According to general relativity theory (GRT), the clock at a position with lower geopotential ticks slower than an identical one at a position with higher geopotential. Here, we provide a geopotential comparison using a non-transportable hydrogen clock and a transportable hydrogen clock for altitude transmission based on the two-way satellite time and frequency transfer (TWSTFT) technique. First, we set one hydrogen clock on the fifth floor and another hydrogen clock on the ground floor, with their height difference of 22.8 m measured by tape, and compared the time difference between these two clocks by TWSTFT for 13 days. Then, we set both clocks on the ground floor and compared the time difference between the two clocks for seven days for zero-baseline calibration (synchronization). Based on the measured time difference between the two clocks at different floors, we obtained the height difference 28.0 $\pm$ 5.4 m, which coincides well with the tape-measured result. This experiment provides a method of height propagation using precise clocks based on the TWSTFT technique.

In November 2022, the China Space Station (CSS) was equipped with a cold atomic microwave clock and a Sr optical lattice clock with stabilities of $5\times {10}^{-14}/\sqrt{\tau }$ and $2\times {10}^{-15}/\sqrt{\tau }$ (where $\tau$ is the integration time in seconds), respectively, which provides an excellent opportunity to test gravitational redshift (GRS) with higher accuracy than previous results. Based on high-precision frequency links between the China Space Station and a ground station, we formulated a model and provided simulation experiments to test GRS. Simulation results suggest that this method could test the GRS at the accuracy level of $5\times {10}^{-7}$, more than 2 orders in magnitude higher than the result of the experiment of a hydrogen clock on board a flying rocket more than 40 years ago.

Based on the general relativity theory, the geopotential difference can be determined by comparing the change in time difference between precise clocks using the precise point positioning (PPP) time transfer technique, referred to as the relativistic PPP time comparison approach. We focused on high-precision time comparison between two high-performance clocks for determining the geopotential difference using this approach, and conducted the simulation experiments to validate this approach. In the experiment, we consider three cases for evaluating the performance of this approach with different types of atomic clocks, namely, the fractional frequency stabilities of the clocks equipped at three selected ground stations (BRUX, OPMT, and PTBB) are $4.0 \times 10^{-13} / \sqrt {\tau }$ (Case 1), $2.3 \times 10^{-14} / \sqrt {\tau }$ (Case 2), and $2.8 \times 10^{-15} / \sqrt {\tau }$ (Case 3) at averaging time ${\tau }$ , respectively, and the accuracy of these clock have been evaluated to be $5.3 \times 10^{-16}$ , $7.8 \times 10^{-17}$ , and $8.6 \times 10^{-18}$ . Two main conclusions can be drawn from the experimental results. First, high-performance clocks can significantly improve the precision for GNSS PPP time transfer. Compared to Case 1, the long-term stabilities of the time link OPMT-BRUX as well as PTBB-BRUX are improved in Cases 2 and 3. Second, the geopotential difference between any two stations can be determined at the decimeter level, and the accuracy of geopotential difference is consistent with the stabilities of the time links in Cases 1–3. In Case 3, the determined geopotential differences between OPMT and BRUX deviate from the EIGEN-6C4 model values by −0.64, m ² / ² with an uncertainty of 1.11 m ² /s ² , whereas the deviation error between PTBB and BRUX is 0.76 m ² /s ² with an uncertainty of 1.79 m ² /s ² . The results of this study suggest that a one decimeter-level geopotential difference between two arbitrary stations can be determined based on this approach.

The present Global Navigation Satellite System (GNSS) can provide at least double-frequency observations, and especially the Galileo Navigation Satellite System (Galileo) can provide five-frequency observations for all constellation satellites. In this contribution, precision point positioning (PPP) models with Galileo E1, E5a, E5b, E5 and E6 frequency observations are established, including a dual-frequency (DF) ionospheric-free (IF) combination model, triple-frequency (TF) IF combination model, quad-frequency (QF) IF combination model, four five-frequency (FF) IF com-bination models and an FF uncombined (UC) model. The observation data of five stations for seven days are selected from the multi-GNSS experiment (MGEX) network, forming four time-frequency links ranging from 454.6 km to 5991.2 km. The positioning and time-frequency transfer performances of Galileo multi-frequency PPP are compared and evaluated using GBM (which denotes precise satellite orbit and clock bias products provided by Geo Forschung Zentrum (GFZ)), WUM (which denotes precise satellite orbit and clock bias products provided by Wuhan University (WHU)) and GRG (which denotes precise satellite orbit and clock bias products provided by the Centre National d’Etudes Spatiales (CNES)) precise products. The results show that the performances of the DF, TF, QF and FF PPP models are basically the same, the frequency stabilities of most links can reach sub10⁻¹⁶ level at 120,000 s, and the average three-dimensional (3D) root mean square (RMS) of position and average frequency stability (120,000 s) can reach 1.82 cm and 1.18 × 10⁻¹⁵, respectively. The differences of 3D RMS among all models are within 0.17 cm, and the differences in frequency stabilities (in 120,000 s) among all models are within 0.08 × 10⁻¹⁵. Using the GRG precise product, the solution performance is slightly better than that of the GBM or WUM precise product, the average 3D RMS values obtained using the WUM and GRG precise products are 1.85 cm and 1.77 cm, respectively, and the average frequency stabilities at 120,000 s can reach 1.13 × 10⁻¹⁵ and 1.06 × 10⁻¹⁵, respectively.

Due to the influence of pseudo-range noise, traditional GNSS common view method is difficult to improve the accuracy of time-frequency transfer. GNSS carrier phase precise point positioning (PPP) time-frequency transfer has become a research hotspot because of its high accuracy. In this paper, a time-frequency transfer model of GNSS carrier phase single difference (SD) and Integer Single Difference (ISD) between any two ground stations is studied. In order to solve the problem that the SD ambiguity cannot be fixed due to the influence of the phase biases at the receivers, a method of SD ambiguity fixing is proposed, that is the SD ambiguity is fixed with the constraint of the fixed double difference ambiguity among several stations and satellites. Here taking four time-frequency links between pairs of ground stations, BRUX-OPMT (262.3km), BRUX-PTBB (454.6km), BRUX-WTZR (637.7km) and BRUX-CEBR (1331.6km) as examples, the multi-GNSS time-frequency transfer experiment of SD, ISD and PPP method is carried out. The results show that the SD and PPP time-frequency transfer accuracy is equivalent, the stability of ISD is improved compared with SD, and the difference RMS between epochs is less than 10 ps. High precision carrier phase SD, ISD and PPP technology can be applied to the study of determining the gravity potential based on time-frequency measurements. This study is supported by the National Natural Science Foundations of China (NSFC) under Grants 42030105, 41721003, 41804012, 41631072, 41874023, Space Station Project (2020)228, and the Natural Science Foundation of Hubei Province of China under Grant 2019CFB611.