A Relativistic Geodetic Approach to Unify the Height System for A Relativistic Geodetic Approach to Unify the Height System for Africa Africa

This study focuses on the establishment of a uni ﬁ ed height system for Africa called African Uni ﬁ ed Height System (AFRUHS) by utilizing atomic eight clocks and clock networks. The International Association of Geodesy has nine long aims to construct an International Height Reference Frame, but the lack of accurate and globally harmonized vertical coordinates, particularly in Africa, has posed a challenge. To overcome this, the researchers propose using clock networks to determine geopotential or elevation differences between distant stations by measuring the gravitational redshift through clock frequency comparisons. The research uses simulation studies using the atomic clock ensemble in space and microwave signal communication to calculate the geopotential differences between atomic clock ensemble in space and three selected ground stations in Africa: Alexandria, Egypt (northeast); Dakar, Senegal (west); and Windhoek, Namibia (south). The study provides speci ﬁ c gravity level values and corresponding elevation differences for each of the three stations. For Alexandria Station, the gravity level is determined as 62636440.270 m 2 /s 2 , with an elevation difference near the ground of 0.4 e 1.1 m compared with the standard gravity level value. For the Dakar Station, the gravity level is calculated as 62636652.310 m 2 /s 2 , with an elevation difference of 0.2 e 1.2 m near the ground. Finally, Windhoek Station ' s gravity level is estimated as 62620192.794 m 2 /s 2 , with an elevation difference of 0.6 e 0.9 m near the ground. These ﬁ ndings highlight the potential of using clock networks and atomic clocks to establish an accurate and uni ﬁ ed height system for Africa, contributing to the construction of the International Height Reference Frame.


