in: F. Sans`o (ed.), A Window on the Future of Geodesy, IAG Symposium Series, Vol. 128, pp. 288–293, Springer Verlag, 2004
One year of time-variable CHAMP-only gravity field models using kinematic orbits Nico Sneeuw University of Calgary, Department of Geomatics Engineering,
[email protected] ˇ Christian Gerlach, L´or´ant F¨oldv´ary, Thomas Gruber, Thomas Peters, Reiner Rummel, Draˇzen Svehla Technische Universit¨at M¨unchen, Institut f¨ur Astronomische und Physikalische Geod¨asie Abstract. A full year of CHAMP gravity field solutions has been calculated using the energy integral approach. The monthly solutions in the time frame 03.2002–02.2003 were based solely on kinematic orbits from CHAMP GPS orbit tracking and accelerometry. These kinematic orbits have not been contaminated by a priori gravity field information. Recovery of medium-wavelength time-variable gravity signal from CHAMP is expected to be at the edge of feasibility. This expectation is validated by comparison to an SLR solution of seasonal gravity variations and by comparison to oceanographic and meteorological models. It is shown that the error level of monthly CHAMP solutions is insufficient for revealing these time variations. Orbit decay—and consequently ground track variation—is a main contributor to this effect. Keywords.
gravity field
CHAMP ,
Jacobi integral, time variable
1 Introduction Gravity field determination from CHAMP data using the energy balance approach is successfully demonstrated, e.g. (Han et al., 2002) or (Gerlach et al., 2003c). First attempts of recovering time-variations in the gravity field from CHAMP data were published by Sneeuw et al. (2003) and Cheng et al. (2003). The former made use of the energy balance approach too. These studies made clear that time-variable gravity recovery from CHAMP will be at the edge of feasibility. C HAMP’s sensitivity is too low to reveal the full spectrum of seasonal and sub-seasonal variability. With proper care, perhaps the lowest spherical harmonic degrees can be revealed. In the aforementioned studies, two main obstacles were still present: i) Insufficient length of CHAMP time series. At least one full year of high quality orbits and accelerometry should be available. ii) Use of reduced-dynamic orbits. In (Gerlach et al., 2003a) it is demonstrated that gravity solutions from (reduced-) dynamic orbits are biased towards the a priori gravity field.
Recently, a full year of kinematic CHAMP orbits ˇ were published, (Svehla & Rothacher, 2003a), that strictly avoid the dependence on prior gravity information. Based on this data set a stationary CHAMPonly gravity field was determined by Gerlach et al. (2003a), denoted TUM -1 S. Validation showed TUM 1 S to be of equal or better quality than EIGEN -2, (Reigber et al., 2003; Gerlach et al., 2003b). This orbital data set would overcome the two aforementioned obstacles. It is optimally suited to test CHAMP’s sensitivity to temporal gravity changes. The next section will briefly describe the methodology and data used. After that the gravity recovery results are presented in spatial and spectral domain. As will be shown, the results are disappointing. The final section will focus on the role of orbit height and ground-track variation.
2 Data and methodology Energy balance approach. Using the energy integral—or Jacobi integral—is a well documented and proven method for gravity field recovery from orbit data by now, e.g. (Han et al., 2002; Gerlach et al., 2003c). Its main merits are: i) simplicity, ii) separation of orbit and gravity field information (and corresponding processing), and iii) avoidance of initial state problems. It relies, though, on precise orbit determination (POD), preferably as a long and uninterrupted time-series of orbit positions and velocities. In particular the resulting energy profile is sensitive to velocity errors, cf. (Visser et al., 2003). The energy integral (per unit mass) which is the basis for gravity field determination, reads: Z T + c = Ekin − U − Z − f · dr (1) At the left we have the unknown disturbing potential T , up till an unknown constant c. All terms at the right are determined from CHAMP data: Ekin = 12 v·v, the kinetic energy, requires orbit velocities v, U , the normal gravitational potential, requires satellite positions r,
δN March 2002
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Geoid variability, L=60, ψcap=30 −3
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Fig. 1. Monthly CHAMP solutions, in terms of geoid residuals, smoothed with spherical cap of 30◦ radius.
