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Accounting BDS3–BDS2 inter-system biases for precise time transfer
Measurement 156 (2020) 107566
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Accounting BDS3–BDS2 inter-system biases for precise time transfer Weijin Qin a,b,c,⇑, Yulong Ge a,b,c, Zhe Zhang a,b,c, Hang Su a,b,c, Pei Wei a,b,c, Xuhai Yang a,b,d a
National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China Key Laboratory of Precise Positioning and Timing Technology, Chinese Academy of Sciences, Xi’an 710600, China c University of Chinese Academy of Sciences, Beijing 100049, China d School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China b
a r t i c l e
i n f o
Article history: Received 26 December 2019 Received in revised form 16 January 2020 Accepted 29 January 2020 Available online 7 February 2020 Keywords: BDS-3 Time delay bias (TDB) Stochastic modeling Precise point positioning (PPP) Time transfer
a b s t r a c t The rapid development of BDS not only brings opportunities but also new problems. This article has been verified that the biases exist between BDS-2 and BDS-3 precise point positioning time transfer. We defined internal biases of BDS as Time Delay Bias (TDB) for a distinguish with inter system biases (ISB) exists in different systems. Motivated by the bias, we make comprehensive analysis for determining the characteristics of TDB referring to all the receivers capable of BDS-2 along with BDS-3 signals, furtherly, three kinds of TDB stochastic model are designed for parameter estimation. Later, the influence on time transfer with different TDB estimation strategy is compared in the end. We have some findings: (1) It is confirmed that TDB indeed exists in JAVAD TRE_3, CETC-54-GMR-4016, GNSS GGR and UB4B013478 receivers. Without external frequency source, the characteristics of TDB is related to the receiver itself. With the same external frequency source, TDB tendency of two receivers show the same evolution. (2) It is concluded that the STD (Standard Deviation) and stability of TDB estimated with random walk model and constant model is superior to white noise model because of strong correlation among TDB time series. (3) It is verified that the stability of time transfer estimated as random walk model and constant model is smaller than that of white noise model, what is more, when the two sides were receivers of different brands, the significant deviation exists in the clock offset with TDB estimation and without TDB estimation. When the two sides are receiver of the same brand, marginal deviation exists in the clock offset with TDB estimation and without TDB estimation. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction The appearance of Global Navigation Satellite Systems (GNSS) allows worldwide positioning &navigation &timing with unprecedented precision and accuracy. In the current status, there are four fully operational GNSS, namely the United States Global Positioning System (GPS), the Russian Globalnaja Nawigazionnaja Sputnikowaja Sistema (GLONASS), the European Galileo positioning system (Galileo) and the Chinese BeiDou navigation System (BDS). According to the actual condition, three steps of construction are employed by BDS, including the demonstration system (BDS-1), the regional system (BDS-2) and the global system (BDS-3) [1]. The BDS-1 is out of date, while BDS-2 is still active with its superior performance. Hence, an interesting phenomenon appears that BDS-2 and BDS-3 will both contribute to serve users located in the everywhere in future [2].
⇑ Corresponding author at: National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China. E-mail address: [email protected] (W. Qin). https://doi.org/10.1016/j.measurement.2020.107566 0263-2241/Ó 2020 Elsevier Ltd. All rights reserved.
Joint BDS-2 and BDS-3 precise point positioning (PPP) can enhance the positioning accuracy and reduce the convergence time [3]. However, the new confusion came out that the non-negligible existed discrepancy between BDS-2 and BDS-3, which attributes to the signal modulation mode and signal characteristics [4,5]. Jiao et al. [6] mentioned that a bias named TDB (Time Delay Bias) at the receiver end was caused by receiving or processing BDS-3 and BDS-2 signals with different unit. The bias should be considered in the observation equation. To my knowledge, plenty of studies focus their attention on the inter-system biases (ISBs) of BDS, GLONASS, Galileo relative to GPS. Paziewski et al. [7] investigated a methodology of accounting for GPS-Galileo receiver intersystem biases in precise relative positioning. Gao et al. [8] designed the approach that combined GPS and BDS for single-frequency continuous real-time kinematic (RTK) positioning through realtime estimation of differential inter-system biases. Zeng et al. [9] analyzed BDS-GPS ISBs of code observation. Zhou et al. [10] presented the modeling for ISBs in multi-GNSS PPP processing with undifferenced and uncombined model. Mi et al. [11] discussed the characteristics of GPS, BDS2, BDS3 and Galileo inter-system biases and the influence on RTK positioning, Jiang et al. [12]
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proposed a short-term station-dependent ISB model, Tu et al. [13] pointed that the Multi GNSS ISBs in precise time transfer. While, few studies referred to the discrepancy between BDS-2 and BDS3. Jiao et al. merely discussed the influence of TDB estimation on BDS-2 and BDS-3 precise point positioning not referred to the timing performance. The performance of BDS time transfer is critical for supporting precise time service. Tu et al. and Zhang et al. [14,15] modelled and assessed the precision of time transfer with triple frequency and multi-GNSS observations. Ge et al. [16] verified the precision of BDS real-time time transfer. Despite achieved high level of accuracy of BDS, there is still room for improvement through advanced observation modelling. Timing service is one of the core function for GNSS. Therefore, this paper analyzes the TDB characteristic and provides users the BDS-2/BDS-3 combination PPP strategy considering TDB parameter to improve the time transfer performance, and gives a beneficial reference for the receiver manufacture and BDS users. The structure of article is arranged as follows: section 2 is devoted to the fundamentals of BDS data processing as well as the TDB estimation strategy. Section 3 validates the existence of TDB on all the IGS (International GNSS System) Service and iGMAS (international GNSS Monitoring and Assessment Service) receiver applicable to BDS-2/BDS-3 and gives a detailed analysis on the characteristics of TDB. Based on the precise products, section 4 focus on the parameter estimation methods of TDB. In preparation of the practical application of the TDB model, Section 5 introduces the influence on time transfer with three kinds of TDB model. Section6 concluded with a summary of the findings and a suggestion of the future BDS2-/BDS-3 application. 2. Methodology This part mainly discusses the mathematical model employed in BDS TDB estimation. The absolute positioning model with dual-frequency ionosphere-free BDS PPP model has been modified here: except the common estimated parameters, such as the coordinate, zenith troposphere delay and so on, the new parameter called TDB has been added to the observation equation. It should be noted that, the BDS receiver clock offset has been divided into two parts: the BDS-3 receiver clock offset and the TDB of BDS-2 relative to the BDS-3, in other words, the BDS-3 receiver clock is chosen as the reference, the BDS-2 receiver clock offset is composed of BDS-3 receiver clock offset plus TDB relative to BDS-3. Hence, the BDS-2 receiver clock offset can be rewritten as:
t BDS
-2
¼ t BDS
-3
þ TDB
ð1Þ
Theoretically, we assumed that the TDB value should be zero which can be attributed to BDS-2 and BDS-3 is a part of BDS, whereas, actually, the TDB exists all the time since the BDS-2 and BDS-3 provide service together. The satellite orbits and clock errors are corrected by applying WHU precise satellite products, currently, BDS-3 satellite information is exclusively available from WHU (WuHan University) precise products. Taken one BDS-2/BDS-3 satellite for example, some errors can be corrected with experienced model, observation equation will be available after linearization. The design matrix is listed here with all the unknown parameter:
2
X Y
3
7 2 6 3 7 36 PBDS - 3 7 1 1 1 1 0 1 0 6 6 Z 7 6 s 7 6 /BDS - 3 7 6 1 1 1 1 1 1 0 76 7 7 76 6 s 7 76 ZTD 7 ¼ 6 6 BDS - 2 7 7 6 4 1 1 1 1 0 1 1 56 P 5 4 s 6 N 7 7 BDS - 2 1 1 1 1 1 1 1 6 7 6 /s 4 tBDS3 5 TDB 2
ð2Þ
