Numerical simulation of the gravity field recovery from GOCE mission data. Sean Bruinsma, Jean-Charles Marty, Georges Balmino. CNES-CT/SI/GS, 18, avenue ...
Numerical simulation of the gravity field recovery from GOCE mission data Sean Bruinsma, Jean-Charles Marty, Georges Balmino CNES-CT/SI/GS, 18, avenue E. Belin, 31401 Toulouse cedex 4, France ABSTRACT Numerical simulations of the gravity field parameter recovery using the direct method, with satellite rectangular coordinates as pseudo observations, instead of simulating GPS Satellite-to-Satellite (SST) tracking data, and with gravity gradients (SGG data), were done and are ongoing in the framework of the European GOCE Gravity Consortium test and validation plan for GOCE mission data processing. The GOCE orbit perturbation analysis revealed a high sensitivity to sectorials and order 16, 32, 48 and 96 resonances of the gravity field, as well as very strong resonant effects due to the ocean tide, order 18 of the M2 constituent in particular. The simulations using pseudo SST observations demonstrated the known deteriorating effect of the polar gap on the determination of the zonal coefficients. The cumulated geoid error due to the SST part of the GOCE combined SST+SGG data gravity field solution is at the few mm level, at least for these simplified simulations. 1. INTRODUCTION This study pertains to the direct method of gravity field recovery in the framework of the European GOCE Gravity Consortium (EGGC) High Level Processing Facility (HPF), of which it is one of three methods selected (direct, timewise, and space-wise approaches). The direct method fistly requires either precise satellite positions, provided by kinematically-determined ephemerides, as pseudo-observations, or the tracking data. In the latter case, reduction and evaluation of the GPS satellite-to-satellite (SST) tracking data and the linear non-gravitational accelerations provided by the gradiometer (SGG common mode), and the employment of gravitational force models are required in order to determine precise orbits in a dynamical approach. After the iterative least-squares orbit adjustment procedure has converged to the highest attainable precision level, the gravity field normal equations are computed in a subsequent step. These SST normal equations, representing the long wave-length gravity field signal, are then reduced for arcdependent parameters (i.e. state vector at epoch, empirical parameters) and cumulated over the entire observation period. Secondly, the gravity gradient measurements (SGG) are processed and yield (high resolution) normal equations that are combined with the SST normal equations. Finally, the dynamical, gravity field and gradiometer common mode calibration parameters are simultaneously estimated, the errors of which may be estimated through the variancecovariance matrix. The software packages that use the direct method are GINS from CNES/GRGS, and EPOS from GFZ. These were used in the framework of the GRIM [1] and EIGEN [2] gravity field modelling projects. They offered the possibility of deriving, from any kind of satellite tracking or geodetic measurement, so-called observation and normal equations for almost any type of parameter of the force model underlying the dynamical reconstruction of the orbit (of one or several spacecraft at a time). New pieces of software have been written and implemented recently to process SGG data. The entire software package will be tested through increasingly realistic numerical simulations, as part of the HPF review and validation program, in order to verify the proper implementation of new software modules as well as to identify the best SST and SGG processing strategy. This paper concerns only the very beginning of the test program, which started in 2004. Only SST and very simple SGG simulation results are presented in Section 3. Knowledge of the sensitivity of the GOCE orbit to the gravity field and the ocean tide is necessary to the design of a realistic test program. Therefore, the next section presents an orbit sensitivity analysis. 2. GOCE ORBIT SENSITIVITY ANALYSIS The SELECT software, which uses the Kaula formulation [3] for the gravity and ocean tide potentials, is used to compute the amplitudes and periods of the orbit perturbation of each individual Stokes or ocean tide coefficient. The following mean GOCE orbit parameters were used : semimajor axis a=6628 km, eccentricity e=0.002, and inclination i=96.5°. 2.1 Sensitivity to the Gravity Field Two recent gravity field models up to degree and order 119 have been used in the analysis, EIGEN-1s [2], based mainly on CHAMP data, and the first GFZ model based on GRACE data only, EIGEN-GRACE01S (available at: http://op.gfz-
potsdam.de/grace/index_GRACE.html). This was done in order to have an upper limit (since EIGEN-1s is significantly less accurate than EIGEN-GRACE01S) due to gravity model uncertainty in view of its low altitude, and in particular due to resonances. The amplitudes of the perturbation of the orbit position computed with EIGEN-GRACE01S are shown per degree and order in Fig. 1, in which the resonant bands are also clearly visible. The amplitudes obtained by subtracting the EIGEN-1s minus EIGEN-GRACE01S perturbations are large, particularly so for the zonals up to about degree 75 and for the resonant orders 16, 32, 48 and 96. The periods of the resonances are given in Table 1.