Introduction
T he ESA-CNES atomic clock ensemble in space (ACES) experiment, which will be sent onboard the International Space Station (ISS), was primarily designed to examine gravitational redshift (GR) to an order of 2 Â 10 À6 , using atomic clocks with fractional frequency instability and error of (1 À3 ) Â 10 16 .It is worth pointing out that the mentioned level is 1.5 orders greater than the Gravity Probe A (GPA) experiment.The principal onboard instrumentation (PHARAO) includes a space-active hydrogen maser and a cold cesium atom clock.The PHARAO clock has fractional frequency stability of 1.1 Â 10 13 √t, where t represents the accuracy of a few parts in 10 16 (Cacciapuoti and Salomon, 2009).
However, after 10 000 s of integration time, spaceactive hydrogen maser exhibits an instability with a fractional frequency of 1.5 Â 10 15 .An onboard time scale will be created by combining the accuracy and long-term stability of the cesium clock with the stability of the H-short-term maser.ACES uses the European Laser Timing optical link and the microwave hyperlinks (MWL) as two distinct time and frequency transfer connections to study general relativity and create applications in geodesy (relativistic geodesy) and time and frequency metrology.These scientific goals are strongly related to the efficiency of the MWL, and their effectiveness is critical in the current study.To transfer time/frequency, MWL uses an uplink of provider frequency 13.475 GHz (Ku-band) and downlinks of carrier frequencies 14.70333 GHz (Ku-band) and 2248 M Hz (S-band).MWL will outperform 0.3 ps at 300 s, 7 ps at 1 day, and 23 ps at 10 days in terms of time discrepancy.After a few days of integration, these results, which are 12 orders of magnitude better than current approaches (TWSTFT and GPS), will allow ground clock comparisons with 10e17 frequency resolution (Hartmann and Wenzel, 1995).
Regarding the ACES objective, previous research focused on the examination of GR utilizing time comparisons (Blanchet et al., 2001;Cacciapuoti et al., 2017;Chou et al., 2010); however, there is relatively limited research regarding frequency comparing.
The benefits of frequency comparison over time comparison are as follows: (a) since frequency monitoring is the result of measuring during a brief period and cannot be used for ranging purposes, it can decrease the effects of phase ambiguity and (b) it can evaluate the gravitational potential (GP) immediately, whereas time comparison requires more data to extract the GR value and solve for the time-shifting rate.Nevertheless, because the precision of determining the immediate frequency is highly confined, more data must be gathered to get more precise results.The trifrequency combination (TFC) approach will be used in this research project to combine three frequency measurements to get the GP difference (Blanchet et al., 2001).Formulation in free space with a medium, which is precise to a c3 order is used for the one-way frequency transfer model with a required precision equal to 10À16.Contributions to theory were made by extending the Blanchet et al. (2001) model from free to real space with media and formulating a method to remove the Doppler frequency shift (the term Doppler impact or Doppler frequency shift used in the current study indicating the first Frontier Doppler impact) while taking into account the time balance out among three links.The final TFC model is capable of effectively eliminating all types of shifts ranging in value from 1016.Simulation tests that took into account the actual orbit, realistic clock noises, real atmosphere, and real gravity are performed to evaluate the model in this research and identify the magnitude of conditions that were required to be met.
To achieve the objective of this research, initially, a gravity field model is needed to evaluate the GP at the ground station (in the simulation process), so the current study begins by identifying an accurate geoid model for Africa using the shallow-layer technique.Then, this geoid is used as a gravity model in the simulation process.Because it is based on the geoid's definition itself, the shallow-layer technique differs from the conventional Stokes and Molodenesky geoid computation techniques.In the current research, the shallow-layer technique is applied to determine the 5 Â 5 geoid model for Egypt which ranges from 21 f 32 e24 l 37 , where f and l are the latitude and the longitude, respectively.The Global Digital Topographic Model DTM2006.0, the EGM2008 gravity field model, the Danish National Space Center DNSC08 model, and the CRUST2.0crust model were used to identify the boundaries for the shallow layer and identify its internal formation.The recently established AFRgeo2019 gravimetric geoid model (Abd-Elmotaal et al., 2020), which was established by the initiatives of the International Association of Geodesy (IAG) Sub-Commission on the Gravity and Geoid in Africa, has been used to validate the computed shallow-layer geoid model for Africa.These geoids' differences revealed the significant effect of using local gravity data.In most regions of Africa, these differences are found to be significantly low (<0.5 m).In addition, a comprehensive study has been done to make a comparison between the calculated shallow-layer geoid and the geoid obtained using several global reference (geopotential) models.According to this study, it is revealed that they all produce equivalent geoid outcomes.The accuracy of the input models used determines the accuracy of the calculated shallow-layer geoid.Gross errors in EGM2008, DTM2006.0, and CRUST1.0models affected the result in geoid's accuracy.Large errors in geoidal heights may be exhibited by errors in the elevations of DTM2006.0, which is supplemental to EGM2008 (Kasdin NJ Discrete simulation of, 1995; Kiamehr and Sj€ oberg, 2005).

