Abstract
For low-Earth-orbit (LEO) satellites, high-precision clock estimation often depends on high-precision real-time global navigation satellite system (GNSS) products. Thus, service providers often choose to downlink observation data to the ground to achieve high accuracy. To relieve this burden for future LEO mega-constellations, this study investigates the performance of relative clocks and orbits determined between LEO satellites using the phase common-view (PCV) method. The PCV results are compared with results from three other single-satellite-based clock and orbit determination methods. Using real data from the Gravity Recovery and Climate Experiment (GRACE) Follow-On satellites and three different types of real-time GNSS products, the PCV method can deliver a relative clock precision below 0.2 ns and a relative orbital user range error of approximately 5 cm, even when using the broadcast ephemeris, whereas all three other methods encountered sharp degradations in their results when using degraded real-time GNSS products.
1 INTRODUCTION
In recent years, low-earth-orbit (LEO) satellites have been considered an essential augmentation to traditional global navigation satellite systems (GNSSs) and are expected to bring revolutionary improvements to satellite-based high-precision positioning, navigation, and timing (PNT) services (Reid et al., 2018, 2022; Yang, 2019; Wang, El-Mowafy et al., 2022, Wang et al., 2022). The low altitudes of these satellites (a few hundred kilometers to approximately 1500 km (Montenbruck & Gill, 2000)) directly lead to a much higher signal strength at the user end, faster speeds, and shorter latencies for signal transmission. Compared with GNSS satellites in medium earth orbits (MEOs) or even geostationary orbits (GEOs), LEO satellites enable an improved anti-jamming capability (GPS World Staff, 2017), aid in the whitening and modeling of multipath effects in urban areas, and generate much faster satellite geometry changes. As another beneficial characteristic for filter-based positioning and timing, these satellites can shorten the convergence time in ambiguity-float scenarios and the time to first fix in integer ambiguity resolution for precise point positioning (PPP) (Ge et al., 2018; Li et al., 2018) and PPP real-time kinematic (RTK) positioning (Wang, El-Mowafy et al., 2022).
However, using high-precision LEO satellite orbital and clock products in real time is a pre-condition for realizing real-time LEO-GNSS-integrated PNT services. In contrast to GNSS satellite precise orbit determination (POD), which uses data collected from a ground network (Zeng et al., 2021; Peng et al., 2022; Zhang et al., 2021), LEO satellites have much smaller footprints on the ground (Cakaj et al., 2014; Wang & El-Mowafy, 2020) and are rarely able to be continuously observed by a ground network. Thus, their POD relies on GNSS observations tracked onboard. For high-accuracy LEO single-satellite POD and clock determination, i.e., at a few centimeters or even better with ambiguities fixed in a post-processing mode (Mao et al., 2021; He et al., 2022; Wang, El-Mowafy et al., 2023), this situation often requires the downlinking of GNSS observation data tracked onboard, so that data processing can be performed on the ground to ensure that sufficient computational power and high-accuracy real-time GNSS products transferred via the Internet are available (Wang, Liu et al., 2023). Onboard LEO satellite POD and clock determination have been studied for over a decade (Hauschild et al., 2016, 2022; Gong et al., 2022). On the one hand, these processes require a compromise between model complexity and computational power; on the other hand, these processes must handle receiving high-precision real-time GNSS products in space while LEO satellites orbit the Earth.
