The analysis on time transfer of GPS/Galileo /BDS PPP with integer ambiguity resolution

GPS precise point positioning (PPP) approach has been considered for achieving time transfer for a long time. By virtue of GPS/Galileo/BDS FCB products, PPP model has the possibilities for changing phase ambiguities from ‘float’ value to ‘integer’ value. In this study, PPP time/frequency transfer model has been presented and performance of seven links equipped with Hydrogen Masers (H-Masers) and cesium atomic clocks are compared in static and kinematic modes. With partial ambiguity resolution (PAR) enabled, in contrast to GPS, results show that multi-GNSS’s fixing rate is much higher and TTFF(Time To First Fixing) is much shorter. It is verified that the fixing rate and TTFF has nothing to do with the atomic clock type but has strong correlation with the quality of observation. We find that frequency stability of time link is seriously dependent on the type of atomic clock. As far as H-Masers, it has reached the order of 1E-16/1E-15 at the averaging time of 122880 s, respectively. As far as Cesium clock, it has reached the order of 1E-15/1E-14 at the averaging time of 122880 s, respectively. For H-Maser, the long-term frequency stabilities of integer PPP (IPPP) have been improved by roughly 3% at the static mode and 4% at the kinematic mode on average, respectively. For positioning, compared to PPP solutions, the stabilities of the IPPP coordinates are improved after an averaging time of 7680 s in static or kinematic mode.


