodetic satellites are less sensitive to the variations in the odd zonal harmonics, and only a linear combination of the odd zonal variations can be observed using a single satellite. Tables 3 and 4 show that the Starlette orbit contains strong zonal variation signals, but it cannot separate the effects of the low-degree terms because it is also sensitive to the effects of the higher-degree terms. The separation of the lower-degree zonal terms can only be achieved by using Starlette orbit information along with that from other satellites at different inclinations and altitudes. LAGEOS 1 is a prime candidate for this combination. LAGEOS 1 is 10 times less sensitive to the zonal variations than Starlette, and the sensitivity is limited to lower-degree (< 5) variations. Both satellites have a SLR tracking history on the order of 20 years. However, the two eigenvalues of the spectral norm of the matrix formed by.(-2[ (l = 2, 4) for the LAGEOS 1 and Starlette combinations are 48.1 and 1.9, respectively. The weak condition can be improved by including observations from other satellites, including Ajisai, LAGEOS 2, Etalon 1 and 2, and earlier satellites such as BE-C. After examining the possible combinations using the normal matrix of E ' and.(2f, the result suggest that except for Starlette and LAGEOS 1, significant contributions to separation of the higher-degree zonal rates, for example, the J4, J6, and J7, can be obtained from Stella and BE-C data. However, the contributions to the determination of long-term changes in the Earth's gravity field from each satellite are limited by weakness in tracking data distribution and the ability to model the tidal and nonconservative forces for respective satellites; the details will be discussed in the next section. 3. Satellite Data and Orbit Analysis Among the space geodetic measurements, the SLR data represent the most accurate and unambiguous range measurement. The SLR tracking technology supports passive spherical SLR targets for minimizing the nongravitational effects, which is particularly important in retaining the zonal signals in the high-accuracy orbit fits. Table 1 provides the SLR data histories, the number of the normal point observations, and the tracking stations for the data sets used in this investigation. To study the long-term zonal variations using SLR data, the first step is to determine an accurate reference orbit for each satellite by fitting the SLR data with the best available force and measurement models. The dynamic force models used in this study and the observed orbit perturbations for those satellites are discussed in the following sections. 3.1. Force Modeling The requirements for an accurate reference orbit for studying the temporal variations in the Earth's gravity field place stringent demands on the completeness and accuracy of the force and measurement models. The nominal models used in this study are based on the models defined by Tapley et a/.[1994] with exceptions related to the gravity and tide models, and the model for the zonal rates. The joint gravity model 3 (JGM-3) [Tapley et al., 1996] has significantly reduced the error perturbation spectrum in the geodetic satellite orbits at both low- and high-frequency bands, and it has produced range residual rms with an average of 5 cm or better for a typical Stadette 5-day arc. The modeling of the tidally induced variations in the Earth's external gravitational potential is crucial for isolating the satellite temporal signals arising from other causes. A "background" ocean tide model is used to (1) avoid the aliasing effects from the significant sideband tides and minor tides; (2) achieve a high level of accuracy and efficiency in computing the ocean tide perturbation; and (3) ensure uniform modeling to all satellites. On the basis of linear analytical ocean tide perturbation theory, the background ocean tide model used in this investigation was selected from the 474 ocean tide constituents using 30x30 fields, including the long-period equilibrium tides and the diurnal and semidiurnal tides from CSR 3.0 ocean tide height model, which was estimated from the TOPEX/Poseidom satellite altimeter data [Eanes and Bettadpur, 1995]. This "background" ocean tide model contains 1563 perturbation lines for 173 ocean tidal constituents and includes both prograde and retrograde terms. It accounts for all orbit perturbations with arms amplitude greater than 1 mm in the radial and normal components and 1 cm in the along-track component for the 12 satellites, including all of the satellites used in this investigation, and produces a smaller SLR range residual rms than previously developed ocean tide models. The ocean tides are modeled in the presence of the frequencydependent solid Earth tide model developed by Wahr [1981]. The improved free core nutation (FCN) period of 430 solar days, based on the analysis of very long baseline interferometry (VLBI) measurements [Herring et al., 1991], was adopted in the model for computing the frequency dependent Love numbers associated with the diurnal tides. The model, however, is free of anelasticity. In addition, the fourth-degree terms in the free-space potential due to the effects of the ellipticity of the Earth have been modeled in terms of the second-degree term [Cheng et al., 1995].

CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS 22,381 0.4 8 o.o -0.2 <1-0.4 -- -0.6-0.8, 4z778 43508 44238 44968 45698 46428 47158 47888 48618 49348 50078 Modified.luli Figure i. Residuals of the ascending node for LAGF. OS-1 long-arc orbit using the riomina! force mode]s, an eclui]ibrium long-period tide mode] and Jn = 0. 'the period includes data from May?,]9?6 to December. 3 ], ] 995. Date 3.2. SLR data and Reference Orbit Analysis LAGEOS 1. The LAGEOS 1 satellite is a critical resource for the study of geodesy and geodynamics. The study of its The long-arc orbit determination technique is well suited for studying the long-term changes in the Earth's gravity field by the analysis of satellite orbit perturbations. The long-arc technique requires the precise computation of a single dynamically consistent satellite trajectory using the best available dynamic force and measurement models to fit the SLR tracking data over a long time span [Tapley et al., 1985, orbit has occupied a central role in satellite geodesy during the last two decades, and details of the analysis are given, for example, by Tapley et al., [1985, 1993] and Eanes [1995]. The LAGEOS 1 SLR data set is one of the primary data sets in this investigation because of its long time span and high orbit accuracy. The LAGEOS 1 node residuals obtained using the nominal dynamic model during the period from May 1976 to 1993]. Any error in the dynamic force model used to propagate December 1995 are shown in Figure 1. The large curvature in the satellite orbit will result in a systematic laser range residual. At present, a single, dynamically consistent orbit for the node residuals is caused by a combination of errors in the ocean tide model, unmodeled secular changes in the even zonal LAGEOS 1, over a 20-year period, can be determined using the Encke integration technique [Lundberg et al., 1991]. However, owing to the coupling between errors in the dynamic model and the numerical integration process, it is difficult to converge a single orbit over a time span longer than 5 years for low-altitude satellites such as Starlette. Consequently, in this study, the long-arc technique was used for the entire tracking history of LAGEOS 1, LAGEOS 2, Etalon 1, and Etalon 2, while a multiple 1-year arc technique was used to piecewise fit the long-period variations in the orbit for Starlette, Ajisai, Stella, and BE-C. Owing to smaller sensitivities, the arc length using the entire data span is employed to strengthen the signal from the zonal variations in the LAGEOS and Etalon satellite orbits. The 1-year arc length is the minimal length required to separate the annual and semiannual variations in the Earth's gravity field associated with the seasonal meteorological mass transport within the Earth system. As a starting point for this investigation, the reference orbit for each satellite is determined by estimating the satellite initial position and velocity at the arc epoch, along with a constant along-track acceleration (C r) in each 15- day interval for the LAGEOS and Etalon type satellites, or a drag coefficient (Ca) in each 3-day interval for Starlette, Ajisai, Stella, and BE-C, to reduce the nongravitational model errors and ensure that the mismodeled zonal gravitational signal is not absorbed. The laser range residual rms for each of the four high-altitude geodetic satellites are given in Table 1 using the nominal harmonics, and the effects of the uncertainties in modeling the response of the inelastic Earth to long-period (18.6 year) tidal forces. The 18.6-year tidal signal and the secular variation in the Earth's gravity field become separable using the 19.5 years of LAGEOS 1 SLR tracking data. The anomalous eccentricity excitations for the LAGEOS 1 orbit resulting in an irregular fluctuation in the LAGEOS 1 determined third-degree zonal harmonic at near-annual and 561-day periods were reported by Tapley et al. [1993]. Recent studies have demonstrated that the LAGEOS 1 eccentric anomaly is mainly due to a mismodeling of the LAGEOS 1 surface radiation reflectivity coefficient, the surface temperature anomaly, and the evolution of the satellite spin axis [Metris et al, 1997; Rubincam et al., 1997; Martin and Rubincam, 1996]. At present, the exact models for these effects are not available due to lack of physical information on the LAGEOS 1 satellite. The parameters for the third-degree annual tide (Sa) are adjusted over 1-year intervals to reduce the nongravitational effects on the eccentricity vector for LAGEOS 1 orbit. LAGEOS 2. The LAGEOS 2 satellite was launched in October 1992. Its altitude is 110 km lower than LAGEOS 1. With an inclination of 52.6 degrees, the tidal perturbation spectra of LAGEOS 2 are different from that of LAGEOS 1. Analysis of the orbit residuals from a 3.2-year arc of LAGEOS 2 data shows that the dominant variations in the eccentricity and node residuals are due to tidal perturbations at the K, P, S, K2, and S2 tidal frequencies, and the seasonal annual and semiannual variations in the zonal harmonics. However, the observation history for LAGEOS 2 is short when compared with LAGEOS 1. Although the 3.2-year SLR data for LAGEOS dynamic models discussed above. Plate 1 shows the rms for 1- year arc orbit fits for Starlette, Ajisai, Stella and BE-C. The larger rms is an indication of errors in the nominal models, 2 do not provide a strong observation equation for determining including the tide models. The observed perturbations in the reference orbit for each of the eight geodetic satellites used in the zonal rates, they are capable of improving the tidal perturbations and reducing the variance in the combination this investigation are briefly discussed as follows. solution for the zonal rates.

22,382 CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS Starlette. Starlette has been tracked extensively by the the parameters for the S and S2 tides, which have to be adjusted global SLR network since it was launched in March 1975. Because of its significant sensitivity to the gravitational signals, its spherical shape with a small area-to-mass ratio and the long tracking history, the Starlette SLR data set is an to reduce the errors in the S and S2 tide models. The laser range residual rms for the Stella orbit fit over a 6- day interval was 4.5 cm using a tuned JGM-3 gravity model, referred to as JGM-3b (J. Ries, personal communication, important resource for studying the long-term changes in the 1996). The accuracy of the 6-day orbit fit for Stella is Earth's gravity field. Details of the analysis of the Starlette comparable with Starlette, but better than Ajisai, and indicates orbit have been reported by Williamson and Marsh [1985] that the accuracy of the current modeling of the and Cheng et al. [1990, 1992]. The large laser range residual rms for the 1-year arc shown in Plate 1 is an indication of the errors in the dynamic force models, primarily the atmospheric drag and ocean tide models. The magnitude of the residual rms can be reduced significantly by solving for appropriate tide model parameters, and