Assessment of the impact of one-way laser ranging on orbit determination of the Lunar Reconnaissance Orbiter
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Abstract
From June 2009 to September 2014, ten stations of the International Laser Ranging Service acquired more than 5000 h of one-way ranging data to NASA’s Lunar Reconnaissance Orbiter (LRO). The LRO campaign was intended to support the determination of precise science orbits for which S-band Doppler and range measurements are the primary source. The laser ranges were finally not used for the NASA orbits, since the quality of the radiometric data exceeded the expectations. In this contribution, we explore the potential of the LRO laser ranges using an independent software which is able to process all types of LRO tracking data. It is shown that the laser ranges agree with the radiometric data at the level of 5 cm, provided a proper modeling of the LRO clock error is done, including relativistic effects. This accuracy does not fully attain that of the Doppler data, hence the impact of the laser ranges on orbit determination is small, improving the orbit on average by no more than 7 cm. Additional tests without the Doppler data show, however, that the laser ranges would have been of great use if radiometric tracking to LRO were less successful.
Keywords
Lunar Reconnaissance Orbiter Laser ranging Precise orbit determination1 Introduction
The Lunar Reconnaissance Orbiter (LRO, Chin et al. 2007) was launched in 2009 as a pathfinder mission for the resumption of manned lunar spaceflights considered at that time as a priority in the US space programme. Its main objective is the detailed survey of the lunar topography required for the identification of future landing sites. This survey is accomplished by photographic images taken by the Lunar Reconnaissance Orbiter Camera (LROC, Robinson et al. 2010) and by altimetric ranges collected by the Lunar Orbiter Laser Altimeter (LOLA, Smith et al. 2010). The LRO instrument suite is completed by further devices measuring the lunar surface temperature or looking for signs of water ice.
Some of the LRO instruments have a spatial resolution in the range of meters; the precision of the LOLA ranges even is specified with 10 cm. To retain as much as possible of these accuracies in the derived maps and terrain models, highly accurate and consistent orbits are required. The primary source for the orbits are S-band Doppler and range measurements acquired by an LRO-dedicated station at White Sands, New Mexico, and four stations in Hawaii, Australia and Europe associated with the commercial Universal Space Network (USN). The initial strategy of NASA also included to constrain the orbits by altimetric crossovers as practiced successfully in the Mars Global Surveyor mission. Even the first release of the NASA science orbits thus attained a precision of 20 m in total position and 1 m radially, both assessed by an analysis of overlaps of adjacent arcs (Mazarico et al. 2012). After a reprocessing in 2013, the precision is reported to be around 10 and 0.5 m, respectively (Mazarico et al. 2013, 2017), the improvement achieved by using a lunar gravity field from the Gravity Field and Interior Laboratory (GRAIL, Zuber et al. 2013) mission.
The performance of the LRO radio network was initially not believed to enable such accurate orbits. It was decided, therefore, to extend the LOLA capabilities to acquire additional tracking data from the Earth. To this end, a laser optics was mounted onto the LRO radio antenna and connected to the LOLA detector by a fiber optic cable. The spacecraft was thus equipped for one-way ranging from laser stations which are capable to reach the Moon. The first laser shot successfully received from Earth was from the station GO1L located at NASA’s Goddard Space Flight Center (GSFC) on June 30, 2009, less than two weeks after LRO’s launch (McGarry et al. 2013). In the following months, nine more stations joined the laser tracking campaign to LRO which was pursued over more than five years until September 2014.
Since the radiometric data turned out to be better than anticipated, the laser ranges to LRO were never used for the official NASA orbits. The LRO laser campaign thus produced de facto an experimental data set for investigating the properties of a novel data type. As such, the LRO laser ranges were intensively studied by various research groups. A topic repeatedly addressed is their use for characterizing the LRO clock behavior and the related problem of time transfer from station clock to spacecraft clock (Bauer et al. 2017; Mao et al. 2017). Most research, of course, was focused on the potential of the laser ranges for orbit determination. In Bauer et al. (2016) this was studied exclusively with orbits from laser ranges alone, while Buccino et al. (2016) and Mao et al. (2017) also computed orbits with mixed data sets from radio and laser observations. In all studies, the quality of the orbits is assessed by comparisons with the NASA science orbits. For the laser-only orbits, the differences reported vary strongly, from 7 m averaged from four arcs with a length of 1.5 days (Buccino et al. 2016) to 50 m based on 58 arcs each spanning two weeks (Mao et al. 2017). There is also no consensus on how the laser ranges contribute to the mixed solution. Buccino et al. (2016) observe an improvement in the orbit by 2 m, Mao et al. (2017) a degradation by 6 m, though the radial direction is improved by 0.5 m. All these results are affected by considerable uncertainties, since the orbits compared for these assessments differ themselves by 20 m and more from the NASA orbits.
