Elevation-cutoff-dependent convergence and steady-state performance of LEO-augmented multi-GNSS PPP time transfer

1 Introduction

Time transfer enables high-precision time information to be exchanged between two or more locations for clock synchronization and comparison, and provides an important technical basis for communication networks, power systems, and scientific metrology (Levine, 2008). Precise point positioning (PPP), which uses precise satellite orbit and clock products, enables high-precision positioning without relying on a regional reference-station network and has subsequently been widely applied to precise time and frequency transfer (Defraigne and Baire, 2011). The International Bureau of Weights and Measures (BIPM) has adopted PPP for the computation of time links contributing to International Atomic Time (TAI) (Petit and Jiang, 2008), and PPP has also been used for high-precision time-link estimation (Petit et al., 2015). With the continued development of Global Navigation Satellite Systems (GNSS), PPP time transfer has evolved from single-system to multi-GNSS processing, providing increased observation redundancy and improved inter-station time comparison. State-space representation (SSR) corrections and BDS-3 PPP-B2b corrections have been applied to real-time PPP time transfer (Lyu et al., 2026; Ge et al., 2023; Ma et al., 2025). The Galileo High Accuracy Service (HAS), together with its integration with PPP-B2b, has further demonstrated the feasibility of multi-service real-time PPP time transfer (Xu et al., 2025; Wei et al., 2025).

PPP performance depends not only on observation quality but also on satellite geometry and the separability of the estimated parameters (Hong et al., 2025; Zhao et al., 2026). Float PPP generally requires joint estimation of station coordinates, receiver clock offsets, tropospheric wet delays, and carrier-phase ambiguities. Conventional GNSS satellites operate mainly in medium Earth orbit (MEO), and their line-of-sight directions relative to a ground station vary comparatively slowly. Consequently, sufficient time-varying geometric information requires a certain period of accumulation, which contributes to the relatively long initialization process of PPP (Hong et al., 2025). When the elevation cutoff angle is increased or satellite signals are obstructed, both the number of visible satellites and sky coverage decrease, weakening the geometric constraints and further degrading convergence and reconvergence performance (Li et al., 2022).

For PPP time transfer, the primary quantities of interest are the receiver clock estimates and the resulting inter-station time links rather than the station coordinates themselves. Even when station coordinates are fixed, the receiver clock must still be jointly estimated with tropospheric wet delays, inter-system biases (ISBs), and carrier-phase ambiguities. Insufficient time-varying geometry may therefore delay parameter separation and time-link convergence. Under a common time reference and consistent precise products, differencing the receiver clock estimates obtained at two stations forms an inter-station time link and can partially suppress errors associated with the common reference, enabling remote time comparison at the sub-nanosecond level (Defraigne and Baire, 2011; Petit and Jiang, 2008). Previous studies have shown that PPP time transfer generally requires approximately 10 min or longer to reach a stable state (Liu et al., 2024). A standard deviation (STD) of the time-link estimates below 0.1 ns within a short sliding window has also been adopted as a convergence criterion (Xie et al., 2025a). After convergence, time-link performance is commonly evaluated in terms of steady-state dispersion and frequency stability. STD characterizes the overall dispersion of the steady-state time-link series (Liu et al., 2024; Xie et al., 2025a), whereas the modified Allan deviation (MDEV) describes frequency stability over different averaging times (Allan and Barnes, 1981).

Low Earth orbit (LEO) satellites exhibit faster relative motion and more rapid epoch-to-epoch changes in line-of-sight geometry than conventional GNSS satellites. These characteristics can provide additional time-varying geometric information over relatively short intervals and thereby strengthen the observation geometry of PPP and facilitate filter convergence (Hong et al., 2025). In recent years, the benefits of LEO augmentation have been demonstrated mainly in positioning applications. Simulation studies have shown that LEO observations can shorten the initialization and reconvergence times of float PPP (Zhao et al., 2026). Experiments using CENTISPACE™ satellites have further demonstrated that even a limited number of LEO observations can improve multi-GNSS PPP positioning and convergence (Xu et al., 2024). LEO augmentation has also been investigated for real-time kinematic (RTK) positioning and PPP ambiguity resolution (PPP-AR) (Li et al., 2019; He et al., 2024; Ge et al., 2018). In addition, the quality of orbit and clock products, constellation size, and elevation cutoff angle can affect the number of available observations and the geometric contribution provided by LEO satellites (Deng and Pan, 2026). Multi-service integration has also been investigated for real-time PPP processing (Pan and Yang, 2025). For time transfer, receiver-clock treatment has been shown to affect time-link convergence and frequency stability (Xie et al., 2025b). Studies focusing on LEO platforms have further demonstrated that precise products and onboard clock models affect LEO PPP timing performance (Liu et al., 2025; Ge et al., 2025; Wang et al., 2024). These findings support the use of the rapidly varying LEO geometry as an additional source of observation information for PPP time transfer.

Although previous studies have demonstrated the potential of LEO observations to improve PPP time transfer, most have focused on constellation size, real-time processing feasibility, or overall performance enhancement. Systematic investigations of how LEO augmentation benefits change with progressively restricted observation conditions remain limited, particularly with respect to whether convergence and steady-state performance respond in the same manner to changes in elevation cutoff angle. As the elevation cutoff angle increases, the GNSS geometry becomes weaker, and the additional geometric information provided by LEO satellites may become more important during initialization. At the same time, however, a higher cutoff angle also reduces the number of usable LEO satellites and shortens their continuous tracking arcs, thereby limiting the accumulation of additional information during steady-state operation (Li et al., 2022). This raises three questions: (1) how the relative contribution of LEO observations to observation redundancy and timing geometry varies with elevation cutoff angle; (2) how changes in observation geometry are reflected in time-link convergence, steady-state dispersion, and frequency stability; and (3) whether convergence time, steady-state STD, and MDEV exhibit consistent responses to elevation cutoff angle.

