Adaptive clock modeling for real-time GPS/Galileo PPP timing of LEO satellites

1 Introduction

Low Earth orbit (LEO) satellites have become an important component of modern space infrastructure because of their rapid revisit capability, low signal propagation delay, and potential for high-resolution Earth observation and integrated communication, navigation, and remote-sensing services (Yang et al., 2024; Hui et al., 2025). With the planning and deployment of large LEO constellations, the increasing number of satellites and growing mission complexity place higher demands on autonomous onboard operation and real-time processing. Autonomous navigation and precise time synchronization have therefore become important capabilities for supporting LEO constellation operations (Montenbruck et al., 2021a; Kunzi and Montenbruck, 2022). Accurate and stable space–time references are required not only for orbit determination and precise time-tagging of payload data, but also for inter-satellite coordination, information exchange, and future LEO-based navigation augmentation.

Spaceborne Global Navigation Satellite System (GNSS) observations provide an effective means of autonomously determining the orbit and onboard time of LEO satellites, thereby reducing their dependence on continuous support from globally distributed ground-tracking networks. Precise point positioning (PPP) jointly processes pseudorange and carrier-phase observations together with precise GNSS orbit and clock products, enabling the simultaneous estimation of the LEO satellite position, receiver clock offset, carrier-phase ambiguities, and other relevant parameters within a unified estimation framework (Kouba and Héroux, 2001). The receiver clock offset establishes the relationship between the onboard clock and the GNSS system time scale and is therefore a key parameter for autonomous spaceborne timing (Montenbruck et al., 2025). The development of the International GNSS Service Real-Time Service (IGS RTS) and satellite-broadcast precise correction services, particularly the Galileo High Accuracy Service (HAS), has further increased the feasibility of real-time PPP applications (Hadas and Bosy, 2015; Carlin et al., 2021; Hauschild et al., 2022). Moreover, GPS/Galileo combined processing can improve satellite availability and observational redundancy for LEO satellites (Montenbruck et al., 2021b). Nevertheless, satellite visibility can still become unfavorable in single-system processing. As illustrated in Figure 1, the number of GPS satellites tracked by GRACE-FO C on DOY 258, 2023, and Sentinel-6A on DOY 259, 2023, occasionally falls below four. In conventional PPP, where the receiver clock offset is estimated independently at each epoch, at least four GNSS satellites are generally required to maintain the observability of the three-dimensional position and receiver clock offset. Periods of insufficient satellite availability or unfavorable geometry may therefore degrade the continuity and stability of the PPP solution.

Receiver clock offsets are not necessarily independent between consecutive epochs, and their temporal behavior is closely related to the stability of the underlying frequency reference. Early investigations therefore explored the use of temporal clock information to strengthen GNSS parameter estimation. Weinbach and Schön (2011) showed that the clock behavior of a GNSS receiver connected to an active hydrogen maser could be represented by a deterministic model over a limited time interval, thereby improving kinematic PPP performance. Krawinkel and Schön (2016) subsequently demonstrated the benefits of receiver clock modeling for GNSS navigation using miniaturized atomic clocks. Receiver clock modeling was later extended from deterministic descriptions to stochastic and adaptive approaches. Ge et al. (2018) incorporated between-epoch receiver clock correlation into real-time PPP time and frequency transfer, while Qin et al. (2021) developed a two-state stochastic model accounting for several oscillator noise processes. More recently, Mikoś et al. (2023a) introduced random-walk constraints into Galileo-only and multi-GNSS PPP and showed that the effectiveness of clock modeling depends strongly on oscillator stability and the treatment of inter-system biases. Han et al. (2025) further developed an adaptive clock-constraint model in which clock characteristics and process noise are updated using a sliding window. Recent real-time GPS/Galileo experiments using HAS, IGS, and CNES products have also confirmed that appropriate receiver clock constraints can improve PPP solution stability when sufficiently stable frequency references are available (Mikoś et al., 2026).

