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Time Synchronization and Performance of BeiDou Satellite Clocks in Orbit

Time Synchronization and Performance of BeiDou Satellite Clocks in Orbit Hindawi Publishing Corporation International Journal of Navigation and Observation Volume 2013, Article ID 371450, 5 pages http://dx.doi.org/10.1155/2013/371450 Research Article Time Synchronization and Performance of BeiDou Satellite Clocks in Orbit Han Chunhao, Cai Zhiwu, Lin Yuting, Liu Li, Xiao Shenghong, Zhu Lingfeng, and Wang Xianglei Beijing Satellite Navigation Center, Beijing 100094, China Correspondence should be addressed to Lin Yuting; lyt1108@163.com Received 24 March 2013; Revised 3 July 2013; Accepted 31 July 2013 Academic Editor: Sandro Radicella Copyright © 2013 Han Chunhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The time model of Beidou satellite clocks is analyzed. eTh general relations of satellite clocks with the system time are studied. The error sources of two-way radio time transfer between satellites and uplink stations are analyzed. The uncertainty of type A is about 0.3 ns in Beidou system. All the satellite clocks in orbit of Beidou satellite navigation system are evaluated by the clock offsets observed by the two-way radio time transfer. eTh frequency stabilities at a sample time of 10000 s and 1 day for all the satellite clocks −13 are better than 1.0 × 10 . It means that the performance of Beidou satellite clocks in orbit is consistent with the ground test, and the results in orbit are a little better than those in ground vacuum. 1. Introduction one DCBFS serial number 1010 (Cs 1010) and two RFS serial numbers 27 and 14 (Rb 27 and Rb 14). The frequency Beidou satellite navigation system began to provide regional stabilities of SVN62 Cs 1010 and Rb 27 are, respectively, 5× service since December 2012. eTh constellation of Beidou −14 −15 10 and 7×10 at 1 day in orbit. And the frequency stability system is constituted of 14 satellites in orbit: 5GEO satellites, −15 of GPS Block IIR Rb is about 9×10 at 1 day for October 5IGSOsatellites, and4MEOsatellites. Table 1 shows the basic 2010 (all using NGA data) [2–4] while the frequency stability information of the Beidou satellites. Service area now covers −14 at 1 day for Galileo satellite clock is about 5×10 for GIOVE ∘ ∘ ∘ ∘ latitude 55 S ∼55 Nand longitude 55 E ∼ 180 E. Practical −15 ARbclocksand 8×10 for GIOVE B PHM [5, 6]. operational accuracy of Beidou system is better than 10 m How about the performance of Beidou clocks in orbits? (95%) in horizontal and 15 m (95%) in vertical [1]. It is a very concerned question for many GNSS users. eTh As we all know that time synchronization of satellite system signals and observations of Beidou regional system clock plays a significant role in satellite navigations, accurate areanalyzedbyDeutsches Zentrumfur Lu-ft undRaumfahrt and reliable satellite clock offset parameters are the base of (DLR) using a local monitoring network in March 2012. PNT service. Time synchronization error of satellite clock is The short-term stability and middle-term stability of Beidou mainly caused by the time transfer from the master station satellite clocks are analyzed and compared to other systems. and its offset prediction. The clock prediction error depends −12 Frequency stability of Beidou RAFS is about 7×10 ∼ on its frequency instability. Then the measurement, predic- −11 1×10 at 1 second, Frequency stability of the best Beidou tion and evaluation, of satellite clocks are very important for −13 satellite clock is about 1×10 at 1000 seconds, and GPS a satellite navigation system. −13 Block IIF is not worse than 1×10 [7]. In the following sec- GPS operates a worldwide monitoring stations network, tions,thetimemodelofsatelliteclocksusedbyBeidousystem and includes six USAF stations, eleven NGA stations, and two IGS stations. Geodetic receivers are equipped in these stations is described; then the error source and uncertainty of the two- to monitor the performance of satellite clocks. On 28 May, way radio time transfer (TWTT) are analyzed, which is used to measure clock differences between satellites and the uplink 2010, the first Block IIF satellite, designated SVN62/PRN25, was launched containing three atomic frequency standards, stations. Finally, the results and conclusions are detailed. 