Fig. 1. The amplitude of the position perturbations on the GOCE orbit with the gravity field model EIGEN-GRACE01S (left plot), and the EIGEN-1s minus EIGEN-GRACE01S perturbations (right plot), in m. Table 1. Resonance periods per order, in days, computed with the gravity field model EIGEN-GRACE01S. Even degree Odd degree
Order 16 18.3 15.2
Order 32 7.6 8.3
Order 48 5.4 5.1
Order 64 3.8 4.0
Order 80 3.1 3.0
Order 96 2.5 2.6
Order 112 2.2 2.2
2.2 Sensitivity to Ocean Tide Constituents The FES2002 [4] ocean tide model up to maximum degree and order 50 was used in this study. This is the most recent version of the FES tidal models. An uncertainty estimate of the tide modelling will be obtained by subtracting perturbations of the new FES model when it becomes available in the course of 2004. The GOCE orbit perturbations due to the most important long period tides are given in Table 2. The amplitude of the degree 2 coefficients of the annual and semiannual constituents as well as the monthly and fortnightly constituents is high. The length of the GOCE measurement phases, 6 months each, will make the estimation of the annual and in a lesser degree the semiannual term delicate. However, if the uncertainty of these coefficients is large in the GOCE standards to be adopted, they need to be estimated. Table 2. The GOCE orbit perturbations due to the main long period tide constituents of FES2002. The information per cell is: degree (order=0) / amplitude (m) / period (days). Sa 2 / 0.65 / 365
Ssa 2 / 2.09 / 182 5 / 0.13 / 171
Mm 2 / 0.52 / 73 5 / 0.06 / 40
Mf 2 / 0.45 / 19 5 / 0.04 / 16
Mtm 2 / 0.05 / 11
The main perturbations due to diurnal and semidiurnal tide constituents are presented in Table 3. The P1+ and the K1+ constituents have very large amplitudes with periodicities of about one year, which makes their estimation (if necessary) delicate because of the above mentioned too short GOCE measurement phases. The semidiurnal M2+ constituent has a particularly strong order 18 resonance, with a maximum orbit perturbation amplitude of 188 m and a 551 days period. This study shows that the ocean tide must be developed up to significantly higher degree than 50 for certain constituents, such as O1+, N2+, and M2+ in particular, as can be seen in Fig. 2. This will be done when FES2002 will be compared to its successor.
Table 3. The GOCE orbit perturbations due to the main diurnal and semidiurnal tide constituents. For the resonant orders, the degree interval is given for those amplitudes that are larger than 0.01 m only. Q1+ O1+ P1+ K1+ N2+ M2+ S2+ K2+
Degree
Order
2 5 2 4 5 4 2 5 4 2 5 2 4 5 2 4 5 3 5 7 2 4 3
1 1 1 1 2 2 2 2
Amplitude (m) 0.10 0.05 0.66 0.66 0.35 7.14 6.22 0.94 21.52 18.62 2.52 0.75 0.47 0.11 5.22 3.16 0.67 2.06 2.04 0.71 6.32 3.60 0.89
Period (days) 11 10 20 20 16 364 364 116 366 366 116 12 12 10 22 22 17 88 88 88 183 183 171
Resonant order 17 33 17 33
Resonant degrees 17-31 33-46 17-50 33-46
Maximum amplitude (m) 0.24 0.37 12.76 0.26
Period for odd/even degrees (days) 54 / 33 72 / 399 202 / 147 19 / 25
17
17-31
0.45
24 / 19
17 33
17-47 34
1.20 0.07
21 / 17 9
18 34
18-45 34-50
0.86 0.43
37 / 64 60 / 191
18 34
18-50 34-50
188.41 0.47
551 / 105 18 / 24
18 34
18-43 34-38
1.22 0.05
23 / 18 8/9
18
19-37
0.29
20 / 16
Fig. 2. GOCE orbit perturbations, in m, due to the M2+ constituent of the model FES2002.
