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Journal Pre-proofs Contribution Analysis of QZSS to Single-frequency PPP of GPS/ BDS/ GLONASS/ Galileo Ju Hong, Rui Tu, Rui Zhang, Lihong Fan, Pengfei Zhang, Junqiang Han PII: DOI: Reference:
S0273-1177(20)30006-5 https://doi.org/10.1016/j.asr.2020.01.003 JASR 14600
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
8 October 2019 3 January 2020 6 January 2020
Please cite this article as: Hong, J., Tu, R., Zhang, R., Fan, L., Zhang, P., Han, J., Contribution Analysis of QZSS to Single-frequency PPP of GPS/ BDS/ GLONASS/ Galileo, Advances in Space Research (2020), doi: https:// doi.org/10.1016/j.asr.2020.01.003
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Contribution Analysis of QZSS to Single-frequency PPP of GPS/ BDS/ GLONASS/ Galileo Ju Hong1,2, Rui Tu1,2,3 *, Rui Zhang1,3, Lihong Fan1,3, Pengfei Zhang1,3, Junqiang Han1,3 1 National Time Service Center, Chinese Academy of Sciences, Shu Yuan Road, 710600, Xi'an, China 2 Key Laboratory of precision navigation and timing technology, Chinese Academy of Sciences, Xi’an, 710600, China 3 University of Chinese Academy of Sciences, Yu Quan Road, Beijing, 100049, China *Correspondence: [email protected]; Tel.: +86-029-8389-0326
Abstract:The Quasi-Zenith Satellite System (QZSS) established by the Japan Aerospace Exploration Agency mainly serves the Asia-Pacific region and its surrounding areas. Currently, four in-orbit satellites provide services. Most users of GNSS in the mass market use single-frequency (SF) receivers owing to the low cost. Therefore, it is meaningful to analyze and evaluate the contribution of the QZSS to SF precise point positioning (PPP) of GPS/BDS/GLONASS/Galileo systems with the emergence of GNSS and QZSS. This study compares the performances of three SF PPP models, namely the GRoup and PHase Ionospheric Correction (GRAPHIC) model, GRAPHIC with code observation model, and an ionosphere-constrained model, and evaluated the contribution of the QZSS to the SF PPP of GPS/BDS/GLONASS/Galileo systems. Moreover, the influence of code bias on the SF PPP of the BDS system is also analyzed. A two-week dataset (DOY 013–026, 2019) from 10 stations of the MGEX network is selected for validation, and the results show that: (1) For cut-off elevation angles of 15, 20, and 25º, the convergence times for the static SF PPP of GLONASS + QZSS are reduced by 4.3, 30.8, and 12.7%, respectively, and the positioning accuracy is similar compared with that of the GLONASS system. Compared with the BDS single system, the convergence times for the static SF PPP of BDS + QZSS under 15 and 25º are reduced by 37.6 and 39.2%, the horizontal positioning accuracies are improved by 18.6 and 14.1%, and the vertical components are improved by 13.9 and 21.4%, respectively. At cut-off elevation angles of 15, 20, and 25º, the positioning accuracy and precision of GPS/BDS/GLONASS/Galileo + QZSS is similar to that of GPS/BDS/GLONASS/Galileo. And the convergence times are reduced by 7.4 and 4.3% at cut-off elevation angles of 20 and 25º, respectively. In imitating dynamic PPP, the QZSS significantly improves the positioning accuracy of BDS and GLONASS. However, QZSS has little effect on the GPS-only, Galileo-only and GPS/BDS/ GLONASS/Galileo. (3) The code bias of BDS IGSO and MEO cannot be ignored in SF PPP. In static SF PPP, taking the frequency band of B1I whose multipath combination is the largest among the frequency bands as an example，the vertical component has a systematic bias of approximately 0.4–1.0 m. After correcting the code bias, the positioning error in the vertical component is lower than 0.2 m, and the positioning accuracy in the horizontal component are improved accordingly. (3) The SF PPP model with ionosphere constraints has a better convergence speed, while the positioning accuracy of the three models is nearly equal. Therefore the GRAPHIC model can be used to get good positioning accuracy in the absence of external ionosphere products, but its convergence speed is slower. Keywords Quasi-Zenith satellite system; Single-frequency precise point positioning; GNSS; Analysis of multipath; Ionospheric correction
1