where the subscript s denotes the epoch, P and / are the phase and code observation residuals, superscript BDS-3, BDS-2 means satellite type,ZTDdenotes Zenith Tropospheric Delay. The ambiguity is estimated with float value, the relativistic effect, phase windup and tide displacement, have been precisely corrected with the associated empirical models in advance. As the external constraint, the known coordinate is added to the pseudo-range observation equation to avoid rank deficiency. Considering the WHU product was estimated with the B1I and B3I frequency combination the same as observation equation employed here. The experiments have been performed to analysis the influence of TDB for time transfer. The accurate phase center offset and variation file of BDS-3 can be available for the IGS website (http://www.igs.org/). The measurement error ratio of BDS pseudo-range and carrier phase observations was set to be 500. Considering the poor orbit precision, the GEO satellites weight should be degraded with the ratio of 1:10 between GEO (Geostationary Earth Orbit). satellites and non-GEO satellites [17]. In addition, the satellite-induced code biases of BDS-2 IGSO (Inclined geosynchronous orbit) and MEO (Medium Earth Orbit) satellites were corrected with the empirical value [18]. In this article, the Kalman filter computation is used in parameter estimation rather than batch least-square which is often more efficient and the concept TDB allows a better representation of the time-varying parameters. 3. Validation and characteristic analysis of TDB 3.1. Data collection Restricted by the present condition, of all the IGS stations, the BDS-2 and BDS-3 signals are simultaneously tracked merely by JAVAD receiver. What is more, the home-made iGMAS receiver can receive all the in-orbit BDS satellites. The iGMAS was developed by China and a backbone network has been set up in the mainland of China and around the world. Table 1 showed that some representative stations are selected to participate in the experiment for the validation of existence of TDB, including receiver type, antenna type, manufacture and location. B1I and B3I open service signals are available from the iGMAS station. We categorize the receiver into four groups: JAVAD, CETC54, CETC20 and UNICORECOM. It must be pointed that the WHU clock file is not realigned to broadcast GPS time. BRCH, XIA1 and XIA3 are connected to the frequency source of timing lab (See Table 2). Fig. 1 shows the number of BDS satellites that are tracked at eight stations during four days. All the Y axis show the same labels. The BDS-2 and BDS-3 satellites in view are separated by colour red and black. As far as stations inside China, the number of visible BDS-2 satellite is over eight further than that of BDS-3 all the time. While, the stations outside China shows something different. The number of BDS-3 is equivalent to the number of BDS-2. CNYR is a special case of China station which distributed in the polar region. 3.2. Result analysis of TDB Single day time series of TDB is shown in Fig. 2 for all the stations. The horizontal legend Doy is the abbreviation of Day of year. Inspecting Fig. 2, we can see, that the gap exists in the LHA1 which can attribute to the missing observation. The color red represents JAVAD receiver, the color green represents UNICORE receiver, the color blue represents CETC 54 receiver, the color black represents CETC 20 receiver. Surprisingly, all the sub-figures are non-zero time series, thus the same conclusion can be drawn as Jiao et al. [6]: the bias indeed exists between BDS-2 receiver clock offset and BDS-3 receiver clock offset. Now, we turn to summarize the
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W. Qin et al. / Measurement 156 (2020) 107566 Table 1 BDS PPP Processing strategies. Item
Strategies
Estimator Relativistic effect Sagnac effect Phase wind-up Ionospheric delay Tide displacements Tropospheric delay
Kalman filter IERS conventions 2010 [19] IERS conventions 2010 None IF linear combination [20] IERS conventions 2010 estimated as a continuous piecewise linear function (2 h parameter spacing) [21] Estimated as constant Estimated as white noise process Estimated as constant; float values Estimated as random walk process No estimation Estimated as white noise process Estimated as random walk process Estimated as constant
Receiver coordinate Receiver clock offset Ambiguity Tropospheric wet delay TDB
characteristic of TDB. The fluctuation of 1 ns can be observed in the TDB time series for most of the receivers, while UNICORE receivers show the TDB fluctuation with an amplitude of up to 2 ns. More interestingly, the individual receiver can be observed significantly different even if they belong to the same brand and are installed with the same antenna. Note, another finding is that the XIA3 and XIA1 has the similar evolution only differs in value. The difference in value is caused by hardware delay. It looks that the TDB is related to the external atomic clock which XIA3 and XIA1 are connected with the same frequency source. Considering the analysis of TDB up to now suffer from the limited receivers, it is hard to make the certain conclusion that the TDB characteristic is more sensitive for the external frequency source than the receiver itself. To Confirm this finding, Fig. 3 shows the long-term TDB time series. We have some findings: (1) a small intra-daily variation is presented in each sub-figure; (2) the jump of day boundary is caused by precise products; (3) for multiple days, the evolution of XIA3 and XIA1 is still similar. For seeking a proper TDB model, we should ascertain the TDB characteristics in advance. Initially, the TDB is estimated with the BDS PPP solution and white noise model. Between-epoch TDB differences are demonstrated in the following figure. The TDB differences time series of multiple days is on exhibition in Fig. 4. The color in Fig. 4 has the same indicator with Fig. 3. The STD values of the between-epoch TDB differences with eight receivers are 0.02, 0.04, 0.04, 0.04, 0.17, 0.02, 0.01 and 0.02 ns, respectively. Taken the figure and value together, we have several deductions: first, no matter which kind of receiver, slight fluctuation exists in the TDB differential time series. It is useful for cases when we do expect small variations over time. Second, the STD value of maximal is no more than 0.04 ns. The STD is a statistic that
measures the dispersion of a dataset. The smaller STD, the closer differential value relative to its mean of time series. We have enough grounds to believe that former TDB has some correlation with subsequent TDB. Fig. 4 is clear that the TDB differences vary smoothly from epoch to epoch and can well be approximated by a continuous function. To verify our assumption, auto-correlation coefficient is employed in our study。The theoretical auto-correlation coefficient is defined as:
RðsÞ ¼
E½ðxt lÞðxtþs lÞ
ð3Þ
d2
where: E½ is the expectation operator, xt is the time series, l is its mean and dis its variance, t is time, sis the time interval. The TDB time series xt can be considered as stationary stochastic process, and the auto-correlation function measures the correlation between xt andxtþs . Here, IGS receiver and iGMAS receiver is randomly selected for checking whether epoch-wise TDB value has some relationship with each other. The auto-correlation function (ACF) of the TDB differences is displayed in Figs. 5 and 6. Two discoveries come out. First, the auto-correlation of the TDB difference die downs as time-lag elapses. Second, we can see that R (1) is greater than 0.9. In the statistics, it is the strong correlation when the absolute value of ACF reaches 0.80–1.00. The R (1) is up to 0.9 which reveals that the TDB estimates exhibits pronounced mathematical correlations. Hence, we conceive the random walk model and constant model employed in this contribution. We expect to achieve better results using our approach. 4. TDB analysis with different model The white noise model shows the normal distribution as well as the random walk noise model which differentiates them from the variance value. For the white noise process, random walk process and constant model, the simplest TDB parameter can be described as [10]:
ISBðkÞ Nð0; r2wh Þ
ð4Þ
ISBðkÞ ¼ ISBðk 1Þ þ -ISB; -ISB Nð0; r2rw Þ
ð5Þ
ISBðkÞ ¼ ISBðk 1Þ
ð6Þ
where k denotes the epoch, rwh is the variance of white noise, rrw is the variance of random walk noise. (1) The white noise is considered dis-related to the time variation, thus, it is the general estimation model for determining unknowns. The approach applied here is that the TDB value should be initialized at each epoch. The white noise model is a special case of the random walk model with a quasiinfinite noise.