3. SIMULATION RESULTS 3.1 Pseudo SST Data Simulation Results The simulation procedure for the gravity recovery tests with pseudo SST data is as follows: 1. A GOCE ephemeris of 60 days in rectangular coordinates every 15 seconds is computed with the reference models (EIGEN-2 [5] to degree and order 90, and a truncated FES2002 for the ocean tide). These reference GOCE positions are the pseudo SST data. 2. Using the pseudo SST data, 60 arcs of 1 day are adjusted, estimating the state vector at epoch (SV) and in most tests a bias and 1 cycle per revolution terms per revolution (CPRS), using EIGEN-1s as a-priori gravity field model, and an older FES version, FES98 (also truncated), as a-priori ocean tide model. A normal equation per arc is generated. 3. The 60 normal equation systems are cumulated and the solution of all parameters (gravity field and SV, and depending on the simulation, CPRS and ocean tide coefficients) is obtained by Choleski decomposition. 4. The solution is evaluated by computing the cumulated geoid error (per degree and per order), EIGEN-2 being the reference. The maximum degree of the simulations, 90, is much higher than what the actual contribution of the SST part to the GOCE gravity field solution will be in reality, since the SGG data will have increasing weight in the solution from degree 20 onward and probably attain 100% at about degree 60-70. Therefore, the cumulated geoid error of the solution at degree 90 is not as important in the evaluation as the degree at which the cumulated error reaches 1 cm. The simulations and their results are presented in Table 4. Table 4. Simulation description and results with pseudo SST data. ‘Par./orbit’ is short for the parameterization of the orbit adjustment, is the average RMS of the 60 arcs in m, Σ deg 90 is the cumulated geoid error in cm at degree 90, and Σ 1 cm gives the degree at which 1 cm cumulated geoid error is attained. Case: reference / a-priori Par./orbit Σ deg 90 A: EIGEN-2 / EIGEN-1s SV 4.74 7.2 B: EIGEN-2 / solution case A SV 0.0003 6.4 C: EIGEN-2 / EIGEN-1s SV+CPRS 0.25 3.8 D: case D, but GOCE in polar orbit SV+CPRS 0.11 1.2 E: normal equations case A combined with normal equations LAGEOS SV 4.74 3.6 F: EIGEN-2 / EIGEN-1s with EIGEN-2 zonals SV 4.60 1.7 G: EIGEN-2 and FES20021 / EIGEN-1s and FES981 SV+CPRS 0.25 2.4 H: EIGEN-2 and FES20022 / EIGEN-1s and FES982 SV+CPRS 0.25 1.8 I: EIGEN-2 and FES20022 + drag error3 / EIGEN-1s and FES982 SV+CPRS 0.25 20.8 1 only diurnal and semidiurnal constituents with periods of less than 89 days and maximum degree and order of 8. 2 only diurnal and semidiurnal constituents with periods of less than 89 days and maximum degree and order of 31. 3 residual drag error of ±9 10-8 m/s2 (approximately drag free system error specification), zero on average.
Σ 1 cm 42 44 54 90 58 82 70 47 47
Case A , though simple, presents the worst results. The explanation is found by computing the geoid error cumulated per order, which shows that the error for order 0 is already 6.9 cm. This means that, due to correlations, the zonal coefficients cannot be accurately determined in the not so realistic case of a gravity field only simulation. An iteration of the procedure (case B), using the gravity field solution of case A, does not improve the result by much. The recovery of the zonal coefficients improves much by the estimation of empirical parameters (CPRS), as can be seen in Table 4 for case C. Almost the entire error is still caused by the order zero, 3.7 cm, but it is divided by two. The determination of the zonals is inaccurate due to the inclination of the GOCE orbital plane, leaving a polar gap of 6.5°. This is demonstrated by case D, in which an inclination of 89.7° instead of 96.5° was used. To improve the GOCE solutions, two ways are open: using information from other satellites, such as LAGEOS-I/ II, or not estimating zonals (to some degree) and relying on the accuracy of the a-priori model at the time of launch. In case E, the GOCE normal equations were combined with those from LAGEOS I and II covering the same 60 days. The result is already much improved compared to case A. The impact of combining with more normal equation systems from geodetic satellite such as Starlette and Stella will be studied in the near future. Relying on the a-priori model was simulated in case F, in which all zonals were replaced. The recovery is much more accurate; however, fixing zonal coefficients to the GOCE standards a-priori model may not be the best solution if the temporal gravity corrections are not sufficiently accurate. In cases G and H, besides the gravity field model coefficients, certain coefficients of the ocean tide model are estimated. In these simulations, because of the limited period of 60 days, only those tide constituents causing orbit perturbations of less than 89 days were selected. The coefficients of the resonant ocean tide model constituents in case H were estimated as a lumped sum (one even and one odd degree coefficient per order). When all coefficients related to
resonance are estimated, the result is much worse: the cumulated geoid error reaches 1 cm already at degree 18. The last test case, I, is similar to H but with an additional residual drag error. This RMS error of 9 10-8 m/s2 is in accordance with the drag free system specifications. It can be seen in Table 4 that the drag error has a large effect at degree 90, the cumulated geoid error becomes 20.8 cm, but that the 1 cm cumulated error is still at degree 47.
