GEOPHYSICAL RESEARCH LETTERS, VOL. 29, NO. 16, 10.1029/2002GL015180, 2002
Efficient gravity field recovery using in situ disturbing potential observables from CHAMP S.-C. Han, Christopher Jekeli, and C. K. Shum Laboratory for Space Geodesy and Remote Sensing Research, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, Columbus, Ohio, USA Received 20 March 2002; revised 22 May 2002; accepted 22 May 2002; published 24 August 2002.
[1] We demonstrate an efficient method to determine gravity field model using data from accelerometer- and GPS-equipped satellites, such as CHAMP. On the basis of the conservation of energy principle, in situ (on-orbit) and along track disturbing potential observables were computed using 16-days of CHAMP data. The global disturbing potential observables were then used to determine a 50 50 test gravity field solution (OSU02A) by employing a computationally efficient inversion technique based on conjugate gradient [Han et al., 2002]. An evaluation of the model using independent GPS/leveling heights and Arctic gravity data, and comparisons with existing gravity models, EGM96 and GRIM5C1, and new models, EIGEN1S and TEG4 which include CHAMP data, indicate that OSU02A is commensurate in geoid accuracy and, like other new models, it yields some improvement (10% better fit) in the polar region at wavelengths longer INDEX TERMS: 1214 Geodesy and Gravity: than 800 km. Geopotential theory and determination; 1227 Geodesy and Gravity: Planetary geodesy and gravity (5420, 5714, 6019); 1241 Geodesy and Gravity: Satellite orbits; 1243 Geodesy and Gravity: Space geodetic surveys; 1294 Geodesy and Gravity: Instruments and techniques
simple technique based on the method by Jekeli [1999] that uses precise orbits in the inertial (or non-rotating) frame and the conservation of energy principle, to construct on-orbit disturbing potential observables from the CHAMP data in the along track component. The resulting test gravity field solution, OSU02A, using 16-days of CHAMP data is complete to degree 50 in spherical harmonics. OSU02A is then compared and evaluated against EGM96, GRIM5C1, and the more recent solutions, EIGEN1S (a satellite-only model including 88-days of CHAMP data [Reigber et al., 2001]) and TEG4 (a combination model including 80-days of CHAMP data [Tapley et al., 2002]). The models are also evaluated using independent GPS/leveling data and Arctic gravity anomaly data.
2. Method [4] Jekeli [1999] derived a model relating the Earth’s gravitational potential, VE, to xi, the position vector in the inertial frame; x_ i , the velocity vector; and F, the net vector of all non-conservative forces measured by the accelerometer: 1 2 VE x_ i we xi1 x_ i2 xi2 x_ i1 2
F x_ i dt ðVS þ VM Þ V0 ð1Þ
1. Introduction [2] The CHAllenging Minisatellite Payload (CHAMP) gravity and magnetic mapping satellite mission, launched in July 2000 by the GeoForschungsZentrum (GFZ), Potsdam, Germany, provides the first data set with high-low satellite tracking and accelerometer measurements for gravity field studies. CHAMP’s orbit is at an altitude of 450 km and its 87 inclination enables near-global coverage. Its payload includes geodetic-quality, Blackjack-class, GPS receivers (16-channel, dual-frequency) with multiple antennas for precise orbit determination and atmospheric limb-sounding, and the 3-axis STAR accelerometer (3 109 m/s2 and 3 108 m/s2 precision in the along track or cross track, and radial directions, respectively [Perret et al., 2001]) to measure non-conservative forces including atmospheric drag. [3] The conventional technique for gravity field solutions using satellite tracking data involves a geophysical inversion process that relates the data (e.g., high-low GPS tracking) to the gravity field and estimates its coefficients based on an a priori model using least squares [e.g., EGM96, Lemoine et al., 1998; GRIM5C1, Gruber et al., 2000]. In this paper, we intend to demonstrate a relatively Copyright 2002 by the American Geophysical Union. 0094-8276/02/2002GL015180$05.00
Z
T where xi = [x1i x2i x3i]T and x_ i ¼ x_ i1 x_ i2 x_ i3 . The geopotential coefficients can be estimated from a global distribution of observables, VE. For this derivation, the luni-solar gravitational potential, VS + VM, is the only N-Body effect considered. The N-body effect, for example, could be accurately modeled using a planetary ephemeris, such as JPL DE405. The first term in (1) is the kinetic energy ( per unit mass) of the satellite, determined by its inertial velocity. The second term is the so-called ‘potential rotation’ term that accounts for the (dominant) rotation of the Earth’s potential in the inertial frame. The third term is the dissipating energy due to the atmospheric drag, solar radiation pressure, thermal forces, and other non-conservative forces. The last term is the energy constant of the system including the constant zero-degree harmonic of the gravitational potential. In this study, we ignored the second-order gravitational sources such as the solid Earth and ocean tides, but we included the effect of the permanent tides. Although there are recently available more accurate orbits, the Rapid Science Orbit (RSO), which employed GRIM5C1 improved with CHAMP data for orbit determination, from GFZ was used to calculate xi and x_ i , and it has a RMS of 5 cm in 3-D position differences according to the evaluations using satellite laser ranging (SLR) [Koenig, 2001].