In this work, the laser ranges are studied over the full period and without referring to orbits of external origin. The investigation builds on a set of radio-only orbits computed independently from NASA which is described in detail in Löcher and Kusche (2018). These orbits are close to the NASA science orbits, the averaged difference being 5.56 m in total position and 0.42 m radially. The overall precision, assessed by overlap errors, was found to be 2.81 m and 0.11 m, respectively. The orbits are based on both types of radiometric data, the data weighting being accomplished by a variance component estimation (VCE) on a pass-by-pass basis. To allow for such a procedure, the Doppler range-rates are integrated to biased ranges so that all observations have the same unit of meters. It is made easy by this policy to introduce the laser data as a third type of ranges. The data weights can then again be adjusted by a VCE now applied to all passes of the augmented data set.
The main challenge in using the laser ranges consists in the proper modeling of this observation type. The issues to solve are mostly related to the LRO clock which is affected both by the errors of its oscillator and the relativistic time shift observable for a moving clock in space. In this paper, both these disturbances are treated in a pragmatic way. Since we are interested primarily in precise orbits, no attempt is made to separate the error of the LRO clock from those of the station clocks. The modeling rather aims at the resulting error by estimating one clock polynomial per pass. For the relativistic time shift, a model for near-Earth space is used whose applicability for LRO will be demonstrated.
The paper proceeds in Sect. 2 with the detailed development of the observation model. In Sect. 3, the laser ranges and the chosen model are studied in residual analyses along our radiometric orbits. Section 4, finally, presents the results of orbit determination from the mixed data set. This section also includes some additional computations to assess the accuracies obtainable with less accurate radiometric data.
2 Modeling of the laser data
The laser ranging to LRO is technically a one-way uplink ranging. A laser station transmits laser pulses to the spacecraft and records the transmit times referred to the station clock which is synchronized with UTC. The pulses received by the laser optics and forwarded to the LOLA detector trigger again clock readings, now in the time system of the LRO clock denominated as Mission Elapsed Time (MET). The receive times are later transmitted to Earth via the science telemetry link and matched with the transmit times using preliminary solutions for the LRO orbit and the lag between the time systems involved (Mao et al. 2017). To reduce the data amount and improve the data quality, the matched time tags are finally condensed to normal points with a nominal sampling rate of 5 s.
The first two corrections are familiar from SLR analysis to Earth satellites and can be modeled with high accuracy. For the tropospherical correction, we use the standard model Mendes and Pavlis (2004). The center-of-mass correction is computed using the orientation data for the spacecraft and the antenna to which the laser telescope is mounted. The antenna boom is modeled with two hinges and its dimensions derived from the drawings in Tooley (2009). For both corrections, a coarse position of LRO is needed which is taken from the navigation orbit from the GSFC Flight Dynamics Facility.
The second effect \(\Delta _\mathrm{Rel}^\mathrm{CLK}\) is again specific for one-way ranging. It arises from the fact that the clocks involved in one-way ranging move on different paths in space. According to relativity, such clocks have different proper times. Consequently, a range computed from two readings of the clocks is wrong by an offset continuously changing according to the motion of the clocks and the surrounding masses.
This relativistic clock error has been treated differently in previous studies. While it is neglected in Mao et al. (2017), it is accounted for in Buccino et al. (2016) and Bauer et al. (2017), but computational details are given only in the latter reference. The strategy applied there is to refer both clocks to Barycentric Dynamical Time (TDB). For the station clocks, this is done via Terrestrial Time (TT) using the conventional relation between TT and TDB with periodical terms given by Moyer (1981). The proper time of the LRO clock is transformed via Barycentric Coordinate Time (TCB) which differs from TDB only by a constant rate. The transformation from proper time to TCB is accomplished by numerically solving a differential equation given by Moyer (1971).