The study objective
Based on the theory of general relativity, the running rate and vibration frequency of an atomic (or optical atomic) clock will fluctuate at various positions with various GPs (Duchayne et al., 2009;Weinberg, 1972).On the contrary, the Gravitational Potential (GP) at a point in space or on the ground can be figured out by gauging the changes in the clock's running rates (Bjerhammar, 1985) or by monitoring the changes in the frequency of electromagnetic signals as described by Pound and Snider (Pound and Snider, 1965).
Both of these alternate techniques for figuring out GP (distinction), known as the clock transport comparison and frequency signal transmission comparison need extremely accurate clocks or oscillators, such as 1 Â 10 À18 , which is equal to 1 cm in height.According to different studies (Ashry et al., 2020;Cacciapuoti and Salomon, 2011;Galleani et al., 2003;Pavlis et al., 2008;Pound and Rebka, 1959;Shen et al., 1993;Turneaure et al., 1983) the timeefrequency associated proof of the GR provides a potential technique based on the clock transport comparison and frequency signal transmission comparison to evaluate the GP directly.
By comparing the vibration frequencies or running rate of two optical atomic clocks situated at two separate stations, the GR hypothesis provides an interesting approach to precisely figure out the GP distinction.Recently, a new technique has been proposed to evaluate the GP difference between satellite and ground stations by exchanging microwave signals, known as satellite frequency signal transmission, which is developed based on the TFC strategy or Doppler canceling technique.Here, as a continuation of past research, establishing a computation of GP at specified ground stations is intended.These stations will be used as a benchmark for the unified height system for Africa.
Fig. 1 shows the selected station to generate a unified height system for Africa, these stations have been selected before by IAG.Three stations in Africa were chosen; the first is located in Egypt (Alexandria), the second in Senegal (Dakar), and the third is in Nigeria (Windhoek) (see Fig. 1).
A relativistic approach will be used to define the geopotential difference between these three stations.Then the GP at each point will be determined.The TFC method will be used to determine the GP under a simulation process.
At this time, there is no real data.Simulated exercises are provided to put our theory and formulations to the test.In these experiments, data from the ISS's actual orbit, the ionosphere, the troposphere, GP computed by the widely used gravitational reference model EGM2008 (Meynad et al., 2018) and the shallow-layer geoid for Africa (Ashry et al., 2021a), solid 146 earth tide (Hafele and Keating, 1972), and simulated clock data generated 147 by a generally and widely used accepted stochastic noises prototype (Allan et al., 1991;Einstein, 1915) are used (Table 1).
Fig. 2 shows the general simulation steps to use the gravity information and models to define the geopotential of the used ACES and determine the GP of the selected stations using Africa Geoid.In the following sections, a simulation study will be detailed.

Simulation experiments of the atomic clock ensemble in space project
The ACES is a European Space Agency project whose primary objective is to study gravitation using atomic clocks in both space and on the ground with high performance.The two-way MWL is a critical component of this experiment, which transfers time and frequency using an uplink of carrier frequency 14.70333 GHz (Ku-band) and carrier frequency downlinks of 2248 MHz (S-band) and 14.70333 GHz (Ku-band).The formulation based on time comparison has been researched for more than 10 years.However, there are benefits to employing frequency comparison rather than time comparison to evaluate the GR The ESA-CNES Atomic Clock Ensemble.Thus, in the current study, a TFC approach is established based on the measurements of the frequency changes of three separate MWLs between ACES and a ground station, and then this newly developed approach is applied to the ACES data.
As a result, a variety of factors must be accounted for such as tidal impacts, atmospheric frequency shift (DFS), refraction produced by the atmosphere, the Doppler impact, second-order Doppler impact, and the Shapiro impact, with precision levels in tens of centimeters for the possible scientific item.The ACES payload will launch as schedule in mid-2025, and the solution provided in this paper would allow the GR to be measured with a precision of at least 2 Â 10 À6 , which is a mount of magnitude greater than the existing precision level of 7 Â 10 À5 .
As the ACES plan is planned to be implemented in 2025, simulation experiments can be only used to test our theories.To verify the feasibility of the model more realistically, the clock frequency value of the simulation experiment, the geographical and orbital parameters of the ground station and the space station, and the parameters of the ground and the space stations and the parameters of the atmosphere (ionosphere and troposphere) are simulated carefully.
In the simulation experiment, the one-way frequency comparison link is first analyzed and then the TFC method is analyzed.In the process of analyzing the TFC method, in addition to checking whether the final gravity potential is correct, we also check whether the combination between the two downlinks (link 2 and link 3) can effectively extract the ionosphere because the TFC method needs to calculate the residual quantity, and we will also test the accuracy requirements of the parameters involved in the residual quantity calculation (Ashry et al., 2021).