2 International Journal of Navigation and Observation Table 1: Basic information of Beidou satellite clocks. the eccentricity, 𝐸 the real eccentric anomaly, and 𝑐 the speed of light, respectively. eTh n, Num Type (Num) Date 03 GEO-1 2010.1.17 𝑊 −((3/2) (GM /𝑎)) 0 𝐸 𝜏−𝑡 = [ ](𝑡 − 𝑡 ) 04 GEO-3 2010.6.2 05 IGSO-1 2010.8.1 (2) 06 GEO-4 2010.11.1 − √𝑎 GM ⋅𝑒 sin 𝐸 07 IGSO-2 2010.12.18 08 IGSO-3 2011.4.10 If the satellite clock 𝑇(𝑡) is modeled as 09 IGSO-4 2011.7.27 10 IGSO-5 2011.12.2 (3) 𝑇 (𝑡 ) =𝜏 (𝑡 ) +𝑎 +𝑎 (𝑡 − 𝑡 )+𝑎 (𝑡 − 𝑡 ) +𝜉 (𝑡 ) . 0 1 0 2 0 11 GEO-5 2012.2.25 12 MEO-3 2012.4.30 in which 𝑎 , 𝑎 ,and 𝑎 are clock offset parameters and 𝜉(𝑡) is 0 1 2 13 MEO-4 2012.4.30 the clock phase noise, the offset of satellite clock reeff red to BDT can be written as 14 MEO-5 2012.9.19 15 MEO-6 2012.9.19 𝑥 (𝑡 ) ≡𝑇 (𝑡 ) − BDT (𝑡 ) 16 GEO-6 2012.10.25 (4) =𝑎 +𝑎 (𝑡 − 𝑡 )+𝑎 (𝑡 − 𝑡 ) +Δ𝑡 +𝜉 𝑡 ( ) 0 1 0 2 0 grav Here Δ𝑡 is the periodic term of relativistic eeff ct as follows: grav 2. Time Model of Satellite Clocks Considering the large-scale spacetime involved (about 1× ⃗ ⃗ 2 2𝑥 ⋅ 𝑥 𝑝 𝑆 𝑆 (5) 5 Δ𝑡 =− 𝑒 sin 𝐸=− . grav 2 2 10 km in space and several days, even several months or 𝑐 𝑐 years in time) and the precision requirements (1 m, even 1 cm, eTh relativistic eeff ctmustbetaken into accountfor theeval- 1 mm level), the GNSS data process must be dealt with under uation of clock performance. If not, the stability of frequency the framework of relativity and quantum theory. Two kinds will be influenced. The quasi-half-day periodical terms in of conceptually different time scales are concerned in GNSS, Allan deviations of GPS clocks and Galileo clocks [2, 5], we proper times, and coordinate times. Essentially, for any two guess, maybecausedbythisterm. events, the observed space interval and time interval between them are dependent on the observer. eTh time readings givendirectlybyideal clocks locatedinsatellites, stations,or 3. Two-Way Satellite Time observers are proper times. eTh y are related to the observer, Transfer and Error Analysis or to the spacetime environments of the clocks. This means that different observer has different clock due to its relative In Beidou system, TWTT between satellites and uplink sta- velocity and position in the gravitation efi ld. In order to have tions is used for the satellitetime synchronization. eTh basic a common time reference for all observers, we must choose a principle of TWTT is as follows. The satellites and stations special observer and construct a reference system. A reference generate and transmit pseudo-range signals controlled by system contains a 3-dimensional space reference frame and their local clocks; then the uplink pseudo-range 𝜌 and a time reference. eTh former determines the spatial position downlink pseudo-range 𝜌 are measured by the satellites (3 space coordinates) of an event and the latter gives the and the stations, respectively. The uplink pseudo-range and happening time, which is called coordinate time. For Earth downlink pseudo-range can be written as satellites, a nonrotating geocentric reference system is used to describe their orbits. eTh reference time is usually TCG (the 󵄨 󵄨 𝑟 𝑟 𝑒 𝑒 𝑟 󵄨 󵄨 𝜌 (𝑇 )= 𝑥 ⃗ (𝑡 )− 𝑥 ⃗ (𝑡 ) ⋅ −Δ𝑇 (𝑡 )+Δ𝑇 (𝑡 ) 󵄨 󵄨 𝑢 𝑆 𝑆 𝑆 𝑅 𝑅 𝑅 𝑅 𝑆 𝑆 󵄨 󵄨 geocentric coordinate time) or TT (the terrestrial time) [8, 9]. eTh relationship between the proper time 𝜏 of satellite and 𝑒 𝑟 +𝜏 +𝜏 +𝜏 +𝜏 (𝑓 )+𝜏 , thecoordinatetimeTT(here notedby 𝑡 ) can be modeled as 𝑅 𝑆 tro ion 𝑢 grav (6) [8] 󵄨 󵄨 𝑟 𝑟 𝑒 𝑟 𝑒 󵄨 󵄨 ⃗ ⃗ 𝜌 (𝑇 )= 󵄨 𝑥 (𝑡 )− 𝑥 (𝑡 )󵄨 ⋅ +Δ𝑇 (𝑡 )−Δ𝑇 (𝑡 ) 𝑑 𝑅 𝑅 𝑅 𝑆 𝑆 𝑅 𝑅 𝑆 𝑆 󵄨 󵄨 (𝑊 − (3/2) (𝜇/𝑎)) 𝑟 𝑒 𝑡 = [1 − ](𝜏 − 𝜏 ) +𝜏 +𝜏 +𝜏 +𝜏 (𝑓 )+𝜏 , tro ion 𝑑 grav 2 𝑅 𝑆 (1) 𝑟 𝑒 where 𝑡 and 𝑡 are time of reception and emission of the 𝑆 𝑆 𝑟 𝑒 + √ ⋅ 𝑒 ( sin 𝐸− sin 𝐸 ), satellite signal; 𝑡 and 𝑡 are time of reception and emission of 𝑅 𝑅 the station signal; Δ𝑇 and Δ𝑇 are satellite and