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[5] The STAR accelerometer assembly on board CHAMP measures the non-conservative forces in three dimensions. Its x-axis is aligned along the radial direction, the y-axis is along the direction of the forward boom, and the z-axis is along the (positive) perpendicular direction of the orbital plane. However, only data from the y-axis (along-track) component were used in this study, primarily because it is the dominant and most sensitive component. In addition, the x-axis sensor is less sensitive and one sensor used to compute x-axis (radial) acceleration was discovered to have a malfunction [CHAMP Newsletter No.4, 2001]. The boom direction (along-track) component of the velocity vector dominates the total speed of the satellite, n, as indicated by the following approximate friction energy formula: Z VF ¼
i
F x_ dt
n X
Fy ðti Þvðti Þdt:
ð2Þ
i¼1
Instead of using the CHAMP stellar compass data to determine the precise component, Fy, we rely on the on-orbit stabilization of the satellite, which maintains the y-axis (forward boom) in the direction of the velocity. Again, since the non-conservative accelerations are expected to dominate in the along-track direction, this approximation neglects primarily second-order errors in orientation. [6] The y-axis accelerometer data in GFZ’s current Level-2 products are not calibrated, which requires that at least the bias be estimated or modeled. A bias, df, in the yaxis accelerometer, appears as a linear trend in the friction energy equation. From (2), assuming a constant speed of the satellite and ignoring the accelerometer scale, we have: V F ðt n Þ
n X i¼1
n X Fy ðti Þ df vdt Fy ðti Þvdt ðvdf Þtn ; ð3Þ i¼1
where the last term is the linear trend due to the accelerometer bias. We estimate this bias by comparing the slope, s1, of the friction energy predicted using (1) on the basis of an existing disturbing potential model (EGM96) with the slope, s2, of the friction energy computed according to (2) from the y-axis accelerometer data. The slope difference, s1s2, presumably is due primarily to the last term of (3), and thus is a good indication of the y-axis accelerometer bias. The bias estimation is conducted every 10 14 hours depending on the duration of the RSO and the accelerometer data. The estimate of the y-axis bias is in good agreement with the value of 2.7 mm/s2 obtained by Tapley et al. [2002], if our bias estimate is scaled by 0.74, which is the accelerometer scale obtained by Tapley et al. [2002]. [7] After the y-axis accelerometer bias is removed from the data, we compute Earth’s gravitational potential at the satellite altitude according to (1) and (2). The set of in situ measurements constitutes boundary values for a solution of the geopotential in terms of spherical harmonics. The relationship between the in situ disturbing potential measurements and the harmonic geopotential coefficients is given as a linear function. The unknown spherical harmonic coefficients are estimated in a (linear) least-squares sense by solving the corresponding normal equations. For a lowdegree gravity model such as Nmax = 50, the direct solution is possible through the Cholesky decomposition or the brute-
force inversion of the normal matrix after it is accumulated on the basis of all measurements of VE. However, in this study, the conjugate gradient method is applied to determine the least-squares estimates iteratively, with the block-diagonal part of the normal matrix serving as a pre-conditioner. The accumulation of the block-diagonal part of the normal matrix is considerably faster than that of rigorous inversion of the full normal matrix. This numerical procedure has been validated using simulated gravity field solutions (Nmax 90) to assess polar gap and aliasing problems using GRACE and GOCE data by Han et al. [2002].