Atomic clock ensemble in space plan
ACES uses two high-precision clocks, namely the cold atomic cesium clock and the hydrogen atomic clock.The combination of the two clocks can provide (1 À3 ) Â 10 À16 accuracy and stability.Among them, the relative frequency stability of this cold atom cesium clock has reached 1.1 Â 10 13√ t, where t represents the average time in seconds, and the clock has an accuracy of 10 À16 (Blanchet et al., 2001).
For the timeefrequency transfer between the space and the ground, the ACES plan uses two independent comparison technologies, MWL and laser link (European Laser Timing), to examine the GR effect of GR, the fine structure constant, determination of GP, and studies of timeefrequency metrology (Cacciapuoti and Salomon, 2009;Blanchet et al., 2001).
These scientific goals are strongly linked to the performance of microwave links (Merry, 2003).The two-way MWL consists of a carrier frequency of 13.475 GHz (Ku-band) and two downlinks with carrier frequencies of 2248 MHz (S-band) and 14.70333 GHz (Ku-band) downlink composition; they have a time skew of 0.3, 7, and 23 ps 300 s, 1, and 23, and 10 days, respectively (Hartmann and Wenzel, 1995).These performances outperform the current high-precision microwave transfer technologies (two-way satellite timeefrequency transmission and GPS timeefrequency transmission) by an order of magnitude of two, allowing for space-toground comparison, and can also serve as a 'bridge' to provide two ground stations.Coviewing and noncoviewing alignments between 10 À17 orders of magnitude frequency resolution (average time 1e2 days).The comparison between open spaces and not between the two places is studied in this research.
The ACES program was implemented on the ISS, and the orbit of ISS is at 51.6 inclination, which means that areas with latitudes above 51.6 are difficult to observe.The principle is shown in Fig. 3.There are more observations per day in low-latitude areas, but the time for each observation is very short; although there are fewer observations per day in high-latitude areas, the time for each observation is relatively long.
Because of the low orbit of the ISS (about 400 km), the line of sight with the ground station is far less than that of GNSS and geostationary satellites.In addition, according to the mechanical principle, its speed is also greater than that of high-orbit satellites, and the time it takes to pass over the station is very short, at most 600 s (Merry, 2003).According to the average observation time of 400 s, the same station can only be observed four to six times a day.Therefore, for the same station, it will be interrupted for several hours after each observation.Take the next observation.Although this observation has little effect on the frequency comparison (because the frequency comparison does not need to accumulate for a long time), it seriously affects the accuracy evaluation, especially the frequency stability evaluation of the Allan variance.In the data processing of this study, a hierarchical processing strategy is adopted, first observe the arcs passed by the space station each time, average the processing results, and then process the results of each average again.But when evaluating frequency stability, all the data to a continuous-time period are shifted, and then calculate the Allan variance.

Experimental ideas and parameter settings
There are no real data at present.Simulation experiments were introduced to examine our theory and formulations.In the current study, the data of the ISS real orbit, ionosphere, troposphere, computed GP by the commonly used gravity reference model EGM2008 (Meynad et al., 2018) and the shallow-layer geoid for Africa (Ashry et al., 2021a), solid earth tide (Hafele and Keating, 1972), and simulated clock data by a generally accepted and widely used stochastic noises model (Allan et al., 1991;Einstein, 1915) are used.