1. Introduction The Quasi-Zenith Satellite System (QZSS) is a regional satellite navigation and augmentation system developed and implemented by the Japan Aerospace Exploration Agency (JAXA) (JAXA. 2016; Zhao et al. 2017). The QZSS was originally designed to improve the service capability of GPS in the Asia-Pacific region, particularly for Japan’s mountainous landscape and urban high-rise buildings with severe signal occlusion. The track is a highly inclined ellipse, known as the quasi-zenith orbit (QZO). With similar figure-eight shaped ground tracks, the QZSS satellites spend an extended time at high elevations above Japan with a separation of 8 h passing the same region (Teunissen et al. 2017). In Asia-Oceania region, all the four QZSS satellites are located above an elevation angle of 10º (Safoora et al. 2018). Since the launch of the first satellite of the QZSS in 2010, three IGSO and one GEO satellites have been launched and officially served in November 2018. Table 1 lists the detailed information of the in-orbit satellites. The QZSS is planned to be extended to a seven-satellite system by 2024. Many domestic and foreign scholars have explored and evaluated the QZSS and the contribution of QZSS to other navigation systems with the development of the QZSS. Hauschild et al. analyzed the multipath (MP), yaw attitude, and orbit of QZSS-1 using tracking data obtained from the CONGO monitoring network (Hauschild et al. 2012). Guo (2017) verified the quality of precise orbit and clock products of the QZSS by comparing with those obtained using individual analysis centers (ACs), satellite laser ranging (SLR) residuals, and precise point positioning (PPP) (Zumberge et al. 1997) tests. The orbit and clock of the QZSS have good consistency among ACs, showing agreements of 0.2–0.4 m and 0.4–0.8 ns, respectively, and the accuracy of the QZSS orbit is 0.2 m lower than those of Galileo and BDS orbits. The QZSS can improve the single-point positioning (SPP) accuracy of the GPS/BDS single system, the fixed degree of ambiguity, and the PPP accuracy of BDS; however, the accuracy of GPS PPP has not been improved (Zhang et al. 2018). Most users of the GNSS in the mass market use single-frequency (SF) receivers owing to the low cost (Ola. 2002). The mitigation of the ionospheric delay as the big challenge for SF PPP has attracted wide attention at home and abroad, and several approaches have been developed for correcting the ionospheric delay. Model correction and algorithm correction methods are commonly used. Model correction has gradually developed from the broadcast ephemeris correction model with a low precision, to the grid ionosphere map (GIM) ionospheric model with a higher precision, while GRoup and PHase Ionospheric Correction (GRAPHIC), ionospheric delay parameter estimation, and ionosphere-constrained model are commonly used among algorithm correction methods. There are advantages and disadvantages to each approach which are worthy of further verification and exploration. Meanwhile, for BDS, the multipath (MP) of the code bias is also an important factor affecting the accuracy of SF PPP (Wanninger et al. 2015), which is both elevation- and frequency-dependent bias (Lou et al. 2017; Lin et al. 2017). It must be corrected in SF PPP. With the development of multi-GNSS, extensive researches have focused on using multi-GNSS observation data to accelerate convergence speed and improve positioning accuracy of SF PPP. Lou et al. exploited the contribution of multi-GNSS to SF PPP solution based on raw observations. With more satellites available, the SF PPP multi-GNSS PPP positioning accuracy has shown improvements compared to the GPS results, while combination of BDS did not give better results due to the low GEO orbit accuracy and the negative impacts of code bias (Lou et al. 2016). However, a few scholars evaluated the contribution of the QZSS to the SF PPP of other satellite navigation systems. Based on this background, three SF PPP models commonly used are first given for the mitigation of ionospheric delay to compare different methods, including the GRAPHIC model, the GRAPHIC with code 2
observation model and ionosphere-constrained model. Then, the experimental data and processing strategies are introduced. Finally, contributions of the QZSS to GNSS in PDOP, satellites number and SF PPP are analyzed and compared. The SF PPP results of the three models and impacts of code bias of BDS are also assessed and explored.
