Low Earth orbit (LEO) augmentation in the global navigation satellite system has become a focus in the current satellite navigation field. To achieve high precision in positioning, navigation and timing services, relativistic effects should be considered, as they are difficult to distinguish from LEO satellite clock estimates and disturb their predictions. The relativistic effects on LEO satellite clocks are discussed in detail based on both theoretical and empirical results. Two LEO satellite clock prediction strategies are proposed, with and without removing the relativistic effect, using real data from typical LEO satellites: SENTINEL-3B and Gravity Recovery and Climate Experiment Follow-On (GRACE FO-1). For GRACE FO-1 and SENTINEL-3B, the relativistic effects are both on the order of nanoseconds and after removing the relativistic effects, the modified Allan deviations of the clocks are shown to be significantly improved. Based on the prediction strategies proposed, for SENTINEL-3B at around 810 km, with the prediction period increased from 30 to 3600 s, the root mean square error (RMSE) increases from 0.025 ns to about 1.4–1.6 ns. For the lower LEO satellite GRACE FO-1 at around 500 km, the RMSE of the predicted clocks increases more rapidly, i.e. from 0.012 ns at 30 s to about 4.5 ns at 3600 s. Results showed that the LEO satellite relativistic effects developed based on the theory could correct the majority, but not all of the once- and twice-per-revolution terms in the LEO satellite clocks. Although the corrections have exhibited effective improvements in the clock stability, they do not behave better than simply applying the mathematical model to the clock predictions. The latter model, however, does not have physical foundations as the former one.

The utilization of Low Earth Orbit (LEO) satellites is anticipated to augment various aspects of traditional GNSS-based Positioning, Navigation, and Timing (PNT) services. While the LEO satellite orbital products can nowadays be produced with rather high accuracy in real-time of a few centimeters, the precision of the LEO satellite clock products that can be achieved in real-time is less studied. The latter, however, plays an essential role in the LEO-augmented positioning and timing performances. In real-time, the users eventually use the predicted LEO satellite clocks, with their precision determined by both the near-real-time clock precision and the prediction time needed to match the time window for real-time applications, i.e., the precision loss during the prediction phase. In this study, a real-time LEO satellite clock determination method, consisting of near-real-time clock determination with ultra-short-term clock prediction is proposed and implemented. The principles and strategies of this method are discussed in detail. The proposed method utilized Kalman-filter-based processing, but supports restarts at pre-defined times, thus hampering continuous bias propagation and accumulation from ancient epochs. Based on the method, using Sentinel-3B GNSS observations and the real-time GNSS products from the National Center for Space Studies (CNES) in France, the near-real-time LEO satellite clocks can reach a precision of 0.2 to 0.3 ns, and the precision loss during the prediction phase is within 0.07 ns for a prediction time window from 30 to 90 s. This results in a total error budget in the real-time LEO satellite clocks of about 0.3 ns.

High-precision Low Earth Orbit (LEO) satellite clocks are essential for LEO-augmented Positioning, Navigation, and Timing (PNT) services. Nowadays, high-precision LEO satellite clocks can be determined in real-time using a Kalman filter either onboard or on the ground, as long as the GNSS observations collected onboard LEO satellites can be transmitted to the ground in real-time. While various real-time and high-precision GNSS products are available nowadays in the latter case, their continuity and latencies in engineering reality are not as perfect as expected and will lead to unignorable impacts on the precision of the real-time LEO satellite clocks. In this study, based on real observations of Sentinel-3B, the impacts of different latencies and continuity of the real-time GNSS products on LEO real-time clocks are determined and discussed for two scenarios, namely the “epoch estimation” and “arc estimation” scenarios. The former case refers to the traditional filter-based processing epoch-by-epoch, and the latter case connects LEO satellite clocks from different rounds of filter-based processing under a certain arc length. The two scenarios lead to the “end-loss” and “mid-gap” situations. Latencies of the real-time GNSS products are discussed for the cases of orbit-only latency, clock-only latency, and combined forms, and different handling methods for the missing GNSS satellite clocks are discussed and compared. Results show that the real-time LEO satellite clock precision is very sensitive to the precision of real-time GNSS satellite clocks, and prediction of the latter becomes essential in case of their latencies. For the “end-loss” situation, with a latency of 30 to 120 s for the GNSS real-time clocks, the LEO satellite clock precision is reduced from about 0.2 to 0.28–0.57 ns. Waiting for the GNSS products in case of their short latencies and predicting the LEO satellite clocks instead could be a better option. For “arc-estimation”, when the gap of GNSS real-time products increases from 5 to 60 min, the real-time LEO clock precision decreases from 0.26 to 0.32 ns.

High precision of real-time (RT) low Earth orbit (LEO) satellite clocks is crucial for enabling future LEO-enhanced positioning, navigation, and timing services. For filter-based RT processing, multi-GNSS observations from onboard LEO satellites can strengthen the observation model and improve the LEO precise orbit determination, but have limited improvement effect on the clock estimation. The time references for RT GNSS clocks of different constellations (even from the same analysis center) are different and vary with time. This requires estimating constellation-wise LEO satellite clocks or time-varying inter-system biases (ISBs). This contribution investigates multi-GNSS LEO satellite clock and the corresponding signal-in-space ranging errors (SISRE) using GPS/Galileo observations collected onboard Sentinel-6A satellite, and GNSS products from various analysis centers with different latencies, including the Centre National d’Etudes Spatiales real-time (RT), the GeoForschungsZentrum Multi-GNSS Experimental (MGEX) rapid, the European Space Agency MGEX final, and the Wuhan University MGEX final products. Three ISB stochastic models were implemented, i.e. the constant, the random walk, and the white noise models. Results show that the ISB estimates exhibit systematic effects of orbital periods (ca. 1.85 h) when applying the random walk or white noise models using the post-processing products. The white noise model shows the best performance when applying the RT GNSS products with large time-varying differences in the constellation-wise time references, with improvements of about 40.8% and 12.5% for clock precision, and about 38.1% and 7.9% for SISRE, respectively, compared to the constant and random walk models. When using the post-processing products with very similar time references for different constellations, the three ISB stochastic models do not deliver significant differences in clock precision and SISRE. This contribution provides a reference for ISB estimation strategy when using multi-constellation GNSS measurements to estimate RT LEO satellite clocks.