during 1980-1981 and 1989, the residuals are correlated with higher solar activity and its nongravitational forces, including drag, radiation, and thermal forces, is at the same level as Starlette and Stella. The RMS for the 1-year orbit fits to Stella SLR data using the JGM-3b gravity model are 4.5 and 4.8 m for 1994 and 1995, respectively, which are more than 3 times larger than the rms of the Starlette orbit fits. The laser ranging residual rms for a 1-year orbit arc can be reduced 27% by adjusting a 15- influence on the atmosphere density variations [Schutz et al., day solar radiation reflectivity coefficient. However, by 1993]. Ajisai. Ajisai is a spherical geodetic satellite, which was launched in 1986 by the National Space Development Agency of Japan. The Ajisai satellite is a hollow sphere with a fixing the reflectivity coefficient to the Starlette determined value, the laser ranging residual rms can be reduced by 78% after adjusting the coefficients for the S (degree 3 and 5 ) and S2 (degree 2 and 4) tides. With this approach, the resulting diameter of 2.15 m and an area-to-mass ratio of accuracy of the orbit fit to the 1-year arc of Stella data becomes comparable with the Starlette orbit fit. However, the extremely large values for the estimated coefficients of the S 5.34x10 '3 m2/kg, which is 7.7 and 5.6 times larger than that of LAGEOS 1 and Starlette, respectively. The larger area-to-mass ratio results in larger surface forces, including the radiation- and S2 tides imply errors due to aliasing with the mismodeled induced thermal effects as discussed by Sengoku et al. [1995, solar radiation induced forces on Stella. 1996]. Owing to its higher altitude, the rms for a 1-year orbit The orbital residuals for the node and inclination fits to fit is smaller than Starlette's, as shown in Plate 1, but the Starlette, Ajisai, and Stella during 1995 are shown in Plate 2 averaged rms for orbit fits over a 6-day interval is slightly and 3. Spectral analysis indicates that the dominant larger than Starlette due to the influence of the errors in the perturbations in the orbital node have similar annual and surface force modeling. Although it is at nearly the same semiannual variations. The orbit perturbation spectra for inclination as Starlette, Ajisai is less sensitive to changes in the Earth's gravity field than Starlette due to its higher altitude. The perturbation spectra of the two satellites are Starlette and Ajisai are similar since they have almost the same inclination. The perturbations in Stella's node residuals are larger than those for Starlette and Ajisai due to the aliasing similar due to their similar inclinations. Thus, although the from the K1 and P tides. The perturbations in the orbital Ajisai SLR data set is the third longest SLR data set, it does not inclination are mainly due to the even-degree K and S2 tides provide a unique additional observation equation for the zonal for the three satellites. The perturbations in Stella's rates, but it can strengthen the information for the zonal rates inclination are also attributed to the even-degree K2 and the at the 50-degree inclination and reduce the effects of tidal odd-degree S2 tides. perturbations in a combination solution. Stella. Stella was injected into a Sun-synchronous orbit, B E-C. Most of the current geodetic satellites are orbiting at inclinations ranging from 50 to 110 degrees. To extend the along with the SPOT 3 satellite, by the French Center National inclination coverage, the Beacon-Explorer C (BE-C)satellite d'etudes (CNES) in September 1993. This satellite has the was added in this study in an effort to improve the separation same physical characteristics as Starlette but with a different orbit. The altitude is lower than Starlette by approximately of the zonal rate terms. The BE-C is approximated as an eightsided prism with four large solar arrays equally spaced around 160 km, and it is nearly circular. The perturbation spectra for the main body [Safren, 1975]. The satellite attitude is Stella due to the K1, P1, T2, and K2, tides are all close to the geomagnetically stabilized, and for this reason, the BE-C laser annual and semiannual periods, which results in a significant array was not visible from southern hemisphere stations with aliasing effect on the annual and semiannual variations in the latitudes greater than 20 degrees. The complicated shape and Earth's gravity field from Stella data. The significant attitude of BE-C causes difficulties in modeling both the resonances between the Stella's orbit motion and the apparent nongravitational forces and the correction for the diurnal and semidiurnal motion of the Sun (i.e., S and S2 tides retroreflector offset from the center of mass. Although the BE- C satellite was launched in April 1965, the useful SLR data set covers only the period from January 1980 to June 1986, and after June 30, 1986, BE-C was no longer tracked. The BE-C ) cause perturbations in orbit ascending node and inclination with periods of 156.6 and 78.8 years. Consequently, owing to this deep resonance, the accuracy of the model for these two tides is critical for a Stella long-arc orbit determination. However, the two solar tide constituents are significantly affected by the thermal excitation of the atmosphere, and the difficulty in modeling the thermal effects limits the accuracy of models for these tides. The analysis of residuals from a 2-year orbit fit to the Stella data shows that the tidal perturbations for S and S2 dominate the node and inclination residuals. The zonal rate induced signals in Stella's orbit will be aliased into was tracked with an averaged of two or three passes per day by 10-13 stations during this period. The distribution of the stations tracking BE-C is in the range of latitudes between- 16.45 degrees (Arequipa, Peru) and 52 degrees (Potsdam, Germany). It is difficult to converge a 1-year arc during 1982 and 1983 due to the very sparse SLR data distribution. The larger rms during this period is due primarily to larger measurement errors and sparse data distribution. The B E-C data

... CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS 22,383 6O0 5O0 4O0 3O0 2O0 100 /4 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 Year (+ 1900) Plate 1. RMS of 1-year orbit fits for Starlette, Ajisai, BE-C, and Stella using the nominal force model and adjusting initial conditions and 3-day Ca. '..,,;,a.... 49353 49418 49473 49533 49598 49653 49713 Modified Julian Date Plate 2. The node residuals from a 1-year orbit for Starlette, Ajisai, and Stella during 1995 using the nominal force model and an equilibrium long-period tide model. 0.08 0.02,._.-.,.-... "..:.'i...'..'...-..'.7 i.'-'i -'-' -; "--*'-' Ajisai _ -. _ '"'.'"' s.n, - - - ::': -:: : :'::'... ':![... -'- o.o0-0.02-0.04 --:-,... I...,' --,, I,, ; t, I... ;--, ' 49353 49413 49473 49533 49593 49653 49713 Modified Julian Date Plate 3. The inclination residuals from a 1-year orbit for Starlette, Ajisai, and Stella during 1995 using the nominal force model and an equilibrium long-period tide model.