Receiver clock modeling has also been investigated for LEO orbit determination and timing. Weinbach and Schön (2013) represented the ultra-stable oscillators onboard the GRACE satellites using a piecewise-linear clock model and demonstrated that the additional temporal constraint could maintain position estimation even when the number of usable GPS satellites decreased to three. Yang et al. (2014) subsequently developed two- and three-state receiver clock models for GRACE kinematic orbit determination and reported an improvement of more than 40% in the radial component under a simulated real-time scenario. For data discontinuities, Qin et al. (2019) combined adaptive clock prediction with between-epoch constraints to mitigate clock re-convergence in ground-based real-time PPP time transfer, demonstrating that observation interruptions can be handled by exploiting historical clock information. More recently, Ge et al. (2025) introduced receiver clock modeling into real-time PPP timing of GRACE-FO satellites and showed clear improvements in clock noise and frequency stability. These studies demonstrate the value of temporal receiver clock information, but also reveal an important distinction between ground-based timing and LEO applications. Ground timing experiments generally benefit from fixed station coordinates and stable external frequency references, whereas LEO PPP requires simultaneous kinematic position and clock estimation under rapidly varying satellite geometry and onboard operating conditions (Zehentner and Mayer-Gürr, 2016; Mikoś et al., 2023b; Xie et al., 2024). Existing LEO clock-modeling studies have primarily focused on continuous clock constraints and overall orbit or timing performance, while the coordinated treatment of normal observations, short-term observation outages, and post-outage recovery remains insufficiently investigated for real-time LEO PPP timing (Gao et al., 2023; Wang et al., 2023).

To address this issue, this study develops an adaptive receiver clock model for real-time GPS/Galileo PPP timing of LEO satellites. Rather than applying a fixed stochastic model throughout the entire processing period, the proposed approach adjusts the receiver clock constraint according to the observation status and PPP convergence condition. During short-term observation outages, a sliding-window quadratic polynomial model is used to predict the receiver clock offset, while the prediction uncertainty is increased with the outage duration to account for the decreasing reliability of clock extrapolation. After observations resume, the predicted receiver clock offset is retained as a temporary prior constraint and progressively relaxed, preventing the predicted state from being over-weighted and allowing the recovered observations to gradually regain dominance in the estimation. Under normal observation conditions, the receiver clock is modeled as white noise before PPP convergence and as a random-walk process after stable convergence. In this manner, outage prediction, recovery constraint reconstruction, and convergence-dependent clock estimation are incorporated into a unified Kalman-filter framework to improve the continuity and stability of real-time LEO PPP timing.

The remainder of this paper is organized as follows. First, the methodology is described, including the GNSS PPP timing model for LEO satellites and the proposed adaptive receiver clock model for observation interruption, outage recovery, and normal observation conditions. Next, the experimental data and processing strategy are introduced, together with an accuracy assessment of the real-time GNSS orbit and clock products. The performance of the proposed method is then evaluated using GPS, Galileo, and combined GPS + Galileo PPP timing solutions, with particular emphasis on receiver clock offset estimation and timing stability. Finally, the results and limitations of the proposed method are discussed, followed by the main conclusions.

2 Methodology

2.1 GNSS PPP timing for LEO satellite

The orbital altitude of LEO satellites generally ranges from 300 to 1,500 km, which is much higher than the troposphere (Li et al., 2019). Therefore, when constructing the PPP functional model for LEO satellites, the tropospheric delay error can be ignored. Based on this, the pseudo-range and carrier phase observations (unit: m) can be expressed as:

P i , l e o s a t = ρ l e o s a t + c · d t r l e o − c · d t r s a t + I i s a t + b i , l e o − b i s a t + d a n t i + ε i , p s a t L i , l e o s a t = ρ l e o s a t + c · d t r l e o − c · d t r s a t − I i s a t + B i , l e o − B i s a t + d a n t i + p h w i + λ · N i s a t + ε i , l s a t (1)

In Equation 1, s a t, l e o, and i denote the GNSS satellite, LEO satellite, and signal frequency, respectively. The raw pseudo-range and carrier phase observations are represented by P i , l e o s a t and L i , l e o s a t, both observed in meters. The variable ρ l e o s a t denotes the geometric distance between the GNSS and LEO satellites; d t r l e o and d t r s a t are the clock errors of the LEO satellite receiver and the GNSS satellite, respectively; and I i s a t represents the ionospheric delay. b i , l e o and b i s a t are the uncalibrated code delays (UCD) at the LEO satellite receiver and GNSS satellite ends, respectively, while B i , l e o and B i s a t are the uncalibrated phase delays (UPD) at the LEO satellite receiver and GNSS satellite ends, respectively. The term d a n t i denotes the sum of the antenna phase center offsets for the LEO and GNSS satellites; p h w i represents the phase wind-up error in the carrier phase observation; λ is the wavelength of the carrier phase; N i s a t denotes the integer ambiguity of the carrier phase observation; ε i , p s a t and ε i , l s a t represent the observation noise for the pseudo-range and carrier phase, respectively; and c is the speed of light.