Table 2 The information involved in the TDB estimation. Agency
Station name
Location
Receiver type
Manufacturer
Antenna type
IGS MGEX
POTS URUM BRCH LHA1 XIA1 XIA3 CNYR SHA1
Germany China Germany China China China China China
JAVAD TRE-3 JAVAD TRE-3 CETC-54-GMR-4016 CETC-54-GMR-4016 GNSS-GGR CETC-54-GMR-4016 UB4B0- 13,478 UB4B0- 13,478
JAVAD JAVAD CETC54 CETC54 CETC20 CETC54 UNICORECOM UNICORECOM
JAVRINGANT_G5T JAVRINGANT_G5T NOV750.R4 NOV750.R4 RINT-8CH GNSS-750 NOV750.R4 NOV750.R4
iGMAS
4
W. Qin et al. / Measurement 156 (2020) 107566
Fig. 1. BDS-2 and BDS-3 visible satellites in eight stations.
Fig. 2. The difference on BDS-3 PPP receiver clock offset and BDS-2 PPP receiver clock offset.
(2) The random walk model is the simplest and yet most important models in time series forecasting. This model assumes that in each period the variable takes a random step away from its previous value, while the steps are independently and identically distributed in size. The determination of a random walk model with drift or without drift according to whether the distribution of step sizes has a non-zero mean or a zero mean. It should be pointed that a random walk model without drift is employed in our study. The explanation is that a prior co-variance is hard to constrain the large gap between epoch-wise time series. The variance QW used in this contribution are listed in Table 3, which were calculated by the Allan deviation of the TDB derived
from BDS PPP with WHU final products. The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. The Allan deviation at the averaging time 30 s was employed as variance of random walk model. (3) The random constant model is a special case of the random walk model with a zero value of process noise -ISB . In this part the strategies to incorporate three kinds of TDB model are compared. For acquiring more information about the individual receiver, the STD (Standard deviation) and mean with the form of daily TDB time series are demonstrated in Fig. 7.
W. Qin et al. / Measurement 156 (2020) 107566
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Fig. 3. TDB time series estimated using the PPP method and white noise model.
Fig. 4. Between-epoch TDB differences at eight stations from the BDS PPP solution.
Fig. 5. Auto-correlation function of the TDB time series of XIA3. Fig. 6. Auto-correlation function of the TDB time series of POTS.
The top panel is obtained with white noise model, the middle panel is obtained with random walk model, the bottom panel is obtained with constant model. The TDB mean value is presented with the shape of bar, each period is distinguished with different color. The TDB STD is presented with square symbol connected with broken line.
(1) Receiver: The TDB STD is basically the same for the same brand of receiver. The same type of antenna was equipped with the same type of receiver, respectively. What is more, STD of TDB is more or less for the same brand of receiver. The value tells us that the TDB is enough stable in the receiver referred to the evaluation.
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Table 3 A priori noise variances of eight stations. Station
QW
Station
QW
POTS URUM CNYR SHA1
8.6905e-07 6.1686e-07 9.9475e-07 8.8059e-07
BRCH XIA3 LHA1 XIA1
7.8618e-07 6.2111e-07 5.9870e-07 4.9265e-07
Fig. 7. Std and mean of TDB with three kinds of model.
(2) TDB estimation model: The TDB STD estimated with constant model is smaller than 0.2 ns, which is the smallest among those of models, and the TDB STD estimated with white noise model is the biggest among three kinds of model. In contrast to the white noise result of URUM DOY 16, the random walk model and the constant model can obtain more smooth series, the STD was decreased from 1.4 ns to 0.2 ns. The improvement is that the correlation of adjacent epoch is considered in the random model. For UNICORECOM and CETC20 receiver, the daily TDB means are nearly the same whatever station is, whereas, the TDB means are different in each JAVAD receiver. The graphical analysis intuitively shows that the characteristic of TDB. In Fig. 8, the corresponding Allan deviation analysis is further applied to compare the TDB stability among three models with all the stations. It is not too difficult to find that the constant model is most stable, followed by random walk and white noise model. Normally, the standard version of Microsemi MHM 2010Active Hydrogen Maser is introduced here as the criterion: the Allan deviation of 100 s, 1000 s, 10000 s and 86400 s is 5.0 E-15, 2.0 E-15, 1.5 E-15 and 2.0 E-16, respectively. The TDB obtained from constant model has the equivalent stability with the Hydrogen Maser 2010 in the typical averaging time.