22,384 CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS set was given a low weight in the combination solution. The perturbations, studying the effects of nonzonal variations on period of the dominant K1 ocean tide perturbation is similar to the satellite orbit is difficult. that of Starlette. However, the third-degree S tide produces a significant 14.3-year periodic perturbation on the eccentricity of BE-C. A clear drift is observed in the BE-C eccentricity In this investigation, a statistical parameter estimation approach is used to reduce the observed variations in zip by directly adjusting the pertinent model parameters. These residuals. The fourth-degree S2 tide produces a small-amplitude model parameters include the parameters for the secular change 6.3-year perturbation. The analysis of BE-C node residuals shows that the dominant perturbations for a 1-year arc are at the annual and semiannual periods. the orbit's behavior, as provided by the tracking data, is used, and an appropriate parameterization for the satellite dynamics EtaIon I and EtaIon 2. The Etalon 1 and Etalon 2 is adopted to account for all the perturbation signals in the satellites were launched by the former Soviet Union in January satellite orbit. The weakness in the parameter estimation 1989 into nearly circular orbits at the same altitude and inclination but with different ascending nodes. Owing to their very high altitude (25,500 km), the satellites are only influenced by the variations in J2 and J3. The sparseness of the approach is that there will be aliasing effects due to any unaccommodated signal left in the residuals. However, the effects on the estimated parameters due to errors in the unestimated parameters in the nominal model can be studied tracking data for both Etalon 1 and Etalon 2 satellites leads to using a consider covariance analysis. In addition, by a limitation in the accuracy of the orbit fits. Only 2776 and 2415 acceptable passes were obtained between 1989 and 1995 for Etalon 1 and Etalon 2, respectively. The two satellites are identical in physical characteristics and structure with a mass of 1346 kg and a radius of 64.7 cm, and are covered by 2140 combining the data sets from satellites orbiting at various altitudes and various inclinations, the aliasing effects from the variations in the high-degree terms can be reduced, and the separability of the zonal variations can be improved as well. An optimal weighting estimation procedure is used to obtain cube corner laser reflectors. The large surface of the satellites the best combination of the multisatellite data sets. The results in large solar radiation induced effects on their orbits. The solar radiation reflectivity coefficients estimated using 6.2 years of SLR data are 0.192 for Etalon 1 and 0.228 for Etalon 2. These larger values may reflect the thermal effects on both satellites. The laser range residual rms for Etalon 1 is approximately 2.4 times larger than that for Etalon 2 due to a larger perturbation in the semimajor axis and eccentricity of the Etalon 1 satellite. Analysis of the orbit residuals, using the nominal dynamic force model, shows a drift in node and inclination with values of 9 milliseconds of arc (mas)/yr and - 2.8 mas/yr for Etalon 1, and 13 mas/year and 1 mas/yr for optimal weight factor for each satellite data set is determined, along with the estimate for the selected parameter set, by minimizing a joint probability density function, which depends on both the observation equations and the weight factors [ Yuan, 1991 ]. The static gravity solutions, such as those represented by the JGM type models [Nerem et al., 1994; Tapley et al., 1996], used information from over 30 satellites at various altitudes and inclinations to reduce the aliasing effects among the geopotential coefficients. The requirement for temporal distribution of satellite data is not critical for recovering the Etalon 2. In addition, a very long period variation in the static gravity model, but it is a key requirement for extracting eccentricity residuals is observed. The apparent secular drift is mainly caused by the errors in the K and K2 ocean tide, which produce the perturbations in node and inclination with periods of 29 and 14.5 years, and the perturbations in eccentricity with periods of 32.5 and 15.3 years, respectively. Trajectory comparison indicates that the two orbits are not identical, with an averaged difference of 4 km in semimajor axis and a maximum difference of 0.9 degrees in inclination. Over a 6- the signal due to the temporal changes in the Earth's gravity field. Owing to limitations in the existing SLR data sets, this investigation did not seek to optimize the spatial distribution of the SLR data sets over a particular time period but utilized all of the satellite data sets with continuous tracking history of more than 1 year. The determination of the zonal rates and 18.6-year tide model parameters is largely based on information from the longest data sets, Starlette and LAGEOS year time span, the long-period variations in the inclination 1. The data from satellites with shorter observation intervals and eccentricity are very different for these two satellites. are used to reduce the shorter period (<18.6 years) signals in Further study is required to understand the orbit perturbations the observation residuals and to help improve the solution for these two satellites. conditioning and strengthen the separation of the individual parameter' s contribution. 4. Solution Methodology As discussed in section 3, the contribution of a satellite data set to the parameters estimated in a combination solution To determine the long-term changes in the Earth's gravity depends not only on the perturbation sensitivity, but also on field from analysis of SLR data, there are two approaches the length of tracking data history and the accuracy of the currently used. One approach maps the laser ranging residuals, reference orbit. For example, Ajisai is more sensitive to the zonal variations than LAGEOS 1; however, LAGEOS 1 will Ap, to the time series of orbit element residuals over a short time span (a few days) and uses an analytical perturbation theory to determine the zonal rates from the analysis of the residuals for a set of non-singular orbital elements [Yoder et al., 1983] or from the excitation time series [Eanes, 1995]. This approach takes advantage of time series analysis techniques and the ability to handle the unmodeled signals and stochastic fluctuations contained in the orbital element corrections by filtering techniques. However, since this approach is based on a first-order approximation of the zonal in the zonal harmonics, Jl. The complete information about provide a more accurate solution of Jt (/<4) due to its longer tracking history and more accurate force model. To evaluate the role of a specific data set, the subset solution method has been used by Lerch [1991]. As we have noted, the separation of zonal rates can only be obtained by a combination of the data sets from the Stadette and LAGEOS 1 with the other data sets. A subset solution of Jl (l = 2 to 6) without using Starlette data will lead to an ambiguous interpretation of Starlette's contribution since the solution is not well conditioned without

CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS 22,385 Table 5. Comparison of Satellite Results for J2, J3, J4, is, and 6 J2 J3 J4 J5 J6 Data set Source -2.7+0.4-1.3_+0.5-1.4+1.0 2.1_+0.6 0.3_+0.7 eight satellites this study -2.6+0.3-0.9+1.3 L-l, S Eanes and Bettadpur [1996] -2.8+0.3 1.6_+0.4' 0.2+1.5 L-l, L~2, S, A Nerem and Klosko [1996] -3.0- -_0.5-1.8+0.1' -0.8+1.5 L-l, L~2, A Cazenave et al. [1996] -2.6+0.5-1.2+0.5-1.1+1.0 2.3+1.5 six satellites Cheng et a/.[1993] -2.6 L-1 Nerem et al. [1993] -2.8+0.4 L~I Gegout and Cazenave [1991] -2.5+0.3-0.1+0.3 0.3_+0.6 S Cheng et a/.[1989] -2.6+0.6 L-1 Rubincam [1984] -3. L-1 Yoder et al. [1983] * Denotes the lumped result. Abbreviations are L-I, LAGEOS 1; L-2, LAGEOS 2; S, Starlette; and A, Ajisai. Values are 10-11/yr. Starlette. That is, results from subset method will depend on the accuracy of the estimates for the zonal rates from the combination of other data sets. The following computational efficient algorithm is proposed to evaluate the contribution of each data set to the combination solution. The difference between the complete solution and a subset solution, which lacks the jth data set, defines the contribution of the jth data set to the estimated parameter [Sage and Melsa, 1971]. The covariance for the solution difference, AX, can be computed as E[zlX r] = t_tjpmhjrri I P,n z (4) 18.6-year zonal variations in the observation residuals. However, on the basis of analysis of the reference orbit residuals in section 3, significant long-period tide perturbations are observed for all satellites. In addition, the amplitude and phase of the annual and semiannual variations change from year to year. In order to reduce the tidal and nontidal periodic variations in the observation residuals, the tide model parameters are included in the estimation parameter set. The parameters adjusted in the optimal weighted least squares estimation of multisatellite data sets are classified as (1) global parameters, which are determined using all of the satellite data sets; (2) common parameters, which are determined using the information from Starlette, Ajisai, Stella and BE-C at yearly intervals with the available data set for each satellite; and (3) local or satellite dependent parameters. The global parameters include,]2, J3, J4, is,,]6, the 18.6-year tide, and other selected tidal parameters for Ssa (degree 3), Mm, Mt (degree 2 and 3), Q1, O, P, K, S, N2, M2, r2, S2, K2 (degree 2 to 5), and the constant zonal geopotential coefficients (degree 2 to 6 and 8). The common parameters include the yearly values of the Sa (degree 2 and 3) and Ssa (degree 2) tide parameters, which characterize the seasonal annual and semiannual variations. The satellite-dependent parameters include the satellite position, velocity, and a solar reflectivity coefficient for each arc; 3-day drag coefficients, Ca for Starlette, Ajisai, BE-C, and Stella; and 15-day Cz parameters and the yearly Sa (degree 2, 3) and Ssa (degree 2) tide parameters for LAGEOS 1, LAGEOS 2, Etalon 1, and Etalon 2. The tidal parameters of the yearly Sa (degree 2, 3) and Ssa (degree 2) tides are estimated separately for satellites at low and high altitudes because of the different arc length used for those satellites and the significant aliasing effects due to other where Pm is the covariance of a complete solution, H/and Rj represent the matrix of observation partials and the error covariance matrix for the jth satellite, respectively, and $tj is tides and the nongravitational effects on the yearly Sa (degree 2, 3) and Ssa (degree 2) estimates from the LAGEOS satellites. a weighting factor for the jth satellite. The diagonal element in this matrix provides the error estimate for a solution parameter when the jth data set is absent. After the Pm 5. The Solution for 2, 3, 4,, 6, and the 18.6- covariance matrix is obtained, the contribution to the solution Year Tide parameters from all of the data sets used in the solution can be evaluated based on the Equation (4). (Table 8 shows the Tables 5 and 6 compare the solution for J2, J3, J4, is, J6, and the 18.6-year tide parameter solutions with solutions from dependence of the estimated parameters,]2, J3,]4,]5, and,]6 on previous investigations. For the solution obtained in this each of the specific satellite SLR data sets.) The long-arc technique used in this investigation has the investigation, the correlation coefficient is -0.75 for,]2 and J4, advantage of strengthening the signals for the secular and 0.76 for J2 and J6,-0.86 for J4 and S+20 The other correlations are generally less than 0.5, which J6, and 0.57 for J2 and 18.6. implies a good separation between J3 and is, the zonal rate, and the 18.6-year tide parameters with the data sets and technique used here. Using global tide gauge data, Trupin and Wahr [1990] obtained an amplitude for the 18.6-year ocean tide of 1.13+0.22 times the equilibrium amplitude. The anelasticity effect on the 18.6-year tide is predicted to be 0.44 cm. Combining the anelasticity effects and the ocean tide values, the amplitude of the 18.6-year tide is estimated to be 1.53 cm Table 6. The Estimated and Equilibrium Values for the 18.6- Year Tide C t82.ø6, cm,-, +20 _ Ola.6,cm This study 1.57_-4-0.15-0.08_-4-0.25 Equilibrium 0.967 0.0