PPP techniques can form virtual observations by linearly combining measurements of different frequencies or different observation types, thereby effectively eliminating specific error sources (Wanninger et al., 2017; An et al., 2020). In this study, the dual-frequency ionosphere-free (IF) combination is adopted to eliminate the first-order ionospheric delay. Assuming that the dual-frequency pseudo-range and carrier-phase observations are denoted as P a, P b and L a, L b respectively, the corresponding dual-frequency ionosphere-free (IF) combined observations can be expressed as shown in Equation 2:

P I F = f a 2 f a 2 − f b 2 · P a + − f b 2 f a 2 − f b 2 · P b L I F = f a 2 f a 2 − f b 2 · L a + − f b 2 f a 2 − f b 2 · L b (2)

The parameters to be estimated are:

X = X ¯ , d t r l e o , I F , N I F s a t T (3)

In Equation 3, X ¯ represents the LEO satellite position, d t r l e o , I F and N I F s a t denote the receiver clock offset and the carrier-phase ambiguity of the dual-frequency IF combination, respectively, which can be expressed as:

d t r l e o , I F = d t r l e o + b I F N I F s a t = m · N a s a t + n · N b s a t + B I F − B I F s a t + b I F − b I F s a t (4)

In Equation 4, b I F, b I F s a t, B I F and B I F S a t denote the corresponding parameters after the IF combination. The detailed definitions of these parameters can be referred to Equation 1 and are not repeated here.

2.2 Adaptive clock model enhanced GNSS PPP timing

In PPP processing, the receiver clock offset is typically treated as a white noise parameter (Wang et al., 2015), modeled as a time-uncorrelated stochastic variable. Nevertheless, although the receiver clock offset exhibits inherent temporal correlation, strict time-domain correlation models are rarely adopted in practice. This is mainly due to the frequent interruptions, abrupt changes, and quality fluctuations in onboard GNSS observations from LEO satellites or real-time precise product streams of GNSS. In addition, the noise characteristics of onboard crystal oscillators are strongly affected by environmental factors such as temperature and power supply, rendering them inconsistent with the assumptions of ideal time-domain statistical models. Forcibly applying a strict time-domain model across all epochs may lead to spurious error accumulation at anomalous data points.

To ensure clock model reliability after observation outages and to facilitate effective constraint reconstruction upon data recovery, this work introduces an adaptive clock model incorporating outage prediction and recovery constraint functionality, as illustrated in Figure 2.

2.2.1 Mode 1: observation interruption phase

When an observation outage occurs between the current epoch and the most recent valid observation epoch, and the outage duration does not exceed 60 s, the receiver clock offset and its error variance at the current epoch are predicted using a pre-established sliding-window quadratic polynomial model, expressed as:

c △ t = c 0 + c 1 △ t + c 2 △ t 2 (5)

In Equation 5, c 0, c 1, and c 2 denote the receiver clock offset at the reference epoch, the first-order variation rate, and the second-order term, respectively. The sliding-window quadratic polynomial model is constructed using approximately 15 min of historical receiver clock offset estimates obtained from the valid PPP solutions immediately preceding the observation outage. With the 10 s processing interval adopted in this study, the fitting window contains approximately 90 epochs under continuous observations.

To quantitatively evaluate the fitting performance of the second-order polynomial clock model with respect to historical data and to properly characterize the uncertainty involved in clock extrapolation, both the model fitting accuracy and the prediction uncertainty are modeled in this study. After estimating the parameters of the quadratic polynomial model, the fitting accuracy is assessed using the residuals between the historical clock offset estimates and the model-fitted values, and is quantified by the RMS of the residuals, expressed as:

σ f i t = 1 n ∑ i = 1 n c ^ i − c Δ t i 2 (6)

In Equation 6, c ^ i denotes the estimated receiver clock offset at epoch index i; c Δ t i represents the fitted value of the quadratic polynomial clock model at Δ t i; and n is the number of historical epochs involved in the model fitting.

