## 6 Conclusions  
This study formulated an approach to unify different local vertical height systems at the centimeter level via ultrahigh precision frequency signal links between one satellite and two datum stations separated by oceans. We performed simulation experiments that connect China's VHS and the US' VHS based on the SFST approach. The results show that the deviation between the true value and the calculated result based on \"observations\" was around 3 cm with an accuracy level (STD) of about one decimeter in 1.5 h, provided that the OACs' stability achieved a level of \(4.8\times 10^{-17}\) in 1 s. Increasing the experiment period can also improve the accuracy of the SFST approach.  
A prerequisite for the SFST approach is frequency synchronization before measurement. The output frequency of the OACs' oscillators should be identical if their locations have the same geopotential value. The error of the initial synchronization will also affect the precision of height unification based on relativistic geodetic methods. When two clocks are separated at a considerable distance, such as in our case, it is very challenging to synchronize them precisely. However, we can realize synchronization by combining a fiber connection and repeated clock transportation. Determining how to precisely synchronize two separated clocks is a particular technical problem that is not entirely relevant to the main topic of this study. Hence, we address that problem in a separate paper.  
With the rapid development of time-frequency science, ultrahigh precision OACs (say at \(1\times 10^{-18}\) level or better within one hour) have been developed and are still under improvement, which makes the SFST approach possible for the unification of VHSs at the centimeter-level. The SFST approach is also possible for realizing the IHRS. As a preliminary study, we only connected two stations in this work. However, if a globally covered SFST network is established, the VHSs worldwide could be unified.  
The SFST method offers several advantages over conventional methods, primarily its ability to quickly and accurately determine the geopotential difference between any two stations, making it a highly efficient solution. While distance or obstacles, such as mountains or oceans, are not a factor with the SFST method, it does have some limitations. Firstly, it requires ultrahigh precision clocks and appropriate equipment, making measurements relatively tricky. Additionally, its precision is slightly lower compared to optical fiber link methods. Therefore, the best approach is to use the SFST method as a supplement to conventional methods. For instance, we can utilize the SFST method to connect benchmarks of two distant VHSs and use conventional methods or optical fiber links for local VHS unification.  
Conceptualization, Z.S. and W.S.; methodology, Z.S.; software, Z.S.; validation, C.K.S. and S.Z.; formal analysis, T.Z.; investigation, L.W.; resources, L.H.; data curation, L.H.; writing--original draft preparation, Z.S.; writing--review and editing, Z.C.; visualization, S.X.; supervision, W.S. and S.Z.; project administration, W.S.; funding acquisition, W.S. and Z.S. All authors have read and agreed to the published version of the manuscript.  
This study is supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 41721003, 42030105, 41631072, 41874023, 42274011, 41974034, 42204006), and the Space Station Project (Grant No. 2020-228).  
The data obtained in the study are available from the corresponding author upon reasonable request.  
We would like to thank [PERSON] and [PERSON] for providing the data patterns of the OACs. We would also like to thank the anonymous reviewers for their valuable comments and suggestions.