To address these questions, this study evaluates LEO augmentation of multi-GNSS PPP time transfer under observation-constrained conditions represented by different elevation cutoff angles. In contrast to studies focusing primarily on constellation size, overall augmentation performance, or real-time processing feasibility, the LEO constellation configuration and PPP processing strategy are kept unchanged while the elevation cutoff angle is systematically varied, allowing the influence of observation constraints on LEO augmentation benefits to be examined under controlled conditions. A Walker 288/24/1 LEO constellation with an orbital altitude of approximately 850 km and an inclination of 55° is simulated, and elevation cutoff angles of 10°, 20°, 30°, and 40° are considered. GPS, BDS-3, and Galileo observations constitute the GCE scheme, while the addition of LEO observations forms the GCE + LEO scheme. The two schemes are compared in terms of the number of visible satellites, time dilution of precision (TDOP), cold-start time-link convergence, reconvergence following a complete observation outage, steady-state STD, and MDEV over different averaging times. Particular attention is given to whether convergence and steady-state performance exhibit consistent responses to changes in elevation cutoff angle, with emphasis on how the role of LEO augmentation shifts between time-link establishment/recovery and steady-state performance as the observation constraint becomes more restrictive.

2 Methods

This section first formulates the multi-GNSS PPP time-transfer model and then extends it by incorporating LEO observations into the joint estimation framework. The construction of inter-station time links, the LEO constellation and observation simulation, the elevation-cutoff strategy, and the performance evaluation procedures are subsequently described.

2.1 Multi-GNSS static PPP time-transfer model

A dual-frequency ionosphere-free (IF) PPP model is adopted for time-transfer estimation (Zumberge et al., 1997; Kouba and Héroux, 2001). The station coordinates are fixed to their known values and are not estimated in the filter, while satellite orbit and clock corrections are introduced from external precise products. After applying deterministic corrections for relativistic effects, satellite and receiver antenna phase-center effects, carrier-phase wind-up (Wu et al., 1993), solid Earth tides (Petit and Luzum, 2010), and ocean tide loading (Scherneck, 1991), the receiver clock is referenced to GPS system time.

For GPS, BDS-3, and Galileo, the ionosphere-free pseudorange observation equations are given in Equation 1.and the corresponding ionosphere-free carrier-phase observation equations are given in Equation 2.

Δ P i , k j , G = c δ t i , k G + M i , k j , G δ T i , k w + ε P , i , k j , G Δ P i , k j , C = c δ t i , k G + c b i , k C / G + M i , k j , C δ T i , k w + ε P , i , k j , C Δ P i , k j , E = c δ t i , k G + c b i , k E / G + M i , k j , E δ T i , k w + ε P , i , k j , E (1)

Δ Φ i , k j , G = c δ t i , k G + M i , k j , G δ T i , k w + λ I F G N i , k j , G + ε Φ , i , k j , G Δ Φ i , k j , C = c δ t i , k G + c b i , k C / G + M i , k j , C δ T i , k w + λ I F C N i , k j , C + ε Φ , i , k j , C Δ Φ i , k j , E = c δ t i , k G + c b i , k E / G + M i , k j , E δ T i , k w + λ I F E N i , k j , E + ε Φ , i , k j , E (2)

Here, i, j and k denote the station, satellite, and observation epoch, respectively, while the superscripts, G, C and E refer to GPS, BDS-3, and Galileo. Δ P i , k j , S and Δ Φ i , k j , S denote the ionosphere-free pseudorange and carrier-phase residuals after correction for geometric range, satellite clock offset, and other deterministic effects. c is the speed of light in vacuum; δ t i , k G is the receiver clock offset with respect to GPS system time; b i , k C / G and b i , k E / G are the receiver-side BDS-3/GPS and Galileo/GPS inter-system biases (ISBs), respectively; M i , k j , S is the wet-delay mapping function; and δ T i , k w denotes the residual zenith wet delay. λ I F S and N i , k j , S are the ionosphere-free wavelength and carrier-phase ambiguity for system, respectively. ε P , i , k j , S and ε Φ , i , k j , S represent observation noise and residual unmodeled errors. Carrier-phase hardware delays that are not explicitly parameterized are absorbed into the ambiguity parameters.

GPS is used as the reference system, and therefore no ISB is introduced for GPS itself. For the GCE scheme comprising GPS, BDS-3, and Galileo, the estimated state vector is given in Equation 3.

x i , k GCE = δ t i , k G , b i , k C / G , b i , k E / G , δ T i , k w , N i , k G T , N i , k C T , N i , k E T T (3)

Here, N i , k S denotes the ambiguity vector associated with the currently valid continuous tracking arcs of system. Pseudorange observations provide the absolute ranging information and receiver-clock constraint, whereas carrier-phase observations contribute substantially higher measurement precision. Their joint use supports the estimation of ISBs, tropospheric delay, and ambiguities, thereby facilitating PPP convergence and improving the stability of the time link after convergence.

2.2 LEO-augmented multi-GNSS PPP time-transfer model

LEO is treated as an independent satellite system, denoted by L. Because LEO and GPS observations are processed through different receiver signal channels, system-dependent receiver hardware delays may arise. A LEO/GPS receiver-side ISB, b i , k L / G, is therefore included in the joint estimation model.

The ionosphere-free pseudorange and carrier-phase observation equations for LEO are given in Equation 4.where b i , k L / G denotes the receiver-side LEO/GPS ISB, and λ I F L and N i , k j , L are the LEO ionosphere-free wavelength and ambiguity, respectively.

Δ P i , k j , L = c δ t i , k G + c b i , k L / G + M i , k j , L δ T i , k w + ε P , i , k j , L Δ Φ i , k j , L = c δ t i , k G + c b i , k L / G + M i , k j , L δ T i , k w + λ I F L N i , k j , L + ε Φ , i , k j , L (4)

At epoch t k, observations from GPS, BDS-3, Galileo, and LEO can be expressed in the unified form given in Equation 5.where z i , k is the stacked observation vector containing pseudorange and carrier-phase measurements from all participating systems, H i , k is the corresponding design matrix, and ε i , k is the combined observation-error vector.

z i , k = H i , k x i , k GCE + LEO + ε i , k (5)

The state vector for the GCE + LEO scheme is given in Equation 6.

x i , k GCE + LEO = δ t i , k G , b i , k C / G , b i , k E / G , b i , k L / G , δ T i , k w , N i , k G T , N i , k C T , N i , k E T , N i , k L T T (6)

Relative to the GCE scheme, the additional states introduced by the LEO observations are given in Equation 7.