Introduction
The availability of accurate and reliable time information has become an essential need for the function of critical infrastructure.GPS Precise Point Positioning (PPP) has been participated in time and frequency transfer as a routine solution since 2009 [1].Bcause of the characteristic of GNSS carrier phase measurements, they are more precise than the GNSS code measurements and not easily influenced by the multipath effect [2][3][4][5][6].It can be explained easily that Common View (CV)/All in View(AV) results are more nosiy than those of PPP results.Finally, PPP results present much better short-term stability benefiting from the more precise phase measurements.When the averaging time lasts one day or even longer, the uncertainty of PPP frequency transfer can reach 10 -15 and 10 -16 , respectively.The statistical uncertainties of GPS PPP time links contribute to the formation of time scale is 0.3 ns [7,8].Accompanied by the evident advantages, the inherent weakness has been exposed gradually.The raw observations without differenting have been attending the PPP solution which result in the uncalibrated phase delay (UPD) will step into the ambiguity [9].It is hard to fix the ambiguities into integer value.It is not hard, but actually impossible to fix the ambiguities to their integers in the absence of satellite phase bias information.Conventionally, the ambiguity can be estimated with float value due to the effectiveness and simplicity.As a special parameter existing in the GNSS carrier-phase measurements, the ambiguity is highly correlated with other parameters like the clocks, the Zenith Tropospheric Delays (ZTDs) and the receiver coordinates.As a result, its processing strategy directly affects the precision of these correlated parameters.Therefore, recovering the integer nature of the ambiguities is of great importance to their decorrelations with these parameters and improving the performance of the estimated clocks and positional results.Specially, for time communities, it will shorten the convergence time and access the true stability of the compared clocks more rapidly.The statistical uncertainty of inter-continental time link can be degraded which can contribute to the calculation of Coordinated Time Universal (UTC).
The key to restore integer PPP(IPPP) nature is to remove the various hardware biases from the float ambiguity.The hardware biases can be corrected with the help of satellite augmentation product so as to restore the integer nature of ambiguity.The discrepancy of these approachs is in that the content of satellite augmentation product, such as the Uncalibrated Phase Delay(UPD)/Fractional Cycle Bias (FCB) model, Integer-Recovery Clock (IRC) model and the Decoupled Clock model (DC) as well as PPP-RTK model achieved by integer ambiguity resolution-enabled precise point positioning [10][11][12][13].IRC/DC model conducts with IF (Ionosphere-Free) combinations and uses the same S-basis as the CC (Common Clock model) with different parametrization.UPD/FCB model can be obtained as a reparametrized version of the CC model using IF observations.Besides, the reparametrization model is even simpler as it merely includes the wide-lane transformation.Ge et al uses the satellite wide-lane (WL) and narrow-lane (NL) FCBs with a satellitedifferencing model for recovering the integer ambiguity [14].By introducing just LAMBDA, one cannot simply resolve the ambiguities as integers.After recovering the integer nature of ambiguities, Geng et al obtained IPPP results by introducing the least-square ambiguity decorrelation adjustment (LAMBDA) algorithm [12,15].The IRC clock method has been introduced by Laurichesse [12].France CNES CLS analysis center of the IGS released GRG precise clock products that can obtain IPPP result.The NL FCB at the satellite end was absorbed by the satellite clock difference and the ambiguity can thus be fixed [16,17].The identical effects on FCB model and IRC model have been verified by Geng lately [18].A Khodabandeh infers the effect of GNSS IPPP on atmospheric sounding, instrumental calibrations or time transfer and obtains different response on IPPP stategy [19].Geng et al improves the reliability of partial AR dramatically with the combination of GPS and GLONASS PPP-AR [20].Li Bofeng reviews the gain on the ambiguity resolution of triple-frequency GNSS signals compared to the dual-frequency GNSS signals [21].On the basis of the IRC method, the decoupled clock (DC) method was developed by Collins.The integer nature of ambiguity is recovered with the pseudorange clock difference and phase clock difference.
Currently, available literatures have focused their interests on IPPP time transfer with the IRC products.Petit et al has presented that stability of IPPP time link is 10 -16 at 5 days and is 10 -17 at 10 days [22,23].Lyu et al has showed that GRE (G IPPP) PPP time transfer had the smallest standard deviations (STDs) comparing to the processing mode of G PPP/GRCE PPP/GRCE PPP (G IPPP) /GRCE PPP(E IPPP), what is more, it has showed that the variation of frequency larger than 6.1 × 10 -16 and clock jumps larger than 0.6 ns can be observed in GRE (GPS IPPP) real-time (RT) IPPP processing [24,25].Xu et al has noted that IPPP is implemented using the CNES products and preferable for very long baseline time transfer [26].GPS and Galileo perfoms better and BDS pefroms performs worse.Ou Yang et al has employed GNSS time transfer with three kinds of ambiguity-fixed strategies: OSB, FCB and IRC and points that Multi-GNSS PPP AR play a slight role in short-term stability [27].Without the outernal input, Wang Kan et al broadcasts the satellite augment products (the satellite clocks and satellite phase biases) generated by the PPP-RTK regional network and the time transfer can be achieved [28].Geng invetigates the ambiguity resolution has the potential in near real-time PPP-RTK model [29].It should be noted that the WL FCBs present in the first section of the CNES GRG precise clock file and the NL FCBs are absorbed in the satellite clocks.Algorithms and products must be used together so as to fix ambiguity.FCB products and precise clock/orbit product can be used separately which provide flexible choice for the IPPP users, so, it is more useful for focusing on the IPPP time transfer with FCB products.
Making use of the FCB products, this study focuses on the investigation of the performance of time transfer with the ambiguity-fixed solution using Cesium atomic clock and Hydrogen Maser.First, the characteristic of GPS/Galileo FCB products generated by the Wuhan University (SGG) are evaluated.Next, the IPPP strategy is explained with Multi-GNSS IPPP time transfer model reviewed.Subsequently, the performance of time transfer and positioning of GPS/Multi-GNSS PPP/IPPP is compared and analyzed using different frequency standards.Conclusions and perspectives are provided at the end.