To avoid underestimation of the fitting residuals due to low noise levels or limited data within a short-time window, a minimum bound is imposed on the fitting accuracy, as given in Equation 7:

σ f i t = max σ f i t , σ min , σ min = 0.01 m (7)

During clock offset extrapolation to future epochs, the prediction uncertainty tends to increase with the duration of the extrapolation. To describe this effect, an empirical epoch-index-based uncertainty inflation model is employed, given by:

σ p r e d Δ t = σ f i t 1 + α Δ t (8)

In Equation 8, σ p r e d Δ t denotes the clock offset uncertainty at the predicted epoch, and α is the uncertainty inflation coefficient, representing the growth of the prediction error over time (set to 0.1 in this study). The predicted clock offset and its associated variance at the outage epoch are then used as a prior constraint to initialize the PPP model.

2.2.2 Mode 2: outage recovery phase

Upon completion of an observation outage, if the number of consecutively recovered epochs does not exceed 100 and the standard deviation of the clock offset error remains below 10 m, the period is defined as the outage recovery phase. During this phase, the receiver clock offset is extrapolated using a second-order polynomial model. The uncertainty of the predicted clock offset is characterized by its standard deviation, which is initialized to 0.3 m at the first recovered epoch and progressively increased by 0.03 m per epoch as the recovery proceeds. The corresponding variance used in the Kalman filter is obtained by squaring the resulting standard deviation. The predicted receiver clock offset and its corresponding variance are subsequently employed as prior constraints in the PPP estimation.

2.2.3 Mode 3: normal observation phase

During the normal observation phase, the receiver clock offset model is selected based on the convergence status of the PPP solution. The convergence criterion is defined such that the posterior standard deviation of the receiver clock offset remains below 0.3 m for 20 consecutive epochs. If this criterion is not met, the white noise model is adopted, and the prior variance of the clock offset at the current epoch is reset to 60 m². Conversely, if the convergence criterion is satisfied, the random walk model is employed, and the prior variance of the clock offset at the current epoch is updated as the sum of the posterior variance and the process noise:

C L K k = C L K k − 1 + ω R W , ω R W ∼ N 0 , σ R W 2 (9)

In Equation 9, ω R W denotes the process noise of the receiver clock offset in the random walk model. Based on empirical experience, σ R W 2 is set to 5 × 10⁻⁶ (m/s)² in this study.

Overall, the three modes are adaptively switched according to the observation status and PPP convergence condition. When an observation outage is detected, the model enters Mode 1, in which the receiver clock offset is predicted using the quadratic polynomial model for short-term outages not exceeding 60 s. Once the observations resume, the model switches to Mode 2, where the predicted clock offset is retained as a temporary prior constraint and its uncertainty is progressively relaxed during the recovery process, allowing the recovered observations to gradually regain dominance in the estimation. After the recovery phase is completed, or under normal observation conditions, the model operates in Mode 3. In this mode, the white-noise model is adopted before stable PPP convergence, whereas the random-walk model is activated once the convergence criterion is satisfied. In this manner, the proposed strategy enables adaptive transitions among clock prediction, recovery constraint reconstruction, and normal clock estimation according to the current observation and solution status.

The key parameters of the adaptive clock model were empirically selected based on the datasets and processing conditions used in this study. The outage threshold and prediction-uncertainty parameters were set to limit clock extrapolation to short interruptions and to avoid over-constraining increasingly uncertain predictions. The recovery-stage parameters were used to gradually relax the predicted clock constraint after data recovery, while the convergence criterion determines the transition from the white-noise model to the random-walk model. These settings are not intended to be universal and may require adjustment for different onboard oscillators, receiver characteristics, sampling intervals, or orbital conditions.

2.3 Experimental data processing strategy

This study employs the PPP technique based on a Kalman filter, with a primary focus on evaluating the impact of receiver clock modeling on the estimated receiver clock offsets of LEO satellites. Real-time orbit and clock products provided by the Centre National d’Études Spatiales (CNES) were adopted for data processing. The LEO observation targets include the twin-satellite configuration of the Gravity Recovery and Climate Experiment Follow-On (GRACE-FO) mission and the Sentinel-6A satellite. The experiments were conducted in a replay-based real-time mode. During the sequential Kalman-filter processing, only information available at the current and previous epochs was used, and no future observations or products were introduced into the estimation or clock prediction process. The overall experimental flowchart is illustrated in Figure 3.