Δ x i , k L = b i , k L / G , N i , k L T T (7)

The receiver clock offset δ t i , k G retains the same physical definition for all satellite systems and remains referenced to GPS system time. Incorporating BDS-3, Galileo, and LEO observations therefore does not alter the definition of the time-transfer parameter itself. Instead, the additional observations increase observation redundancy and provide more rapidly varying line-of-sight geometry, thereby strengthening the joint estimation model and improving the separability of the receiver clock from the other estimated parameters.

2.2.1 Stochastic models of the estimated parameters

The receiver clock offset and the BDS-3/GPS, Galileo/GPS, and LEO/GPS ISBs are modeled as epoch-wise white-noise processes. No dynamic continuity constraint is imposed between different epochs:

Cov δ t i , k G , δ t i , l G = 0 , k ≠ l Cov b i , k R / G , b i , l R / G = 0 , k ≠ l (8)

The same stochastic treatment is applied to all receiver-side ISBs to avoid imposing system-dependent temporal constraints and to maintain a consistent estimation strategy across the different cutoff scenarios.

The residual zenith wet delay is modeled as a random walk, whereas the carrier-phase ambiguity is treated as constant within a continuous cycle-slip-free tracking arc, as given in Equation 9:where ω T , i , k is the tropospheric process noise and S ∈ G , C , E , L denotes the satellite systems included in the joint solution. The white-noise model in Equation 8 indicates only that the receiver clock and ISBs are estimated independently between epochs; it does not imply that their estimated values are zero. Whenever a cycle slip, loss of lock, or signal reacquisition occurs, the corresponding carrier-phase ambiguity is reinitialized.

δ T i , k w = δ T i , k − 1 w + ω T , i , k N i , k j , S = N i , k − 1 j , S (9)

2.3 Inter-station time-link construction

PPP time transfer is based primarily on the estimated receiver clock offsets. In this study, GPS system time is adopted as the common reference for receiver clocks at all stations. For station i, the receiver clock offset relative to GPS system time is defined by Equation 10.where t i , k denotes the receiver time at station i and epoch k , and t k G denotes GPS system time at the same epoch. The PPP solution provides an estimate of this clock parameter for a given observation scheme and elevation cutoff angle.

δ t i , k G = t i , k − t k G (10)

For any two stations A and B, the inter-station time link under observation scheme and elevation cutoff angle is constructed from the difference between their GPS-referenced receiver clock estimates, as given in Equation 11:where δ t A , k G , M E c and δ t B , k G , M E c denote the estimated GPS-referenced receiver clock offsets at stations A and B , respectively, with M ∈ G C E , G C E + L E O. The construction of the LEO-augmented multi-GNSS PPP time link is illustrated in Figure 1.

l A B , k M E c = δ t A , k G , M E c − δ t B , k G , M E c (11)

BRUX is selected as the reference station. Six time links are formed between BRUX and NIST, MGUE, USUD, MAS1, HRAO, and PARK. For each elevation cutoff angle, the GCE and GCE + LEO schemes are processed independently using identical station pairs and observation periods. Cold-start convergence, reconvergence following a complete observation outage, steady-state STD, and MDEV are all evaluated using these six links, as described in Section 3.3.

3 Simulation and processing

3.1 LEO constellation and observation simulation

Because publicly available LEO navigation observations are still limited, LEO navigation measurements are simulated for the PPP time-transfer experiments. As shown in Figure 2, a Walker 288/24/1 constellation comprising 288 LEO satellites is adopted. The constellation has an orbital altitude of approximately 850 km and an inclination of 55°, providing coverage compatible with the selected low- and mid-latitude station network.

The initial orbit of each LEO satellite is specified by six Keplerian elements and numerically propagated to the required epochs. The onboard clocks are simulated according to the stochastic characteristics of spaceborne rubidium frequency standards. The power spectral density of the fractional-frequency fluctuations is represented by the power-law model in Equation 12.

S y f = h 0 + h − 1 f − 1 + h − 2 f − 2 (12)

Here, h 0, h − 1 and h − 2 correspond to white frequency modulation (WFM), flicker frequency modulation (FFM), and random-walk frequency modulation (RWFM), respectively. All LEO satellites are assumed to carry rubidium clocks, with h 0 = 8.0 × 10 − 22, h − 1 = 2.0 × 10 − 26, h − 2 = 1.0 × 10 − 29. These coefficients are of the same order as those reported in related studies of compact and spaceborne clock systems (Van Dierendonck and McGraw, 1984; Van Buren et al., 2021). Fractional-frequency sequences are generated from the above noise model and subsequently integrated to obtain the corresponding clock-offset series. The Rb clock model is introduced to provide the simulated LEO observations with physically representative satellite-clock variations rather than an idealized constant clock assumption. The corresponding simulated clock series are subsequently used as fixed clock products in the PPP processing, thereby allowing the analysis to focus on the elevation-cutoff-dependent LEO augmentation effect rather than satellite clock-product errors. This controlled setting is intended to isolate the contribution of LEO observation availability and geometry without introducing additional uncertainty from LEO precise products. The simulated LEO orbit products are likewise treated as fixed products, and no additional orbit-product errors are introduced.

For the simulated observations, the LEO downlink frequencies are set equal to GPS L1 (1575.42 MHz) and L2 (1227.60 MHz), with a sampling interval of 30 s. This frequency setting is adopted to construct consistent dual-frequency ionosphere-free observations and does not represent the frequency plan of a specific operational LEO navigation system. The carrier-phase and pseudorange noise levels are set to 5 mm and 0.1 m, respectively.