Analysis of multi-GNSS SGG FCB product
To enable the IPPP solution for users, various institutions have been generating GPS FCB products since a few years ago, e.g., SGG has been providing GPS FCBs since 2015 and GPS, BDS-2/BDS-3, Galileo and QZSS FCBs since 2019 [30].WL FCB files are produced daily on account of their better stabilities, while NL FCBs are produced with the period of 15 min due to their worse stabilities and high accuracy requirements under their short wavelengths.The performance of the IPPP solutions is strongly dependent on the performance of FCBs.In this contribution, the temporal characteristic of the SGG FCBs is evaluated corresponding to the Center for Orbit Determination in Europe (CODE) observable-specific signal bias (OSB for short).CODE OSB provides the satellite code and phase biases for each frequency with the units of ns.Hence, the corresponding WL and NL biases induced by CODE can be formed as follows: where l IF denotes the Ionosphere-Free Afterward, multiply is the real-value NL ambiguity.The average value of the WL/NL FCBs of all the test satellites has been selected as the datum for evaluation of the WL and NL FCBs, respectively.For each system, single-differenced WL/NL FCB series between each WL/ NL FCB and the selected datum have been formed to remove the effect of datum.Some satellites are not shown in figure 1 due to the absence of the corresponding FCB or OSB files.The OSB products released by CODE only support GPS and Galileo during the test period.As such, only the performance of SGG WL/NL for GPS and Galileo is evaluated in this study.
Y axis of figures 1 and 2 has been limited with the same scale for better view.The FCB products of all the satellite are integrally which ensuring the availability of fixing ambiguity.Comparing figure 1 with figure 2, the evolution on WL FCB of Galileo and GPS are stable in general.The WL FCB of Galileo's satellite agrees better with each other that that of GPS.Ionosphere-free (IF) dual-frequency (DF) combination has been employed in the generation of FCB products calculated by the and has strong relation with the square frequencies.GPS noise coefficients are 2.546 and 1.546, Galileo noise coefficients are 2.261 and 1.261.During the seven days, most of the GPS FCBs vary within 0.05 cycles.All the Galileo FCBs vary within 0.02 cycles.Some large values, like G03, G08, G13, G21, G25, G30, are excluded in the average.The average STD of GPS and Galileo WL FCBs is 0.04 and 0.03 cycles, respectively.There are two speculations for the discrepancy of WL FCBs: one is the instability of the satellite hardware biases, and another is the observation quality.The low-quality observations degrade the accuracy of the FCB estimates.
In figures 3 and 4, the evolution on NL FCB of Galileo and GPS are stable in general.Similarly, the NL FCB of Galileo's satellite agrees better with each other that that of GPS.Because of the fast time-varying characteristic of the NL FCBs, daily file (DOY 33) has been investigated for convenience.The daily variation of the GPS/Galileo NL FCBs is nearly 0.15 cycles, and the average STD of GPS and Galileo amounts to 0.03 and 0.02 cycles, respectively.The weekly variation of GPS and Galileo NL is 0.25 cycles and 0.2 cycles, respectively.The variation of daily NL is almost the same as that of weekly WL.Therefore, it is necessary for estimating NL frequently aim at obtaining precise results.
Overall, the performance of NL FCBs is slightly worse than that of WL FCBs.This is caused by the much smaller wavelength of the NL ambiguity and the resulting higher sensitivity to the mis-modeled biases.The determination of NL FCBs has been divided into two steps: first, time series of NL ambiguity are obtained by differencing WL ambiguities and IF ambiguities.Second, NL FCBs are obtained by averaging the fractional parts of NL ambiguity time series.The accuracy of NL FCBs could be effected by the estimation accuracy of WL FCBs.

PPP partial ambiguity fixing solution(PAR)
Undoubtedly, fixing ambiguity is contributive for GNSS solutions.However, a multi-dimensional matrix costs much time to decorrelate and search.In this study, partial ambiguity resolution (PAR) has been applied, which allows resolving a subset of the ambiguity vector that passes the pre-defined success rate, i.e. [30], and the WL_ratio is 1.5, NL_ratio is 3 [31].The biggest elevation angle has been choosed as the reference for forming inter-satellite ambiguities.The ambiguity-fixed solution of WL and NL has been implemented with different criteria [32].The criteria for fixing WL ambiguity are described here: (1) The elevation angle is higher than than 15 degrees to avoid large mutlipath effects; (2) The difference between the corrected WL float ambiguity (with FCBs) and its nearest integer is less than 0.25 cycles;  (3) The success rate for this nearest integer is larger than 0.999.
Based on the fixed WL ambiguity and IF ambiguity, NL ambiguity can be obtained.The NL PAR algorithm is shown in figure 5.For NL ambiguities, the PAR is stopped when the subset of the fixed ambiguities is smaller than 4. The satellites experiencing initial ascending or re-initialization, or with an elevation angle smaller than 15°, or with a standard deviation (STD) large than 1.5 cycles are not enabled for NL ambiguity fixing.LAMBDA method was employed for ambiguity fixing.The linear ambiguity combinations have been sorted by variance after decorrelating.The ambiguity can then be fixed when bootstrapping success rate and ratio-test threshold meet the criteria.Otherwise, the last ambiguity is droped and the previous action is executed again until the remaing number is less than 4. The flow of PAR is listed here:

Data selection and strategy implementation
Observations from eight MGEX stations equipped with Hydrogen Maser and cesium atomic clocks during DOY 033-039 in 2022 are processed in PPP/IPPP dual-frequency IF scenario in static/dynamic modes.The samping interval is 30 s. Figure 6 shows the stations' distribution, more details are recorded in table 1.For kinematic PPP, a white noise process is used to model the dynamics of the vehicle in the Kalman filter.