Regarding the receiver clock modeling strategy, to systematically assess the impact of different models on the estimated clock offsets, this study adopts both the conventional white-noise model (Scheme 1) and the proposed adaptive clock model (Scheme 2) for parameter estimation. The white-noise model is selected as the reference scheme because it is commonly adopted in conventional PPP processing, in which the receiver clock offset is estimated independently at each epoch without considering temporal correlations between consecutive epochs. This comparison therefore provides a direct assessment of the improvement achieved by the proposed adaptive clock modeling strategy over the conventional PPP clock estimation approach. The results obtained from the two schemes are subsequently compared and analyzed. Other key strategies employed in the PPP processing are summarized in Table 1.

TABLE 1. Specific experimental data processing strategy.

3 Results

3.1 Real-time products accuracy assessment

The accuracy of the CNES real-time products over the study period (DOY 258–290, 2023) was assessed with the WHU final products at 5-min intervals. The double-differencing method was employed to eliminate the systematic biases arising from different clock product time references, and the influence of orbit radial errors on the clock offsets was also removed. For the double-differencing clock comparison, a reference satellite was selected separately for each GNSS constellation. The reference satellite was required to have continuous and valid clock information in both the CNES real-time products and the WHU final products over the corresponding comparison period, so as to ensure the continuity and consistency of the differenced clock series. Figure 4 presents the root mean square errors (RMSE) of the CNES real-time GPS and Galileo orbit and clock products in the radial (R), along-track (A), and cross-track (C) directions, as well as the corresponding clock products, with the average RMSE values indicated in the upper-right corner. Each data point represents the RMSE of a specific GPS or Galileo satellite product for a given day within the study period.

For the CNES real-time orbit products, the mean RMSE of Galileo satellites was 6.68 cm, 5.83 cm, and 5.15 cm in the radial, along-track, and cross-track directions, respectively, while the corresponding values for GPS satellites were 3.77 cm, 4.87 cm, and 3.90 cm. The statistics indicate that, over the study period, the GPS orbit products exhibited slightly higher overall accuracy than those of the Galileo system. For the clock products, the average RMSE was 0.15 ns for Galileo and 0.11 ns for GPS, suggesting that the CNES clock products perform comparably for both systems.

3.2 Adaptive clock model enhanced GPS PPP timing result

The receiver clock offset primarily reflects the difference between the LEO satellite time reference and the GNSS system time. Its magnitude is influenced by multiple factors, including not only hardware delays but also the performance of the onboard oscillator, the intrinsic stability of the receiver, the satellite thermal control environment, and the accuracy of the real-time orbit and clock products employed.

Figures 5, 6 illustrate the PPP receiver clock offset estimates of GRACE-FO C and D satellites based on CNES real-time GPS products, obtained under Scheme 1 (white noise model) and Scheme 2 (adaptive clock model), respectively. It can be observed that, under Scheme 1, the clock offset estimates of both satellites exhibit frequent divergence. This behavior is likely attributable to the relatively high orbital inclination of the GRACE-FO satellites and the comparatively lower quality of the onboard GNSS observations. In contrast, under Scheme 2, the adaptive clock model enhances the ability to capture the actual characteristics of the receiver clock offsets and maintains reasonable state predictions during observation outages. As a result, it effectively suppresses the divergence of the clock estimates and improves the overall stability of the time series. Only minor divergence is observed at a few epochs, which is primarily attributable to extended observation outages during this period that exceed the tolerance of the model’s predictive capability, resulting in the temporary ineffectiveness of the prediction function.

To quantitatively assess the stability of the LEO satellite receiver clock offset solutions obtained under Scheme 1 and Scheme 2, the Modified Allan Deviation (MDEV) (Allan, 1987) was computed for the receiver clock offset time series over three selected study periods.

σ x 2 τ = 1 2 m 2 M − 2 m + 1 ∑ j = 1 M − 2 m + 1 ∑ i = j j + m − 1 x i + m − x i 2 (10)

In Equation 10, τ denotes the length of the time window, M denotes the number of frequency deviation values involved in the calculation, m is the averaging factor, and x i represents the corresponding frequency deviation value. The MDEV results of the GPS-based PPP timing solutions for the GRACE-FO C and D satellites under both schemes are illustrated in Figures 7, 8, respectively.