Z = 12 (ω×r)·(ω×r), the centrifugal potential at the satellite’s location, R f· dr is the dissipated energy, which is an integral of CHAMP’s accelerometer data f along the orbit, Boldface symbols denote vectors. Note that the dissipative energy mainly integrates the along-track force. Data sets. One full year of CHAMP GPS data (10 March 2002 – 9 February 2003) has been processed ˇ according to the methodology of Svehla & Rothacher (2003b). Zero-difference ambiguities and epochwise CHAMP positions and clock parameters were estimated from zero-difference carrier phase using CODE derived GPS satellite clock and earth rotation parameters. As a result, a full year of high quality 3D orbit positions has become available with a reported accuˇ racy of 1–3 cm, (Svehla & Rothacher, 2003a). Kinematic orbit determination does not provide orbit velocities. They were determined in a separate post-processing step from the orbit positions using smoothing splines, cf. (Gerlach et al., 2003a). The sampling rate for both r and v is 30 s. The accuracy of the velocity is estimated to be 0.1 mm/s or less C HAMP accelerometry was used to evaluate the dissipative energy integral in (1). However, due
to mismodelled scale factor and in particular to accelerometer bias, this integral potential would show a strong drift. Thus, the resulting disturbing potential is high-pass filtered above 1.2 · 10−4 Hz. Moreover, ancillary data (e.g. attitude quaternions) and models (e.g. tides) are used in the evaluation of the disturbing potential. Any known time-variable gravity contributions can be dealt with similarly in the dissipation integral. Spherical harmonic analysis. The resulting time series of disturbing potential T is reduced to a constant altitude, e.g. mean orbital altitude, and subsequently gridded onto a torus, cf. (Sneeuw, 2001). Spherical harmonic analysis is performed using inclination functions. The radial continuation and the gridding induce errors that are iteratively eliminated. Pre-mission error analyses and current CHAMP solutions indicate that one can expect a degree of resolution around L = 60. Then a Nyquist-type rule of thumb says that we need at least twice as much orbital revolutions, i.e. 120. With the given orbital revolution rate—around 16 revolutions per day—we would need approximately one week of data. Previous experience, (Sneeuw et al., 2003), has shown that high redundancy reduces sampling problems.
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Geoid variability, L=4, ψcap=0 −3
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Fig. 2. Monthly CHAMP solutions, in terms of geoid residuals, up to maximum degree L = 4.
Thus, the data have been separated into monthly batches, leading to monthly gravity field solutions up to L = 60. The solution March 2002 actually stands for the time frame 10.03.2002–9.04.2002, and so on. Each monthly solution is considered to be stationary. The time-series of all solutions provides the timevariable gravity field. Naturally, all shorter period phenomena will alias into the individual solutions. It is believed, though, that with the present accuracy of CHAMP data, this effect is insignificant.
3 Results Spatial domain. Although the monthly gravity fields were solved up to L = 60, sensitivity analyses and previous studies have demonstrated that only the very long wavelengths of time-variable signal may be revealed by CHAMP. Thus the results have to be filtered. We choose two approaches here: i) spatial filtering with a smoothing cap of 30◦ radius, and ii) spectral filtering by spherical harmonic degree cutoff at L = 4. Both filter techniques represent extremes. The chosen cap size and cut-off degree are somewhat arbitrary. However, they serve the purpose of focusing on the long wavelenghts, where timevariable signal is expected. Figure 1 shows the monthly geoid solutions, cor-
rected for a mean geoid, with the 30◦ -cap smoothing. The variations are in the ±3 cm range. A lot of high frequency content is still present in the filtered geoid residuals because of the wide spectral transfer of Pellinen coefficients. Figure 2 avoids this problem by rigorously lowpass filtering up till L = 4. The resulting monthly residuals are very smooth indeed. The range of variations, roughly ±3 cm, compares well to reported and expected values of geoid change, due to atmospheric, oceanographic or hydrological loading. Whether the patterns in figure 2 are a physically meaningful expression of mass transports is a different question. Spectral domain. Various signal and error spectra are represented in figure 3. The dashed TUM -1 S signal spectrum is overlain over Kaula’s curve. The TUM -1 S error curve represents the error spectrum of the static field, derived from the full year of data.√Every monthly solution will be worse by a factor 12, indicated by the dotted line. The thin lines represent the signal spectra of the monthly solutions after subtraction of the mean. In principle, whenever a signal curve is above the error curve, it implies that this signal is recovered by CHAMP . One should be suspicious for two reasons,
tent over a time frame longer than the one spanned by the SLR analysis. This expectation is confirmed, indeed, by the time series of residual coefficients in figure 5. This figure extends the SLR derived annual and semi-annual signals over the time frame of the aforementioned combined oceanographic-meteorological data set. For the selected coefficients, the correlation is convincingly high. For other coefficients—not shown here—the correlation can be lower.
