Chengpan Tang

Accurate low Earth orbit (LEO) prediction is significantly affected by uncertainties in atmospheric drag, which is driven by complex space weather variations. Existing physics-based and data-driven methods often fail to fully capture multi-scale and time-lagged dependencies between space environment indices and drag coefficient dynamics. This study proposes a multi-scale lag-aware correlation modeling and stochastic residual learning method for LEO orbit prediction. First, empirical mode decomposition (EMD) is applied to decompose the atmospheric drag coefficient and space weather indices into intrinsic mode functions to capture scale-dependent dynamics. Second, a lag-dependent correlation model is constructed to quantify nonlinear relationships between solar/geomagnetic activity and drag coefficient variations, forming a deterministic prediction component. Finally, a long short-term memory (LSTM) network is employed to model the residual errors that are not explained by the correlation structure, providing stochastic correction. The reconstructed drag coefficient is incorporated into the orbital dynamics model for state propagation. The method is validated using Sentinel-3A and GRACE-FO1 data over 180 days. Compared with traditional numerical propagation, STE-LSTM reduced the 24-hour SISRE to 1.10–3.35 m for Sentinel-3A and 2.50–4.20 m for GRACE-FO1, corresponding to accuracy improvements of 69.1%–89.4% and 64.2%–83.3%, respectively. These results demonstrate that combining multi-scale lag-dependent correlation modeling with stochastic residual learning provides an accurate and interpretable method for LEO orbit prediction under varying space weather conditions.

Autonomous onboard orbit determination of low earth orbit (LEO) satellite using global navigation satellite system (GNSS) broadcast ephemerides has become increasingly important for time-critical space geodetic applications, such as GNSS radio occultation and satellite altimetry. With the full operational capability of the BDS-3 constellation, the use of BDS-3 broadcast ephemerides for autonomous LEO orbit determination has attracted growing interest. This study proposes an improved method to enhance the autonomous orbit determination accuracy of LEO satellites by incorporating real-time compensation of BDS-3 broadcast clock. The analysis of broadcast clock characteristics reveals pronounced stochastic variations in BDS-3 clock drift. To mitigate this, a clock-drift correction method is proposed. The method utilizes historical clock offset data to correct clock drift through quadratic fitting, then extrapolates the clock bias for the subsequent epoch based on the corrected clock drift. The results demonstrate that the average broadcast clock errors of BDS-3 satellites are reduced from 70.91 to 48.91 cm. Allan deviation (ADEV) analysis reveals a marked improvement in clock frequency stability, with the mean frequency stability at 10,000 s improved from 1.75×10–¹³ to 3.27×10–¹⁴. The proposed method is further integrated into LEO orbit determination processing and evaluated using onboard BDS-3 observations from four LEO satellites. Experimental results show that the average three-dimensional (3D) orbit accuracy is improved by 39.34% across all test arcs. Specifically, all satellites exhibit average improvement rates exceeding 30%, with satellite SATD achieving the most substantial enhancement, where its 14-day average error is reduced from 23.55 cm to 9.96 cm. Therefore, the proposed method can significantly enhance the autonomous orbit determination accuracy of LEO satellites.

In global navigation satellite systems (GNSS), the system time reference is maintained by the ground control segment and kept traceable to UTC, enabling inter-system compatibility and interoperability. Advances in onboard atomic-clock stability and inter-satellite time transfer accuracy make it feasible for a constellation to autonomously realize a space-based time reference, with periodic traceability updates and steering via satellite–ground links to enhance resilient time maintenance. BeiDou-3 (BDS-3) carries high-performance onboard hydrogen masers and Ka-band inter-satellite links (ISL) for time transfer, providing stable frequency sources and high-precision time transfer capability for establishing a space-based time reference. Using in-orbit BDS-3 clock offset data, we propose a space-based autonomous timekeeping approach that combines high-precision ISL synchronization with timekeeping by a small ensemble of hydrogen masers, together with a space–ground cooperative strategy with BeiDou time (BDT). The approach first performs constellation-wide synchronization using ISL, then selects a timekeeping ensemble based on in-orbit clock performance to generate a space-based ensemble atomic timescale, denoted TA(SPACE); when satellite–ground links are available, TA(SPACE) is steered to BDT to maintain consistency with the ground time reference. Based on this space-based time reference, satellite clock offsets are predicted to generate clock-parameter products. Experiments show that, in the autonomous mode, the time offset between TA(SPACE) and BDT is kept within 25.06 ± 41.47 ns over 90 days, whereas in the space–ground cooperative mode, satellite–ground steering stabilizes the offset within 10 ns. The proposed approach provides a practical solution for constellation time maintenance under disruptions such as anomalous ground injection, improving the resilience and reliability of GNSS services.