For the GRACE-FO C satellite, the stability analysis of the receiver clock offset reveals that, under Scheme 1 (white noise model), the short-term stability at a time scale of 1,280 s, characterized by the Modified Allan Deviation (MDEV), ranges from 1.04 × 10⁻¹² to 6.29 × 10⁻¹³. In contrast, Scheme 2 (adaptive clock model) improves the short-term stability to the range of 1.51 × 10⁻¹³ to 2.17 × 10⁻¹³. This improvement indicates that the adaptive clock model effectively suppresses short-period noise and enhances the short-term determinacy of the receiver clock offset estimates. With respect to long-term stability at a time scale of 10,240 s, Scheme 1 yields MDEV values ranging from 1.93 × 10⁻¹³ to 3.28 × 10⁻¹³, whereas Scheme 2 further improves the stability to the range of 4.06 × 10⁻¹⁴ to 5.40 × 10⁻¹⁴, representing an improvement of nearly one order of magnitude. This result demonstrates that the adaptive clock model, through dynamic constraint enforcement and state prediction mechanisms, effectively mitigates long-term error accumulation, enhancing the continuity and reliability of the receiver clock offset time series over extended time intervals.

The stability characteristics of the receiver clock offset solutions for the GRACE-FO D satellite are generally consistent with those observed for GRACE-FO C. In terms of short-term stability, the MDEV values under Scheme 1 (white noise model) fluctuate between 5.32 × 10⁻¹³ to 7.47 × 10⁻¹³, whereas Scheme 2 (adaptive clock model) yields substantially improved stability within the range of 1.25 × 10⁻¹³ and 2.17 × 10⁻¹³. For long-term stability, Scheme 1 produces MDEV values ranging from 2.02 × 10⁻¹³ to 3.06 × 10⁻¹³, while Scheme 2 further enhances the performance to 3.62 × 10⁻¹⁴-6.00 × 10⁻¹⁴. These results confirm that the adaptive clock model consistently outperforms the white noise model in both short- and long-term stability for the GRACE-FO D satellite.

During the study period, the receiver clock offset of the Sentinel-6A was estimated using the GPS PPP. A comparison of the results obtained under Scheme 1 and Scheme 2 is presented in Figure 9. As illustrated in the figure, the receiver clock offset of Sentinel-6A demonstrates evident periodic fluctuations, which are presumably related to orbital periodicity, cyclic thermal variations, and the inherent behavior of the onboard receiver clock. Similar to the GRACE-FO satellites, Scheme 2 effectively suppresses the divergence tendency in the receiver clock offset estimates through the adaptive clock model. Nevertheless, unavoidable deviations can still be observed during certain intervals with prolonged observation outages. Notably, during DOY 290, the receiver clock offset estimates are characterized by frequent data interruptions and significant divergence, preventing effective convergence even with the use of the adaptive clock model. This indicates that, under the current parameter settings, the robustness of the proposed model against long-duration abnormal disturbances remains limited. Future work may focus on further optimizing the process noise parameters of the random-walk model to enhance the stability and continuity of receiver clock offset estimation under extreme conditions.

Figure 10 illustrates the receiver clock stability of Sentinel-6A under the two schemes. At the 1,280 s time scale, Scheme 1 shows large variability in MDEV, ranging from 1.27 × 10⁻¹¹ to 7.23 × 10⁻¹³, whereas Scheme 2 achieves a more consistent short-term stability, with values confined to 5.14 × 10⁻¹³-9.48 × 10⁻¹³. At the 10,240 s time scale, the long-term stability of Scheme 1 spans 3.64 × 10⁻¹³ to 7.23 × 10⁻¹³, while Scheme 2 yields more tightly clustered results between 3.56 × 10⁻¹³ and 4.41 × 10⁻¹³, demonstrating superior long-term stability performance. Table 2 presents the short- and long-term stabilities of GPS PPP timing results for the three satellites used in the experiments under schemes 1 and 2.

TABLE 2. Summarizes the mean MDEV values at averaging times of 1,280 s and 10,240 s for the three satellites under Schemes 1 and 2.