Relativistic effects, carrier-phase wind-up, and Earth-rotation effects are included according to the corresponding models. The satellite antenna phase-center offset and variation (PCO/PCV) for the simulated LEO signals are set to zero, representing the case in which these effects are assumed to be known and fully corrected. Ionospheric delays are generated using CODE global ionosphere maps (GIMs) (Hernández-Pajares et al., 2009). The slant tropospheric delay is divided into hydrostatic and wet components and modeled using the Saastamoinen model (Saastamoinen, 1972) and the Global Mapping Function (GMF) (Boehm et al., 2006), respectively. Station coordinates are taken from the IGS weekly solutions. The receiver clock offsets used in the LEO observation simulation are obtained from the corresponding station clock products and interpolated to the simulation epochs, thereby preserving the temporal behavior of the actual receiver clocks and maintaining a consistent receiver-clock reference between the real GNSS and simulated LEO observations. Receiver PCO/PCV values for the simulated LEO signals are also set to zero. The LEO/GPS inter-system bias (ISB) is set to zero when generating the simulated observations, but is estimated as an epoch-wise white-noise parameter in the PPP processing.

3.2 Station network and processing strategy

Seven Multi-GNSS Experiment (MGEX) stations are selected for the LEO-augmented multi-GNSS PPP time-transfer experiments. Their geographical distribution is shown in Figure 3, and the receiver and clock configurations are summarized in Table 1. All stations are equipped with high-performance external atomic frequency standards. BRUX, equipped with an external hydrogen maser, is selected as the reference station, while MAS1, NIST, HRAO, USUD, MGUE, and PARK form six inter-station time links with BRUX. The corresponding baseline lengths range from approximately 3,034 to 12,250 km, covering medium- to ultra-long-distance time links.

TABLE 1. Experimental stations and clock configurations.

Five days of real GNSS observations and simulated LEO observations from DOY 252–256 of 2025 (9–13 September) are used, with a sampling interval of 30 s. GPS L1/L2, BDS-3 B1/B3, and Galileo E1/E5a dual-frequency observations are combined to form ionosphere-free observables. The simulated LEO observations use the same frequencies as GPS L1/L2. CODE final orbit, clock, and code-bias products are used for the GNSS processing (Dach et al., 2026), whereas the simulated LEO orbit and clock products described in Section 3.1 are fixed in the PPP solution.

All solutions are processed in static ionosphere-free PPP mode using the GREAT software and a square-root information filtering implementation. Because receiver clock estimation is the primary objective of PPP time transfer, the station coordinates are held fixed to reduce coupling between position and receiver-clock parameters. The estimated parameters include the receiver clock offset, residual zenith tropospheric delay, ISBs, and carrier-phase ambiguities. Receiver clock offsets are treated as epoch-wise white-noise parameters. The Saastamoinen model provides the a priori tropospheric delay, while the residual zenith delay is estimated as a random walk. GPS is used as the reference system, and the Galileo/GPS, BDS-3/GPS, and LEO/GPS ISBs are estimated as white-noise parameters. Carrier-phase ambiguities remain float throughout the processing. The main processing settings are summarized in Table 2.

TABLE 2. PPP processing strategies for the GCE and GCE + LEO schemes.

Two processing schemes are considered: GCE, comprising GPS, Galileo, and BDS-3 observations, and GCE + LEO, in which LEO observations are added to the GCE solution. To investigate the dependence of LEO augmentation on the elevation cutoff angle, both schemes are independently processed at elevation cutoff angles of 10°, 20°, 30°, and 40°, with the same cutoff applied to both GNSS and LEO observations.

For satellite system, the set of satellites available at station and epoch is defined aswhere E i , k j , S denotes the elevation angle of satellite j observed at station i . A single E c is maintained throughout each processing session, and the filter is independently initialized for every cutoff scenario to avoid state transfer between different observation conditions.

V i , k S E c = j ∣ E i , k j , S ≥ E c (13)

For pseudorange and carrier-phase observations satisfying Equation 13, elevation-dependent weighting is applied according to Equation 14.where σ q , 0 S denotes the a priori zenith standard deviation for observation type q and satellite system S . The corresponding weight is therefore proportional to sin 2 E , assigning lower weights to observations at lower elevations. As the elevation cutoff angle increases, progressively more low-elevation GNSS and LEO observations are excluded, reducing satellite availability and observation redundancy and shortening continuous LEO tracking arcs.

σ q , i , k j , S = σ q , 0 S sin ⁡ E i , k j , S (14)

For each scheme M , a conventional four-parameter geometry matrix G i , k M E c , consisting of three line-of-sight direction-cosine columns and one receiver-clock column, is constructed using the union of satellites satisfying Equation 13. The time dilution of precision (TDOP) is defined by Equation 15.

TDOP i , k M E c = G i , k M E c T G i , k M E c − 1 4 , 4 . (15)

Although station coordinates are fixed in the PPP processing, TDOP is used only as an independent four-parameter geometry metric. It is not included in the actual filter and does not account for elevation-dependent weighting, ISBs, tropospheric parameters, carrier-phase ambiguities, or their stochastic models. TDOP therefore characterizes the satellite geometry under different elevation cutoff angles and processing schemes and should not be interpreted as the actual variance of the PPP receiver-clock estimate.

For the GNSS observations common to both schemes, GCE and GCE + LEO use identical precise products, correction models, constraints, weighting strategies, and filtering settings. GCE + LEO differs only by the addition of LEO observations, the corresponding orbit and clock products, and the associated LEO/GPS ISB and carrier-phase ambiguity parameters. Therefore, comparisons between the two schemes at the same cutoff angle primarily reflect the contribution of LEO observations to observation availability and satellite geometry.

3.3 Time-link performance evaluation

Cold-start convergence and recovery following a complete observation outage are evaluated using the sliding standard deviation (STD) of the estimated time-link series. A 10 min sliding window is used. With the 30 s sampling interval, each window contains 20 epochs. The mean of each window is removed before calculating the STD. Convergence is declared when the sliding STD remains below 0.1 ns for the first five consecutive epochs, and the beginning of this continuous interval is defined as the convergence time.