3. Results [8] Two friction energy values were computed by (1) and with EGM96 (Nmax = 360) for VE, as well as by (2) using the CHAMP data. If EGM96 is the true gravitational field, the difference between these two values would be primarily due to a bias in the y-axis accelerometer and everything else which has been ignored in this work. After removing the linear trend in both values, Figure 1 shows that the medium and long wavelength components of both the data (CHAMP) and the model (EGM96) agree well. [9] Next, we computed the differences between the in situ potentials from EGM96 and from CHAMP data with and without the (bias-calibrated) accelerometer data. The long wavelength variation in the difference increased from 0.77 m2/s2 to 1.36 m2/s2 (standard deviation), indicating that the y-axis STAR accelerometer data are valid and useful for gravity field determination. However, the difference still contains once-per-revolution type of components that could be due to the unmodeled forces, such as tides and residual accelerometer and other data errors. [10] Using 16 days of CHAMP RSO data and STAR accelerometer data, we computed the in situ potentials and determined the harmonic geopotential coefficients up to degree and order 50 using the conjugate gradient iterative method. The final solution was obtained after 5 iterations. The solution also, as EGM96 and TEG4 did, adopted C21 and S 21 values according to the IERS2000 Standard [McCarthy, 2000] and the indirect permanent tide in the estimate, C20, is consistently handled as specified by the IERS2000 Standard. [11] The determined CHAMP gravity model (called OSU02A model) is compared with other recent gravity models. Figure 2 presents the degree variances of the differences among various pairs of gravity models. The GRIM5C1 model is closest to OSU02A, probably due to the fact that the CHAMP RSO is a dynamic orbit computed using the GRIM5C1 model and CHAMP GPS satellite-tosatellite tracking data [Koenig, 2001]. As the degree increases, the difference between GRIM5C1 and OSU02A gradually approaches the differences between other models and OSU02A. From degrees 43 to 50, the differences between OSU02A and GRIM5C1, EGM96, and TEG4, respectively, are almost identical. Whereas the latter three models are based on satellite tracking as well as terrestrial gravity data, the satellite-only model, EIGEN1S, combined 88 days of CHAMP data with satellite laser ranging data, and is based on the GRIM5S1 normal equation system [Reigber et al., 2001]. The OSU02A differences relative to EIGEN1S are similar with those of EGM96 and TEG4
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HAN ET AL.: GRAVITY FIELD RECOVERY FROM CHAMP 4 3
2 2
friction energy [m /s ]
2 1 0 1 2 soild: based on accelerometer data
3
dotted: based on EGM96 and orbit data 4 5 497.9
498
498.1
498.2 time [day]
498.3
498.4
498.5
Figure 1. Difference between kinetic energy and the a priori model, EGM96; and friction energy computed from STAR accelerometer data (mean and trend have been removed).
below degree 35, however, they become relatively larger beyond degree 35. The fact that OSU02A follows the models GRIM5C1, EGM96, and TEG4 more closely rather than EIGEN1S beyond degrees 35, indicates that OSU02A is significantly affected by terrestrial data. That is, this test model is constructed on the basis of the RSO, that depends on GRIM5C1; and hence, OSU02A, is not a satellite-only model based solely on in situ accelerometry and GPS tracking data. [12] In the spatial domain, we compared the models (all truncated at degree 50) in terms of the RMS geoid differences over all longitudes per latitude. Figure 3 shows these RMS differences for the cases of EGM96-OSU02A, TEG4OSU02A, GRIM5C1-OSU02A, and EIGEN1S-OSU02A. OSU02A is closest to GRIM5C1 over the middle latitude, while the difference between them increases toward the poles. EGM96 and GRIM5C1 do not include CHAMP data and this reflects their relatively larger differences with respect to OSU02A over the polar regions. On the other hand, EIGEN1S, TEG4 and OSU02A models include CHAMP data, and the difference between OSU02A and EIGEN1S as well as TEG4 is not as pronounced over the poles, specifically the Antarctic region. [13] We assessed the performance of OSU02A against the other models in terms of geoid accuracy by comparing each
Figure 3. Global geoid difference (RMS over longitude bands) between different gravity models. model to geoid undulations obtained from GPS and leveled heights in various parts of the world [Shum et al., 2001]. Each of the four gravity models, EGM96, GRIM5C1, TEG4, and EIGEN1S, was truncated at degree 50 to be consistent with OSU02A. Means and standard deviations of geoid differences between the gravity models and the GPSleveling heights are shown in Table 1. The performance of the OSU02A model is comparable to that of the other models, and one would expect even better results with more than 16 days of satellite data. [14] It should be emphasized that, although terrestrial data do not enter directly into OSU02A, they are indirectly involved via the CHAMP RSO (orbit), computed using GRIM5C1 that includes terrestrial data. To show that, in fact, OSU02A improves the inherent a priori model by virtue of the CHAMP satellite data (GPS tracking and accelerometry), we compared it (and the other models) to 150 150 mean free-air gravity anomaly data in the Arctic, determined from airborne, marine, surface gravimetry, and satellite altimetry [Kenyon and Forsberg, 2001]. The areas not covered by CHAMP’s ground tracks (due to its nonpolar orbit) and where gravity anomalies are derived from EGM96 have been excluded from the evaluation. In Area 1 (see Table 2), all models except GRIM5C1 perform at a
Table 1. Gravity Models Assessment Using GPS-Leveling Data (unit: cm) Canada(1443pt)
Australia(59pt)
Models
mean
US(6169pt) s.d.