station’s clock 𝑆 𝑅 𝑟 𝑒 offset; 𝜏 and 𝜏 are time delay of reception and emission of 𝑅 𝑅 𝑒 𝑟 where 𝜇= GM is the geocentric gravitation constant, 𝑊 the station equipment; 𝜏 and 𝜏 are time delay of reception 𝐸 0 𝑆 𝑆 the gravity potential of the geoid, 𝑎 the orbit main axis, 𝑒 and emission of the satellite equipment; 𝜏 and 𝜏 are tro ion 𝜇𝑎 𝜇𝑎 International Journal of Navigation and Observation 3 troposphere delay and ionosphere delay; 𝑓 and 𝑓 are uplink 𝑢 𝑑 Residual errors of GEO-3 satellite from frequency and downlink frequency; 𝜏 relativistic time January 1, 2012 to February 21, 2012 grav delaycausedbyEarth gravitation. The clock differences between satellites and stations are computed in the master station by using the uplink pseudo- ranges and the downlink pseudo-ranges. The satellite clock offset can be given by the observed uplink pseudo-range and downlink pseudo-range as follows: 0 0 𝑖 𝑖 Δ𝑇 (𝑡 )= Δ𝑇 (𝑡 )+ 𝑆 𝑅 −100 −2000 𝑖 𝑖 Second-order polynomial: ×{[𝜌 (𝑇 )−𝜌 (𝑇 )] 𝑢 𝑆 𝑑 𝑅 −9.0644e − 010t + 0.016545t − 924685.4626 (7) −200 −4000 ̇ ̇ 0 10 20 30 40 50 60 ⃗ ⃗ ⃗ − (𝑥 − 𝑥 )⋅ 𝑛 (Δ𝑇 −Δ𝑇 −𝜏 ) 𝑆 𝑅 𝑆 𝑅 Time (days) +Δ𝜏 −Δ𝜏 −Δ𝜏 }+⋅ ⋅ ⋅ , 𝑅 𝑆 ion Figure 1: Residual of GEO-3 satellite clock. where 𝑟 𝑒 Δ𝜏 ≡𝜏 −𝜏 , 𝑅 𝑅 𝑅 Residual errors of GEO-4 satellite from January 15, 2012 to February 14, 2012 𝑟 𝑒 Δ𝜏 ≡𝜏 −𝜏 , 1000 20 𝑆 𝑆 (8) Δ𝜏 ≡𝜏 (𝑓 )−𝜏 (𝑓 ), ion ion 𝑢 ion 𝑑 500 10 (𝑥 ⃗ − 𝑥 ⃗ ) 𝑆 𝑅 𝑛 ⃗ ≡ . 0 0 󵄨 󵄨 󵄨 󵄨 𝑥 ⃗ − 𝑥 ⃗ 󵄨 󵄨 󵄨 𝑆 𝑅 󵄨 The random error of satellite clock difference includes the −500 −10 noiseofpseudo-rangeobservableandthesatelliteclockphase noise. In short term (≤1000 s), the influence of the frequency −1000 −20 drift and phase noise of satellite clock to clock offset can be Second-order polynomial: neglected. So the uncertainty of type A of satellite-board clock 1.058t + 0.12814t + 474545.7275 −1500 −30 offset measurement can be calculated by the fluctuation of 2205 2210 2215 2220 2225 2230 2235 2240 clock difference. Analysis shows that the uncertainty of type Time (days) Aislessthan0.3ns [10]. In middle or long term (≥10000 s), the influence of the pseudo-range noise can be neglected, and Figure 2: Residual of GEO-4 satellite clock. theresults of theAllan variance of satelliteclocksare reliable. 4. Performance Evaluation of Residual errors of IGSO-2 satellite from Beidou Satellite Clocks in Orbit January 13, 2012 to March 31, 2012 400 2000 Satellites that include GEO satellites of serial number 03, 04, 300 1000 06, and 11, IGSO satellites of serial number 07, 08, 09 and 10, and MEO satellites of serial number 13 and 14 are evaluated. In 200 0 order to ensure the reliability of the evaluation result, the time interval of satellite clock data is no less than 15 days. eTh time 100 −1000 scale reference for analysis is the high performance hydrogen clock in ground. 0 −2000 Figures 1, 2,and 3 show the linear residuals and second- order polynomial residuals of the observed satellite clock off- −100 −3000 Second-order polynomial: sets. −4.2549e − 010t + 0.010565t − 124063.5676 The green curves are plots of the linear residuals of satel- −200 −4000 lite clocks. All of the linear residual of GEO-3, GEO-4, and 0 10 20 30 40 50 60 70 80 IGSO-2 are smooth, which mean that the rubidium clocks Time (days) have significant frequency drifts. The blue curves are the second-order polynomial residuals of satellite clocks, which Figure 3: Residual of IGSO-2 satellite clock. Second-order polynomial residuals (ns) Second-order polynomial residuals (ns) Second-order polynomial residuals (ns) Linear residuals (ns) Linear residuals (ns) Linear residuals (ns) 𝑅𝑆 𝑅𝑆 𝑅𝑆 4 International Journal of Navigation and Observation Table 2: Frequency stability of Beidou system satellite clocks. GEO-1 GEO-3 GEO-4 GEO-5 IGSO-1 IGSO-2 IGSO-3 IGSO-4 IGSO-5 Stability −14 −14 −14 −14 −14 −14 −14 −14 −14 7.31 × 10 5.52 × 10 7.58 × 10 9.17 × 10 8.13 × 10 5.95 × 10 7.94 × 10 8.53 × 10 8.98 × 10 (10000 s) Stability −14 −14 −14 −14 −14 −14 −14 −14 −14 6.71 × 10 2.90 × 10 3.83 × 10 5.66 × 10 9.38 × 10 3.07 × 10 2.53 × 10 3.91 × 10 4.45 × 10 (1 day) −10 Allan stability −11 −13 −12 𝜎 −13 −14 −15 0 1 2 3 