Because of the 10 min sliding window, the earliest reportable convergence onset is approximately 9.5 min. The five-consecutive-epoch requirement can first be confirmed at approximately 11.5 min. A convergence time close to 9.5 min therefore indicates that the time link satisfies the adopted criterion from the first valid sliding window. Consequently, the finite-window definition may partially compress differences among solutions that reach the prescribed stability level very rapidly.

To evaluate recovery from a complete loss of observations, a 10 min observation outage is introduced at each remote station after the filter has reached steady operation, while the reference station BRUX remains continuously observed. Identical outage start and end epochs are used for GCE and GCE + LEO, and no measurement update is performed during the outage.

When observations resume, the carrier-phase ambiguities associated with the pre-outage arcs are removed, and new ambiguity parameters are initialized for the newly established continuous phase arcs. The estimated receiver clock, zenith tropospheric delay, and ISB states are retained, while the station coordinates remain fixed throughout. During the outage and after observation resumption, the covariance matrices of the retained states are propagated according to their prescribed state-transition and stochastic models. No artificial covariance inflation to initialization-level values or additional state reset is applied.

The post-recovery sliding STD is constructed exclusively from time-link estimates obtained after observation resumption and therefore contains no pre-outage observations. Reconvergence is determined using the same 0.1 ns criterion as that used for cold-start convergence.

Steady-state performance is evaluated after removing the first 1 h of each solution. A constant offset and linear trend are removed from each time-link series before calculating the STD, which is used to characterize the overall dispersion of the steady-state link. The modified Allan deviation (MDEV) is also calculated to assess frequency stability over different averaging times, with the value at τ = 3000 s adopted as the principal metric for medium- to long-term stability comparisons.

4 Results

4.1 Satellite visibility and TDOP

The performance of PPP time transfer is closely related to the available observation geometry, which is characterized here by the number of visible satellites and the time dilution of precision (TDOP). To examine the geometric contribution of LEO augmentation under different elevation cutoff angles, Figure 4 presents the results for three representative stations, MGUE, USUD, and BRUX, over DOY 252–256. The left panels show the time series of visible-satellite counts at cutoff angles of 10°, 20°, 30°, and 40°, while the right panels show the corresponding five-day mean TDOP values; the error bars represent the day-to-day standard deviation. As defined in Section 3.2, TDOP is used solely as an independent indicator of the timing geometry.

The network-averaged satellite visibility and TDOP statistics at the four elevation cutoff angles are summarized in Table 3. As the elevation cutoff angle increases from 10° to 40°, the number of visible satellites in the GCE scheme decreases continuously, accompanied by a pronounced increase in TDOP. Averaged over the three stations and 5 days, the mean number of visible GCE satellites decreases from approximately 26.1 at 10° to 20.4, 15.4, and 11.4 at 20°, 30°, and 40°, respectively. The corresponding mean TDOP values are approximately 0.55, 0.89, 1.58, and 3.48 at cutoff angles of 10°, 20°, 30°, and 40°, respectively. Even at a cutoff angle of 40°, approximately ten GNSS satellites remain available for PPP processing. The deterioration in geometry at high cutoff angles therefore arises primarily from restricted sky coverage rather than from a complete loss of satellite availability. The satellite count and TDOP exhibit a clear inverse relationship: as fewer satellites remain available and their spatial distribution becomes less favorable, TDOP increases.

TABLE 3. Satellite visibility and TDOP statistics at different elevation cutoff angles.

The addition of LEO observations increases the number of visible satellites and reduces TDOP at every cutoff angle. Relative to GCE, GCE + LEO increases the mean satellite count by approximately 45%, 29%, 22%, and 17% at 10°, 20°, 30°, and 40°, respectively, while reducing the mean TDOP by approximately 28%, 19%, 15%, and 21%. These results confirm that LEO observations provide additional line-of-sight directions and strengthen the geometry available for receiver-clock estimation. The relative improvement is not strictly monotonic with cutoff angle. The gain in satellite availability generally decreases as the cutoff angle rises, whereas the relative TDOP reduction increases again at 40° because of station-dependent differences in the underlying GNSS geometry.

The magnitude of the geometric improvement also varies among stations. At BRUX, for example, approximately 9.6 GNSS satellites remain available on average at a cutoff angle of 40°, but the corresponding mean TDOP reaches approximately 4.32. With LEO augmentation, the mean satellite count increases to approximately 12.7, corresponding to about 3.1 additional satellites, while the mean TDOP decreases to approximately 2.78, a reduction of about 36%. Under the same cutoff condition, MGUE and USUD retain more favorable GCE geometry, with mean TDOP values of approximately 3.51 and 2.61; their corresponding reductions after LEO augmentation are only about 12% and 8%, respectively. Thus, when the remaining GNSS observations are still sufficient for PPP processing, the geometric contribution of LEO is more pronounced at stations where the underlying GNSS geometry is already weak, whereas the relative benefit is smaller where the geometry remains favorable.

Figure 5 further shows that increasing the elevation cutoff angle reduces both the mean number of visible LEO satellites and their mean continuous tracking-arc duration. The simultaneous reduction in satellite availability and arc continuity indicates that although LEO observations can compensate for degraded GCE geometry, the amount of usable LEO information itself becomes increasingly restricted as the cutoff angle rises. These geometric characteristics provide the basis for the convergence and steady-state analyses that follow.

4.2 Time-link convergence performance

Using the convergence criterion defined in Section 3.3, the cold-start convergence performance of GCE and GCE + LEO is compared at elevation cutoff angles of 10°, 20°, 30°, and 40°. The analysis includes six BRUX-centered time links: NIST–BRUX, MGUE–BRUX, USUD–BRUX, MAS1–BRUX, HRAO–BRUX, and PARK–BRUX. Both schemes are initialized at the same epoch for each experiment, with all other processing settings kept identical, ensuring a paired comparison under the same initial observation geometry and filter conditions.