Experiments and simulation configuration
In the simulation experiment, three stations in Africa are chosen; the first is located in Egypt (Alexandria), the second in Senegal (Dakar), and the third is in Nigeria (Windhoek) (see Fig. 1).The geographical parameters for the selected stations are listed in Table 2.
The tidal impact of the simulation experiment is nevertheless incorporated in addition to the precision of the ACES design, even if its impact on GR is only about 10 À17 .Table 3 represents other parameters assigned in our experiment.Fig. 4 illustrates the design.
Fig. 4 demonstrates the design process of the simulation experiment.The observed values in the experiment are the received signal frequency values of the three links of ACES, which are counted as f 1 ', f 2 ', and f 3 ', in the purple box in the figure.The combined frequency values f 1 0 /f 1 , f 2 0 /f 2 , f 3 0 /f 3 are used to determine the gravity position and elevation of the ground station.In the experiment, real data to calculate the orbit, ionosphere, troposphere, and gravity field of ISS are used.Also, the internationally universal clock model (Allan et al., 1991) is used to simulate the clock frequency error.Among them, for the simulation of the orbit, the daily number of orbits is used to compute; the ionosphere uses TEC data and the layered structure; the troposphere uses the zenith tropospheric delay calculation of wet and dry constituents.The projection function uses the VMF1 model; the GP model EGM2008 (Meynad et al., 2018) is used as the tidal impacts on ground station coordinates are much less significant than other residual errors which are not taken into account.However, as the effect of these tidal impacts is considered the principal reason for the residual error in GR, it has been considered on the ground gravity level.
To get the received frequency values ( f 1 ', f 2 ', and f 3 '), frequency values as well as different frequency shifts are needed to transit.The transmit frequency value is obtained from the original carrier frequency f 1 , f 2 , f 3 , and clock noise (including equipment noise).The frequency shift consists of Doppler frequency shift, relativistic frequency shift (including GR and second-order Doppler frequency shift), and DFS (including tropospheric and ionospheric frequency shift).The orbit of the space station and the location of the ground station can both provide the researchers with the position, velocity, and acceleration data that can be needed to compute these effects.To obtain good observation data, the restriction that the elevation angle must be more than 15 is imposed.When calculating the frequency emission value, the clock noise is needed to be obtained.According to the random noise model, five kinds of clock noise are required: Random walk FM, flicker FM, white FM, flicker PM, and phase modulation white noise (white PM) (Allan et al., 1991).Their spectral densities of noises are of the order of f À2 , f À1 , f , f 1 , and f 2 .Before experiment simulations, some samples of the analog clock error are shown, as shown in Fig. 4. Five sets of single-type clock frequency noises of the same magnitude are simulated (Fig. 5aee), and their summation is determined (Fig. 5f).The data length is 10 000 groups.Fig. 5a illustrates a clear trend, while Fig. 5b demonstrates a weaker trend, and Fig. 5cee shows an irregular pattern.The mathematical models of white noise and random walk noise are simple and easy to simulate.Research by Galleani et al. (Einstein, 1915) is drawn on.Flicker noise is calculated using an autoregressive model (Hess et al., 2011).In the actual simulation, the FM white noise as the main part is regarded, and the sampling rate is 1 s (the duration is 29 days).Fig. 5 shows the modified Allan variance (MDEV) of our simulated clock.It shows that it is close to the curve of previous studies (Cacciapuoti et al., 2017;Blanchet et al., 2001).Fig. 6 shows that the long-term stability is on the degree of 10 À16 .After the above experiments, the first step of the simulation experiment has been completed: generating the frequency value of the emission.
Fig. 6 depicts the ISS's track of the subsatellite point when it is in orbit with a period of 5400 s and an inclination of 51.6 .The range that can be observed is highlighted in red in this figure.Due to the ISS's low orbit and high speed, every pass over the ground station will last at most 600 s; however, at the current observational elevation threshold of 15 , about 300 s every pass can be observed, approximately.