4 5 6 10 10 10 10 10 10 10 1 2 3 4 5 6 7 8 9 Averaging time (s) In orbit IGSO-3 GEO-1 In ground vacuum GEO-3 IGSO-4 IGSO-5 GEO-4 Figure 5: Clock day stabilities in orbit and in ground vacuum pots. GEO-5 MEO-3 IGSO-1 MEO-4 IGSO-2 show that the performance of satellite clock is steady and in Figure 4: Frequency stability of Beidou system satellite clocks. good condition. The frequency stabilities at a sample time of 10000 s and 1 day for all the satellite clocks are better than 1.0× −13 10 . It means that the performance of Beidou satellite clocks in orbit is consistent with the ground test, and the results in demonstrate that the frequency drasft are changing slowly orbitare alittlebetterthanthose in ground vacuum. and the rubidium clocks in orbit have high-level noise char- acteristic, such as flick and random walk. The frequency stability of Beidou satellite clocks is evalu- Acknowledgments ated by use of the overlapping Allan deviation. Figure 4 shows plots of the frequency stability of Beidou system satellite eTh authors wish to thank the Editor Sandro M. Radicella, the clocks. Table 2 shows the frequency stability at a sample time Editorial Assistant Ms. Joanna, and the anonymous reviewers of 10000 seconds and 1 day. whosecommentshelpedimprove this paperenormously. The frequency stability of Beidou satellite clocks is of the −14 level of 10 at a sample time of 10000 seconds and 1 day. eTh References frequency stability at a sample time of 10000 seconds is about −14 5.95 ∼ 9.17 × 10 , and that at a sample time of 1 day is about [1] H. Qiaohua, “Development of Beidou navigation satellite sys- −14 tem,” in Proceedings of the 5th Meeting of International Commit- 2.53 ∼ 9.38 × 10 . tee on GNSS (ICG-5 ’12), Beijing, China, 2012. Figure 5 gives the comparison of the clock performances [2] F. Vannicola, R. Beard, J. White, and K. Senior, “GPS Block IIF in orbit and in the ground vacuum. The results show that the atomic frequency standard analysis,” in Proceedings of the 42th performances in orbits are conformable with those in ground. Annual Precise Time and Time Interval (PTTI) Meeting,pp. 181– As a whole, the results in orbit are a little better than those in 196, 2010. ground. [3] J. Oaks, J. A. Buisson, and M. M. Largay, “A summary of the GPS constellation clock performance,” in Proceedings of the 39th Annual Precise Time and Time Interval (PTTI) Meeting,pp. 119– 5. Conclusion 130, 2007. The long-term evaluation for Beidou satellite clocks has been [4] D. M. Manning and C. P. Petersen, “AF/NGA GPS monitor done using TWTT between satellites and stations. The results station high-performance cesium frequency standard stability Allan frequency stability (𝜏) Allan frequency stability (86400) y International Journal of Navigation and Observation 5 2007/2008: from NGA kalman filter clock estimates,” in Pro- ceedings of the 40th Annual Precise Time and Time Interval (PTTI) Meeting,pp. 335–348, 2008. [5] P. Waller, F. Gonzalez, S. Binda et al., “eTh in-orbit performan- ces of GIOVE clocks,” IEEE Transactions on Ultrasonics, Ferro- electrics, and Frequency Control,vol.57, no.3,pp. 738–745, 2010. [6] P. Waller, F. Gonzalez, and S. Binda, “Long-term performance analysis of giove clocks,” in Proceedings of the 42th Annual Pre- cise Time and Time Interval (PTTI) Meeting, pp. 171–180, 2010. [7] O. Montenbruck, A. Hauschild, P. Steigenberger, U. Hugento- bler, P. Teunissen, and S. Nakamura, “Initial assessment of the compass/beidou-2 regional navigation satellite system,” GPS Solutions,vol.17, no.2,pp. 211–222,2013. [8] H. Chunhao, “Time measurement within the frame of relativ- ity,” Progress in Astronomy,vol.20, no.2,pp. 107–113,2002. [9] H. Chunhao, C. Zhiwu, L. Yuting, L. Li et al., “Time synchroni- zation and performance evaluation of beidou satellite clocks,” in Proceedings of the 3rd China Satellite Navigation Conference, [10] L. Liu, L.-F. Zhu, C.-H. Han, X.-P. Liu, and C. Li, “The model of radio two-way time comparison between satellite and station and experimental analysis,” Chinese Astronomy and Astrophys- ics,vol.33, no.4,pp. 431–439, 2009. 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Time Synchronization and Performance of BeiDou Satellite Clocks in Orbit