Figure 6a summarizes the distribution of cold-start convergence times across the six links. For GCE, the median convergence time increases from approximately 13.8 min at 10° to 16.0, 18.0, and 19.8 min at 20°, 30°, and 40°, respectively. With LEO augmentation, the corresponding medians are approximately 11.5, 12.0, 14.0, and 13.5 min. The resulting median reductions are 16.4%, 25.0%, 22.2%, and 31.6% at cutoff angles of 10°, 20°, 30°, and 40°, respectively. Although the relative improvement does not vary monotonically with cutoff angle, the compensation provided by LEO is generally more pronounced under the more restrictive cutoff conditions. As shown in Figure 6b, all 24 link–cutoff combinations exhibit a reduction in convergence time, although the magnitude of the improvement varies considerably among individual links.

Figure 7 shows the cold-start convergence process for three representative links, NIST–BRUX, MGUE–BRUX, and PARK–BRUX. Across all 12 link–cutoff comparisons, the 10 min sliding STD for GCE + LEO falls below the 0.1 ns threshold earlier than that for GCE, consistent with the network-level statistics in Figure 6. As the cutoff angle increases, convergence of the GCE solution is generally delayed, whereas the addition of LEO observations mitigates this degradation to varying degrees. At 40°, the convergence times of NIST–BRUX, MGUE–BRUX, and PARK–BRUX are shortened by approximately 4.5, 7.5, and 12.5 min, respectively. This behavior is consistent with the increase in satellite availability and reduction in TDOP identified in Section 4.1, indicating that improved observation geometry is accompanied by earlier cold-start convergence.

Figure 8 provides a link-by-link comparison of the absolute reduction in cold-start convergence time. The median reductions across the six links are approximately 2.3, 4.0, 4.0, and 6.3 min at cutoff angles of 10°, 20°, 30°, and 40°, respectively. The absolute compensation provided by LEO therefore tends to increase as the observation constraint becomes more restrictive. The magnitude of the benefit nevertheless remains strongly link dependent. At 30°, for example, the convergence times of NIST–BRUX and PARK–BRUX are shortened by only 1.0 and 0.5 min, whereas MGUE–BRUX and MAS1–BRUX improve by 6.0 and 5.5 min. At 40°, the reductions for MAS1–BRUX and PARK–BRUX reach 11.5 and 12.5 min, respectively, compared with only about 3.0 min for HRAO–BRUX. The network-wide improvement is therefore not driven by a small subset of links, but the magnitude of the LEO benefit clearly depends on the individual time link.

To examine recovery after a complete short-term loss of observations, reconvergence experiments are conducted at two representative cutoff angles, 10° and 40°. After the filter reaches steady operation, all observations at each remote station are interrupted for 10 min, from 04:00 to 04:10, while BRUX remains continuously observed. The same outage interval and recovery strategy are applied to GCE and GCE + LEO.

Figure 9a compares the reconvergence times for NIST–BRUX, MGUE–BRUX, and PARK–BRUX. At 10°, the GCE reconvergence times are 13.0, 13.0, and 16.0 min, respectively, whereas GCE + LEO reduces them to 10.5, 9.5, and 10.5 min, corresponding to relative reductions of approximately 19%, 27%, and 34%. At 40°, the corresponding GCE values are 24.5, 19.0, and 17.0 min, compared with 14.5, 15.0, and 16.0 min for GCE + LEO, giving reductions of approximately 41%, 21%, and 6%. The median relative reduction across the three links is therefore about 27% at 10° and 21% at 40°. In every case considered, the GCE + LEO solution satisfies the 0.1 ns criterion earlier than GCE.

Figures 9c,d show the corresponding post-outage 10 min sliding STD. The initial post-recovery STD is substantially larger at 40° than at 10°, reflecting the more restrictive observation geometry. With LEO augmentation, the STD decreases more rapidly and crosses the 0.1 ns threshold earlier. The improvement is nevertheless highly link dependent: for NIST–BRUX, the relative reduction in reconvergence time increases from approximately 19% at 10° to 41% at 40°, whereas for PARK–BRUX it decreases from about 34% to only 6%. Thus, unlike the more systematic tendency observed for cold-start convergence, reconvergence gains depend more strongly on the particular time link and on the observation geometry available at the recovery epoch.

Overall, LEO augmentation shortens the time required for the PPP time link to satisfy the adopted usability criterion during both cold start and recovery from a complete observation outage. The cold-start benefit tends to become more pronounced under high cutoff angles, whereas the reconvergence benefit remains positive but shows a stronger dependence on the individual link and the instantaneous geometry at observation resumption.

4.3 Steady-state time-link performance

Convergence time characterizes how rapidly a PPP time link reaches the adopted usability criterion, whereas steady-state performance describes the dispersion and frequency stability of the link after convergence. The first 1 h of each solution is discarded, after which a constant offset and linear trend are removed from each of the six BRUX-centered time-link series. The STD of the resulting residual series is then used to characterize steady-state dispersion.

Figure 10 presents the steady-state STD for the six time links together with the relative reduction achieved by LEO augmentation. At 10°, GCE + LEO yields a lower STD than GCE for all six links. The reductions are 30.9%, 29.0%, and 27.4% for NIST–BRUX, MGUE–BRUX, and USUD–BRUX, respectively, and 17.8%, 8.4%, and 18.0% for MAS1–BRUX, HRAO–BRUX, and PARK–BRUX. The six-link median STD decreases from approximately 0.266 ns for GCE to 0.214 ns for GCE + LEO, corresponding to a median link-wise reduction of 22.7%. Thus, at a cutoff angle of 10°, the addition of LEO observations produces a consistent reduction in steady-state dispersion across the network.

This advantage diminishes rapidly as the elevation cutoff angle increases. At 20°, MAS1–BRUX still exhibits a 13.0% reduction in STD, whereas NIST–BRUX, MGUE–BRUX, USUD–BRUX, and PARK–BRUX show small increases of approximately 1.1%–3.5%; HRAO–BRUX remains essentially unchanged. The median relative reduction across the six links is −1.5%. At 30°, only MAS1–BRUX retains a clear positive reduction, at approximately 5.4%, while changes for the other links remain small; the six-link median is again −1.5%. At 40°, all link-wise changes remain within approximately ±1%, and the median change is only 0.2%. The corresponding six-link median STD values for GCE and GCE + LEO are approximately 0.279 and 0.286 ns at 20°, 0.295 and 0.301 ns at 30°, and 0.331 and 0.329 ns at 40°. The systematic steady-state STD benefit of LEO augmentation is therefore concentrated primarily at the 10° cutoff, whereas the two schemes exhibit similar steady-state dispersion at the higher cutoff angles.