5. Analysis on TDB estimation strategies for precise time transfer The PPP algorithm provides us with a very precise estimate of the receiver clock offset with respect to the timescale imposed by the satellite clock product. The time transfer can be conducted by forming differences between PPP receiver clock time series of two stations. The aim of the investigations presented in this part
is to gain an understanding of the PPP-derived time transfer include TDB estimation or exclude TDB estimation. Fig. 9 demonstrates the clock offset time series with three kinds of TDB estimation strategies as well as without TDB estimation. Based on the above figure, we have several findings: first, with the same receiver, such as link BRCH-XIA3, no system bias appeared in the clock offset whether estimate TDB or not; second, with the different receiver, such as link XIA1-XIA3 or BRCH-XIA1, a system bias appeared in the clock offset if no TDB estimation. In addition, the impact of mixed-receiver on RTK positioning has been verified by some scholars [22]. We are not sure the reason caused by the bias and need further investigation; third, obvious dayboundary jump exists in the BDS time transfer while the jump in GPS time transfer nearly disappeared. BDS model should be still refined for improving the quality of precise product; fourth, the time series with three kinds of model match well, especially in common-clock link XIA3-XIA1. In terms of mitigating dayboundary jump, random walk model and constant model is superior to white noise model. The system error was up to 6 ns which would deteriorate the precision of time transfer, therefore, it was a limitation factor in the use of BDS measurements for time transfer. More attention should be paid when the different receiver was performed in the time transfer. Next, another two links POTS-XIA1 and POTSURUM were calculated for supporting our conclusion. The more result was presented in Fig. 10. The clock offset was estimated with two schemes: Scheme 1: without TDB estimation, Scheme 2: with TDB white noise model estimation. The difference with two schemes were demonstrated in Fig. 10. The time series of line blue looks more stable than that of line red whatever intra-day or cross-day which stays in the small amplitude of 0.3 ns. We can assume that there was nearly no deviation in the link POTS-URUM. It is reasonable that the receiver of same type is installed in POTS and URUM, while the POTS and XIA1 are different. When the two sides were receivers of different brands, the significant deviation exists in the clock offset with Schemes 1 and 2; when the two sides are receiver of the same brand, marginal deviation exists in the clock offset with Schemes 1 and 2. The comparison on time transfer stability with different TDB model. A more detailed comparison of the two approaches with respect to white noise model can be discussed. Figs. 11–13 shows the stability comparison of time transfer of different TDB estimation strategy. The distances between the stations range from 7000 km for the stations BRCH and XIA3 to 0.5 km for the stations XIA1 and XIA3. At present, the suggestion is that the same type receiver was employed in precise time transfer. We hope that no bias appears after all the receiver finish update of hardware. Here, we make an analysis on Allan deviation and improvement percentage of stability with different TDB estimation strategies. The random walk model and constant model is more effective than the white noise model in improving the stability of time link. There is no general rule in the enhancement of stability. At the averaging time of 100 s, 1000 s,10000 s and 86400 s, for random walk model: the improvement percentage of link XIA1-BRCH is 11.35, 3.04, 31.83 and 1.03%; the link XIA3-XIA1 is 23.67, 10.04, 3.43 and 3.04%; the link XIA3-BRCH is 9.49, 2.60, 26.44 and 52.25%; for constant model: the improvement percentage of link XIA1-BRCH is 15.23, 6.52, 30.98 and 42.87%; the link XIA3-XIA1 is 23.08, 3.14, 5.18 and 38.24%; the link XIA3-BRCH is 10.71, 32.83 and 60.07%. The random walk model and constant model indicate a considerably increased frequency Stability which is preferred to describe the bias between BDS-2 and BDS-3. The improvement extent isn’t the same at each averaging time and is different in each link, the minimal is 1% and the maximal is 60%.
W. Qin et al. / Measurement 156 (2020) 107566
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Fig. 8. Stability of TDB with three kinds of model.
Fig. 10. Clock difference with the same type receiver and different type receiver with TDB estimation. Fig. 9. Clock offset with TDB estimation or without TDB estimation.
6. Conclusion TDB is a crucial element for the fusion application of BDS-2 and BDS-3. Triggered by the finding-the bias between BDS-2 and BDS3, this article concentrated on the stochastic modeling of TDB in BDS-2/BDS-3 combination PPP processing and the TDB influence on time transfer performance with the same/different frequency source. We analyze the TDB stability and characteristic of shortterm and long-term derived from iGMAS and IGS receivers, as well as study their impact on BDS time transfer while used three different strategies to estimate TDB. More valuable information can be founded here:
(1) Without common frequency source, the TDB characteristic depends on the receiver itself, regardless of the same antenna and the receiver type. With common frequency source, the external frequency source plays more influence on the TDB characteristic than the receiver type. (2) The TDB time series has been validated that the strong relation exists in the adjacent epochs, hence, the random walk model and constant model are reasonable for TDB estimation. (3) The TDB is stable enough in the investigation period. The discrepancy from TDB estimation strategies can lead to different estimated TDB results. The TDB absolute value was nearly 2–6 ns, thus it must be corrected prior to GNSS application.
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Fig. 11. Allan deviation and improvement percentage in link XIA1-BRCH.
Fig. 12. Allan deviation and improvement percentage in link XIA3-XIA1.
(4) When the two sides were receivers of different brands, the significant deviation exists in the clock offset with TDB estimation and without TDB estimation. When the two sides are
receiver of the same brand, marginal deviation exists in the clock offset with TDB estimation and without TDB estimation.
W. Qin et al. / Measurement 156 (2020) 107566
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Fig. 13. Allan deviation and improvement percentage in link XIA3-BRCH.
(5) In terms of improving stability of time transfer, TDB estimation with the random walk model and constant model are potentially good candidate compared with white noise model. 7. Credit author statement Qin conceived and wrote the article, Ge designed the experiments, Su and Zhang enrich the languages, Wei and Yang contributed to discussions and revisions. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11703033), CAS ‘‘Light of West China” Program and Youth Promotion Committee of the Chinese Academy of Sciences. The authors gratefully acknowledge Wuhan University, iGMAS and IGS for providing the precise products. References [1] Y. Yang, Y. Xu, J. Li, C. Yang, Progress and performance evaluation of BeiDou global navigation satellite system: Data analysis based on BDS-3 demonstration system, Sci. China Earth Sci. 61 (2018) 614–624. [2] A.M. Manzino, P. Dabove, N. Gogoi, Assessment of positioning performances in Italy from GPS BDS and GLONASS constellations, Geodesy Geodyn. 9 (2018) 439–448. [3] G. Jiao, S. Song, Y. Ge, K. Su, Y. Liu, Assessment of BeiDou-3 and Multi-GNSS precise point positioning performance, Sensors 19 (2019). [4] F. Ye, Y. Yuan, J. Ou, Initial orbit determination of BDS-3 satellites based on new code signals, Geod. Geodyn. 9 (2018) 342–346.
[5] O. Montenbruck, A. Hauschild, P. Steigenberger, U. Hugentobler, P. Teunissen, S. Nakamura, Initial assessment of the COMPASS/BeiDou-2 regional navigation satellite system, GPS Solutions 17 (2013) 211–222. [6] G. Jiao, S. Song, W. Jiao, Improving BDS-2 and BDS-3 joint precise point positioning with time delay bias estimation, Meas. Sci. Technol. (2019). [7] J. Paziewski, P. Wielgosz, Accounting for Galileo–GPS inter-system biases in precise satellite positioning, J. Geod. 89 (2014) 81–93. [8] W. Gao, X. Meng, C. Gao, S. Pan, D. Wang, Combined GPS and BDS for singlefrequency continuous RTK positioning through real-time estimation of differential inter-system biases, GPS Solutions 22 (2017). [9] A. Zeng, Y. Yang, F. Ming, Y. Jing, BDS–GPS inter-system bias of code observation and its preliminary analysis, GPS Solutions (2017). [10] F. Zhou, D. Dong, P. Li, X. Li, H. Schuh, Influence of stochastic modeling for inter-system biases on multi-GNSS undifferenced and uncombined precise point positioning, GPS Solutions 23 (2019). [11] X. Mi, B. Zhang, Y. Yuan, X. Luo, Characteristics of GPS, BDS2, BDS3 and Galileo inter-system biases and their influence on RTK positioning, Meas. Sci. Technol. (2019). [12] N. Jiang, Y. Xu, T. Xu, G. Xu, Z. Sun, H. Schuh, GPS/BDS short-term ISB modelling and prediction, GPS Solutions 21 (2016) 163–175. [13] Multiple GNSS inter-system biases in precise time transfer.pdf. [14] R. Tu, P. Zhang, R. Zhang, J. Liu, X. Lu, Modeling and performance analysis of precise time transfer based on BDS triple-frequency un-combined observations, J. Geod. (2018). [15] P. Zhang, R. Tu, R. Zhang, Y. Gao, H. Cai, Combining GPS, BeiDou, and galileo satellite systems for time and frequency transfer based on carrier phase observations, Remote Sens. 10 (2018) 324. [16] Y. Ge, F. Zhou, T. Liu, W. Qin, S. Wang, X. Yang, Enhancing real-time precise point positioning time and frequency transfer with receiver clock modeling, GPS Solutions 23 (2018). [17] X. Li, M. Ge, X. Dai, X. Ren, M. Fritsche, J. Wickert, H. Schuh, Accuracy and reliability of multi-GNSS real-time precise positioning: GPS GLONASS, BeiDou, and Galileo, J. Geodesy 89 (2015) 607–635. [18] L. Wanninger, S. Beer, BeiDou satellite-induced code pseudorange variations: diagnosis and therapy, GPS Solutions 19 (2015) 639–648. [19] G. Petit, B. Luzum, IERS conventions (2010) BUREAU INTERNATIONAL DES POIDS ET MESURES SEVRES (FRANCE). (2010) (No. IERS-TN-36). [20] P. Defraigne, N. Guyennon, C. Bruyninx, GPS time and frequency transfer: PPP and phase-only analysis, Int. J. Navigat. Observat. 2008 (2008) 1–7. [21] W. Ding, F.N. Teferle, K. Kazmierski, D. Laurichesse, Y. Yuan, An evaluation of real-time troposphere estimation based on GNSS precise point positioning, J. Geophys. Res.: Atmosp. 122 (2017) 2779–2790. [22] N. Nadarajah, P.J.G. Teunissen, J.M. Sleewaegen, O. Montenbruck, The mixedreceiver BeiDou inter-satellite-type bias and its impact on RTK positioning, Gps Solutions 19 (2015) 357–368.