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Hindawi Publishing Corporation International Journal of Navigation and Observation Volume 2013, Article ID 371450, 5 pages http://dx.doi.org/10.1155/2013/371450 Research Article Time Synchronization and Performance of BeiDou Satellite Clocks in Orbit Han Chunhao, Cai Zhiwu, Lin Yuting, Liu Li, Xiao Shenghong, Zhu Lingfeng, and Wang Xianglei Beijing Satellite Navigation Center, Beijing 100094, China Correspondence should be addressed to Lin Yuting; lyt1108@163.com Received 24 March 2013; Revised 3 July 2013; Accepted 31 July 2013 Academic Editor: Sandro Radicella Copyright © 2013 Han Chunhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The time model of Beidou satellite clocks is analyzed. eTh general relations of satellite clocks with the system time are studied. The error sources of two-way radio time transfer between satellites and uplink stations are analyzed. The uncertainty of type A is about 0.3 ns in Beidou system. All the satellite clocks in orbit of Beidou satellite navigation system are evaluated by the clock offsets observed by the two-way radio time transfer. eTh frequency stabilities at a sample time of 10000 s and 1 day for all the satellite clocks −13 are better than 1.0 × 10 . It means that the performance of Beidou satellite clocks in orbit is consistent with the ground test, and the results in orbit are a little better than those in ground vacuum. 1. Introduction one DCBFS serial number 1010 (Cs 1010) and two RFS serial numbers 27 and 14 (Rb 27 and Rb 14). The frequency Beidou satellite navigation system began to provide regional stabilities of SVN62 Cs 1010 and Rb 27 are, respectively, 5× service since December 2012. eTh constellation of Beidou −14 −15 10 and 7×10 at 1 day in orbit. And the frequency stability system is constituted of 14 satellites in orbit: 5GEO satellites, −15 of GPS Block IIR Rb is about 9×10 at 1 day for October 5IGSOsatellites, and4MEOsatellites. Table 1 shows the basic 2010 (all using NGA data) [2–4] while the frequency stability information of the Beidou satellites. Service area now covers −14 at 1 day for Galileo satellite clock is about 5×10 for GIOVE ∘ ∘ ∘ ∘ latitude 55 S ∼55 Nand longitude 55 E ∼ 180 E. Practical −15 ARbclocksand 8×10 for GIOVE B PHM [5, 6]. operational accuracy of Beidou system is better than 10 m How about the performance of Beidou clocks in orbits? (95%) in horizontal and 15 m (95%) in vertical [1]. It is a very concerned question for many GNSS users. eTh As we all know that time synchronization of satellite system signals and observations of Beidou regional system clock plays a significant role in satellite navigations, accurate areanalyzedbyDeutsches Zentrumfur Lu-ft undRaumfahrt and reliable satellite clock offset parameters are the base of (DLR) using a local monitoring network in March 2012. PNT service. Time synchronization error of satellite clock is The short-term stability and middle-term stability of Beidou mainly caused by the time transfer from the master station satellite clocks are analyzed and compared to other systems. and its offset prediction. The clock prediction error depends −12 Frequency stability of Beidou RAFS is about 7×10 ∼ on its frequency instability. Then the measurement, predic- −11 1×10 at 1 second, Frequency stability of the best Beidou tion and evaluation, of satellite clocks are very important for −13 satellite clock is about 1×10 at 1000 seconds, and GPS a satellite navigation system. −13 Block IIF is not worse than 1×10 [7]. In the following sec- GPS operates a worldwide monitoring stations network, tions,thetimemodelofsatelliteclocksusedbyBeidousystem and includes six USAF stations, eleven NGA stations, and two IGS stations. Geodetic receivers are equipped in these stations is described; then the error source and uncertainty of the two- to monitor the performance of satellite clocks. On 28 May, way radio time transfer (TWTT) are analyzed, which is used to measure clock differences between satellites and the uplink 2010, the first Block IIF satellite, designated SVN62/PRN25, was launched containing three atomic frequency standards, stations. Finally, the results and conclusions are detailed. 