Numerical analysis of various errors
The orbital inclination of the ISS is 51.6 , and the orbital period is 5400 s.Its subsatellite point trajectory is illustrated in Fig. 6.The range highlighted in red in this figure can be observed.It can be demonstrated that the DFS is the most important part; the Shapiro frequency shift is the smallest part; and the relativistic frequency shift includes the GR and the lateral Doppler effect.As the refraction impact is much larger than both the ionospheric and tropospheric frequency shifts, it cannot be ignored.Because of the error's uncertainty, some residuals in the experiment, particularly the high-order terms of the ionosphere, exceeded 10 16 ; however, simulation experiments showed that they could be less than 10 16 after long-term averaging.For the 15 elevation threshold set in this experiment, the value for 300 s per round can be observed.
In the experiment, 29 days of data are simulated.The ISS flew over the station a total of 130 times, and each arc segment was only 5 min, that is, the entire observation consisted of 130 arc segments.As shown in Fig. 7, the locus of the subsatellite point centered is plotted on (a) Alexandria, Egypt, (b) Dakar, Senegal, and (c) Windhoek, Nigeria.As the orbital inclination is 51.6 , the maximum latitude value of the subsatellite point is this value.
After assessing the orbit of the ISS, the period that can be observed at an elevation angle of more than 15 is selected.Based on the location information of the air-ground station, the downloaded atmospheric parameters, the gravity position values, and the various models mentioned above, the values of the various frequency shifts mentioned in Fig. 2 are calculated, and their magnitudes are listed in Table 4. Table 4 shows that the DFS is the most important part; the Shapiro frequency shift is the smallest part; and the relativistic frequency shift includes the GR and the lateral Doppler effect.The refraction effect cannot be ignored, it is much larger than the ionospheric frequency shift and the tropospheric frequency shift.Due to the errors' unpredictability, some residuals in the experiment (particularly the high-order terms of the ionosphere) are bigger than 10 À16 ; nevertheless, simulated experiments demonstrate that following long-term averaging, they may be less than 10 À16 .
In addition, the various frequency shifts of the unidirectional (link 3) link are canalized, and the residuals of the different frequency shifts after the   4. The relativistic frequency shift seems to remain unchanged, because the GP as well as velocity of the space station change relatively slowly, and the change seems to be smaller in the logarithmic coordinate system.Fig. 8 looks like a singularity exists, but it does not.When the space station flies just above the station, the velocity is perpendicular to the line-of-sight direction, so the Doppler shift and the Shapiro have a minor effect, just like a singularity.
The TFC method can eliminate various errors to a large extent.However, there are still residual errors that are known.The Doppler residual, the ionospheric-related residual (including the ionospheric frequency shift and the ionosphere-induced  refraction component), the lateral Doppler residual, and the GR are calculated.The Doppler, ionospheric-related, and lateral Doppler residuals are induced by the Doppler difference between links 2 and 3, ionospheric higher-order term, and variation in velocity, respectively.and the GR residuals are caused by tidal effects.Fig. 9 demonstrates that the ionospheric residual is the greatest among the residuals of different frequency shifts, which can reach 1 Â 10 À15 , and is at the level of 10 À16 most of the time, followed by the Doppler residual, which can reach the largest 1 Â 10 À16 , while the magnitude of other frequency shifts is much lower than 10 À16 , so there is no need to test.In the TFC technique model, these two frequency shifts should be adjusted; however, simulation experiments reveal that because of their variability, they can be minimized to below 10 16 after long-term averaging, and therefore they can be disregarded.In addition, this demonstrates that the TFC method technique is capable of canceling the biggest order Doppler frequency shift.
Besides, the various systematic residual errors discussed, the inaccuracy of these parameters in the measurement can cause random errors in the results because the TFC method model needs to be substituted into the parameters of the ground station and the space station.The impact of this error is analyzed.Figs. 8 and 9 depicts the frequency shift experienced by the space station during a single flight.The period for a single space station is defined to fly over the station as an epoch.

Materials and methods
The material in this research is real data for the orbit of the ISSs and the published data about the ACES, the data for the selected Earth stations,  and a simulation study has been done to calculate the shift of the atomic clock due to the ionosphere, troposphere, and the signal as detailed in the paper.The tricombinations method, frequency shift, and geopotential difference determination are the methods.