Some studies have investigated real-time LEO satellite POD using satellite-linked GNSS products, e.g., products from the Japanese Multi-GNSS Advanced Demonstration of Orbit and Clock Analysis or the Australian/New Zealand Satellite-Based Augmentation System (Allahvirdi-Zadeh et al., 2021). However, global coverage, which is necessary for onboard processing of LEO satellites throughout their entire orbit, remains a challenge for both products. Commercial services via GEO links may be a possibility for obtaining real-time GNSS products for onboard POD; however, the satellite geometry leads to poor precision in polar areas (Hauschild et al., 2016). Moreover, the Galileo High-Accuracy Service has been discussed for real-time onboard POD for Sentinel-6A, which achieved a sub-decimeter to 1-dm POD accuracy (Hauschild et al., 2022). However, this approach requires the continuous onboard tracking capability of Galileo signals as a condition. In short, the precision of onboard LEO satellite POD and clock determination is sub-optimal compared with that on the ground and requires additional payloads and computation power that would add to the cost of this approach. Downlinking the observation data of the entire constellation in real time is also challenging because of the large amount of data transmitted, the data transmission infrastructure (between LEO satellites and from LEO satellites in view to the ground), and data storage. A situation may arise in which only a limited portion of the reference LEO satellites (see the black reference satellites in Figure 1) can downlink the onboard observations to the ground monitoring stations (GMSs) (see the red dots). The data of other satellites can be exchanged in real time by utilizing inter-satellite links (green lines) and downlinking data (red lines) from satellites in view. Moreover, high-precision orbital and clock products would be needed in real time for the entire constellation. These conditions highlight the need for a strategy for real-time LEO satellite clock and orbital determination that does not rely on the availability of high-precision GNSS products and very high computational power (e.g., for numerical integration) and, for the above-mentioned case, that can operate with only a relative mode, such that single-satellite POD and clock determination can be performed for reference LEO satellites on the ground with high precision.
Conceptual illustration of time synchronization and clock determination for a LEO satellite constellation
The GMSs and master processing center (MPC) are given here only for a demonstration of the concept.
The phase common-view (PCV) method, also known as RTK time transfer, has been used for ground-based time transfer for decades (Feng & Li, 2010; Sun et al., 2021; Tu et al., 2021; Xue et al., 2021). With the use of between-receiver differences, the satellite clock errors and hardware biases are eliminated, and the satellite orbital errors are significantly reduced depending on the baseline length. The estimated between-receiver clock errors can be interpreted as time-transfer results. Similar to ground baselines, time transfer between LEO satellites can also benefit from between-satellite differences to relieve the need for real-time high-precision GNSS satellite orbits and clocks. However, in contrast to ground baselines, which are primarily static during the process of time transfer, LEO satellite time transfer deals with fast-moving objects and changes in baseline lengths. The relative kinematic orbital errors are added as estimable parameters. As an advantage, GNSS observations tracked at the same time by LEO satellites are not influenced by tropospheric delays, thus avoiding the need to estimate the zenith tropospheric delays required for long ground baselines. Nie et al. (2007) applied the common-view method for the Gravity Recovery and Climate Experiment (GRACE) satellites A and B using their code observations and achieved nanosecond-level accuracy for the estimated inter-satellite clock errors. For the PCV method, the precision of both the relative LEO satellite clock and orbital errors that can be achieved in real time needs further investigation. In addition to cases in which available high-precision LEO reference satellite orbits are introduced during processing, this study examines cases in which the orbits of the reference satellites are not sufficiently precise, e.g., when issues arise in data downlinking (see the red lines in Figure 1) or in attaining high-precision GNSS products. In such cases, time synchronization between LEO satellites that are needed in, e.g., formation flying can still be enabled with inter-satellite links (see the green lines in Figure 1) and onboard processing using the PCV method. Thus, the corresponding influences of the reference satellite orbital accuracy on the relative clock and orbital errors are also assessed.
In this contribution, real data from the GRACE Follow-On satellites are used to validate the proposed LEO satellite PCV method for time transfer among satellites to ensure their synchronization. In addition to the PCV method processed with a filter-based sequential least-squares (SLS) adjustment, three other GNSS-based methods for LEO satellite time transfer are tested for comparison, as listed below:
The first method utilizes the difference between LEO satellite clock errors determined in single-satellite-based reduced-dynamic POD mode with a batch least-squares (BLS) adjustment, termed the RD BLS method in this study;
The second method utilizes the difference between LEO satellite clock errors determined in single-satellite-based kinematic POD mode with BLS adjustment, termed the KN BLS method in this study;
The third method utilizes the difference between LEO satellite clock errors determined in single-satellite-based kinematic POD mode with a filter-based SLS adjustment, termed the KN SLS method in this study.
The PCV method and the other three GNSS-based methods will be described in the following section. Next, time-transfer results obtained by the four methods are compared for three different kinds of GNSS products with different precisions. Conclusions are given at the end of the paper.