Results and analysis
The performance of PPP/IPPP positioning and time transfer is analyzed here.IPPP has been evaluated in terms of GPS/Multi-GNSS Time To First Fix (TTFF) and fixing rate.TTFF and fixing rate should be introduced more detail: TTFF is defined as the time taken for the ambiguity-fixed solution to be successfully achieved for at least 5  epochs.The fixing rate is defined as the ratio of the number of fixed epochs to the number of total epochs after TTFF [35].

The fixing rate and TTFF
The fixing rates and TTFF of Multi-GNSS observations equipped with cesium atomic clock s and Hydrogen Masers corresponding to those of GPS-only observations are demonstrated in figures 7 and 8.The results of stations equipped with cesium atomic clocks and Hydrogen Masers have been compared separately to inspect the relation between the stability of atomic clock and IPPP strategy.
In figure 7, it can be easily found that the fixing rate and TTFF of individual day has some discrepancy for the same station.In the experiment period, the processing strategy and the atomic clock are invariable, while the observation is variable.Some important information related with quality of data, for example, the number of observations participating in the calculation, slips and gaps, the SNR and multipath are both different.What is more, the differences between stations are much more obvious.The distribution of station and the number of visible satellite result in the otherness of fixing rate and TTFF.shorter than that of GPS, regardless of the type of atomic clocks.Multi-GNSS scenario has offered a better satellite geometry and, thus, a stronger model available for the IPPP.Besides, the TTFF for stations equipped with cesium atomic clock is longer than the stations with Hydrogen Masers in both GPS-only and multi-GNSS  scenarios.This can be explained by the fact that the performance of Hydrogen Maser is more stable than that of cesium atomic clock , and the frequency signal generated by the superior atomic clock has postive influence for the observation.The average fixing rate of GPS-only stations equipped with cesium atomic clocks over 7 days is 96, 96, 97, 97, 97, 97, 96%, and the average fixing rate of Multi-GNSS stations equipped with cesium atomic clock s over 7 days is 96, 96, 97, 97, 96, 97, 96%.The average fixing rate of GPS-stations equipped with Hydrogen Masers over 7 days 79, 79, 81, 84, 82, 85, 85%.The average fixing rate of Multi-GNSS stations equipped with Hydrogen Masers over 7 days is 82, 79, 83, 84, 84, 86, 85%.It can be found that the increase of the observation number plays a negligible role when the fixing rate is high enough.It is easily to recognize that GPS/Galileo/BDS observations provide more visible satellite and improve PDOP value.Among all the stations, station SYDN shows the lowest fixing rate and the longest convergence time.The satellites of SYDN involved in the processing are merely half of those of other stations.The satellite number are the main influencing factors for the time transfer model strength.The average numbers of GPS, BDS, and Galileo satellites at BRUX are 9, 9 and 8, and at PTBB are 9, 9 and 7, respectively.

Time transfer experiment
In this section, time transfer performance using multi-GNSS PPP and IPPP solutions has been compared and evaluated regarding two aspects: time transfer and frequency transfer.Detailedly, the absolute bias and instability of time series and is my focus for time/frequency transfer.It should be mentioned that instability of frequency transfer has not been affected by the systematical bias.SGG FCB products only contain GPS/Galileo/ BDS presently.Limited by the FCB products, the results obtained by three systems in this study.