The geoid error and cumulated geoid error, per degree, for two test cases 1.00E+00 1.00E-01
(m)
1.00E-02 H(sol-ref) H(cumul) D(sol-ref) D(cumul)
1.00E-03 1.00E-04 1.00E-05 1.00E-06 0
10
20
30
40
50
60
70
80
90
Degree
Fig. 3. The geoid error (solution – reference) and the cumulated geoid error, summed per degree, for test cases H and D of Table 4. The geoid errors and cumulated geoid errors, summed per degree, are shown in Fig. 3 for cases H and D of Table 4. The polar orbit, case D, has an error that is basically flat up to approximately degree 45, after which it slowly increases. The determination of the odd degrees is more accurate than that of the even degrees, as the saw tooth signatures in Fig. 4 indicate. The ocean tide is responsible for the large bump around degree 17, while the residual drag causes the steep increase in geoid error at degree 80. A longer measurement period, and combining with LAGEOS normal equations, will reduce the bump around degree 17 and the initially steep slope in the low degrees, respectively. 3.2 SGG Data Simulation Results The simulation procedure for the gravity recovery tests with SGG data is as follows: 1. Perfect GOCE orbit positions are available every 15 seconds and the reference gradients are computed with EIGEN-2 to degree and order 120 over a period of 20 days. 2. Using an a-priori gravity field model, the gravity gradient residuals are computed at the same GOCE orbit positions. The observation equations are formed and subsequently the normal equation is generated. 3. The solution is obtained by Choleski decomposition; only gravity field coefficients are estimated. 4. The solution is evaluated by means of computing the cumulated geoid error (per degree and per order), EIGEN-2 being the reference. In these simulations, (coloured) noise was not added to the reference SSG data, only the effect of an orbit position error and of the choice of a-priori gravity field model was tested. The effect of a 3 cm RMS orbit error on the accuracy of recovery of the reference gravity field model, using it as a-priori model too, is negligible: the cumulated geoid error at degree 120 is 0.1 mm. This test demonstrates that the gravity gradient subroutines that have been added to the GINS software are correctly written and implemented. A second series of tests consisted in evaluating the sensitivity of the solution to the a-priori model. The cumulated geoid difference between EIGEN-2 and EIGEN-1s at degree 120 is rather large, 83 cm. When the latter model is used as an apriori in the above described simulation procedure, so under no-noise conditions, the solution has a cumulated geoid error of 24 cm. When this solution is used a a-priori model and the recovery procedure is iterated, the cumulated difference at degree 120 with EIGEN-2 becomes 9.5 cm. This error is mainly due to the inaccurate estimation of the coefficients of the low degrees (up to 20), as is shown in Fig. 4, and orders (up to 15). The gravity field recovery with
pseudo SST data is much more accurate for these low degrees. Therefore, and as shown in previous GOCE studies [6], a combined SGG and (pseudo) SST simulation should show improvement in the recovery of the low degrees, while at least preserving the solution accuracy at degree 120. The solution is indeed more accurate at all degrees, as may be seen in Fig. 4. The cumulated geoid error of the combined solution at degree 120 is 7.7 cm, which is much larger than the GOCE objectives of 1-2 cm at degree 200. The difference between the a-priori field and EIGEN-2 cumulated at degree 120 is 24 cm, which is very large in view of the accuracy of gravity field models constructed with GRACE data (currently 5-10 cm, the mission objectives being 1-2 cm). Thus, taking the sensitivity of the recovery to the a-priori model into account, the accuracy of our gravity field estimation will in reality be signifcantly higher. An extended simulation under more realistic conditions will be done before the summer of 2004.
The geoid error and cumulated geoid error, per degree, for SGG and combined data (SST+SGG) 1.00E-01
1.00E-02
(m)
SGG(sol-ref) SGG(cumul) Comb(sol-ref) Comb(cumul) 1.00E-03
1.00E-04 0
20
40
60
80
100
120
Degree
Fig. 4. The geoid error (solution – reference) and the cumulated geoid error, summed per degree, for an SGG data only simulation and a combined (pseudo SST+SGG) data solution. References 1. Biancale R., et al. A new global Earth's gravity field model from satellite orbit perturbations: GRIM5-S1, Geophysical Research Letters, No. 27, 3611 - 3614, 2000. 2. Reigber C., et al. A high-quality global gravity field model from CHAMP GPS tracking data and accelerometry (EIGEN-1S). Geophysical Research Letters, No. 29(14), 10.1029/2002GL015064, 2002. 3. Kaula W.M. Theory of satellite geodesy, Blaisdell Press, Waltham, Mass., 1966. 4. LeProvost C., et al. FES 2002 – A new version of the FES tidal solution series, Abstract Volume. Jason-1 Science Working Team Meeting, Biarritz, France, 2002. 5. Reigber C., et al. The CHAMP-only earth gravity field model EIGEN-2, Advances in Space Research, Vol. 31, No. 8, 1883 – 1888, 2003. 6. From Eötvös to milligal, Final Report, ESA/ESTEC Contract No. 13392/98/NL/GD, 2000.