Magnitude of laser range residuals along radiometric orbits for different setups
Case | Orbit | Relativity applied | Drift rate adjusted | RMS of residuals (m) |
---|---|---|---|---|
1 | IGG | No | No | 0.48 |
2 | IGG | No | Yes | 0.06 |
3 | IGG | Yes | No | 0.06 |
4 | IGG | Yes | Yes | 0.05 |
5 | NASA science | Yes | Yes | 0.10 |
6 | NASA navigation | Yes | Yes | 0.15 |
It should be noted, however, that our LRO processing is quite different from that of SLR to Earth satellites. The orbit computation is performed in a Moon-centered reference frame, since LRO moves in a Moon-centered gravity field. To achieve this change of frame, the station vectors associated to the ranges are reduced by the geocentric position of the Moon. For this, the lunar ephemeris from DE421 (Williams et al. 2008) is used, adapted to the geocentric frame by applying the transformation between TT- and TDB-compatible coordinates given in Petit and Luzum (2010). In order to be consistent with the time scale of the ephemeris, the time tags of the ranges are finally transformed to TDB.
3 Residual analyses along radiometric orbits
To investigate the properties of the laser ranges and the suitability of the observation model developed, we first consider the residuals along our orbits from radiometric data. The analysis is carried out with different setups: with and without applying the relativistic clock correction and with different degrees of the clock polynomial. In the first setup, the observation model is simplified by omitting both the relativistic correction and the clock drift rate as estimable parameter (Case 1). Another two setups each keep one of these features, Case 2 the clock drift rate (omitting relativity), and Case 3 the relativistic correction (with no drift rate adjusted). Case 4 implements the complete model from the previous section.
Observation volumes of the laser stations and station-averaged residuals. Data statistics based on the start/end time tags of normal point files as listed in index.tab in the LRO Radio Science Archive. Residuals from Case 4 in Table 1
Station | Location | Time observed | RMS of residuals (m) | |
---|---|---|---|---|
h | % | |||
MDOL | McDonald, Texas, USA | 638.3 | 13.3 | 0.05 |
YARL | Yarragadee, Australia | 836.8 | 16.2 | 0.04 |
GODL | Greenbelt, Maryland, USA | 418.5 | 8.1 | 0.04 |
MONL | Monument Peak, California, USA | 1375.2 | 26.7 | 0.06 |
GO1L | Greenbelt, Maryland, USA | 1530.3 | 29.7 | 0.06\(^\mathrm{a}\) |
HARL | Hartebeesthoek, South Africa | 50.0 | 1.0 | 0.04 |
ZIML | Zimmerwald, Switzerland | 81.5 | 1.6 | 0.02 |
HERL | Herstmonceux, UK | 70.9 | 1.4 | 0.11 |
GRSM | Grasse, France | 97.1 | 1.9 | 0.03 |
WETL | Wettzell, Germany | 9.2 | 0.2 | 0.02 |
Figure 1 allows to watch how this effect is absorbed by the clock parameters. The diagrams display the coefficients of the degree 2 polynomial obtained with the uncorrected ranges (Case 2, blue) and with the ranges reduced by the relativistic model (Case 4, red). In the first case, the estimates for the drift rate are spread almost uniformly in a range of \(\pm 9 \times 10^{-15} \; \text {s}/\text {s}^{2}\) (Fig. 1, bottom). A closer look reveals that the estimates ascend in periods of 14 days from the lower to the upper boundary and then suddenly drop down by a change of sign. This conspicuous pattern is absent when the relativistic model is applied. It can thus be identified as the fingerprint of the relativistic clock error—one of the rare signals from relativity which can be isolated unambiguously from satellite data.
The lower degrees of the clock polynomial are less affected by relativity. The clock bias, almost identical in both cases, is governed by the drift of MET with respect to UTC caused by the drift of the LRO clock (Fig. 1, top); the discontinuities occurring are possibly due to offsets applied to MET during the pairing of transmit and receive times (Mao et al. 2017). The slope of the bias between these discontinuities allows to assess the clock drift to be around \(-7 \times 10^{-8} \; \text {s}/\text {s}\). This is confirmed and specified by the dedicated estimates for the clock drift given by the linear term of the polynomial (Fig. 1, center). The shift between the time series depicted is due to the change of the reference time scale in Case 4 and equals the rate \(L_{G}\, (6.97 \times 10^{-10}\), Petit and Luzum 2010) included in the relativistic model.