Results and discussion
Our observations in simulation experiments are the received frequencies ( f 1 ', f 2 ', and f 3 ').Using our TFC method model, the GP can be calculated as shown in Fig. 10, which is very close to the GP calculated by EGM2008.Fig. 10a only shows one calendar.The GP value of each epoch is averaged, and Fig. 10b is the GP value drawn after averaging the epoch-by-epoch gravity potential value.In this figure, the dotted line indicates the addition range of error is plus or minus one time.This result shows that although the calculated gravity level is not very accurate for every second of data, equivalent to an elevation of 250 m, this deviation is within the error range, but after epoch-by-epoch averaging, the accuracy is improved to about 218 m 2 /s 2 (equivalent to about 22.2 m of elevation).Finally, after the average of all the data, the value of the geopotential level for Alexandria station can be obtained as 62636440.16m 2 /s 2 , which is 4 m 2 /s 2 compared with the standard geopotential level value of 62636444.164m 2 /s À2 , and the corresponding elevation difference near the ground is 0.4 ± 1.1 m.The value of the geopotential level for the Dakar station can be obtained as 62636652.55m 2 /s 2 , which is 5 m 2 /s 2 compared with the standard geopotential level value of 62636657.55m 2 /s 2 , and the corresponding elevation difference near the ground is 0.2 ± 1.2 m.The value of the gravity level for Windhoek station can be obtained as 62620192.5m 2 /s 2 , which is 9 m 2 /s À2 compared with the standard value of 62620201.5m 2 /s 2 at geoid, and the corresponding elevation difference near the ground is 0.6 ± 0.9 m.Here, only 29 days of data is used.If the data length is longer, the effect will be better.As the current ACES plan is deployed, there is no actual observation data, so a simulation study can only be used.

Conclusion
Using the TFC method, the GP differences between these stations are determined.The results demonstrate that the clock networks, particularly those involving ACES and the China Space Station after their launch, offer an accurate technique for establishing the International Height Reference System.Constructing an International Height Reference Frame has been a considerable aim of the IAG for a long time.One obstacle is obtaining the recommended stations' vertical coordinates, that is, geopotential numbers, with high accuracy and global harmony.The IAG began a massive job to focus on and service this concern to set up a global reference height system.Africa struggles with bad gravity data and general practitioner benchmarks.This study aims to utilize the benefits of atomic clocks to develop and specify an African Unified Height System (AFRUHS).
An appealing approach to establishing and defining a unified height system is using clock networks, which are effective in specifically obtaining geopotential or elevation differences between far-off stations by determining the GR results by contrasting clock frequencies.Our research will simulate the use of the ACES atomic clock set preceding microwave signal communication to figure out the geopotential difference between the ACES and the selected ground stations in Africa.In the beginning, three stations in Africa with good geographical distribution were chosen, the first station at Alexandria, Egypt (Northeast), the second was at Dakar, Senegal (West), and the third station was at Windhoek, Namibia (South).These stations have been selected before by the IAG community to establish a global height reference system, see https://ggos.org/item/height-reference-frame/.Then, the TFC method is used to determine the GP differences between these ground stations.Simulation experiments and studies have been done using the ACES.A simulation study has been done to determine the GP difference between the ACES and the three stations in Africa.Therefore, from these differences, the GP of these stations has been determined.The results show that using the clock networks shortly after the launch of ACES and the China Space Station will help and give an accurate technique to establish the height systems.

CRediT author contributions statement
Mostafa Ashry: writing the draft and calculations.Wen-Bin Shen: main idea and supervisor.Abdelrahim Ruby: writing review and edit.Zhang Pengfei: calculations and programming.Ziyu Shen: calculations and programming.Hussein A. Abd-Elmotaal: calculations and edit.Mostafa Abd-ElBaky: review and edit.Atef A. Makhloof: review and edit.

Fig. 3 .
Fig. 3.The currently observable seven ground stations and the flight trajectory of the ISS.

Fig. 5 .
Fig. 5. Sample of analog clock error: (a) FM random walk, (b) FM flicker noise, (c) FM white noise, (d) modulation phase flicker noise, (e) phase modulation white noise, and (f) the sum of the above five noises.

Fig. 10
Fig. 10.(a) Frequency shifts in a one-way connection between two consecutive epochs; (b) frequency shifts in a 29-day observation.

Table 1 .
Datasheet for the selected stations in Africa to establish the FRUHS.
Fig. 2. Flowchart of the simulation experiment.

Table 2 .
Ground station gravity position and elevation measurement results and errors.

Table 3 .
Other parameters of the simulation experiment.

Table 4 .
Magnitude of various frequency shifts.
e14 Fig.8.Various frequency shifts in a one-way link.