3.3 Adaptive clock model enhanced Galileo/GPS + Galileo PPP timing result

As Sentinel-6A is equipped with a dual-system GNSS receiver supporting both GPS and Galileo, this study further conducts experiments using a Galileo-only configuration as well as a combined GPS + Galileo solution. The receiver clock offset estimates derived from the Galileo system are presented in Figure 11. The results exhibit pronounced periodic characteristics, consistent with those observed in the GPS-based solutions, while the adaptive clock model adopted in Scheme 2 demonstrates superior overall performance compared to Scheme 1. Furthermore, the observation interruptions observed during DOY 290 are substantially alleviated in the Galileo-based results, suggesting improved satellite availability and enhanced signal continuity of the Galileo constellation on that day.

As shown in Figure 12, the clock stability analysis of Sentinel-6A indicates that Scheme 2 consistently outperforms Scheme 1 over shorter time scales. Specifically, for short-term stability (1,280 s), MDEV values under Scheme 1 range from 7.47 × 10⁻¹³ t o 8.58 × 10⁻¹³, whereas Scheme 2 achieves a narrower and lower range of 5.24 × 10⁻¹³ to 7.47 × 10⁻¹³. For long-term stability (10240 s), Scheme 1 spans 3.69 × 10⁻¹³ to 4.68 × 10⁻¹³, while Scheme 2 converges further to 3.62 × 10⁻¹³-4.49 × 10⁻¹³.

Figure 13 presents the receiver clock offset estimates based on the combined GPS + Galileo solution, with the GPS system serving as the reference. The significant increase in the number of observable satellites provides more robust observational support for the estimation model, thereby substantially enhancing the stability and continuity of the results. Although Scheme 1 exhibits occasional re-convergence and divergence during certain periods, indicating residual fluctuations, the overall clock estimates under both Scheme 1 and Scheme 2 remain largely consistent and demonstrate satisfactory stability under the multi-system configuration. These findings suggest that the increased number of observed satellites in a multi-GNSS solution effectively improves the robustness of the estimation, rendering the performance of the two schemes largely comparable, with only minor differences arising under high-dynamics or complex observational conditions.

Figure 14 illustrates the stability performance of the receiver clock offset estimates under the dual-system solution. For long-term stability (10,240 s), both schemes exhibit comparable performance, with MDEV values ranging from 3.61 × 10⁻¹³ to 5.20 × 10⁻¹³ for Scheme 1 and 3.61 × 10⁻¹³ to 4.44 × 10⁻¹³ for Scheme 2. In contrast, Scheme 2 demonstrates superior short-term stability (1,280 s), with MDEV values confined to 4.90 × 10⁻¹³–9.48 × 10⁻¹³, compared to 1.21 × 10⁻¹²–5.96 × 10⁻¹³ for Scheme 1. The observed short-term fluctuations under Scheme 1 are likely attributable to the introduction of inter-system biases (ISB), which compromise the stability of conventional models. By incorporating the adaptive clock model, Scheme 2 effectively mitigates the impact of ISBs, resulting in enhanced robustness and improved short-term stability. Table 3 presents the short- and long-term stabilities of Sentinel-6A under schemes 1 and 2 for GPS, Galileo, and the combined dual-system solutions.

TABLE 3. Summarizes the mean MDEV values at averaging times of 1,280 s and 10,240 s for the Sentinel-6A GPS, Galileo, and combined GPS + Galileo solutions under Schemes 1 and 2.

4 Discussion

In practical operations, LEO satellites are inevitably affected by observation outages and fluctuations in the quality of real-time products. To address this issue, this study focuses on a comparative analysis of receiver clock estimation performance under observation outage conditions. The results demonstrate that, under short-term observation outages, the adaptive clock model maintains the continuity of receiver clock estimation through predictive mechanisms and achieves smooth convergence after data recovery via constraint reconstruction, significantly outperforming the conventional white-noise model. However, when the duration of observation outages becomes prolonged, the predictive capability of the adaptive clock model exhibits inherent limitations. In addition, the performance of the clock model is, to some extent, dependent on the appropriate tuning of the process noise parameters. Future work should incorporate the characteristics of onboard oscillators to further optimize parameter settings, thereby enhancing the robustness of the model under extreme conditions.