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Kaula TUM−1s (signal) TUM−1s (error) monthly sol. (error) monthly − mean (signal) atmosphere/ocean
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though. The monthly error spectrum is within or very close to the ensemble of monthly signal spectra. Moreover, the signal spectra are more or less parallel to the error spectrum, whereas one would expect a drop in power with increasing spherical harmonic degree. This leaves the meaningfulness of the patterns in figures 1 and 2 in doubt. Figure 3 also contains two spectra with the annual and semi-annual signal power of combined oceanographically and meteorologically induced gravity variation. This combined model is derived from ECMWF pressure data and from MIT ocean circulation models over the time frame 01.1992–12.2000, cf. (Gruber & Peters, 2003). The semi-annual signal cannot be picked up in the monthly solutions. Only the annual signal curve rises above the monthly CHAMP error curve at the very lowest degrees. A further analysis of time-series of individual residual spherical harmonic coefficients δ C¯lm (t) confirms the previous suspicion that the monthly geoid solutions are physically not meaningful. Figure 4 is built up of time-series plots of all δ C¯lm (t) coefficients up to L = 5 (multiplied by 1010 ). Each black dot is a monthly (residual) coefficient. Also error bars are plotted. The monthly solutions are compared to grey curves, that represent annual and semiannual variations derived from two years of SLR data from Cheng et al. (2003). The data set contains laser ranging to 6 laser satellites (Lageos-1, Lageos2, Starlette, Ajisai, Stella and B EC) from January 2000 to December 2001. Note that the time spans of the CHAMP and SLR solutions do not overlap. Nevertheless one would expect the annual and semi-annual signals to be consis-
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Fig. 3. Monthly CHAMP signal and error spectra compared to TUM 1 S and to seasonal gravity signal from atmosphere and ocean.
lm
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Fig. 5. Selected coefficients—(l, m)=(2,1), (2,2) and (3,1)—of a combined oceanographic-meteorological model (01.1992–12.2000) compared to an annual and semiannual SLR solution (01.2000–12.2001).
Thus, the SLR solution seems to reflect physically meaningful gravity field variations. Consequently, the present CHAMP results lack meaning. It can be concluded that the accuracy of the monthly CHAMP solutions is insufficient to recover low degree time variations. The main contribution to the error budget of the gravity solutions are the errors in the orbit data themselves. Apparently, even the kinematic orbits are not sufficient for these purposes. Another main driver of the error budget is temporal and spatial aliasing. As mentioned before, time variations with periods shorter than a month will cause temporal alias into the monthly solutions. But also spatial aliasing, due to changing sampling configuration—read: changing ground tracks—will play an important role. Orbit characteristics. C HAMP’s orbit is slowly decaying. Apart from orbit manoeuvres on June 9– 10 and December 9–10, 2002, the satellite gradually comes down, see figure 6. As a consequence the ground track pattern changes continuously. Figure 7 shows a dense coverage (two weeks in June
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Fig. 4. Low degree spherical harmonic coefficients as time series (03.2002–02.2003) compared to annual and semi-annual SLR solution (01.2000–12.2001).
4 Conclusions
CHAMP orbital height
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Fig. 6. CHAMP height profile in terms of daily mean/min/max. Note the orbit manoeuvres on June 9–10 and December 9–10.
2002) and a repeat-mode ground-track, that existed briefly before the first orbit manoeuvre. The satellite was effectively performing 31 revolutions in 2 days. The two orbit manoeuvres cause the satellite to go through this commensurability again twice. In repeat mode, certain so-called lumped coefficients can be determined with high accuracy. In general, however, a repeat mode means low spatial resolution and a correspondingly low performance in overall gravity recovery. Figure 8 displays several repeat modes through which CHAMP can and will go. Later in its lifetime, at approximately 255 km height, the satellite will exactly perform 16 revolutions per day, leading to a very sparse spatial sampling.
One year of monthly CHAMP-only solutions from kinematic orbits has been derived using the energy balance approach. These orbits guarantee independence from a priori gravity field information. They are probably the best starting point for any CHAMPonly attempt to derive time-variable gravity. Nevertheless, our attempt has not been successful so far. Although their magnitude is right, the monthly results do not seem to reflect physically meaningful gravity variations. Comparison to SLR-derived annual and semi-annual gravity changes and to gravity changes implied by a combined oceanographicmeteorological model has corroborated the above conclusion. Although the time frames for these three different data sets were distinct, the SLR solution was shown to be consistent with many, though not all, spherical harmonic coefficients of the oceanographic-meteorological model. One reason for the lack of CHAMP sensitivity to time-variable gravity is the error level of the orbit data themselves. Most geophysical signal spectra are below the CHAMP error level. Also, the groundtrack variability due to the orbit decay contributes greatly to the lack of sensitivity. Certain months show good spatial coverage. During other weeks or months, the satellite can be in repeat mode, leading to a loss of spatial coverage. One basically tries to solve a time-variable field from a time-variable sampling
DOY 137 − 150
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Fig. 7. Two-weekly ground track patterns: second half of May (top) and second half of June (bottom), 2002.