mean
s.d.
mean
OSU02A GRIM5C1 EGM96 TEG4 EIGEN1S
16.2 20.3 16.0 19.0 13.4
106.4 104.5 103.6 102.6 100.8
24.3 22.0 21.8 28.0 23.8
92.8 92.2 91.7 93.9 97.5
42.1 40.8 43.4 40.4 41.6
Europe(63pt)
Figure 2. Degree variance of differences between different gravity models.
Germany(42pt)
s.d. 140.7 139.7 143.9 145.5 157.0
Doppler(850pt)
Models
mean
s.d.
mean
s.d.
mean
s.d.
OSU02A GRIM5C1 EGM96 TEG4 EIGEN1S
5.1 5.4 8.4 6.7 6.2
136.3 128.6 137.8 132.5 131.1
130.3 113.7 109.7 101.7 91.4
63.6 64.1 62.3 57.8 59.2
51.1 49.8 47.5 50.5 48.9
240.3 237.9 233.7 234.3 241.0
The zero-degree undulation and the local vertical datum shift in the mean differences of GPS/leveling tests have been accounted for N. American and Australia [Shum et al., 2001]. For data in Europe, Germany, and Doppler stations, only the zero-degree undulation has been corrected.
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Table 2. Gravity Models Assessment Using Arctic Gravity Anomalies Data Provided by NIMA (unit: mgal) Models EGM96 TEG4 GRIM5C1 EIGEN1S OSU02A
Area [1]
Area [2]
mean
s.d.
mean
s.d.
0.81 1.05 1.18 1.08 0.18
23.35 23.50 25.23 23.42 23.64
3.36 1.86 1.50 4.49 1.55
16.12 15.49 15.35 15.11 14.73
[1] covers latitude: 80N 87N, longitude: 0E 360E; [2] covers latitude: 76N 80N, longitude: 30E 90E.
comparable level. Notably, EGM96 yields the best comparison even though it doesn’t include CHAMP data. Area 2 is chosen to cover only ocean, where the gravity anomalies are determined from satellite altimetry by NOAA and KMS [Shum et al., 2001]. Here, OSU02A and EIGEN1S show better performance than the other models. In both areas OSU02A is better than the (indirectly) a priori model, GRIM5C1. [15] Based on the model assessment with independent Arctic gravity anomaly data and the geoid height comparison among different models, we conclude that OSU02A is commensurate in accuracy with other models, and probably has improvements, specifically over polar regions.
4. Conclusions and Discussions [16] Based on 16 days of CHAMP rapid science orbit (RSO) and accelerometer data, a test gravity field model OSU02A was determined in an efficient way. We compared OSU02A to other recent models, EGM96, GRIM5C1, TEG4, and EIGEN1S, and found it to be closest to the gravity model for the RSO, i.e., GRIM5C1. The significant difference in the shorter wavelengths between the satellite-only model, EIGEN1S, and the combination (satellite and terrestrial) models is also apparent in the comparison to the OSU02A model. OSU02A differs significantly from GRIM5C1 over the polar regions, and the assessment with the Arctic gravity anomalies data seems to indicate that OSU02A yields a better fit to the geopotential field than GRIM5C1 in these areas. It is due to the fact that OSU02A contains GPS tracking and accelerometer data from CHAMP up to ±87 in latitude, while GRIM5C1 does not. [17] In the comparison with independent geoid undulations, derived from GPS and leveled heights, and the Arctic gravity anomalies, OSU02A (with 16 days of data) is comparable in performance to the other models in terms of mean and standard deviation of the differences. Based on these results, we have demonstrated the feasibility of the in situ measurement method to determine the geopotential field from the CHAMP mission. The tests relied on orbits determined by a dynamic solution (i.e., RSO), rather than the use of pure kinematic or reduced dynamic orbits obtainable with GPS. It is clear that the methodology demonstrated here, being independent of the process that generates the orbit, is applicable to this, as well as the other satellite gravity mapping missions, GRACE and GOCE.