Common clock zero baseline time transfer experiments
Common-clock zero baseline experiment can be considered for a kind of ideal status which used for evaluating the theoretical precision of algorithm.The two same receivers are equipped with the same clock signal which excludes the influence of frequency source and the discrepancy of receiver.The pulse per second (1 PPS) and 10 MHz signals are connected in the receiver.The upper panel of figure 9 shows transient oscillations less than one hour.It is typical for a Kalman-Filter estimation.A filter solution instead of a least square solution is employed here for enchancing IPPP/PPP realtime time transfer efficiency.
The clock offset is about 0.2 ns.When a common-clock experiment is performed, theoretically, the clock difference should be close to zero.The receiver's internal delay maybe the cause of the offset as well as the measurement noise and multpiath noise.We estimate frequency instability for the common-clock comparison with Modified Allan deviations (MDEV) rather than Allan deviations (ADEV).More high frequency noise, such as, white and flicker Phase Module (PM) noise has been identified with MDEV.In the lower panel of figure 9, when the averaging time reaches 7680 s and even longer, IPPP solution performs better than PPP solution.Of all the averaging time, the largest improvement on stability is 12.81%.At the averaging time of 61440 s (the last value), IPPP/PPP obtain 6.54-16/6.63E-16,respectively.The results demonstrate that the frequencies stability of zero baseline can reach 1E-16 at an averaging time of 86400 s.The results provided by two solutions show a divergence of 0.05 ns .It is verified that IPPP and PPP solution agrees well with each other.CEBR, BRUX-ONSA and BRUX-PTBB links have reached the order of 10^-16 at the averaging time of 61440 s.In the static mode, the improvement of IPPP corresponding to PPP amounts to 3%, 3%, 5% at the averaging time of 122880 s.In the kinematic mode, the improvement of IPPP corresponding to PPP is 4%, 4%, 6% at the averaging time of 122880 s.
In terms of cesium atomic clock , no matter what static or kinematic mode, the stability of IPPP turns more stable than that of PPP after 7680 s.The frequency stabilities of BRUX-DAEJ, BRUX-KIRU, BRUX-REDU and BRUX-SYDN links have reached 10^-15 or 10^-14 at the averaging time of 122880 s.The stability of the cesium atomic clock link is worse than that of the Hydrogen maser.Negligible improvement can be inspected in the cesium atomic clock link under both the static or kinematic modes.

Coordinate experiments
Stations BRUX and PTBB are selected for validating influences of the IPPP on the kinematic coordinates (see figure 12).When the averaging time exceeds 7680s, the stability of the kinematic coordinates has been enhanced gradually.The direction Y can increase 50%.It corresponds to the conclusion of the time links: IPPP is beneficial for the stability of kinematic coordinates in the long term.
As a summary of the tests, IPPP enables a stronger model than that of the PPP solution by reducing the number of the estimable ambiguity parameters and their correlations with other parameters.However, the mismodeled biases have less chance to be absorbed by the original float ambiguities and thus bias the other estimable parameters, like the clocks and the coordinates.During PAR, the number of resolved ambiguities could vary over time, which leads to disturbances in the short-term stability of the estimates clock biases.As such, PAR surely improves the convergence of both the coordinates and the clocks, especially in the kinematic mode.However, under Kalman-filter-based processing as discussed in this study, ambiguity resolution brings only marginal benefits in the long-term stability of the time transfer based on the results presented in this contribution.In contrast, it could degrade the short-term stability due to the different numbers of PAR at each epoch and potentially wrongly fixed ambiguities.

Conclusion and discussion
In this article, ambiguity can be restored from 'float' to 'integer' with the SGG FCB products.The accuracy of SGG FCB products is crucial for ambiguity-fixed solution.As such, the SGG FCB products were evaluated corresponding to the CODE bias product.Next, the ambiguity fixing rate and TTFF were assessed for GPS-only and GPS/Galileo/BDS scenarios, and for stations using H-Masers and cesium atomic clock s.The positional and time transfer performances were also evaluated under different scenarios in terms of stability.Some conclusions can be drawn here: (1) The SGG FCB product has good consistency with the CODE bias products.The average STD of WL FCBs amounts to 0.04 and 0.03 cycles for GPS and Galileo, respectively.The average STD of NL FCBs is about 0.03 and 0.02 cycles for GPS and Galileo, respectively.SGG FCB product is stable enough to support ambiguity fixing.
(2) Performance of the IPPP time transfer perfoms better than that of PPP in the long baseline, i.e., after an averaging time of 7680 s.The improvement of kinematic mode is more obvious than that of the static mode.
In the static mode, the improvement of IPPP corresponding to PPP amounts to 3%, 3%, 5% at the averaging time of 122880 s.In the kinematic mode, the improvement of IPPP corresponding to PPP is 4%, 4%, 6% at the averaging time of 122880 s.
(3) In the long baseline time transfer, the stability of time transfer is strongly related to the performances of the clocks.Hydrogen maser has reached the order of 1E-16/1E-15 at the averaging time of 122880 s, respectively.Cesium clock has reached the order of 1E-15/1E-14 at the averaging time of 122880 s, respectively.
(4) The positioning stability of IPPP turns better than that of PPP after an averaging time of 7680 s, which has the same conclusion with time transfer.
(5) In contrast to GPS, GPS/Galileo/BDS combination can largely shorten the TTFF and enhance the ambiguity fixing success rate.Besides, the TTFF and ambiguity fixing success rate is independent of the type of atomic clocks.Furtherly, the TTFF and ambiguity fixing success rate has seriously correlated with the quality of observation.
(IF) wavelength, ̅ N r IF s , and ̅ N r WL s , are the real-value IF ambiguities.N r IF s , is the integer IF ambiguity.f f 1, 2 represent the frequencies used for IF combination, d d , receiver and satellite code biases, respectively, N r wl s , denotes WL ambiguity, N r s ,1 denotes NL ambiguity.The subscript r and superscript s denotes receiver r and satellite s.