configuration. This spatial-temporal sampling problem comes on top of the temporal aliasing of short period effects into the monthly solutions. The latter problem is discussed in the context of GRACE as well, e.g. (Velicogna et al., 2001). It is expected that ground-track variability will impede the consistency of the monthly GRACE solutions greatly.
References Cheng, M. K., Gunter, B., Ries, J., Chambers, D. P., & Tapley, B. D. (2003). Temporal variations in the earth’s gravity field from SLR and CHAMP GPS data. In I. N. Tziavos (Ed.), Gravity and Geoid 2002 (pp. 424–431). 3rd meeting of the IGGC, Thessaloniki, Greece.
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commensurabilities during orbit decay, I = 90
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Fig. 8. Commensurabilities (repeat orbits) as a function of orbit height.
ˇ Gerlach, C., F¨oldvary, L., Svehla, D., Gruber, T., Frommknecht, B., Oberndorfer, H., Peters, T., Rothacher, M., Rummel, R., Sneeuw, N., Steigenberger, P., & Wermuth, M. (2003a). A CHAMP only gravity field model from kinematic orbits using the energy balance approach. Poster presented at EGS-AGU-EUG Joint Assembly, 06–11 April 2003, Nice, France. ˇ Gerlach, C., F¨oldvary, L., Svehla, D., Gruber, T., Wermuth, M., Frommknecht, B., Peters, T., Rothacher, M., Rummel, R., Sneeuw, N., & Steigenberger, P. (2003b). A CHAMP only gravity field model from kinematic orbits using the energy integral. Geoph. Res. Letters. subm. ˇ Gerlach, C., Sneeuw, N., Visser, P., & Svehla, D. (2003c). Champ gravity field recovery using the energy balance approach. Adv. Space Res., 1, 73–80. Gruber, T. & Peters, T. (2003). Identification of mass variations from a series of global gravity field models, a simulation study. Poster presented at IUGG 2003, Sapporo, Japan. Han, S.-C., Jekeli, C., & Shum, C. K. (2002). Efficient gravity field recovery using in situ disturbing potential observables from CHAMP. Geophys. Res. Letters, 29(16). DOI 10.1029/2002GL015180. Reigber, C., Schwintzer, P., Neumayer, K.-H., Barthelmes, F., K¨onig, R., F¨orste, C., Balmino, G., Biancale, R., Lemoine, J.-M., Loyer, S., Bruinsma, S., Perosanz, F., & Fayard, T. (2003). The CHAMP-only earth gravity field model EIGEN-2. Adv. Space Res. accepted. Sneeuw, N. (2001). Satellite geodesy on the torus: Blockdiagonality from a semi-analytical approach. In M. G. Sideris (Ed.), Gravity, Geoid and Geodynamics, volume 123 of IAG Symposia (pp. 137–142).: IAG SpringerVerlag. GGG2000, Banff, Canada. ˇ Sneeuw, N., Gerlach, C., Svehla, D., & Gruber, C. (2003). A first attempt at time-variable gravity recovery from CHAMP using the energy balance approach. In I. N. Tziavos (Ed.), Gravity and Geoid 2002 (pp. 237–242). 3rd meeting of the IGGC, Thessaloniki, Greece. ˇ Svehla, D. & Rothacher, M. (2003a). Kinematic and reduced-dynamic precise orbit determination of CHAMP satellite over one year using zero-differences. Poster presented at EGS-AGU-EUG Joint Assembly, 06–11 April 2003, Nice, France. ˇ Svehla, D. & Rothacher, M. (2003b). Kinematic and reduced-dynamic precise orbit determination of low earth orbiters. Adv. Space Res., 1, 47–56. Velicogna, I., Wahr, J., & Van den Dool, H. (2001). Can surface pressure be used to remove atmospheric contributions from GRACE data with sufficient accuracy to recover hydrological signals? J. Geophys. Res., 106(B8), 16 415–16 434. Visser, P. N. A. M., Sneeuw, N., & Gerlach, C. (2003). Energy integral method for gravity field determination from satellite orbit coordinates. J. Geodesy, 77, 207– 216. DOI 10.1007/s00190-003-0315-8.