2 PROCESSING STRATEGY
This section first introduces an observation model of the PCV method used for LEO satellite time transfer. Then, the three other GNSS-based methods, mentioned in Section 1, are described with their general processing strategies, advantages, and disadvantages.
2.1 PCV Method for LEO Satellite Time Transfer
To form the observation model of the LEO satellite PCV method, the ionosphere-free (IF) observed-minus-computed (O-C) terms of the between-receiver code 
and phase 
observations for GNSS satellite s at time ti can be expressed as follows:
1
2
The estimable terms 
and 
are expressed as follows:
3
4
5
where the subscripts r and v denote the reference and rover LEO satellite, respectively, and rv denotes the between-receiver single difference with the item of LEO satellite r subtracted from that of LEO satellite v. The superscript T denotes the transpose of a matrix. 
represents the orbital vector in the Earth-centered Earth-fixed (ECEF) system of GNSS satellite s, and 
and 
denote the introduced ECEF orbits of the reference satellite and the estimable orbital difference between the two LEO satellites, respectively. c is the speed of light, and 
denotes the estimable between-receiver clock difference, which contains the true between-receiver clock difference (Δtrv), the relativistic effects between the two LEO satellites (Δθrv), and the IF code biases between the two LEO satellite receivers (drv,IF), as shown in Equation (3). f1 and f2 denote the first and second frequencies used for the IF combination, respectively. 
and 
indicate the between-receiver ambiguity on the two frequencies, and δrv,IF and drv,IF denote the between-receiver IF phase and code bias, respectively. 
and 
contain the between-receiver IF noise and mis-modeled errors for code and phase observations, respectively. The mis-modeled errors include the GNSS satellite orbital errors projected on the baseline, the between-receiver multipath effects, and the term 
due to errors in the introduced reference LEO satellite orbits, which can be expressed as follows:
6
where 
denotes the true orbital vector of the reference LEO satellite (as the user). When the LEO satellite baseline length is much shorter than its distance to the GNSS satellites, the term 
is generally ignored. However, 
could increase slightly with increasing baseline length and increasing errors in the reference satellite orbits 
. This issue will be further discussed in Section 3.
In Equations (1) and (2), all of the modeled errors are corrected in the O-C terms, including the phase windups, antenna sensor offsets, phase center offsets and variations of the LEO and GNSS satellites, and various tidal effects. The estimable parameters are 
, 
, and 
. Their full expressions are given in Equations (3)–(5) to better present their contents. In the case of multi-GNSS processing, inter-system biases (Nadarajah et al., 2013) must be further considered in Equations (1) and (2).
Processing is performed in a filter-based mode via the SLS method (Verhagen et al., 2017). In addition to the observation model based on Equations (1) and (2), the time-update step includes a time-constraint set of the estimable ambiguity vector 
, as long as a cycle slip does not occur between the time points ti and ti+1, as follows:
7
8
where 
denotes the time-updated ambiguity vector at ti+1. QÑ(ti) denotes the variance-covariance matrix of the ambiguity vector at ti, and QÑ(ti|i+1) denotes its time-updated counterpart at ti+1.
2.2 Other GNSS-Based Time Transfer Methods for Comparison
In addition to the PCV method introduced in Section 2.1, three additional GNSS-based methods, i.e., the RD BLS, KN BLS, and KN SLS methods, were used for comparison purposes.
The RD BLS method determines the single-satellite clock errors in the reduced-dynamic POD mode using the BLS adjustment. Here, the GNSS observations tracked onboard LEO satellites are used to improve the estimable dynamic parameters, and stochastic velocity pulses or piece-wise constant stochastic accelerations are used to compensate for forces that cannot be fully described by the model, such as air drag effects (for more details, see the work by Wang, Liu et al. (2023)). Differences are then formed between the estimated single-satellite clock errors, resulting in between-receiver clock errors that have the same expectation form as in Equation (3). This method usually delivers good precision of the estimated orbits and clocks, i.e., at a few centimeters, but relies heavily on high-precision GNSS products in real time. In addition, the method typically requires relatively high computational power and requires a slightly longer processing time for 24-h BLS adjustment compared with filter-based processing. The latter implies a relatively longer prediction time to obtain real-time clock values. For example, on a modern X86 processing unit with a main frequency of 3.0 GHz, 24-h arc BLS processing takes approximately 5 min, and at least the same prediction time is required to provide users with real-time clock products. Because of the diverse systematic effects contained in the LEO satellite clock estimates, their prediction precision degrades considerably with increasing prediction time, e.g., from 30 s to 5 min or longer (Wang & El-Mowafy, 2021; Ge et al., 2023; Wu et al., 2023).