Figure 1 .
Figure 1.Time series of the GPS WL FCBs relative to CODE products from DOY 033 to 039 in 2022.G denotes GPS.

Figure 2 .
Figure 2. Time series of the Galileo WL FCBs noise coefficients are to CODE products from DOY 033 to 039 in 2022.E denotes Galileo.

Figure 3 .
Figure 3.Time series of the NL GPS FCBs relative to CODE from DOY 033 to 039 in 2022.The upper panel denotes the NL FCBs of GPS satellites, the lower panel denotes those of Galileo satellites, respectively.

Figure 4 .
Figure 4. Time series of the NL Galileo FCBs relative to CODE from DOY 033 to 039 in 2022.The upper panel denotes the NL FCBs of GPS satellites, the lower panel denotes those of Galileo satellites, respectively.

Figure 5 .
Figure 5.The PAR flow based on bootstrapping success rate and ratio-test.

Figure 6 .
Figure 6.Distribution of stations involved in the experiment.

Figure 7 .
Figure 7. (a) Fixing rate using GPS observations from stations equipped with cesium atomic clock.(b) Fixing rate using GPS observations from stations equipped with H-Masers.(c) TTFF using GPS observations from stations equipped with cesium atomic clock.(d) TTFF using GPS observations from stations equipped with H-Masers.

Figure 8 .
Figure 8.(a) Fixing rate using Multi-GNSS observations from stations equipped with cesium atomic clock.(b) Fixing rate using Multi-GNSS observations from stations equipped with H-Masers.(c) TTFF using Multi-GNSS observations from stations equipped with cesium atomic clock.(d) TTFF using Multi-GNSS observations from stations equipped with cesium atomic clock.

4. 5 .
Long baseline time transfer experiments Figuring out long baseline PPP time transfer is practical.Seven links are established with station BRUX as the reference station.The length of baseline is ranging from 454. 88 km to 1230 km.Results of IPPP/PPP based on all the links are shown in figures 10 and 11.Whatever in static mode or dynamic mode, it can be observed that the overall trends agree well with each other.Since ALBH only supports GPS/Galileo, the station has been abandoned.The stabilities of the static/ kinematic time links are demonstrated in figures 10 and 11.The results of Hydrogen Maser and cesium atomic clock are exhibited separately in two lines.In terms of Hydrogen Maser, the stability of IPPP is better than that of PPP from an averaging time of 7680 s in both the static and kinematic modes.The frequency stabilities of BRUX-

Figure 10 .
Figure 10.The comparison of stability between the IPPP and PPP in static mode.

Figure 9 .
Figure 9. Zero baseline time transfer using IPPP and PPP solutions.

Figure 11 .
Figure 11.The comparison of stability between IPPP and PPP in kinematic mode.

Figure 12 .
Figure 12.The stability of kinematic coordinates using IPPP and PPP.

Table 1 .
The detailes of stations.

Table 2 .
The average Fixing rate and TTFF of GPS/Multi-GNSS H-Maser/ Cesium atomic clock.

Table 3 .
The improvement in the stabilities of the kinematic coordinates using IPPP over PPP in the middleto long-term.