However, care must be exercised in interpreting the solution as a satellite-only model if that orbit is dynamically determined using a priori gravitational models. The dependence of the solution on the orbit determination remains an area of further study. Also, improved procedures to better account for errors (bias, scale, and their possible drifts) in the full 3-axis accelerometer data should be further investigated. [18] Acknowledgments. We thank Ch. Reigber and the German CHAMP Project for providing the CHAMP data products for this study. We acknowledge two anonymous reviewers, Ch. Reigber and R. Koenig for their constructive comments to the original manuscript. This study is partially supported by NASA’s GRACE Project via a grant from the University of Texas Center for Space Research.
References CHAMP Newsletter No.4, http://op.gfz-potsdam.de/champ/more/ newsletter_CHAMP_004.html, 2001. Gruber, T., A. Bode, Ch. Reigber, P. Schwintzer, G. Balmino, R. Biancale, and J.-M. Lemoine, GRIM5-C1: Combination solution of the global gravity field to degree and order 120, Geophys. Res. Lett., 27, 4005 – 4008, 2000. Han, S.-C., C. Jekeli, C. K. Shum, Aliasing and Polar Gap Effect on Geopotential Coefficient Estimation: Space-wise Simulation Study of GOCE and GRACE, in Vistas for Geodesy in the New Millennium, International Association of Geodesy Symposia, vol. 125, edited by J. Adam and K.-P. Schwarz, Springer-Verlag, 2002. Jekeli, C., The determination of gravitational potential differences from satellite-to-satellite tracking, Celestial Mechanics and Dynamical Astronomy, 75, 85 – 100, 1999. Kenyon, S., and R. Forsberg, Arctic Gravity Project — a status, in Gravity, Geoid and Geodynamics 2000, International Association of Geodesy Symposia, vol. 123, edited by M. Sideris, 391 – 395, Springer – Verlag, 2001. Koenig, R., ftp://cddisa.gsfc.nasa.gov/pub/gps/products/leopp/gfz/ README_2.txt, 2001. Lemoine, F., S. Kenyon, J. Factor, R. Trimmer, N. Pavlis, D. Chinn, C. Cox, S. Klosko, S. Luthcke, M. Torrence, Y. Wang, R. Williamson, E. Pavlis, R. Rapp, and T. Olson, The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA Technical Paper NASA/TP-1998-20,6861, Goddard Space Flight Center, Greenbelt, 1998. McCarthy, D., IERS Conventions (2000), IERS Technical Note 2, 2000. Perret, A., R. Biancale, A. Camus, J. Lemoine, T. Fayard, S. Loyer, F. Perosanz, and M. Sarrailh, CHAMP mission: STAR commissioning phase calibration/validation activities by CNES, vol. 1&2, CNES, Toulouse, 2001. Reigber, Ch., P. Schwintzer, R. Koenig, K.-H. Neumayer, A. Bode, F. Barthelmes, G. Balmino, R. Biancale, J.-M. Lemoine, S. Loyer, and F. Perosanz, Earth gravity field solutions from several months of CHAMP satellite data, Eos Trans American Geophysical Union, 82(47), Fall Meet. Suppl. G41C-02, 2001. Shum, C. K., S.-C. Han, C. Jekeli, Y. Yi, C. Zhao, P. Dumrongchai, S. Kenyon, D. Roman, K. Zhang, Y. Lu, and Y. Zhu, Accuracy Assessment of Current Gravity Field Models, paper presented at International Association of Geodesy Symposia, Budapest, Hungary, 2001. Tapley, B., S. Bettadpur, D. Chambers, M. Cheng, K. Choi, B. Gunter, Z. Kang, J. Kim, P. Nagel, J. Ries, H. Rim, P. Roesset, and I. Roundhill, Gravity field determination from CHAMP using GPS tracking and accelerometer data: initial results, paper presented at the first CHAMP Science Meeting, Potsdam, Germany, 2002.
S.-C. Han, C. Jekeli, and C. K. Shum, Laboratory for Space Geodesy and Remote Sensing Research, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470 Hitchcock Hall, 2070 Neil Avenue, Columbus, Ohio 43210-1275, USA. (
[email protected])