22,386 CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS Table 7. Comparison of Estimates of J2, J3, J4, is, and J6 J2 J3 J4 J5 J6 Data Set Source -2.7+0.4-1.3+0.5-1.4+1.0 2.1 +0.6 0.3 +0.7-3.07-1.27-1.21 2.15-0.38-2.37-1.33-1.58 2.20 0.50-2.76-1.30-2.01 2.18 0.51-2.68-1.24-2.50 2.52 0.76-2.75-1.26-1.88 2.32 1.11-2.83-1.31-1.74 2.20 0.93-2.84-1.16-1.84 2.80 0.94-2.75-1.22-1.94 2.49 1.14-2.68-1.15-2.27 2.80 1.31 recommended case 1 from eight satellite case 2 from eight satellites case 3 from eight satellites case 3 without LAGEOS 2 (L-2) case 3 without BE-C case 3 without Stella case 3 without L-2/Stella case 3 without BE-C/Stella case 3 without BE-C/Stella/L-2 Values are 10'l /yr. [Zhu et al., 1996]. This theoretical prediction of anelasticity effects on the Earth tide at the 18.6 year period is in good agreement with the estimate obtained in this study and the results reported by Eanes and Bettadpur [ 1996]. The accuracy of the estimates of the zonal rates determined from long time series of multisatellite SLR data has been Stelland BE-C. The changes in the estimates for J2, J3, J4, changes in J4 and J5 is significant. The recommended solution and J6 are less than 0.1xl0 ' yr - if the tide model parameters and the quoted uncertainties for J2, J3, J4, is, J6, and the 18.6- other than K, S, and 9.3-year tides are not adjusted. The changes in the solution for J2, J4 and J6 are 0.34x10 ' x,- 1.3x10 ', and 0.73x10 - x yr - if the parameters for K tide are not adjusted; -0.14x10 ' 0.46x10 - -0.12x10 -l yr 'l if the parameters for S tide are not adjusted. The changes in the solution for J2, J3, J4, is, J6, and the 18.6-year tide are estimates for J3 and J5 are stable within the quoted uncertainties. However, in the current solution, there are significant variances in the estimates for the J2, J4, and J6. These effects can be reduced only by adjusting the zonal coefficient, Js, Jm, and J12, and the 9.3-year tide parameters. Table 7 compares the estimates for,]2, J3, J4, is, and J6 for several cases. The parameters adjusted in case 1 are those defined in section 4, along with Jt (l = 2... 6)and Js. Case 2 is case 1 with the 9.3-year tide adjusted. Case 3 is case 1 with the Jm and J12 adjusted. The changes in the JGM-3 values for Js, Jm, and J12 are smaller than the variance given by the JGM-3 gravity model. The significant effects may be an indication of the variations in the observation residuals due to the signals of the high-degree (l = 8, 10) even zonal harmonics, although their rates are not separable with the current data sets. As shown in Table 7, adjusting the 9.3-year tide model parameters will cause changes in the estimates for J2, J4, and J6 and result in a small correlation between the estimated parameters. The equilibrium ocean tide at a 9.3-year period has a small difficult to verify. Only J2, has been evaluated with amplitude of 0.0097 cm. The amplitude is estimated to be 0.43 confidence. For the parameter estimation approach, the equilibrium value. Further study is required to understand the uncertainties in the estimated parameters depends on the correlation between the estimated model parameters and the signal absorbed by the 9.3 year tide parameters. With respect to case 3, the changes in the solution for J4 and J6 are - aliasing effects due to errors in the unestimated parameters. 0.5x10 -, 0.6xl0- yr - without BE-C, and -0.34x10 ', The variations in the value of the estimated parameters, when 0.4x10 - yr ' without Stella. Other cases were also tested to different parameter sets are adjusted or different data combinations are used, lead to some measure of the uncertainty verify the effects of BE-C and Stella shown in Table 7. in the estimated parameters. The tidal model parameters for K1, Clearly, the estimates of J4 and,]6 are sensitive S, and S2 along with other tides were adjusted to reduce the and B E-C data, as has been shown in section 2, but their tidal perturbation in the observation residuals, especially for differences with and without using BE-C and Stella are within the quoted uncertainties. The effect of LAGEOS 2 on the to the Stella year tide parameters are the results synthesized from analysis of a number of cases. Using the algorithm described by Equation (4) and neglecting the correlations between the global and local parameters, the square roots of the covariance of the solution difference for J2, J3, J4, is, and J6 are listed in Table 8 for each negligible when adjusting a set of low-degree and low-order data set. Etalon 1 and Etalon 2 data, which have only very nonzonal coefficients complete through the fifth degree and small magnitude, are omitted. The values in Table 8 indicated order. The corresponding zonal coefficients are also adjusted the contribution of each satellite data set to the estimates for simultaneously in the solution for the zonal rates. The J2, J3, J4, is, and J6 obtained the combination of all of data Table 8. Solution Difference Variance for Satellite Date Set Satellite J2 J3 J4 J5 J6 Starlette 0.113 0.019 0.348 0.071 0.289 LAGEOS 1 0.056 0.032 0.043 0.031 0.073 LAGEOS 2 0.019 0.003 0.031 0.009 0.033 Ajisai 0.022 0.009 0.052 0.035 0.060 BE-C 0.009 0.002 0.027 0.007 0.023 Stella 0.003 0.001 0.007 0.006 0.014 Values are 10 ' /yr.

CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS 22,387 sets. The solutions, except for,]3, are more sensitive to the thathe values of J3 and J5 will not change significantly if J? Starlette data set. The LAGEOS 1 data provide a significant can be adjusted. As an example, the sensitivity for,]2 and,]4 to contribution to,]3. Primary contributions from the BE-C data an error in,]6 is 0.4x10 - and -0.8x10 - yr '. This sensitivity set are to,]4 and J6. Only a small contribution to the zonal indicates that if the J6 is not estimated and has an error of rates comes from Stella. The estimate for J4 is more sensitive 0.7x10 - yr -, it could causerrors in the estimates of J2 and,]4 to Starlette and Ajisai data. Since the major orbital of 0.28x!0 - and-0.56x10 - yr -, respectively. It is quite information of,}4 for Starlette and Ajisai is in their along-track clear that the effects of J6 on the estimates of J2 and J4 are component as shown in Table 3, the accuracy of the estimate within their quoted uncertainties. The solutions for J2 and J4, for J4 is limited by the accuracy of the modeling of the presented here, have the same value when J6 is not estimated. nongravitational forces on Starlette and Ajisai. A more important point illustrated in Table 8 is the dependency of the sensitivities of the solution for the zonal rates on the satellite tracking data span. The Ajisai orbit is more sensitive to the zonal variations. However, Table 8 than LAGEOS 1 due to its shorter tracking history. The same conclusion holds for LAGEOS 2. With extension of the SLR tracking data, the contributions from LAGEOS 2, Ajisai, and Stella will increase. Consequently, improvements in the accuracy of the solutions for the zonal rates can be expected. At present, we are still unable to estimate values for Jl (l > 7 ) from the current SLR data sets. The temporal extension of the current SLR data sets will potentially enhance the solution of zonal rates beyond degree 6. The errors in those unestimated high-degree coefficients, Jl J2, J3, J4, and J5 given by Cheng et al. [1993] was obtained by (l > 7 ), lead to reduced accuracy and biased estimates of the In general, the solution of the estimated parameters with the aliasing effects from the errors in the unsolved parameters can be expressed as x = x +S, where x is the estimate of the state vector x; c is the a priori value vector for the unsolved parameters; and S is the sensitivity matrix defined as S =- PxHxrR- Hc, where Px is the solution covariance matrix, Hx is the partial matrix for the estimated parameters, R is the error covariance matrix of observations, and the Hc is the partial matrix for the unestimated parameters [Bierman, 1977]. The results from an optimal estimation procedure show that the solution of J2, J3, J4, is, J6, and 18.6-year tide parameters is mostly affected by the errors in the higher-degree Jl (/> 6), and the third-degree 18.6-year tide parameters. When for J3 and the values predicted from mantle viscosity structure the sensitivity (>0.1) of J2, J3, J4, is, and J6 to the higher- using the ICE models of late Pleistocene-Holocene melting degree Jl are included, the so-called effective value of J2, J3, J4, [Ivins et al., 1993; Mitrovic and Peltlet, 1993]. The J3 is sensitive to present-day glacial mass balance. The maximum is, and J6 can be expressed as follows (with units of xlo - yr - for the zonal rates and units of xlo - yr - cm - for tides): amplitude for J3 predicted from the global ice model ICE-2G was -0.36x10 - yr -. However, the ICE-3G model predicts the :r-, +30 _ +30 J2 = -2.7+0.4-0.36/a + 0.26J 0 + 0.26J 2-11 oc a.6 + 26S a.6 bounds for J3 to be 2.2x10 - yr - [Ivins et al., 1993]. From +30 J3 = -1.3_+0.5 + 0.36J7 + 0.1 gill- 236C.3-0.66S18.6 analysis of satellite SLR data, the estimates for,]3 given by Nerem and Klosko [1996] and by Cazenave et al. [1996] are J4 = -1.4_+1.0 + 1.16J8 + 0.36J 0-0.56J 2-326Ci a - _ +30 _ +30 opposite and represent a lumped-sum effect of J3 and is. The J3 J5 = 2.1_+0.6 + 0.26J7 + 0.46J9 - O.16J - 546C18.6-296S18.6 and J5 are inseparable with the technique an data sethey used. J6 = 0.3+0.7 + 0.86J8 + 0.26/ 0 + 0.46J12 The solution of J3 given by Cazenave et al. [1996] generally _ +30 _ +30-13.86C 8.6-7.6 S 8.6 (5) The third-degree 18.6-year ocean tide is assumed to be in r-, +30 o +30 equilibrium. The values of 8.6 and o 8.6 are 0.01609 and 0.0, respectively, which induces a variation of 0,16x10 - yr - in J3. It is not known whether this tide departs from its equilibrium. shows that the contributions from Ajisai to,]2 and J6 are less The solutions of J3, J4, and J5 will change by approximately 0.25x10 ' yr ', and J2 does not change when adjusting the third-degree 18.6-year tide parameters. Although different time spans were used, the comparison of the results in Table 5 shows that the solutions of J2 and J4 are in general agreement. Cazenave et al. [1996] only used 11 years of SLR data for LAGEOS 1, 5 years for Ajisai, and 2 years for LAGEOS 2. Nerem and Klosko [1996] used the data sets for LAGEOS 1, Starlette, and Ajisai over the period from 1986 to 1994, and two years data from LAGEOS 2. The solution for adjusting the 18.6-year tide parameters and the mean values for the second- and third-degree annual tidal parameters using the SLR data sets for six satellites covering the period through the end of 1992. An additional 3 years of data from LAGEOS 1, Starlette, Ajisai, Etalon 1, Etalon 2; the 3 years of data from LAGEOS 2; and 2 years of data from Stella were also included in the current analysis. With these data sets, J6 is separable for the first time, although it has a large uncertainty and high correlation with J2. The values of J4 and J6 predicted from the global ice model ICE-3G [Ivins et al., 1993] are in the range of (-2.4 to 0.4) and (-0.22 to -0.5) and are in good agreement with the satellite results from this it vestigation. There is a discrepancy in comparing the satellite solution agrees with the results in this investigation. We have estimated,]2,]3,]4,]5, and the 18.6-year tide parameters using The results in Equation (5) show that significant changes can the 20.8 years of Starlette SLR data only and havestimated J2, occur in the estimates for,]4 and,]6 if the neglected value of,]6,]3, and 18.6-year tide parameters using the 19.6 years of has the same order of magnitude as,]4 or J6. However, the small LAGEOS 1 SLR data only. J2 and J4 or J3 and J5 are inseparable sensitivity of J3 and J5 to the neglected value of J? indicates using this 19.6-year LAGEOS 1 data set only.

22,388 CHENG ET AL.' DETERMINATION OF LONG-TERM CHANGES IN ZONALS The estimate for J3 from LAGEOS 1 data only is - 2. lxl 0 -ll yr -1, which is the lumped sum value of J3, is, and? as shown in Table 4. Neglecting the effect of J7 and with the sensitivity coefficient of-0.27 for the 19.6-year LAGEOS 1 data set, J3 is estimated to be -1.53x10 - yr - from the LAGEOS estimate for J3 from LAGEOS 1 dat agrees with the Starlette result as discussed before. This agreement indicates that the 1 data set using the value of J5 from the multisatellite solution. effect on the estimate of J3 due to the LAGEOS 1 anomalous On the other hand, the simultaneous solutions of.j3 and.]5 from eccentricity excitation is insignificant in this investigation. Starlette data only are -1.45x10 - and 3.33x10 ' yr -. Thus In any case, further efforts on modeling the nongravitational the estimates for J3 from Stafiette and LAGEOS 1 alone are in forces on satellites are required to improve the determination of the odd zonal rates. good agreement. This estimate for J5 is questionable, even though 20.8 years of Starlette SLR data were used. This follows from the weak separability of the odd-degree zonal variations from a single satellite as discussed in section 2. The estimates for J3 and J5 became -1.2x10- and 2.87x10 ' yr - when combined with the LAGEOS 1 data set. The combination of Starlette and LAGEOS 1 data with other satellite data further improved the separation of J3 and J and results in the values reported in this paper. When only J3 is estimated along with J2, J4, and 18.6-year tide parameters using the Starlette data alone, the estimate for J3 is 2.01x10 - yr -. This value of J3 as an effective "J3" is the lumped sum of the odd-degree (>3) zonal rates. However, it turns outhat it is just the lumped sum of J3 (-1.45x10 'll yr -1) and J3 (3.33x10 - yr - ) with the sensitivity coefficient of 1.04 from Starlette 20.8-years data alone. This fact may imply that the signals arising from J? and the high-degree zonal rates are small or not detectable from the multi-one-year-arc Starlette 0.4+0.7 mm/yr from Antarctica and -0.3+0.3 mm/yr from orbits over the 20.8-year