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Measurement journal homepage: www.elsevier.com/locate/measurement
Accounting BDS3–BDS2 inter-system biases for precise time transfer Weijin Qin a,b,c,⇑, Yulong Ge a,b,c, Zhe Zhang a,b,c, Hang Su a,b,c, Pei Wei a,b,c, Xuhai Yang a,b,d a
National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China Key Laboratory of Precise Positioning and Timing Technology, Chinese Academy of Sciences, Xi’an 710600, China c University of Chinese Academy of Sciences, Beijing 100049, China d School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China b
a r t i c l e
i n f o
Article history: Received 26 December 2019 Received in revised form 16 January 2020 Accepted 29 January 2020 Available online 7 February 2020 Keywords: BDS-3 Time delay bias (TDB) Stochastic modeling Precise point positioning (PPP) Time transfer
a b s t r a c t The rapid development of BDS not only brings opportunities but also new problems. This article has been verified that the biases exist between BDS-2 and BDS-3 precise point positioning time transfer. We defined internal biases of BDS as Time Delay Bias (TDB) for a distinguish with inter system biases (ISB) exists in different systems. Motivated by the bias, we make comprehensive analysis for determining the characteristics of TDB referring to all the receivers capable of BDS-2 along with BDS-3 signals, furtherly, three kinds of TDB stochastic model are designed for parameter estimation. Later, the influence on time transfer with different TDB estimation strategy is compared in the end. We have some findings: (1) It is confirmed that TDB indeed exists in JAVAD TRE_3, CETC-54-GMR-4016, GNSS GGR and UB4B013478 receivers. Without external frequency source, the characteristics of TDB is related to the receiver itself. With the same external frequency source, TDB tendency of two receivers show the same evolution. (2) It is concluded that the STD (Standard Deviation) and stability of TDB estimated with random walk model and constant model is superior to white noise model because of strong correlation among TDB time series. (3) It is verified that the stability of time transfer estimated as random walk model and constant model is smaller than that of white noise model, what is more, when the two sides were receivers of different brands, the significant deviation exists in the clock offset with TDB estimation and without TDB estimation. When the two sides are receiver of the same brand, marginal deviation exists in the clock offset with TDB estimation and without TDB estimation. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction The appearance of Global Navigation Satellite Systems (GNSS) allows worldwide positioning &navigation &timing with unprecedented precision and accuracy. In the current status, there are four fully operational GNSS, namely the United States Global Positioning System (GPS), the Russian Globalnaja Nawigazionnaja Sputnikowaja Sistema (GLONASS), the European Galileo positioning system (Galileo) and the Chinese BeiDou navigation System (BDS). According to the actual condition, three steps of construction are employed by BDS, including the demonstration system (BDS-1), the regional system (BDS-2) and the global system (BDS-3) [1]. The BDS-1 is out of date, while BDS-2 is still active with its superior performance. Hence, an interesting phenomenon appears that BDS-2 and BDS-3 will both contribute to serve users located in the everywhere in future [2].
⇑ Corresponding author at: National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China. E-mail address: [email protected] (W. Qin). https://doi.org/10.1016/j.measurement.2020.107566 0263-2241/Ó 2020 Elsevier Ltd. All rights reserved.
Joint BDS-2 and BDS-3 precise point positioning (PPP) can enhance the positioning accuracy and reduce the convergence time [3]. However, the new confusion came out that the non-negligible existed discrepancy between BDS-2 and BDS-3, which attributes to the signal modulation mode and signal characteristics [4,5]. Jiao et al. [6] mentioned that a bias named TDB (Time Delay Bias) at the receiver end was caused by receiving or processing BDS-3 and BDS-2 signals with different unit. The bias should be considered in the observation equation. To my knowledge, plenty of studies focus their attention on the inter-system biases (ISBs) of BDS, GLONASS, Galileo relative to GPS. Paziewski et al. [7] investigated a methodology of accounting for GPS-Galileo receiver intersystem biases in precise relative positioning. Gao et al. [8] designed the approach that combined GPS and BDS for single-frequency continuous real-time kinematic (RTK) positioning through realtime estimation of differential inter-system biases. Zeng et al. [9] analyzed BDS-GPS ISBs of code observation. Zhou et al. [10] presented the modeling for ISBs in multi-GNSS PPP processing with undifferenced and uncombined model. Mi et al. [11] discussed the characteristics of GPS, BDS2, BDS3 and Galileo inter-system biases and the influence on RTK positioning, Jiang et al. [12]
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W. Qin et al. / Measurement 156 (2020) 107566
proposed a short-term station-dependent ISB model, Tu et al. [13] pointed that the Multi GNSS ISBs in precise time transfer. While, few studies referred to the discrepancy between BDS-2 and BDS3. Jiao et al. merely discussed the influence of TDB estimation on BDS-2 and BDS-3 precise point positioning not referred to the timing performance. The performance of BDS time transfer is critical for supporting precise time service. Tu et al. and Zhang et al. [14,15] modelled and assessed the precision of time transfer with triple frequency and multi-GNSS observations. Ge et al. [16] verified the precision of BDS real-time time transfer. Despite achieved high level of accuracy of BDS, there is still room for improvement through advanced observation modelling. Timing service is one of the core function for GNSS. Therefore, this paper analyzes the TDB characteristic and provides users the BDS-2/BDS-3 combination PPP strategy considering TDB parameter to improve the time transfer performance, and gives a beneficial reference for the receiver manufacture and BDS users. The structure of article is arranged as follows: section 2 is devoted to the fundamentals of BDS data processing as well as the TDB estimation strategy. Section 3 validates the existence of TDB on all the IGS (International GNSS System) Service and iGMAS (international GNSS Monitoring and Assessment Service) receiver applicable to BDS-2/BDS-3 and gives a detailed analysis on the characteristics of TDB. Based on the precise products, section 4 focus on the parameter estimation methods of TDB. In preparation of the practical application of the TDB model, Section 5 introduces the influence on time transfer with three kinds of TDB model. Section6 concluded with a summary of the findings and a suggestion of the future BDS2-/BDS-3 application. 2. Methodology This part mainly discusses the mathematical model employed in BDS TDB estimation. The absolute positioning model with dual-frequency ionosphere-free BDS PPP model has been modified here: except the common estimated parameters, such as the coordinate, zenith troposphere delay and so on, the new parameter called TDB has been added to the observation equation. It should be noted that, the BDS receiver clock offset has been divided into two parts: the BDS-3 receiver clock offset and the TDB of BDS-2 relative to the BDS-3, in other words, the BDS-3 receiver clock is chosen as the reference, the BDS-2 receiver clock offset is composed of BDS-3 receiver clock offset plus TDB relative to BDS-3. Hence, the BDS-2 receiver clock offset can be rewritten as:
t BDS
-2
¼ t BDS
-3
þ TDB
ð1Þ
Theoretically, we assumed that the TDB value should be zero which can be attributed to BDS-2 and BDS-3 is a part of BDS, whereas, actually, the TDB exists all the time since the BDS-2 and BDS-3 provide service together. The satellite orbits and clock errors are corrected by applying WHU precise satellite products, currently, BDS-3 satellite information is exclusively available from WHU (WuHan University) precise products. Taken one BDS-2/BDS-3 satellite for example, some errors can be corrected with experienced model, observation equation will be available after linearization. The design matrix is listed here with all the unknown parameter:
2
X Y
3
7 2 6 3 7 36 PBDS - 3 7 1 1 1 1 0 1 0 6 6 Z 7 6 s 7 6 /BDS - 3 7 6 1 1 1 1 1 1 0 76 7 7 76 6 s 7 76 ZTD 7 ¼ 6 6 BDS - 2 7 7 6 4 1 1 1 1 0 1 1 56 P 5 4 s 6 N 7 7 BDS - 2 1 1 1 1 1 1 1 6 7 6 /s 4 tBDS3 5 TDB 2
ð2Þ
where the subscript s denotes the epoch, P and / are the phase and code observation residuals, superscript BDS-3, BDS-2 means satellite type,ZTDdenotes Zenith Tropospheric Delay. The ambiguity is estimated with float value, the relativistic effect, phase windup and tide displacement, have been precisely corrected with the associated empirical models in advance. As the external constraint, the known coordinate is added to the pseudo-range observation equation to avoid rank deficiency. Considering the WHU product was estimated with the B1I and B3I frequency combination the same as observation equation employed here. The experiments have been performed to analysis the influence of TDB for time transfer. The accurate phase center offset and variation file of BDS-3 can be available for the IGS website (http://www.igs.org/). The measurement error ratio of BDS pseudo-range and carrier phase observations was set to be 500. Considering the poor orbit precision, the GEO satellites weight should be degraded with the ratio of 1:10 between GEO (Geostationary Earth Orbit). satellites and non-GEO satellites [17]. In addition, the satellite-induced code biases of BDS-2 IGSO (Inclined geosynchronous orbit) and MEO (Medium Earth Orbit) satellites were corrected with the empirical value [18]. In this article, the Kalman filter computation is used in parameter estimation rather than batch least-square which is often more efficient and the concept TDB allows a better representation of the time-varying parameters. 3. Validation and characteristic analysis of TDB 3.1. Data collection Restricted by the present condition, of all the IGS stations, the BDS-2 and BDS-3 signals are simultaneously tracked merely by JAVAD receiver. What is more, the home-made iGMAS receiver can receive all the in-orbit BDS satellites. The iGMAS was developed by China and a backbone network has been set up in the mainland of China and around the world. Table 1 showed that some representative stations are selected to participate in the experiment for the validation of existence of TDB, including receiver type, antenna type, manufacture and location. B1I and B3I open service signals are available from the iGMAS station. We categorize the receiver into four groups: JAVAD, CETC54, CETC20 and UNICORECOM. It must be pointed that the WHU clock file is not realigned to broadcast GPS time. BRCH, XIA1 and XIA3 are connected to the frequency source of timing lab (See Table 2). Fig. 1 shows the number of BDS satellites that are tracked at eight stations during four days. All the Y axis show the same labels. The BDS-2 and BDS-3 satellites in view are separated by colour red and black. As far as stations inside China, the number of visible BDS-2 satellite is over eight further than that of BDS-3 all the time. While, the stations outside China shows something different. The number of BDS-3 is equivalent to the number of BDS-2. CNYR is a special case of China station which distributed in the polar region. 3.2. Result analysis of TDB Single day time series of TDB is shown in Fig. 2 for all the stations. The horizontal legend Doy is the abbreviation of Day of year. Inspecting Fig. 2, we can see, that the gap exists in the LHA1 which can attribute to the missing observation. The color red represents JAVAD receiver, the color green represents UNICORE receiver, the color blue represents CETC 54 receiver, the color black represents CETC 20 receiver. Surprisingly, all the sub-figures are non-zero time series, thus the same conclusion can be drawn as Jiao et al. [6]: the bias indeed exists between BDS-2 receiver clock offset and BDS-3 receiver clock offset. Now, we turn to summarize the
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W. Qin et al. / Measurement 156 (2020) 107566 Table 1 BDS PPP Processing strategies. Item
Strategies
Estimator Relativistic effect Sagnac effect Phase wind-up Ionospheric delay Tide displacements Tropospheric delay
Kalman filter IERS conventions 2010 [19] IERS conventions 2010 None IF linear combination [20] IERS conventions 2010 estimated as a continuous piecewise linear function (2 h parameter spacing) [21] Estimated as constant Estimated as white noise process Estimated as constant; float values Estimated as random walk process No estimation Estimated as white noise process Estimated as random walk process Estimated as constant
Receiver coordinate Receiver clock offset Ambiguity Tropospheric wet delay TDB
characteristic of TDB. The fluctuation of 1 ns can be observed in the TDB time series for most of the receivers, while UNICORE receivers show the TDB fluctuation with an amplitude of up to 2 ns. More interestingly, the individual receiver can be observed significantly different even if they belong to the same brand and are installed with the same antenna. Note, another finding is that the XIA3 and XIA1 has the similar evolution only differs in value. The difference in value is caused by hardware delay. It looks that the TDB is related to the external atomic clock which XIA3 and XIA1 are connected with the same frequency source. Considering the analysis of TDB up to now suffer from the limited receivers, it is hard to make the certain conclusion that the TDB characteristic is more sensitive for the external frequency source than the receiver itself. To Confirm this finding, Fig. 3 shows the long-term TDB time series. We have some findings: (1) a small intra-daily variation is presented in each sub-figure; (2) the jump of day boundary is caused by precise products; (3) for multiple days, the evolution of XIA3 and XIA1 is still similar. For seeking a proper TDB model, we should ascertain the TDB characteristics in advance. Initially, the TDB is estimated with the BDS PPP solution and white noise model. Between-epoch TDB differences are demonstrated in the following figure. The TDB differences time series of multiple days is on exhibition in Fig. 4. The color in Fig. 4 has the same indicator with Fig. 3. The STD values of the between-epoch TDB differences with eight receivers are 0.02, 0.04, 0.04, 0.04, 0.17, 0.02, 0.01 and 0.02 ns, respectively. Taken the figure and value together, we have several deductions: first, no matter which kind of receiver, slight fluctuation exists in the TDB differential time series. It is useful for cases when we do expect small variations over time. Second, the STD value of maximal is no more than 0.04 ns. The STD is a statistic that