2. Single-Frequency PPP Model The original observation equations for the code and carrier phase of satellites and receivers can be expressed as follows (Zhou et al. 2018): 𝑃𝑆𝑟,𝑔 = 𝜌𝑠𝑟 +𝑐 ∙ 𝑑𝑡𝑟 ―𝑐 ∙ 𝑑𝑡𝑆 +𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝛾𝑠𝑔 ∙ 𝐼𝑆𝑟,1 + 𝑑𝑆𝑟,𝑔 ― 𝑑𝑆𝑔 + 𝜀𝑆𝑟,𝑔 Φ𝑆𝑟,𝑔
=
𝜌𝑆𝑟
+𝑐 ∙ 𝑑𝑡𝑟 ―𝑐 ∙ 𝑑𝑡 +𝑚( 𝑆
𝑒𝑙𝑒𝑆𝑟
) ∙ 𝑍𝑤 ―
𝛾𝑆𝑔
∙
𝐼𝑆𝑟,1
+
𝜆𝑆𝑔
∙
𝑁𝑆𝑟,𝑔
+
𝑏𝑆𝑟,𝑔
―
(1) 𝑏𝑆𝑔
+
Here, 𝑔, r and S represent the frequency, receiver, and satellite system, respectively, 𝜙𝑆𝑟,𝑔
are the code and carrier phase measurements, respectively,
𝜌𝑆𝑟
𝜉𝑆𝑟,𝑔 𝑃𝑆𝑟,𝑔
(2) and
is the geometric distance between
the phase centers of the satellite and receiver antennae (at the signal transmission and reception times), 𝑑𝑡𝑟 is the receiver clock error, c is the speed of light, and 𝑑𝑡𝑆 is the satellite clock error. 𝑚(𝑒𝑙𝑒𝑆𝑟) is the projection function from the tropospheric zenith direction of the station to the inclined direction of the station and satellite, 𝑒𝑙𝑒𝑆𝑟 is the elevation angle of the satellites, 𝑍𝑤 is the tropospheric wet delay error of the zenith component (ZWD), 𝐼𝑆𝑟,1 is the slant ionospheric delay at the frequency band of 𝑓𝑆1, 2
𝛾𝑆𝑔 is the frequency-dependent multiplier factor (𝛾𝑆𝑔 = (𝑓𝑆1/𝑓𝑆𝑔) ), 𝑑𝑆𝑟,𝑔 and 𝑑𝑆𝑔 are uncalibrated code delays (UCDs) at the satellite and receiver ends, respectively. 𝑏𝑆𝑟,𝑔 and 𝑏𝑆𝑔 are the uncalibrated phase delays (UPDs) at the satellite and receiver ends, respectively, 𝜆𝑆𝑔 is the wavelength of the carrier phase, 𝑁𝑆𝑟,𝑔 is the integer ambiguity, which is observed at the frequency band of 𝑓𝑆𝑔. 𝜀𝑆𝑟,𝑔 and 𝜉𝑆𝑟,𝑔 denote the code and carrier phase measurement noises, respectively. The phase center offset and variation, relativistic effects, tidal loading, ocean tides, earth rotation effects, and phase wind-up can be corrected using existing models (Dach et al. 2007) and are therefore not explicitly included in Equations (1) and (2). The satellite clock error is corrected using precise clock error products, which contain the satellite UCD of the ionosphere-free code observations. Thus, we have, 𝑐 ∙ 𝑑𝑡𝑆𝐼𝐹12 = 𝑐 ∙ 𝑑𝑡𝑆 + (𝛼𝑆12 ∙ 𝑑𝑆1 + 𝛽𝑆12 ∙ 𝑑𝑆2) = 𝑐 ∙ 𝑑𝑡𝑆 + 𝑑𝑆𝐼𝐹12
(3)
In this study, the observation value at the frequency band of 𝑓𝑆1 was used. Equation (3) is substituted into Equations (1) and (2) and linearized to obtain the following after applying precise clock error products:
{