While our modeling does not allow any statement about the behavior of the station clocks, the overall performance of the stations can be assessed by further analyzing the residuals. Table 2 summarizes their averages for the stations, together with the volumes of observation time. The largest residuals (in a rather homogeneous field) are found for the station Herstmonceux, probably due to a peculiarity in the emitted pulse sequence which is known to cause problems for the LOLA detector (Mao et al. 2017). The most favorable averages result for Zimmerwald, Grasse and Wettzell, in the last case, however, on a poor statistical basis. Another hardware-related issue can be observed at the station GO1L at GSFC, the most productive one in the network. For about one year, until October 2010, the residuals found for this station are on average 8 cm, then decrease to 5 cm. This coincides with the change of the station’s time standard from a cesium to a H-maser clock (Mao et al. 2017), which has obviously led to much more precise records of the transmit times.
Two final analyses will show that the residuals reveal much less about the orbit. For this, the residual computation is carried out along the NASA science orbits (Case 5 in Table 1) and the NASA navigation orbits (Case 6). In the former case, the total average is 10 cm, an increase of 5 cm with respect to our orbits. This increase does not reflect at all the actual difference between the orbits which is 5.56 m (see Sect. 1). The mismatch is even larger with the navigation orbits along which the residuals are on average 15 cm, while the difference to our orbits is 74.70 m. It is obvious that, in both cases, most of the difference is absorbed by the clock polynomials. The residuals are thus nearly blind for large-scale orbit errors which makes them unusable for orbit validation.
4 Orbit determination with combined data set
The size of the residuals found in the previous section may suggest that the laser ranges should be of great use in orbit computation. Indeed, when merged with the radiometric data, they compete with other precise observations which are typically collected in parallel at locations close to the laser stations. Moreover, the radiometric data clearly prevail over them only through their volume. Though the laser network succeeded on many days in observing all possible passes, the observation time during the five years did not exceed 5130 h, while the radio stations acquired 16,567 h. The ratio of data volumes is thus 1 : 3.2, or even 1 : 6.4 when considering that the radio stations produce two types of observations at each epoch.
Mean RMS of overlap errors of orbit arcs from radio and laser observations, in meters. Nominal arc length is 2.5 days. In italics: difference to radio-only orbits
Mission phase\(^\mathrm{a}\) | Total | Along | Cross | Radial |
---|---|---|---|---|
CO | 4.21 | 3.15 | 2.31 | 0.21 |
+0.04 | +0.03 | +0.01 | 0.00 | |
NO | 2.02 | 1.45 | 1.10 | 0.06 |
0.00 | 0.00 | +0.01 | 0.00 | |
SM1 | 2.04 | 1.50 | 1.13 | 0.08 |
−0.34 | −0.34 | −0.01 | 0.00 | |
SM2 | 3.19 | 2.69 | 1.19 | 0.13 |
0.00 | +0.01 | 0.00 | 0.00 | |
ES1 | 2.91 | 2.37 | 1.23 | 0.14 |
−0.01 | −0.01 | −0.01 | 0.00 | |
ES2 | 3.49 | 2.76 | 1.54 | 0.13 |
−0.01 | −0.01 | 0.00 | 0.00 | |
All phases | 2.74 | 2.15 | 1.27 | 0.11 |
−0.07 | 0.08 | −0.01 | 0.00 |
Mean RMS of overlap errors from different range types and range type combinations
Range types | Arcs solved | Overlaps | Overlap error (m) | |
---|---|---|---|---|
Total | Radial | |||
Doppler | 882 | 740 | 3.10 | 0.12 |
Doppler, two-way | 883 | 740 | 2.81 | 0.11 |
Doppler, two-way\(^\mathrm{a}\) | 704 | 427 | 12.76 | 2.98 |
Doppler, laser | 880 | 737 | 3.01 | 0.12 |
Doppler, two-way, laser | 883 | 740 | 2.74 | 0.11 |
Two-way | 883 | 738 | 22.09 | 0.58 |
Two-way\(^\mathrm{b}\) | 432 | 19.42 | 0.52 | |
Two-way, laser | 883 | 740 | 12.30 | 0.38 |
Two-way, laser\(^\mathrm{b}\) | 432 | 6.09 | 0.23 | |
Laser | 637 | 391 | 82.81 | 3.25 |
Laser\(^\mathrm{c}\) | 686 | 428 | 34.53 | 1.86 |