2 International Journal of Navigation and Observation Table 1: Basic information of Beidou satellite clocks. the eccentricity, 𝐸 the real eccentric anomaly, and 𝑐 the speed of light, respectively. eTh n, Num Type (Num) Date 03 GEO-1 2010.1.17 𝑊 −((3/2) (GM /𝑎)) 0 𝐸 𝜏−𝑡 = [ ](𝑡 − 𝑡 ) 04 GEO-3 2010.6.2 05 IGSO-1 2010.8.1 (2) 06 GEO-4 2010.11.1 − √𝑎 GM ⋅𝑒 sin 𝐸 07 IGSO-2 2010.12.18 08 IGSO-3 2011.4.10 If the satellite clock 𝑇(𝑡) is modeled as 09 IGSO-4 2011.7.27 10 IGSO-5 2011.12.2 (3) 𝑇 (𝑡 ) =𝜏 (𝑡 ) +𝑎 +𝑎 (𝑡 − 𝑡 )+𝑎 (𝑡 − 𝑡 ) +𝜉 (𝑡 ) . 0 1 0 2 0 11 GEO-5 2012.2.25 12 MEO-3 2012.4.30 in which 𝑎 , 𝑎 ,and 𝑎 are clock offset parameters and 𝜉(𝑡) is 0 1 2 13 MEO-4 2012.4.30 the clock phase noise, the offset of satellite clock reeff red to BDT can be written as 14 MEO-5 2012.9.19 15 MEO-6 2012.9.19 𝑥 (𝑡 ) ≡𝑇 (𝑡 ) − BDT (𝑡 ) 16 GEO-6 2012.10.25 (4) =𝑎 +𝑎 (𝑡 − 𝑡 )+𝑎 (𝑡 − 𝑡 ) +Δ𝑡 +𝜉 𝑡 ( ) 0 1 0 2 0 grav Here Δ𝑡 is the periodic term of relativistic eeff ct as follows: grav 2. Time Model of Satellite Clocks Considering the large-scale spacetime involved (about 1× ⃗ ⃗ 2 2𝑥 ⋅ 𝑥 𝑝 𝑆 𝑆 (5) 5 Δ𝑡 =− 𝑒 sin 𝐸=− . grav 2 2 10 km in space and several days, even several months or 𝑐 𝑐 years in time) and the precision requirements (1 m, even 1 cm, eTh relativistic eeff ctmustbetaken into accountfor theeval- 1 mm level), the GNSS data process must be dealt with under uation of clock performance. If not, the stability of frequency the framework of relativity and quantum theory. Two kinds will be influenced. The quasi-half-day periodical terms in of conceptually different time scales are concerned in GNSS, Allan deviations of GPS clocks and Galileo clocks [2, 5], we proper times, and coordinate times. Essentially, for any two guess, maybecausedbythisterm. events, the observed space interval and time interval between them are dependent on the observer. eTh time readings givendirectlybyideal clocks locatedinsatellites, stations,or 3. Two-Way Satellite Time observers are proper times. eTh y are related to the observer, Transfer and Error Analysis or to the spacetime environments of the clocks. This means that different observer has different clock due to its relative In Beidou system, TWTT between satellites and uplink sta- velocity and position in the gravitation efi ld. In order to have tions is used for the satellitetime synchronization. eTh basic a common time reference for all observers, we must choose a principle of TWTT is as follows. The satellites and stations special observer and construct a reference system. A reference generate and transmit pseudo-range signals controlled by system contains a 3-dimensional space reference frame and their local clocks; then the uplink pseudo-range 𝜌 and a time reference. eTh former determines the spatial position downlink pseudo-range 𝜌 are measured by the satellites (3 space coordinates) of an event and the latter gives the and the stations, respectively. The uplink pseudo-range and happening time, which is called coordinate time. For Earth downlink pseudo-range can be written as satellites, a nonrotating geocentric reference system is used to describe their orbits. eTh reference time is usually TCG (the 󵄨 󵄨 𝑟 𝑟 𝑒 𝑒 𝑟 󵄨 󵄨 𝜌 (𝑇 )= 𝑥 ⃗ (𝑡 )− 𝑥 ⃗ (𝑡 ) ⋅ −Δ𝑇 (𝑡 )+Δ𝑇 (𝑡 ) 󵄨 󵄨 𝑢 𝑆 𝑆 𝑆 𝑅 𝑅 𝑅 𝑅 𝑆 𝑆 󵄨 󵄨 geocentric coordinate time) or TT (the terrestrial time) [8, 9]. eTh relationship between the proper time 𝜏 of satellite and 𝑒 𝑟 +𝜏 +𝜏 +𝜏 +𝜏 (𝑓 )+𝜏 , thecoordinatetimeTT(here notedby 𝑡 ) can be modeled as 𝑅 𝑆 tro ion 𝑢 grav (6) [8] 󵄨 󵄨 𝑟 𝑟 𝑒 𝑟 𝑒 󵄨 󵄨 ⃗ ⃗ 𝜌 (𝑇 )= 󵄨 𝑥 (𝑡 )− 𝑥 (𝑡 )󵄨 ⋅ +Δ𝑇 (𝑡 )−Δ𝑇 (𝑡 ) 𝑑 𝑅 𝑅 𝑅 𝑆 𝑆 𝑅 𝑅 𝑆 𝑆 󵄨 󵄨 (𝑊 − (3/2) (𝜇/𝑎)) 𝑟 𝑒 𝑡 = [1 − ](𝜏 − 𝜏 ) +𝜏 +𝜏 +𝜏 +𝜏 (𝑓 )+𝜏 , tro ion 𝑑 grav 2 𝑅 𝑆 (1) 𝑟 𝑒 where 𝑡 and 𝑡 are time of reception and emission of the 𝑆 𝑆 𝑟 𝑒 + √ ⋅ 𝑒 ( sin 𝐸− sin 𝐸 ), satellite signal; 𝑡 and 𝑡 are time of reception and emission of 𝑅 𝑅 the station signal; Δ𝑇 and Δ𝑇 are satellite and station’s clock 𝑆 𝑅 𝑟 𝑒 offset; 𝜏 