This behavior differs from that observed for convergence. Section 4.2 shows that the median reduction in convergence time generally becomes larger as the cutoff angle is increased, rising from about 16% at 10° to about 32% at 40°. By contrast, the steady-state STD benefit is concentrated at 10° and becomes negligible at the higher cutoff angles. The distinction indicates that under restrictive elevation masks, the principal contribution of LEO observations is to accelerate the establishment of a usable time link rather than to reduce the overall dispersion of the subsequent steady-state series.

The STD behavior is also consistent with the LEO visibility results in Section 4.1. As the cutoff angle increases, both the number of usable LEO observations and the duration of continuous LEO tracking arcs decrease, reducing the additional information available during steady-state operation. Although several links show small increases in STD at 20° and 30°, these changes are limited to only a few percent and do not exhibit a consistent link-wide pattern. Consequently, the steady-state STD values of GCE and GCE + LEO converge at the higher cutoff angles, indicating that the additional contribution of LEO to overall link dispersion has become limited.

The modified Allan deviation (MDEV) is further evaluated after removing the first 1 h of each solution. MDEV is calculated from the detrended inter-station time-link series to characterize frequency stability over different averaging times. The value at τ = 3000 s is used as the primary metric for medium- to long-term stability, while the values at τ = 30 s and 300 s are used to provide complementary information at shorter averaging times.

Figure 11a shows the six-link median MDEV at τ = 3000 s as a function of elevation cutoff angle. At every cutoff angle, the median MDEV of GCE + LEO is lower than that of GCE. The corresponding median reductions are 48.9%, 30.5%, 13.9%, and 2.1% at cutoff angles of 10°, 20°, 30°, and 40°, respectively. Thus, the improvement in medium- to long-term frequency stability provided by LEO augmentation is strongest at low cutoff angles and progressively decreases as the cutoff angle increases. The relative reductions in convergence time and MDEV at the four elevation cutoff angles are summarized in Table 4.

TABLE 4. Relative reductions in convergence time and MDEV at different elevation cutoff angles. Elevation cutoff Convergence-time reduction MDEV reduction at τ = 3,000s.

The response of medium- to long-term frequency stability differs from that of convergence time. Whereas the relative convergence benefit is generally greater under high cutoff angles, the MDEV improvement after convergence is concentrated in the less restrictive observation conditions. The individual-link results in Figure 11b are broadly distributed around the network median and show a largely consistent direction of improvement, although the magnitude varies among links.

Notably, the GCE median MDEV at τ = 3000 s does not vary monotonically with elevation cutoff angle in the same manner as TDOP. Medium- to long-term frequency stability is therefore not governed solely by instantaneous satellite geometry, but also reflects the observation combination, filter states, and noise characteristics of the individual time links. For this reason, the comparison here focuses primarily on the relative difference between GCE and GCE + LEO under the same cutoff condition. As the cutoff angle increases, the number of additional visible satellites contributed by LEO decreases substantially and the continuous LEO tracking arcs become shorter, reducing the amount of additional information available during steady-state operation. This behavior is consistent with the progressively smaller MDEV difference between GCE and GCE + LEO at higher cutoff angles.

Figure 12 shows the MDEV of the six time links over different averaging times. At short averaging times, particularly at τ = 30 s and 300 s, the MDEV values of the two schemes are generally similar, indicating a relatively limited effect of LEO augmentation on short-term frequency stability. As the averaging time increases, the difference between GCE and GCE + LEO becomes more apparent under the lower cutoff conditions. At cutoff angles of 10° and 20°, the GCE + LEO curves generally remain below the corresponding GCE curves, whereas the two schemes become progressively closer as the cutoff angle increases. Although the magnitude varies among individual links, the overall behavior is consistent with the six-link median statistics at τ = 3000 s.

Taken together, the steady-state results reveal a strong dependence of LEO augmentation on both the evaluation metric and the elevation cutoff angle. At τ = 3000 s, the median MDEV reduction decreases from 48.9% at 10° to 2.1% at 40°. In conjunction with the convergence results in Section 4.2, this indicates two complementary manifestations of LEO augmentation: under restrictive cutoff conditions, its primary benefit is the reduction in convergence time and the faster establishment of a usable time link; under lower cutoff angles, where LEO observations remain more continuously available, the benefit is more evident in steady-state dispersion and medium- to long-term frequency stability. STD and MDEV describe different aspects of the steady-state behavior: STD represents the overall dispersion of the detrended time-link series, whereas MDEV characterizes frequency stability as a function of averaging time. Accordingly, the small STD increases observed for some links at 20°–30° are not inconsistent with the positive medium- to long-term MDEV improvements obtained under the same conditions.

The cutoff-angle response of LEO augmentation is therefore distinctly different between the cold-start and steady-state stages. Under more restrictive cutoff conditions, LEO observations continue to provide a clear reduction in cold-start convergence time. In contrast, the systematic improvement in steady-state STD is concentrated at 10°, while the MDEV reduction at τ = 3000 s decreases progressively from 48.9% at 10° to 2.1% at 40°.

These results indicate that the contribution of LEO observations is stage dependent: under restrictive observation conditions, the principal advantage is faster establishment of a usable time link, whereas under lower cutoff angles, where LEO observations can be accumulated more continuously, the benefit is more evident in steady-state link dispersion and medium- to long-term frequency stability.