This finding may appear disappointing, but it should be kept in mind that the laser ranging to LRO was intended to support radiometric data with an increased error level. According to the numbers given in Mazarico et al. (2012), the S-band tracking error was overestimated by a factor 3 during the mission planning stage. To learn a lesson about the LRO laser campaign, it would be instructive to know what the laser ranges would have contributed in that case. An approximate answer can be found when considering the errors of the radiometric range types as assessed above. A comparison of the figures shows that the errors of the two-way ranges exceed those of the Doppler ranges by a factor 3 at White Sands, and by a factor 5 on average for all stations. The two-way ranges in the quality provided should thus emulate rather well the Doppler ranges with the anticipated reduced accuracy. The resulting orbit precision can be found in Table 4 summarizing all solutions from the different range types and the combinations of them. With the noise of the two-way ranges, the orbit precision degrades from 3.10 to 22.09 m in total position and from 0.12 to 0.58 m in radial direction. When combining two-way and laser ranges, however, the degradation is only half the size, the precision achieved 12.30 and 0.38 m, respectively. Considering only the arcs for which laser data actually are provided, the precision is even 6.09 and 0.23 m. The orbit quality in these arcs is thus already close to that from the Doppler data with their actual noise, degraded only by a factor 2.
For completeness, Table 4 also includes the solution from laser ranges only. From a total of 883 arcs, only 637 arcs can be solved in this case, with an error of 82.81 m in total position and 3.25 m radially. These figures are reduced to 34.53 and 1.86 m by a more restrictive parametrization omitting the drift rate for the clock. Even these results do not match the supposed quality of the laser ranges, obviously due to the lesser density of the observations. This is easily verified with the radiometric ranges by trimming them exactly to the periods of the laser passes. Only due to the sparser coverage, the overlap errors then increase to similar orders, from 2.81 and 0.11 m obtained with the full data set to 12.76 and 2.98 m with the reduced data.
It is difficult to predict from that, which precision would be achieved, if the laser ranges had, inversely, the coverage of the radiometric data. Considering the accuracies assessed by the VCE, it is obvious, however, that it should lie between the precisions obtained with the two radiometric types. Assuming that, in this interval, the orbit error increases linearly with the range error, a prediction of 12 m for the position vector and 30 cm for the radial direction would be a reasonable choice.
5 Conclusions
With the one-way ranges to LRO, the ILRS network produced a set of tracking data which is unique both for the type of observation and for the distance of the spacecraft. It is shown in this paper that this data set is of very good quality, with a mean error of 5 cm, assessed in a residual analysis along independently derived radiometric orbits. To attain this level of agreement, a proper modeling of the range is needed which is complicated, in particular, by the relativistic delay of the LRO clock. It was demonstrated that this effect can be calculated conveniently using a model for near-Earth space, but that it can be absorbed with similar accuracy by a clock polynomial of degree 2.
The high quality of the laser data turns out to be less effective when compared with the radiometric observations. The laser ranges clearly outperform the radiometric ranges, but do not fully attain the accuracy of the Doppler data. Their impact on the orbits is thus small, limited to situations where they fill gaps in the Doppler data or compensate large Doppler noise. Averaged over the mission, the improvements achieved this way were found to be not larger than 7 cm.
This result does fully agree with the expectations since the LRO laser ranging was intended to support radiometric data with a larger error than actually observed. It could be shown that, in this case, the laser ranges would have contributed significantly to the orbit precision, reducing the effect of the additional noise by half. A further improvement would have been prevented by the lesser density of the laser data.
Notes
Acknowledgements
This research was funded by the German Research Foundation (DFG) within the research unit FOR 1503 “Space-Time Reference Systems for Monitoring Global Change and for Precise Navigation in Space.” We thank three anonymous reviewers for their helpful comments.
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