Under the combined GPS + Galileo processing, the number of available satellites increases significantly, providing enhanced observational redundancy for the PPP model and thereby improving the overall stability and continuity of the solutions. The results indicate that, in multi-GNSS fusion scenarios, the performance gap between the two clock modeling schemes is reduced, particularly in terms of long-term stability, where comparable performance is observed. This suggests that, under conditions of high observational redundancy, the measurement information itself can partially compensate for deficiencies in stochastic clock modeling, leading to increasingly similar solutions across different clock models. Nevertheless, in terms of short-term stability and under high-dynamic or complex observation conditions, the adaptive clock model still exhibits superior robustness, highlighting that improvements in clock modeling remain of practical significance, especially when observational conditions are limited or system complexity is high.

Taken together, these results indicate that the proposed adaptive clock model is particularly advantageous for LEO PPP timing under challenging observation conditions, including limited satellite visibility, short-term observation outages, and temporary degradation of real-time product quality. Under such conditions, the temporal information of the receiver clock provides additional constraints for maintaining the continuity and stability of clock estimation, whereas the benefit of clock modeling becomes less pronounced when sufficient observational redundancy is available. Nevertheless, the effectiveness of the proposed model is still limited by the duration of observation outages, as the reliability of clock prediction gradually decreases with increasing extrapolation time. In addition, several key model parameters are empirically determined and have not yet been specifically optimized for different onboard oscillators, receiver characteristics, or orbital conditions, which may restrict the general applicability of the current parameter settings. Future work will therefore focus on adaptive or clock-specific parameter optimization to further improve model robustness under prolonged observation outages and diverse LEO operating environments.

5 Conclusion

This study focuses on receiver clock modeling in LEO PPP timing and presents a systematic comparative analysis of the conventional white-noise clock model (Scheme 1) and the adaptive clock model (Scheme 2) across different satellite platforms and observation conditions. Based on experimental results from the GRACE-FO C/D and Sentinel-6A satellites during DOY 258–290, 2023, the two models are quantitatively evaluated in terms of clock estimation stability, time-scale characteristics, and robustness under observation outage scenarios. The main conclusions are summarized as follows:

• Overall, the adaptive clock model outperforms the conventional white-noise model in receiver clock estimation stability.

In PPP time transfer experiments with three LEO satellites, the adaptive clock model effectively suppresses divergence in clock solutions, significantly enhancing continuity and smoothness. Its stability advantage is particularly evident under discontinuous observations or fluctuating real-time product quality. Quantitative evaluation shows that, at a 1,280 s short-term scale, the MDEV is reduced relative to the white-noise model, while at the 10240 s long-term scale, the adaptive model suppresses error accumulation and maintains stable convergence of the clock time series.

• 2. The adaptive clock model demonstrates stronger resilience to observation outages, although limitations remain for prolonged interruptions.

Leveraging clock prediction and dynamic constraint mechanisms, the model enables smooth state propagation during short-term outages and achieves stable recovery after observations resume, outperforming the conventional model. However, when outages exceed the effective prediction horizon, clock estimates may exhibit bias or re-convergence phenomena, indicating that model performance is sensitive to the process noise parameter settings under extreme conditions.

• 3. Multi-GNSS processing enhances overall PPP time transfer stability and mitigates performance differences between clock models.

Under GPS + Galileo combined processing, increased satellite availability provides stronger observational constraints, resulting in comparable long-term stability across both models. Nevertheless, for short-term stability and under high-dynamic or complex observation conditions, the adaptive clock model maintains superior robustness, highlighting its practical significance when observation conditions are limited or system complexity is high.

In summary, the adaptive clock model effectively improves receiver clock estimation stability and reliability in LEO PPP time transfer, particularly for real-time applications and challenging observation scenarios. It should be noted that some model parameters are empirically set and have not been systematically optimized for different onboard clock types or orbital characteristics. Future work may incorporate adaptive parameter estimation or physics-informed clock modeling to further enhance robustness under extreme conditions and to extend applicability to multi-constellation, multi-frequency PPP time transfer scenarios, thereby supporting the development of high-precision and robust space-based timing systems.

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Footnotes

Edited by: Ke Su, Space Engineering University, China

Reviewed by: Pengli Shen, National Time Service Center, Chinese Academy of Sciences (CAS), China

Ju Hong, Shandong University of Science and Technology, China