measures the dispersion of a dataset. The smaller STD, the closer differential value relative to its mean of time series. We have enough grounds to believe that former TDB has some correlation with subsequent TDB. Fig. 4 is clear that the TDB differences vary smoothly from epoch to epoch and can well be approximated by a continuous function. To verify our assumption, auto-correlation coefficient is employed in our study。The theoretical auto-correlation coefficient is defined as:
RðsÞ ¼
E½ðxt lÞðxtþs lÞ
ð3Þ
d2
where: E½ is the expectation operator, xt is the time series, l is its mean and dis its variance, t is time, sis the time interval. The TDB time series xt can be considered as stationary stochastic process, and the auto-correlation function measures the correlation between xt andxtþs . Here, IGS receiver and iGMAS receiver is randomly selected for checking whether epoch-wise TDB value has some relationship with each other. The auto-correlation function (ACF) of the TDB differences is displayed in Figs. 5 and 6. Two discoveries come out. First, the auto-correlation of the TDB difference die downs as time-lag elapses. Second, we can see that R (1) is greater than 0.9. In the statistics, it is the strong correlation when the absolute value of ACF reaches 0.80–1.00. The R (1) is up to 0.9 which reveals that the TDB estimates exhibits pronounced mathematical correlations. Hence, we conceive the random walk model and constant model employed in this contribution. We expect to achieve better results using our approach. 4. TDB analysis with different model The white noise model shows the normal distribution as well as the random walk noise model which differentiates them from the variance value. For the white noise process, random walk process and constant model, the simplest TDB parameter can be described as [10]:
ISBðkÞ Nð0; r2wh Þ
ð4Þ
ISBðkÞ ¼ ISBðk 1Þ þ -ISB; -ISB Nð0; r2rw Þ
ð5Þ
ISBðkÞ ¼ ISBðk 1Þ
ð6Þ
where k denotes the epoch, rwh is the variance of white noise, rrw is the variance of random walk noise. (1) The white noise is considered dis-related to the time variation, thus, it is the general estimation model for determining unknowns. The approach applied here is that the TDB value should be initialized at each epoch. The white noise model is a special case of the random walk model with a quasiinfinite noise.
Table 2 The information involved in the TDB estimation. Agency
Station name
Location
Receiver type
Manufacturer
Antenna type
IGS MGEX
POTS URUM BRCH LHA1 XIA1 XIA3 CNYR SHA1
Germany China Germany China China China China China
JAVAD TRE-3 JAVAD TRE-3 CETC-54-GMR-4016 CETC-54-GMR-4016 GNSS-GGR CETC-54-GMR-4016 UB4B0- 13,478 UB4B0- 13,478
JAVAD JAVAD CETC54 CETC54 CETC20 CETC54 UNICORECOM UNICORECOM
JAVRINGANT_G5T JAVRINGANT_G5T NOV750.R4 NOV750.R4 RINT-8CH GNSS-750 NOV750.R4 NOV750.R4
iGMAS
4
W. Qin et al. / Measurement 156 (2020) 107566
Fig. 1. BDS-2 and BDS-3 visible satellites in eight stations.
Fig. 2. The difference on BDS-3 PPP receiver clock offset and BDS-2 PPP receiver clock offset.
(2) The random walk model is the simplest and yet most important models in time series forecasting. This model assumes that in each period the variable takes a random step away from its previous value, while the steps are independently and identically distributed in size. The determination of a random walk model with drift or without drift according to whether the distribution of step sizes has a non-zero mean or a zero mean. It should be pointed that a random walk model without drift is employed in our study. The explanation is that a prior co-variance is hard to constrain the large gap between epoch-wise time series. The variance QW used in this contribution are listed in Table 3, which were calculated by the Allan deviation of the TDB derived
from BDS PPP with WHU final products. The Allan variance (AVAR), also known as two-sample variance, is a measure of frequency stability in clocks, oscillators and amplifiers. The Allan deviation at the averaging time 30 s was employed as variance of random walk model. (3) The random constant model is a special case of the random walk model with a zero value of process noise -ISB . In this part the strategies to incorporate three kinds of TDB model are compared. For acquiring more information about the individual receiver, the STD (Standard deviation) and mean with the form of daily TDB time series are demonstrated in Fig. 7.
W. Qin et al. / Measurement 156 (2020) 107566
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Fig. 3. TDB time series estimated using the PPP method and white noise model.
Fig. 4. Between-epoch TDB differences at eight stations from the BDS PPP solution.
Fig. 5. Auto-correlation function of the TDB time series of XIA3. Fig. 6. Auto-correlation function of the TDB time series of POTS.
The top panel is obtained with white noise model, the middle panel is obtained with random walk model, the bottom panel is obtained with constant model. The TDB mean value is presented with the shape of bar, each period is distinguished with different color. The TDB STD is presented with square symbol connected with broken line.
(1) Receiver: The TDB STD is basically the same for the same brand of receiver. The same type of antenna was equipped with the same type of receiver, respectively. What is more, STD of TDB is more or less for the same brand of receiver. The value tells us that the TDB is enough stable in the receiver referred to the evaluation.
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W. Qin et al. / Measurement 156 (2020) 107566
Table 3 A priori noise variances of eight stations. Station
QW
Station
QW
POTS URUM CNYR SHA1
8.6905e-07 6.1686e-07 9.9475e-07 8.8059e-07
BRCH XIA3 LHA1 XIA1
7.8618e-07 6.2111e-07 5.9870e-07 4.9265e-07
Fig. 7. Std and mean of TDB with three kinds of model.
(2) TDB estimation model: The TDB STD estimated with constant model is smaller than 0.2 ns, which is the smallest among those of models, and the TDB STD estimated with white noise model is the biggest among three kinds of model. In contrast to the white noise result of URUM DOY 16, the random walk model and the constant model can obtain more smooth series, the STD was decreased from 1.4 ns to 0.2 ns. The improvement is that the correlation of adjacent epoch is considered in the random model. For UNICORECOM and CETC20 receiver, the daily TDB means are nearly the same whatever station is, whereas, the TDB means are different in each JAVAD receiver. The graphical analysis intuitively shows that the characteristic of TDB. In Fig. 8, the corresponding Allan deviation analysis is further applied to compare the TDB stability among three models with all the stations. It is not too difficult to find that the constant model is most stable, followed by random walk and white noise model. Normally, the standard version of Microsemi MHM 2010Active Hydrogen Maser is introduced here as the criterion: the Allan deviation of 100 s, 1000 s, 10000 s and 86400 s is 5.0 E-15, 2.0 E-15, 1.5 E-15 and 2.0 E-16, respectively. The TDB obtained from constant model has the equivalent stability with the Hydrogen Maser 2010 in the typical averaging time.