and 𝜏 are time delay of reception and emission of 𝑅 𝑅 𝑒 𝑟 where 𝜇= GM is the geocentric gravitation constant, 𝑊 the station equipment; 𝜏 and 𝜏 are time delay of reception 𝐸 0 𝑆 𝑆 the gravity potential of the geoid, 𝑎 the orbit main axis, 𝑒 and emission of the satellite equipment; 𝜏 and 𝜏 are tro ion 𝜇𝑎 𝜇𝑎 International Journal of Navigation and Observation 3 troposphere delay and ionosphere delay; 𝑓 and 𝑓 are uplink 𝑢 𝑑 Residual errors of GEO-3 satellite from frequency and downlink frequency; 𝜏 relativistic time January 1, 2012 to February 21, 2012 grav delaycausedbyEarth gravitation. The clock differences between satellites and stations are computed in the master station by using the uplink pseudo- ranges and the downlink pseudo-ranges. The satellite clock offset can be given by the observed uplink pseudo-range and downlink pseudo-range as follows: 0 0 𝑖 𝑖 Δ𝑇 (𝑡 )= Δ𝑇 (𝑡 )+ 𝑆 𝑅 −100 −2000 𝑖 𝑖 Second-order polynomial: ×{[𝜌 (𝑇 )−𝜌 (𝑇 )] 𝑢 𝑆 𝑑 𝑅 −9.0644e − 010t + 0.016545t − 924685.4626 (7) −200 −4000 ̇ ̇ 0 10 20 30 40 50 60 ⃗ ⃗ ⃗ − (𝑥 − 𝑥 )⋅ 𝑛 (Δ𝑇 −Δ𝑇 −𝜏 ) 𝑆 𝑅 𝑆 𝑅 Time (days) +Δ𝜏 −Δ𝜏 −Δ𝜏 }+⋅ ⋅ ⋅ , 𝑅 𝑆 ion Figure 1: Residual of GEO-3 satellite clock. where 𝑟 𝑒 Δ𝜏 ≡𝜏 −𝜏 , 𝑅 𝑅 𝑅 Residual errors of GEO-4 satellite from January 15, 2012 to February 14, 2012 𝑟 𝑒 Δ𝜏 ≡𝜏 −𝜏 , 1000 20 𝑆 𝑆 (8) Δ𝜏 ≡𝜏 (𝑓 )−𝜏 (𝑓 ), ion ion 𝑢 ion 𝑑 500 10 (𝑥 ⃗ − 𝑥 ⃗ ) 𝑆 𝑅 𝑛 ⃗ ≡ . 0 0 󵄨 󵄨 󵄨 󵄨 𝑥 ⃗ − 𝑥 ⃗ 󵄨 󵄨 󵄨 𝑆 𝑅 󵄨 The random error of satellite clock difference includes the −500 −10 noiseofpseudo-rangeobservableandthesatelliteclockphase noise. In short term (≤1000 s), the influence of the frequency −1000 −20 drift and phase noise of satellite clock to clock offset can be Second-order polynomial: neglected. So the uncertainty of type A of satellite-board clock 1.058t + 0.12814t + 474545.7275 −1500 −30 offset measurement can be calculated by the fluctuation of 2205 2210 2215 2220 2225 2230 2235 2240 clock difference. Analysis shows that the uncertainty of type Time (days) Aislessthan0.3ns [10]. In middle or long term (≥10000 s), the influence of the pseudo-range noise can be neglected, and Figure 2: Residual of GEO-4 satellite clock. theresults of theAllan variance of satelliteclocksare reliable. 4. Performance Evaluation of Residual errors of IGSO-2 satellite from Beidou Satellite Clocks in Orbit January 13, 2012 to March 31, 2012 400 2000 Satellites that include GEO satellites of serial number 03, 04, 300 1000 06, and 11, IGSO satellites of serial number 07, 08, 09 and 10, and MEO satellites of serial number 13 and 14 are evaluated. In 200 0 order to ensure the reliability of the evaluation result, the time interval of satellite clock data is no less than 15 days. eTh time 100 −1000 scale reference for analysis is the high performance hydrogen clock in ground. 0 −2000 Figures 1, 2,and 3 show the linear residuals and second- order polynomial residuals of the observed satellite clock off- −100 −3000 Second-order polynomial: sets. −4.2549e − 010t + 0.010565t − 124063.5676 The green curves are plots of the linear residuals of satel- −200 −4000 lite clocks. All of the linear residual of GEO-3, GEO-4, and 0 10 20 30 40 50 60 70 80 IGSO-2 are smooth, which mean that the rubidium clocks Time (days) have significant frequency drifts. The blue curves are the second-order polynomial residuals of satellite clocks, which Figure 3: Residual of IGSO-2 satellite clock. Second-order polynomial residuals (ns) Second-order polynomial residuals (ns) Second-order polynomial residuals (ns) Linear residuals (ns) Linear residuals (ns) Linear residuals (ns) 𝑅𝑆 𝑅𝑆 𝑅𝑆 4 International Journal of Navigation and Observation Table 2: Frequency stability of Beidou system satellite clocks. GEO-1 GEO-3 GEO-4 GEO-5 IGSO-1 IGSO-2 IGSO-3 IGSO-4 IGSO-5 Stability −14 −14 −14 −14 −14 −14 −14 −14 −14 7.31 × 10 5.52 × 10 7.58 × 10 9.17 × 10 8.13 × 10 5.95 × 10 7.94 × 10 8.53 × 10 8.98 × 10 (10000 s) Stability −14 −14 −14 −14 −14 −14 −14 −14 −14 6.71 × 10 2.90 × 10 3.83 × 10 5.66 × 10 9.38 × 10 3.07 × 10 2.53 × 10 3.91 × 10 4.45 × 10 (1 day) −10 Allan stability −11 −13 −12 𝜎 −13 −14 −15 0 1 2 3 4 5 6 10 10 10 10 10 10 10 1 2 3 4 5 