Similar to the RD BLS method, the KN BLS method determines the single-satellite clock errors based on BLS adjustment and forms the between-receiver difference using the estimated single-satellite clock errors. Different from the RD BLS method, the KN BLS method directly estimates the LEO satellite orbits in Cartesian coordinates instead of integrating the orbits numerically to the desired time points using existing and estimated dynamic models. This approach greatly reduces the required computational power, but results in a slightly lower precision compared with the RD BLS method (Allahvirdi-Zadeh et al., 2021). Similar to the RD BLS method, the KN BLS method relies heavily on high-precision GNSS products in real time and requires a longer processing time, and thus a longer prediction time, for the clocks than filter-based solutions.
Lastly, the KN SLS method is a more efficient method for determining single-satellite-based clock errors compared with the two above-mentioned methods. The KN SLS method allows for a short prediction time of the clock errors in real time. However, this method is sensitive to the precision of the real-time GNSS orbital and clock errors. It is worth noting that all three methods introduced in this section form the between-receiver clock errors based on the difference between estimated single-satellite clock errors, whereas the PCV method directly estimates the between-receiver clock errors, which generally enhances the model strength.
3 TEST RESULTS
This section evaluates the relative clock and orbital results of the proposed method and the three other GNSS-based methods using dual-frequency L1/L2 Global Positioning System phase and code observations from the GRACE Follow-On satellites, i.e., the GRACE C and D satellites (Flechtner et al., 2014), for December 1–7, 2019. The two GRACE Follow-On satellites were flying at an altitude of approximately 503 km during the test week, separated by a distance of approximately 181 km. GRACE C serves as the reference satellite, and GRACE D is considered the rover satellite. The four methods (see Section 2) were applied in the processing, with a processing time of 24 h and a sampling interval of 30 s.
For comparison purposes, three types of real-time GNSS products with different precision were used for the tests, i.e., the real-time stream from the French National Centre for Space Studies (CNES) (Kazmierski et al., 2018), denoted as CNES RT in Table 1; the predicted part of the International GNSS Service (IGS) ultra-rapid products (Noll, 2010; Johnston et al., 2017), denoted as IGS UR in Table 1; and the broadcast ephemeris from the navigation message, denoted as BRDC in Table 1. Details of the three real-time GNSS products are given in Table 1, with the general accuracies of the orbital and clock products summarized based on that reported by Kazmierski et al. (2018) and IGS Products (2023). Because the sampling interval of the IGS UR clocks is larger than the observation sampling interval of 30 s, clock interpolation is necessary, which further degrades the precision.
Real-Time GNSS Products Used for Processing
In the following three subsections, the calculated relative clock errors between LEO satellites, the relative orbital errors between LEO satellites, and the influence of the reference satellite orbital accuracy on the relative clock and orbital errors are discussed in detail. Post-processed high-accuracy orbits provided by the Jet Propulsion Laboratory (Wen et al., 2020) were used as reference orbits. Using the final products provided by the Center for Orbit Determination in Europe (Dach et al., 2009), the clocks obtained from a post-processed BLS reduced-dynamic POD process were used as reference clocks. For an adequate comparison between different methods, the same pre-processed data were used before the final orbit and clock determination. The data were screened by applying fault detection and exclusion using a 4.42-sigma threshold before calculating the root mean square errors (RMSEs) and standard deviations (STDs).
