The BeiDou Navigation Satellite System (BDS-3) has officially provided services worldwide since July 2020. BDS-3 has added new signals for B1C, B2a and B2b based on old BDS-2 B1I and B3I signals, which brings opportunities for achieving high-precision time transfer. In this research, the BDS-3/BDS-2 combined penta-frequency common-view (CV) single-differenced (SD) precise time transfer model is established with B1I, B3I, B2I, B1C, B2a and B2b signals, including dual-, triple-, quad- and penta-frequency (abbreviated as DF, TF, QF and PF) ionosphere-free (IF) combination CV SD models. Taking four long baseline time links (from 637.6 km to 1331.6 km) as examples, the accuracy and frequency stability of the BDS-3/BDS-2 combined DF, TF, QF and PF SD time transfer models were evaluated. The experimental results show that the frequency stability of the TF, QF and PF SD models were improved by 2.5%, 5.3% and 8.5%, on average, over the DF SD model. Compared with the traditional DF (B1I/B3I IF combination) SD model, the standard deviation (STD) of the multi-frequency SD model was reduced by 5.9%, on average, and the frequency stability was improved by 4.0% on average, which had the most apparent effect on the improvement of short-term frequency stability. Specifically, the DF1 (B1C and B2a DF IF combination), TF1 (B1C, B2a and B2b TF IF combination), QF1 (B1C, B1I, B2a and B2b QF IF combination) and PF4 (B1C, B1I, B2a, B2b and B3I PF IF combination) SD models had better performance in timing, and the PF4 SD model had the best performance. Considering that the PF4 (one PF signal IF combination) SD model does not require an estimated inter-frequency bias and that its noise factor is minor compared with the PF1 (four DF signal IF combination), PF2 (three TF signal IF combination) and PF3 (two QF signal IF combination) SD models, we recommend the PF4 SD model for multi-frequency time transfer and the use of the PF2, PF2 or PF3 SD model to supplement the PF4 SD model in cases of penta-frequency observation loss.

Conventional Global Navigation Satellite System (GNSS) time transfer algorithms typically model receiver clock offsets as white noise for estimation, neglecting the physical characteristics of atomic clocks, which consequently limits the performance of GNSS time transfer. To overcome this limitation, this study proposes an undifferenced GPS/Galileo combined Precise Point Positioning (PPP) time transfer model, incorporating both one-state (only clock offset parameter) and two-state (both clock offset and frequency offset parameters) refined clock models with clock instantaneous re-initialization (CIR) strategy at the day boundary epoch. Using observations from International GNSS Service (IGS) Multi-GNSS Experiment (MGEX) stations equipped with external hydrogen masers, precise time transfer performance under refined clock models was evaluated based on undifferenced GPS/Galileo combined PPP float solutions and PPP ambiguity resolutions. Experimental results demonstrate that, compared to traditional models, the refined clock models improve time transfer accuracy and frequency stability by an average of 6.7% and 25.8%, respectively. The improvement is most significant for short term frequency stability, with a maximum enhancement exceeding 85%. As the averaging time increases, the improvement in long term frequency stability gradually diminishes. Notably, the two-state refined clock model slightly outperforms the one-state model in time transfer performance, with the two-state refined clock model improving short-, medium-, and long term frequency stability by 11.5%, 8.0%, and 0.2%, respectively, compared to the one-state refined clock model. These findings strongly advocate adopting the two-state refined clock model to optimize both time transfer precision and short term stability in high-accuracy applications.

The China Space Station (CSS) is currently in orbit and carries the high-precision optical atomic clock with stability of approximately $2.0 \times 10^{-15} / \sqrtτ$ in its experiment module. We have developed a model to determine the gravity potential (GP) based on the gravity frequency shift equation and have created both one-way and dual-frequency transfer models up to $c^{-4}$. These models consider effects from the troposphere, ionosphere, and solid Earth tides. The proposed model is suitable for measurements at the magnitude of $10^{-19}$. Based on the CSS mission, we conducted the simulation experiments. The results indicate that when processing the simulation frequency signal using the proposed model, we can obtain the GP with the accuracies of $ (1.13\pm0.71)\,\mathrm{m^2/s^2}$, $ (0.09\pm0.89)\,\mathrm{m^2/s^2}$, and $(0.66\pm1.18)\,\mathrm{m^2/s^2}$ for cutoff elevation angles of $5^{\circ}$, $10^{\circ}$ and $15^{\circ}$, respectively. With the high-precision optical atomic clock onboard the CSS, the proposed model enables us to measure the GP differences in the magnitude of centimeter-level accuracy.