5. Analysis on TDB estimation strategies for precise time transfer The PPP algorithm provides us with a very precise estimate of the receiver clock offset with respect to the timescale imposed by the satellite clock product. The time transfer can be conducted by forming differences between PPP receiver clock time series of two stations. The aim of the investigations presented in this part
is to gain an understanding of the PPP-derived time transfer include TDB estimation or exclude TDB estimation. Fig. 9 demonstrates the clock offset time series with three kinds of TDB estimation strategies as well as without TDB estimation. Based on the above figure, we have several findings: first, with the same receiver, such as link BRCH-XIA3, no system bias appeared in the clock offset whether estimate TDB or not; second, with the different receiver, such as link XIA1-XIA3 or BRCH-XIA1, a system bias appeared in the clock offset if no TDB estimation. In addition, the impact of mixed-receiver on RTK positioning has been verified by some scholars [22]. We are not sure the reason caused by the bias and need further investigation; third, obvious dayboundary jump exists in the BDS time transfer while the jump in GPS time transfer nearly disappeared. BDS model should be still refined for improving the quality of precise product; fourth, the time series with three kinds of model match well, especially in common-clock link XIA3-XIA1. In terms of mitigating dayboundary jump, random walk model and constant model is superior to white noise model. The system error was up to 6 ns which would deteriorate the precision of time transfer, therefore, it was a limitation factor in the use of BDS measurements for time transfer. More attention should be paid when the different receiver was performed in the time transfer. Next, another two links POTS-XIA1 and POTSURUM were calculated for supporting our conclusion. The more result was presented in Fig. 10. The clock offset was estimated with two schemes: Scheme 1: without TDB estimation, Scheme 2: with TDB white noise model estimation. The difference with two schemes were demonstrated in Fig. 10. The time series of line blue looks more stable than that of line red whatever intra-day or cross-day which stays in the small amplitude of 0.3 ns. We can assume that there was nearly no deviation in the link POTS-URUM. It is reasonable that the receiver of same type is installed in POTS and URUM, while the POTS and XIA1 are different. When the two sides were receivers of different brands, the significant deviation exists in the clock offset with Schemes 1 and 2; when the two sides are receiver of the same brand, marginal deviation exists in the clock offset with Schemes 1 and 2. The comparison on time transfer stability with different TDB model. A more detailed comparison of the two approaches with respect to white noise model can be discussed. Figs. 11–13 shows the stability comparison of time transfer of different TDB estimation strategy. The distances between the stations range from 7000 km for the stations BRCH and XIA3 to 0.5 km for the stations XIA1 and XIA3. At present, the suggestion is that the same type receiver was employed in precise time transfer. We hope that no bias appears after all the receiver finish update of hardware. Here, we make an analysis on Allan deviation and improvement percentage of stability with different TDB estimation strategies. The random walk model and constant model is more effective than the white noise model in improving the stability of time link. There is no general rule in the enhancement of stability. At the averaging time of 100 s, 1000 s,10000 s and 86400 s, for random walk model: the improvement percentage of link XIA1-BRCH is 11.35, 3.04, 31.83 and 1.03%; the link XIA3-XIA1 is 23.67, 10.04, 3.43 and 3.04%; the link XIA3-BRCH is 9.49, 2.60, 26.44 and 52.25%; for constant model: the improvement percentage of link XIA1-BRCH is 15.23, 6.52, 30.98 and 42.87%; the link XIA3-XIA1 is 23.08, 3.14, 5.18 and 38.24%; the link XIA3-BRCH is 10.71, 32.83 and 60.07%. The random walk model and constant model indicate a considerably increased frequency Stability which is preferred to describe the bias between BDS-2 and BDS-3. The improvement extent isn’t the same at each averaging time and is different in each link, the minimal is 1% and the maximal is 60%.
W. Qin et al. / Measurement 156 (2020) 107566
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Fig. 8. Stability of TDB with three kinds of model.
Fig. 10. Clock difference with the same type receiver and different type receiver with TDB estimation. Fig. 9. Clock offset with TDB estimation or without TDB estimation.
6. Conclusion TDB is a crucial element for the fusion application of BDS-2 and BDS-3. Triggered by the finding-the bias between BDS-2 and BDS3, this article concentrated on the stochastic modeling of TDB in BDS-2/BDS-3 combination PPP processing and the TDB influence on time transfer performance with the same/different frequency source. We analyze the TDB stability and characteristic of shortterm and long-term derived from iGMAS and IGS receivers, as well as study their impact on BDS time transfer while used three different strategies to estimate TDB. More valuable information can be founded here:
(1) Without common frequency source, the TDB characteristic depends on the receiver itself, regardless of the same antenna and the receiver type. With common frequency source, the external frequency source plays more influence on the TDB characteristic than the receiver type. (2) The TDB time series has been validated that the strong relation exists in the adjacent epochs, hence, the random walk model and constant model are reasonable for TDB estimation. (3) The TDB is stable enough in the investigation period. The discrepancy from TDB estimation strategies can lead to different estimated TDB results. The TDB absolute value was nearly 2–6 ns, thus it must be corrected prior to GNSS application.
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Fig. 11. Allan deviation and improvement percentage in link XIA1-BRCH.
Fig. 12. Allan deviation and improvement percentage in link XIA3-XIA1.
(4) When the two sides were receivers of different brands, the significant deviation exists in the clock offset with TDB estimation and without TDB estimation. When the two sides are
receiver of the same brand, marginal deviation exists in the clock offset with TDB estimation and without TDB estimation.
W. Qin et al. / Measurement 156 (2020) 107566
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Fig. 13. Allan deviation and improvement percentage in link XIA3-BRCH.
(5) In terms of improving stability of time transfer, TDB estimation with the random walk model and constant model are potentially good candidate compared with white noise model. 7. Credit author statement Qin conceived and wrote the article, Ge designed the experiments, Su and Zhang enrich the languages, Wei and Yang contributed to discussions and revisions. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11703033), CAS ‘‘Light of West China” Program and Youth Promotion Committee of the Chinese Academy of Sciences. The authors gratefully acknowledge Wuhan University, iGMAS and IGS for providing the precise products. References [1] Y. Yang, Y. Xu, J. Li, C. Yang, Progress and performance evaluation of BeiDou global navigation satellite system: Data analysis based on BDS-3 demonstration system, Sci. China Earth Sci. 61 (2018) 614–624. [2] A.M. Manzino, P. Dabove, N. Gogoi, Assessment of positioning performances in Italy from GPS BDS and GLONASS constellations, Geodesy Geodyn. 9 (2018) 439–448. [3] G. Jiao, S. Song, Y. Ge, K. Su, Y. Liu, Assessment of BeiDou-3 and Multi-GNSS precise point positioning performance, Sensors 19 (2019). [4] F. Ye, Y. Yuan, J. Ou, Initial orbit determination of BDS-3 satellites based on new code signals, Geod. Geodyn. 9 (2018) 342–346.
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