3.1 Relative Clock Errors Between LEO Satellites
Relative clock errors represent a major concern for studying time transfer between LEO satellites. Figure 2 illustrates the relative clock errors between GRACE C and D on December 1, 2019, using the CNES RT products as an example. From Figure 2, it can be observed that for the case with high-precision GNSS products available, the RD BLS and KN BLS methods provide better results than the two SLS methods. Compared with the single-satellite clock errors, shown in Figure 3, it can be seen that the differencing step partially reduces the common mis-modeled errors, especially for the BLS methods. From Figure 2, it can also be observed that, compared with the KN SLS method, which determines differences between the estimated single-satellite clock errors, the PCV SLS method shows better relative clock precision because of its greater model strength, as mentioned in Section 2.2.
Relative clock errors between GRACE C and D on December 1, 2019, determined using the CNES RT products
STD represents the STD of the relative clock errors.
Satellite clock errors of GRACE C (left) and D (right) on December 1, 2019, determined using the CNES RT products
STD represents the STD of the satellite clock errors.
When real-time GNSS products of lower precision are utilized, the results differ. Figure 4 shows the relative clock errors between the two GRACE Follow-On satellites obtained using the CNES RT, IGS UR, and BRDC products (see Table 1). It can be seen that in contrast to the PCV SLS method, the other three methods are significantly influenced by the precision of the used GNSS products. Here, the superiority of the PCV SLS method, i.e., its independence of high-precision GNSS products, becomes visible. When the broadcast ephemeris is used (see the bottom panel of Figure 4), the STD of the relative clock errors obtained for the PCV SLS method remains at approximately 0.2 ns, whereas all other methods deliver STDs above 1 ns.
Relative clock errors between GRACE C and D on December 1, 2019, determined using the CNES RT products (top left), the predicted part of the IGS UR products (top right), and the broadcast ephemeris (bottom)
STD represents the STD of the relative clock errors.
These differences in relative clock stabilities can be better illustrated by the modified Allan deviation (MDEV) (Allan & Barnes, 1981), as shown in Figure 5. When the CNES RT products are used (left panel of Figure 5), the relative clock stabilities obtained using the PCV SLS method (red line) only ranked third among the four methods. However, when the broadcast ephemeris is used (right panel of Figure 5), the MDEVs of all methods increase, except for the PCV SLS method. Although the short-term stability of the PCV SLS is worse than that of the RD BLS method, its mid-to long-term stability over an averaging time of 1000 s shows the best performance.
MDEV of the relative clock errors between GRACE C and D on December 1, 2019, determined using the CNES RT products (left) and broadcast ephemeris (right)
The average STDs of the relative clock errors are illustrated by colored bars in Figure 6. The average STDs are calculated as the square root of the mean of the squared STD. This averaging strategy also applies to the diverse root mean squares (RMSs) in the analysis of the relative orbital errors in the next subsection. It is obvious that the PCV SLS method (red bars) has an advantage, particularly when high-precision GNSS products are not available, e.g., in cases of in-orbit processing or without the Internet. As shown in Table 2, the average STD remains below 0.2 ns when different GNSS products are used. In addition, Table 2 also shows a sharp rise in the MDEV at 120, 1200, and 12000 s for all methods when GNSS products with degraded precision are used, except for the PCV SLS method. MDEVs for the PCV SLS method are at approximately 1.5 × 10−12, 1.9 × 10−13, and 9.2 × 10−15 for an averaging time of 120, 1200, and 12000 s, respectively, even when the broadcast ephemeris is used.
Average STDs of relative clock errors between GRACE C and D for December 1–7, 2019, determined using different real-time GNSS products
Average STDs and MDEVs of the Relative Clock Errors Between GRACE C and D for December 1–7, 2019
In addition to the precision of the relative clock errors, the processing efficiency of the SLS methods appears to be better than, e.g., the BLS methods. To process 24 h of data with a sampling interval of 30 s, the SLS methods require less than 1 min (for single-central processing unit [CPU] processing), whereas the BLS methods require approximately 5 min (for two-CPU parallel processing) on a typical X86 Linux processing unit with an operating frequency of 2.7 GHz.