5 Discussion

Previous studies have shown that LEO observations can improve GNSS PPP initialization and reconvergence by increasing observation redundancy and providing rapidly varying line-of-sight geometry, with corresponding benefits also reported for PPP time transfer (Li et al., 2022; Liu et al., 2024; Xie et al., 2025a). The present results are consistent with these findings, but further indicate that the benefit of LEO augmentation does not respond uniformly to elevation cutoff angle across different performance metrics. The median cold-start convergence-time reduction increases from 16.4% at 10° to 31.6% at 40°, although the variation is not strictly monotonic. In contrast, the systematic reduction in steady-state STD is concentrated mainly at 10°, while the median MDEV improvement at τ = 3000 s decreases from 48.9% at 10° to 2.1% at 40°. LEO augmentation of PPP time transfer therefore exhibits a clear dependence on both the observation condition and the processing stage, and its overall benefit cannot be adequately characterized by a single metric.

This contrasting behavior is consistent with the different roles of LEO observations during initialization and steady-state operation. Raising the elevation cutoff reduces GNSS satellite availability and sky coverage and increases TDOP, leaving less geometric information available for separating the receiver clock from the other estimated parameters during initialization. Under these conditions, the rapidly changing LEO line-of-sight directions provide additional time-varying geometry, which is consistent with the generally larger convergence benefit obtained at high cutoff angles. The improvement is not strictly monotonic because it also depends on the instantaneous satellite configuration. At the same time, increasing the cutoff removes an increasing fraction of LEO observations and shortens their continuous tracking arcs, thereby reducing the additional information available during steady-state operation. This provides a plausible explanation for the progressive weakening of the STD and medium- to long-term MDEV benefits. TDOP should nevertheless be interpreted only as a geometric indicator, because it does not account for elevation-dependent weighting, inter-system biases, tropospheric parameters, carrier-phase ambiguities, or their stochastic models.

The reconvergence experiments show a less systematic cutoff response than the cold-start results. LEO augmentation shortens reconvergence at both 10° and 40°, but the median relative reductions for the three representative links are approximately 27% and 21%, respectively, with substantial differences among individual links. Recovery after an observation outage is therefore more sensitive to the satellite geometry and tracking arcs available at the resumption epoch than cold-start convergence alone. Differences in the external frequency standards used at the selected stations may also contribute to link-to-link variations in absolute STD and MDEV. For this reason, the analysis emphasizes paired comparisons between GCE and GCE + LEO for the same time link rather than absolute differences among different links.

The reported results should be interpreted within the adopted simulation and processing assumptions. The LEO observations are simulated, and the corresponding orbit and clock products are treated as fixed in the PPP solution without additional product errors or latency, while the GNSS processing uses CODE final products. The results therefore represent the potential performance of LEO-augmented PPP time transfer under idealized product conditions rather than the absolute performance of an operational real-time system. The uniform elevation cutoff provides a controlled way to vary satellite availability, but it does not reproduce azimuthally asymmetric blockage, multipath, non-line-of-sight propagation, or intermittent loss of lock in real obstructed environments. The magnitude of the reported cutoff response may also depend on the adopted Walker 288/24/1 constellation, observation weighting, and ISB stochastic model. In addition, the outage analysis is based on one representative 10 min interruption interval, and the five-day data span limits the statistical support available at long MDEV averaging times. Further validation using realistic LEO orbit and clock errors, alternative constellation configurations, asymmetric blockage, and measured LEO observations will therefore be needed to assess the extent to which the observed cutoff-dependent behavior persists under operational conditions.

6 Conclusion

This study investigated the elevation-cutoff dependence of LEO augmentation for multi-GNSS PPP time transfer using 5 days of MGEX GNSS observations and simulated observations from a Walker 288/24/1 LEO constellation. The GCE and GCE + LEO schemes were evaluated at elevation cutoff angles of 10°, 20°, 30°, and 40° in terms of observation geometry, time-link convergence, steady-state dispersion, and frequency stability.

Increasing the elevation cutoff angle substantially degraded the multi-GNSS timing geometry. For GCE, the mean number of visible satellites decreased from 26.06 at 10° to 11.42 at 40°, while the mean TDOP increased from 0.547 to 3.482. LEO augmentation increased satellite availability by 17.2%–44.9% and reduced TDOP by 14.6%–27.6% across the four cutoff angles. The magnitude of the geometric improvement depended on the underlying GNSS geometry and station location and did not vary monotonically with cutoff angle.

LEO observations shortened cold-start convergence for all 24 link–cutoff combinations. The six-link median convergence-time reductions were 16.4%, 25.0%, 22.2%, and 31.6% at cutoff angles of 10°, 20°, 30°, and 40°, respectively. The benefit was generally greater under more restrictive cutoff conditions, although the response was not strictly monotonic. The representative 10 min outage experiments also showed shorter reconvergence with LEO augmentation at both 10° and 40°, but the magnitude of the recovery benefit was more strongly dependent on the individual link and the geometry available when observations resumed.

The steady-state response differed from the convergence behavior. At 10°, all six links showed lower STD with LEO augmentation, with a median link-wise reduction of 22.7%. At 20° and 30°, the median changes were both approximately −1.5%, while at 40° the median change was only 0.2%. The systematic contribution of LEO observations to steady-state link dispersion therefore became limited as the cutoff angle increased.

A similar cutoff dependence was observed for medium- to long-term frequency stability. At τ = 3000 s, the six-link median MDEV reductions were 48.9%, 30.5%, 13.9%, and 2.1% for cutoff angles of 10°, 20°, 30°, and 40°, respectively. The difference between GCE and GCE + LEO was comparatively small at short averaging times and became more evident at longer averaging times, particularly under the lower cutoff conditions.

Overall, the results demonstrate that LEO augmentation of multi-GNSS PPP time transfer has a stage-dependent response to elevation cutoff angle. Under more restrictive observation conditions, its primary benefit is to accelerate the establishment and recovery of a usable time link, whereas under lower cutoff angles, where LEO observations remain more continuously available, the benefit is more clearly reflected in steady-state dispersion and medium- to long-term frequency stability. These findings indicate that the effectiveness of LEO augmentation for PPP timing should be assessed jointly from convergence and steady-state perspectives rather than from a single performance metric.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Ke Su, Space Engineering University, China

Reviewed by: Xiao Gongwei, Xi’an University of Posts and Telecommunications, China

Jian Chen, Anhui University of Science and Technology, China