Pseudorange bias refers to the receiver-dependent and satellite-dependent constant bias in the pseudorange resulting from the nonideal characteristics of a signal. The impact of pseudorange bias on high-precision satellite navigation services has long been ignored. This paper proposes a pseudorange bias calibration method for two collocated receivers. Then, we calibrate pseudorange biases for two types of collocated receivers at a monitoring station within China and evaluate their impact on two high-precision services: BeiDou Navigation Satellite System 3 (BDS-3) dual-frequency pseudorange augmentation and precise point precision (PPP). Theoretical analysis reveals that the calibrated pseudorange biases contribute 17.2% and 7.7% to the user equivalent ranging error (UERE) of BDS-3 and Global Positioning System (GPS) dual-frequency pseudorange augmentation, respectively, and that the convergence time of the GPS static and kinematic PPP increases from 6 min and 26 min to 19 min and 58 min, respectively. The experimental results indicate that the calibrated pseudorange biases are consistent as the receiver location and time vary. The spatial distribution consistency is generally better than 0.1 m, and the temporal consistency is better than 0.15 m. The pseudorange biases for BDS-3 B1C and B2a are approximately 0.7 m and 0.1 m, respectively, whereas those for GPS L1C/A and L2P are both approximately 0.25 m. Furthermore, The results show that after correction of the pseudorange biases, the average convergence time for BDS-3/GPS static PPP decreases from 48.83/24.03 min to 38.54/21.12 min, respectively, a decrease of approximately 21%/12%. For BDS-3/GPS/BDS-3 + GPS kinematic PPP, the average convergence time decreases from 109.53/45.10/39.15 min to 62.99/40.83/22.94 min, respectively, a decrease of approximately 42%/41%/9%. Similarly, the three-dimensional positioning accuracy for BDS-3/GPS/BDS-3 + GPS dual-frequency pseudorange augmentation improves from 3.25/3.94/2.49 m to 2.65/3.69/2.16 m, respectively, increasing by approximately 6.3%, 18.5%, and 13.3%, respectively. The above analysis and experiments demonstrate that pseudorange bias is an important error source affecting both dual-frequency pseudorange augmentation and PPP services.

High-precision Global Navigation Satellite System (GNSS) orbit and clock products are crucial for precise applications. An evenly distributed global network enables continuous tracking for GNSS satellites, while a regional network may result in tracking gaps in the areas where monitoring stations are not deployed. This also means that the orbit determination accuracy based on a regional network is not comparable to that with a global network. Integrating the measurements from regional ground stations and Low Earth Orbit (LEO) satellites onboard receivers is a potential approach for generating GNSS orbit and clock products with centimeter-level accuracy, which is particularly important for BDS and the local commercial providers relying on a regional network. In the integrated Precise Orbit Determination (POD), LEO satellites are used to compensate for the drawback of regional ground stations in the precise orbit and clock determination of GNSS satellites. To validate the role of LEO satellites in the orbit determination with a regional network, 6 International GNSS Service stations around China and 13 LEO satellites from January 20 to 26, 2019, including GRACE-C/D, SWARM-A/B/C, Jason-3, Sentinel-3A/B, and SAT-A/B/C/D/E are selected in this study to perform the integrated POD. The orbit and clock accuracies of GPS and LEO satellites are evaluated by comparison with precise products. The average Root Mean Square (RMS)of GPS orbit errors in the radial (R), along-track (T) and cross-track (N) directions are 2.27 cm, 3.45 cm, and 3.08 cm, respectively, and the clock accuracy is better than 0.15 ns based on a comparison with the final products provided by Center for Orbit Determination in Europe (CODE). The LEO orbit accuracy is better than 2 cm in the R direction, and the position errors are mostly within 4 cm. The results indicate that the integrated POD can generate high-precision orbit and clock products for GPS and LEO satellites based on regional network stations. Finally, the integrated POD products are assessed for Precise Point Positioning (PPP). Simulated kinematic PPP has a comparable performance in terms of the convergence time and positioning accuracy. With more LEO satellites available, the orbit and clock determination accuracy and PPP positioning accuracy can be improved.