3.2 Relative Orbital Errors Between LEO Satellites
As an extension to the relative clock errors investigated in Section 3.1, the relative orbital errors are also of great interest for maintaining proper satellite configuration. Using the four different methods (see Section 2) and the three real-time GNSS products (see Table 1), this section analyzes and compares the accuracy of the relative orbital errors in the radial (R), along-track (S), and cross-track (W) directions. The ECEF orbital errors were transformed to the RSW system with respect to the rover satellite, i.e., GRACE D in our case. In addition, the RMS of the orbital user range error (OURE), which here describes the averaged projection of the relative orbital errors to the Earth, is evaluated as follows:
9
where σR,rv, σS,rv, and σW,rv represent the RMS of the radial, along-track, and cross-track relative orbital errors. The coefficient ωR decreases and ωSW increases with decreasing orbital height (Chen et al., 2013). For GRACE Follow-On satellites at approximately 503 km, ωR and ωSW are approximately 0.4564 and 0.6286, respectively.
Considering the LEO satellite relative clock errors, σSISRE,rv was also compared by applying different methods and using different GNSS products, where σSISRE,rv is expressed as follows:
10
with:
11
where RMS(·) calculates the RMS value. δXR,rv and 
represent the relative radial orbital errors and relative clock errors, respectively. Note that a daily mean has been removed from 
via the calculation of σRΔt,rv.
As an example, Figure 7 shows the relative along-track orbital errors computed on December 1, 2019, for the three types of GNSS products. Similar to the relative clock errors, the RMS for all of the single-satellite-based methods (RD BLS, KN BLS, KN SLS) rises sharply as the precision of the GNSS products decreases, whereas the RMS of the relative along-track orbital errors remains nearly constant for the PCV SLS method.
Relative along-track orbital errors between GRACE C and D on December 1, 2019, determined using the CNES RT products (top left), the predicted part of the IGS UR products (top right), and the broadcast ephemeris (bottom)
The along-track direction was not the only direction that exhibited these behaviors. Figure 8 shows the average OURE of the relative orbital errors. Similar to the relative clock errors in Section 3.1, the PCV SLS method ranked third when high-precision CNES RT products were available. However, the OURE is approximately 5 cm for the PCV SLS method, even when the broadcast ephemeris is used, whereas the OURE for the three other tested methods ranged from decimeter-level to approximately 0.8 m. Table 3 gives a more complete overview of the different RMS values obtained by applying the different tested methods with different GNSS products. It can be observed that σOURE,rv and σSISRE,rv are approximately 5–7 cm when the PCV SLS method is applied with the three types of GNSS products.
Average OURE of the relative orbital errors between GRACE C and D for December 1–7, 2019
Average RMS of the Relative Orbital Errors Between GRACE C and D for December 1–7, 2019 σOURE,rv denotes the RMS of the relative OURE not considering the relative clock errors (see Equation (9)), and σSISRE,rv denotes the relative SISRE considering the relative clock errors (see Equation (10)).
3.3 Influence of the Accuracy of the Reference Satellite Orbits
As inferred from Equation (6), the introduced orbits of the reference satellite would lead to a mis-modeled bias in the O-C terms. In this section, the relative clock and orbital errors are compared when centimeter-level and sub-meter-level reference satellite orbits are used. Figure 9 shows the errors of the low- and high-accuracy reference satellite orbits for GRACE C on December 1, 2019. The three-dimensional RMSs of the two types of orbital errors are approximately 0.7 m and 3 cm, respectively.
Errors of low- (left) and high-accuracy (right) reference satellite orbits for GRACE C on December 1, 2019
The estimated relative clock errors do not exhibit significant differences for the tested example. Figure 10 shows the relative clock errors obtained by using the high-accuracy (right panel in Figure 9) and low-accuracy (left panel in Figure 9) reference satellite orbits with the PCV SLS method. The CNES RT and BRDC products were used in the left and right panels of Figure 10. It can be observed that the different qualities of the reference satellite orbits do not lead to large differences in the resulting relative LEO satellite clock errors, regardless of which GNSS products are used. The same trend is observed for the relative orbital errors. As shown in Table 4, the differences in the RMS of the relative orbital errors are generally a few millimeters in each direction. The jumps at approximately 20 h are caused by resetting of the solution convergence during processing.
Relative clock errors between GRACE C and D on December 1, 2019, with the introduction of different reference satellite orbits when the CNES RT products (left) and broadcast ephemeris (right) are used
The PCV SLS method was applied in the processing. Reference orbits A and B refer to the low-accuracy reference orbits in the left panel of Figure 9 and the high-accuracy reference orbits in the right panel of Figure 9, respectively.