6 7 8 9 Averaging time (s) In orbit IGSO-3 GEO-1 In ground vacuum GEO-3 IGSO-4 IGSO-5 GEO-4 Figure 5: Clock day stabilities in orbit and in ground vacuum pots. GEO-5 MEO-3 IGSO-1 MEO-4 IGSO-2 show that the performance of satellite clock is steady and in Figure 4: Frequency stability of Beidou system satellite clocks. good condition. The frequency stabilities at a sample time of 10000 s and 1 day for all the satellite clocks are better than 1.0× −13 10 . It means that the performance of Beidou satellite clocks in orbit is consistent with the ground test, and the results in demonstrate that the frequency drasft are changing slowly orbitare alittlebetterthanthose in ground vacuum. and the rubidium clocks in orbit have high-level noise char- acteristic, such as flick and random walk. The frequency stability of Beidou satellite clocks is evalu- Acknowledgments ated by use of the overlapping Allan deviation. Figure 4 shows plots of the frequency stability of Beidou system satellite eTh authors wish to thank the Editor Sandro M. Radicella, the clocks. Table 2 shows the frequency stability at a sample time Editorial Assistant Ms. Joanna, and the anonymous reviewers of 10000 seconds and 1 day. whosecommentshelpedimprove this paperenormously. The frequency stability of Beidou satellite clocks is of the −14 level of 10 at a sample time of 10000 seconds and 1 day. eTh References frequency stability at a sample time of 10000 seconds is about −14 5.95 ∼ 9.17 × 10 , and that at a sample time of 1 day is about [1] H. Qiaohua, “Development of Beidou navigation satellite sys- −14 tem,” in Proceedings of the 5th Meeting of International Commit- 2.53 ∼ 9.38 × 10 . tee on GNSS (ICG-5 ’12), Beijing, China, 2012. Figure 5 gives the comparison of the clock performances [2] F. Vannicola, R. Beard, J. White, and K. Senior, “GPS Block IIF in orbit and in the ground vacuum. The results show that the atomic frequency standard analysis,” in Proceedings of the 42th performances in orbits are conformable with those in ground. Annual Precise Time and Time Interval (PTTI) Meeting,pp. 181– As a whole, the results in orbit are a little better than those in 196, 2010. ground. [3] J. Oaks, J. A. Buisson, and M. M. Largay, “A summary of the GPS constellation clock performance,” in Proceedings of the 39th Annual Precise Time and Time Interval (PTTI) Meeting,pp. 119– 5. Conclusion 130, 2007. The long-term evaluation for Beidou satellite clocks has been [4] D. M. Manning and C. P. Petersen, “AF/NGA GPS monitor done using TWTT between satellites and stations. The results station high-performance cesium frequency standard stability Allan frequency stability (𝜏) Allan frequency stability (86400) y International Journal of Navigation and Observation 5 2007/2008: from NGA kalman filter clock estimates,” in Pro- ceedings of the 40th Annual Precise Time and Time Interval (PTTI) Meeting,pp. 335–348, 2008. [5] P. Waller, F. Gonzalez, S. Binda et al., “eTh in-orbit performan- ces of GIOVE clocks,” IEEE Transactions on Ultrasonics, Ferro- electrics, and Frequency Control,vol.57, no.3,pp. 738–745, 2010. [6] P. Waller, F. Gonzalez, and S. Binda, “Long-term performance analysis of giove clocks,” in Proceedings of the 42th Annual Pre- cise Time and Time Interval (PTTI) Meeting, pp. 171–180, 2010. [7] O. Montenbruck, A. Hauschild, P. Steigenberger, U. Hugento- bler, P. Teunissen, and S. Nakamura, “Initial assessment of the compass/beidou-2 regional navigation satellite system,” GPS Solutions,vol.17, no.2,pp. 211–222,2013. [8] H. Chunhao, “Time measurement within the frame of relativ- ity,” Progress in Astronomy,vol.20, no.2,pp. 107–113,2002. [9] H. Chunhao, C. Zhiwu, L. Yuting, L. Li et al., “Time synchroni- zation and performance evaluation of beidou satellite clocks,” in Proceedings of the 3rd China Satellite Navigation Conference, [10] L. Liu, L.-F. Zhu, C.-H. Han, X.-P. Liu, and C. Li, “The model of radio two-way time comparison between satellite and station and experimental analysis,” Chinese Astronomy and Astrophys- ics,vol.33, no.4,pp. 431–439, 2009. 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