time span. The value of "J3" is more Greenland, although the results for sea level change need to be than 2 times the lumped sum of J3 + 1.04 J5 with the values of that further efforts are warranted to improve both J3 and J5 from the multisatellite solution in this investigation determination and interpretation of the odd-degree zonal rates. (Table 5) because of the aliasing effects on J5 from other longperiod tide effects on the Starlette orbit. The lumped sums of 3 6. Conclusion and J are very different, with opposite signs for LAGEOS 1 The sensitivity of the geodetic satellite's orbit to the zonal and Starlette, because of the different orbit characteristic, as shown in Tables 2 and 4. Consequently, the positive value of the effective "J3" estimated from Starlette data only, which agrees with results from the ICE-3G model and the result given by Nerem and Klosko [1996], does not represent the actual value for J3. In fact, with and without solving for is, the difference between the estimates of J3 is less than - 0.2x10 - yr - using the multiple satellite data sets, since with the multi-satellite data, J3 and J5 are less correlated. The determination of the odd zonal variations depends on the accuracy of the satellite orbit eccentricity and the argument of perigee. The radiation-induced large signal in the orbital eccentricity excitation vector of LAGEOS 1 results in an irregular fluctuation in the LAGEOS 1 determined third-degree annual tide parameters [Tapley et al., 1993]. However, such a phenomenon does not appear in the Starlette analysis. The harmonics, Jl (l = 2, 3, 4, 5, 6), and the 18.6 year tide possible effect on the solution for J3 from multi-satellite data parameters are J2 = -2.7+0.4(10' yr- ), J3 = -1.3+0.5 sets due to LAGEOS 1 "anomalous eccentricity" excitation is a (10- yr- ), J4 =-1.4+1.0 (10' yr- ), J5 = 2.1+0.6 (10' yr- ), J6 principal concern. To reduce the aliasing effects on LAGEOS 1-0.3+0.7 (10' yr- ),,-+2o -18.6 = 1.56 +0.2 (cm) and o18.6,+2o = - and, in turn, on the solution of 3 from combination of the multisatellite data sets, the yearly values for the second- and third-degree tide, Sa, are estimated for LAGEOS and Etalon type satellites, while the parameters associated with the diurnal and semidiurnal solar tides, S and S2, are also estimated using all data sets over the entire time span in this investigation. On the basis of this parameterization, the The importance of the consideration on satellite results follows from the fact that the odd zonal rates estimated from satellite orbit perturbation representhe integrated effects of the mass changes occurring in the polar areas, and the discrepancy between the satellite observations and the prediction from the Earth's mantle viscosity with the ICE model may provide insight into the complex geophysical process occurring within the Earth (see, for example, Nerem and Klosko [1996] and James and Ivins [1997]). The consideration of the contributions from higher-degree zonal rates may be required to draw a rigorous conclusion [Milnc et al., 1996]. On the other hand, the predictions of the zonal rates are sensitive to relatively small variations in the models of ice sheet loading history [Mitrovica and Peltier, 1993]. On the basis of a model for the surface accumulation and thickness over Antarctica and Greenland, Trupin and Panfili [1997] have shown that the low-degree zonal rates predicted from this model agree well with the satellite results given in this paper, and the predicted sea level contributions are confirmed from observations. In any case, this result indicates variations in the Earth's gravity field and the observability of zonal variations from single- and multiple-satellite orbit analysis were examined using analytical perturbation theory. The results confirm that the odd zonal variations are less observable than even zonal variations from the current geodetic satellites due to weakness in the orbital geometry. The along-track orbit component is insensitive to the odd zonal variations but contains information related to the even zonal variations. The determination of the long-term change in the Earth's gravity field is largely based on the SLR data from the geodetic satellites Starlette and LAGEOS 1, combined with data sets from Ajisai, LAGEOS 2, Etalon 1, Etalon 2, BE- C, and Stella. The overall solution is based on an optimal weighting estimation procedure for the combination of the inhomogeneous multisatellite data sets. The multisatellite solution for the secular variations in the low-degree zonal 0.1+0.2 (cm). The uncertainties in the estimates of these parameters were established by examining the effects of various dynamical model parameters and data combinations.

CHENG ET AL.: DETERMINATION OF LONG-TERM CHANGES IN ZONALS 22,389 The solution of J6 is relatively weak, and further study is required. The value of the amplitude for 18.6-year tide is in good agreement with the predictions based on model for the Earth's mantle anelasticity. The lumped sum of J3 and J5 is very different for single satellite solutions, for example, from LAGEOS 1 and Starlette. The good agreement in the estimates for J3 from individual LAGEOS 1 and Starlette SLR data suggests that the effect of the LAGEOS 1 "anomalous eccentricity" excitation is insignificant for the multisatellite solution of J. A positive value of the effective "J3" can only be explained from Starlette information. Analysis of the SLR data from the multisatellite combination provides important insights and global constraint on the geophysical models for current interpretation of long-term changes in Earth system dynamics. However, the determination of secular zonal variations are limited to degree 6 from the current SLR data sets. Analysis of the sensitivity of the solution for the zonal rates to the satellite tracking data span suggests that separation of the higher-degree terms can be expected from the extension of the current multisatellite SLR tracking data. Finally, the addition of data from geodetic satellites in different orbital inclinations would be especially valuable. Acknowledgments. The authors express their appreciation to John Ries for reading the manuscript and for the helpful discussion with Richard Eanes at CSR, Dahning Yuan at JPL, and Yunxi Yang at Zhengzhou Institute of Surveying and Mapping in China. We also thank Steve Klosko, Francois Barlier, and Hans Kahle for their careful review and constructive comments. This research was supported by the National Aeronautics and Space Administration under grants NAGW- 2615 and NAGW-2941. 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