RMS of the Relative Orbital Errors Obtained Via the PCV SLS Method
The results for low- and high-accuracy reference satellite orbits are separated by “/”
4 DISCUSSION
This paper has investigated the relative orbital and clock results obtained by using the GRACE Follow-On LEO satellite baselines of approximately 180 km. If the distance between LEO satellites increases, e.g., to 500 or 1000 km, the accuracy of the reference satellite orbits could play a larger role in the time-transfer results. A 1-m bias in the reference orbits could result in a bias of a few centimeters in the O-C terms. The number of commonly viewed GNSS satellites could also be reduced in such cases, resulting in a weaker model strength. These aspects will be investigated in the near future when onboard GNSS observation data for real LEO mega-constellations become available.
5 CONCLUSIONS
Currently, LEO satellite clock values can be determined by applying a reduced-dynamic POD process with high precision in near-real time. The precision of the resultant orbital and clock products, however, relies heavily on high-precision real-time GNSS products, while requiring sufficient computational power for processes such as numerical integration. These considerations have prompted the use of ground processing, which entails a heavy load of observation data downlinking for LEO constellations with hundreds to thousands of satellites.
This study investigated the performance of LEO satellite time transfer based on the PCV method, relieving the strict requirement on the precision of real-time GNSS products, which may lead to a computational burden. High-precision time synchronization can be enabled in this way, and high-precision LEO satellite clock values for the entire constellation can be obtained via single-satellite clock determination of only a few reference satellites and time transfer between the reference satellites and other LEO satellites. In-orbit processing with SLS using broadcast ephemeris should fulfill the precision requirements for the time transfer.
Using real data from GRACE Follow-On satellites, i.e., GRACE C and D, this study compared the time-transfer results obtained by the proposed PCV SLS method and three other GNSS-based single-satellite clock determination methods, i.e., the RD BLS, KN BLS, and KN SLS methods. Three real-time GNSS products were tested, including the CNES RT products, the predicted part of the IGS UR products, and the broadcast ephemeris. Results showed that when the high-precision CNES RT products are available, the RD BLS method delivers the best time-transfer precision, i.e., below 0.05 ns. However, as the precision of the real-time GNSS products degrades, the STD of the relative clock errors rises sharply to above 1 ns, whereas the PCV SLS method maintains the STD below 0.2 ns.
The relative orbital errors were also analyzed for the four methods using the three types of GNSS products. Similarly, although good OURE and signal-in-space range error (SISRE) at the sub-decimeter level can be obtained by applying all four methods when CNES RT products are available, the OURE/SISRE values rise to a few decimeters when broadcast ephemeris is used for all methods except the PCV SLS method, which remains at the sub-decimeter level. The PCV SLS method is found to be insensitive to the precision of the GNSS products and delivers good precision of the relative clock errors, at less than 0.2 ns, and of the relative orbital errors, with an OURE of approximately 5 cm. Moreover, it was found that the quality of the reference satellite orbits does not significantly influence the precision of the relative clock and orbital errors, at least for the tested GRACE baseline of approximately 180 km. This relieves the need for high-accuracy single-satellite POD of the reference satellites to achieve proper time transfer between LEO satellites in such cases.
HOW TO CITE THIS ARTICLE
Wang, K., Sun, B., El-Mowafy, A., & Yang, X. (2024). High-precision time transfer and relative orbital determination among LEO satellites in real-time. NAVIGATION, 71(3). https://doi.org/10.33012/navi.659
ACKNOWLEDGMENTS
This work was supported by the International Partnership Program of the Chinese Academy of Sciences (CAS), Grant No. 021GJHZ2023010FN. This research was also funded by the National Time Service Center, CAS (No. E167SC14), the National Natural Science Foundation of China (No. 1207303), the CAS “Light of West China” Program (xbzg-zdsys-202308), and the Australian Research Council— Discovery Project No. DP240101710. We would like to acknowledge the support of the International GNSS Monitoring and Assessment System at the National Time Service Center and the National Space Science Data Center, National Science & Technology Infrastructure of China (http://www.nssdc.ac.cn).
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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