With the rapid development of technologies such as autonomous driving and robotic navigation, the demand for accurate and reliable positioning results with BDS has increased. The current status and future development of BDS high precision services are reviewed. The operational PPP-B2b of BDS-3 achieves positioning accuracy at decimeter-level within 14 min and has the drawbacks in regional coverage and long convergence time compared with the international counterparts, such as CLAS of QZSS and HAS of Galileo. A development frame for multi-layer BDS high-precision services is proposed by considering its construction cost, compatibility of user terminal, operating maintenance burden, and service level. The global high-precision service with the improved orbit and clock accuracy enhanced with the inter-satellite link is taken as the most fundamental layer. By incorporating inter-satellite link observations, the orbit errors URE and clock errors are significantly reduced to approximately 0.05 m and 0.16 ns, respectively, which will expand PPP service to global. Based on this, the regional PPP-AR and PPP-RTK services with improved convergence performance are taken as the second layer. With PPP-AR, the convergence time drops to 10 min. With PPP-RTK, the convergence time would be further shortened to less than 3 min. The LEO enhanced PPP service is taken as the third layer which overcomes both drawbacks of the long convergence time of the inter-satellite link enhanced service (the first layer) and the regional coverage of the PPP-AR/PPP-RTK services (the second layer). The simulation results show that by introducing a LEO constellation of 288 satellites, the LEO enhanced PPP can achieve positioning accuracy better than 5 cm within approximately 1 min. In addition, the integration of LEO constellation and PPP-RTK is further proposed to enable instantaneous convergence. The implementation challenges are also presented.

Starting from February 2023, the International Laser Ranging Service (ILRS) began releasing satellite laser ranging (SLR) data for all BeiDou global navigation satellite system (BDS-3) medium earth orbit (MEO) satellites. SLR data serve as the best external reference for validating satellite orbits, providing a basis for comprehensive evaluation of the BDS-3 satellite orbit. We utilized the SLR data from February to May 2023 to comprehensively evaluate the orbits of BDS-3 MEO satellites from different analysis centers (ACs). The results show that, whether during the eclipse season or the yaw maneuver season, the accuracy was not significantly decreased in the BDS-3 MEO orbit products released from the Center for Orbit Determination in Europe (CODE), Wuhan University (WHU), and the Deutsches GeoForschungsZentrum (GFZ) ACs, and the STD (Standard Deviation) of SLR residuals of those three ACs are all less than 5 cm. Among these, CODE had the smallest SLR residuals, with 9% and 12% improvement over WHU and GFZ, respectively. Moreover, the WHU precise orbits exhibit the smallest systematic biases, whether during non-eclipse seasons, eclipse seasons, or satellite yaw maneuver seasons. Additionally, we found some BDS-3 satellites (C32, C33, C34, C35, C45, and C46) exhibit orbit errors related to the Sun elongation angle, which indicates that continued effort for the refinement of the non-conservative force model further to improve the orbit accuracy of BDS-3 MEO satellites are in need.

A new concept of a space-based synchronized reference network is proposed with the development of an optical frequency reference and laser inter-satellite link. To build such time reference, three clock ensemble algorithms, namely the natural Kalman timescale (NKT) algorithm, the reduced Kalman timescale (RKT) algorithm, and the two-stage Kalman timescale (TKT) algorithm are considered. This study analyzes and compares the performance of these algorithms using BDS, GPS, and Galileo satellite clock data from the GFZ GNSS clock corrections, which will be used in constructing future space-based time references. The study shows that the NKT algorithm improves frequency stability by 0.1–0.2 orders of magnitude in the short and medium term. When the satellite clock is mostly a hydrogen clock, the RKT and NKT are close, and the short and medium-term frequency stability slightly increases. In contrast, the TKT algorithm produces a timescale that improves frequency stability by 1–3 orders of magnitude. A quadratic polynomial model predicts the three timescales, with the results indicating that the short-term prediction accuracy of the satellite clock is within 1ns, and the TKT algorithm’s prediction accuracy is 1–2 orders of magnitude higher than that of the NKT and RKT algorithms. With the deployment of next-generation Low Earth Orbit (LEO) satellites equipped with higher-precision clocks, the space-based time reference system will achieve improved accuracy and greater potential for practical applications.

The inter-satellite link is an important technology to improve the accuracy of clock offset measurement and prediction for BeiDou Navigation Satellite System (BDS). At present, BDS measures clock offsets of invisible satellite mainly through the “one-hop” reduction mode based on the satellite-ground clock offset of the node visible satellite and the inter-satellite clock offset between the two satellites. However, there exists a systematic deviation caused by the node satellite reduction, and there is still a large room for improvement in clock offset measurement and prediction. Therefore, this paper firstly proposes a method of whole-network adjustment for clock offset based on the satellite-ground and inter-satellite two-way data. The least square method is used to realize the whole-network adjustment of clock offset based on the observations of two sources, and to obtain optimal estimates of different clock offset reduction. Secondly, the evaluation method combining internal and external symbols are proposed by the fitting residual, prediction error and clock offset closure error. Finally, experimental verification is completed based on BDS measured data. In comparison with the “one-hop” reduction method, the fitting residual and prediction error of the whole-network adjustment method reduces about 45.06% and 52.15%, respectively. In addition, inter-satellite station closure error and three-satellite closure error are reduced from 0.69 ns and 0.23 ns to about 0 ns. It can be seen that the accuracy of BDS time synchronization is significantly improved.