𝜌𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝐼𝑆𝑟,1 + 𝑑𝑆𝑟,1 ― 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2 + 𝜀𝑆𝑟,1 𝑆
𝑙𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 ― 𝐼𝑆𝑟,1 + 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 + 𝑏𝑟,1 ― 𝑏𝑆1 + 𝑑𝑆𝐼𝐹12 + 𝜉𝑆𝑟,1
(4)
Here, 𝜌𝑆𝑟,1 and 𝑙𝑆𝑟,1 denote the observed minus computed values of the code and carrier phase observables, respectively, 𝝁𝑆𝑟 is the unit vector of the component from the receiver to the satellite, and x is the vector of the receiver position increments relative to the a priori position. The symbols in the equation can be expressed as:
{
2
2
2
2
2
2
𝛼𝑆𝑚𝑛 = (𝑓𝑆𝑚) /((𝑓𝑆𝑚) ― (𝑓𝑆𝑛) ),𝛽𝑆𝑚𝑛 = ― (𝑓𝑆𝑛) /((𝑓𝑆𝑚) ― (𝑓𝑆𝑛) ) 𝑑𝑆𝐼𝐹𝑚𝑛 = 𝛼𝑆𝑚𝑛 ∙ 𝑑𝑆𝑚 + 𝛽𝑆𝑚𝑛 ∙ 𝑑𝑆𝑛 𝐷𝐶𝐵𝑆𝑃1𝑃2 = 𝑑𝑆1 ― 𝑑𝑆2
(5)
2.1 GRAPHIC Model The GRAPHIC model eliminates the ionospheric delay error through the linear combination of 3
code and carrier observations, based on the fact that the ionospheric delay errors in the code and carrier observations are equal in magnitude and opposite in sign (Gold et al. 1994). According to Equation (4), the linearized observation equation obtained by re-parameterization is as follows: 1 𝐺𝑆𝑟,1 = ∙ (𝜌𝑆𝑟,1 + 𝑙𝑆𝑟,1) = 𝝁𝑆𝑟 ∙ 𝒙 +𝑐 ∙ 𝑑𝑡𝑆𝑟 +𝑚 2
(𝑒𝑙𝑒𝑆𝑟) ∙ 𝑍𝑤 +1/2 ∙ 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 +1/2 ∙ (𝜀𝑆𝑟,1 + 𝜉𝑆𝑟,1)
(6)
In the equation,
{
𝑑𝑡𝑟 = 𝑑𝑡𝑟 + [1/2 ∙ (𝑏𝑆𝑟,1 + 𝑑𝑆𝑟,1)]/𝑐 𝑁𝑆𝑟,1 = 𝑁𝑆𝑟,1 + (𝑑𝑆𝐼𝐹12 ― 𝑏𝑆1 ― 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝜆𝑆1
(7)
The estimated unknown parameter vector is: 𝑿 = [𝒙 𝑑𝑡𝑟 𝑍𝑊 𝑁𝑆𝑟,1]
(8)
The GRAPHIC combined observables are dominated by the noise of the code measurements and the GRAPHIC combination from single-frequency code and phase observations produces a rank-deficient mathematical problem, which needs a long convergence time (Montenbruck. 2003;
Shi et al. 2012).
2.2 GRAPHIC with Code Observation Model Compared with GRAPHIC PPP, a code observation equation is added to the model. The equation includes the ionospheric delay error, which can be corrected by the external products. After correcting the ionospheric error, the linearized observation equation is obtained as follows: 𝜌𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝜀𝑆𝑟,1 𝐺𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 1/2 ∙ 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 + 1/2 ∙ (𝜀𝑆𝑟,1 + 𝜉𝑆𝑟,1)
{
(9)
In the equation,
{
𝑑𝑡𝑟 = 𝑑𝑡𝑟 + (𝑑𝑆𝑟,1 ― 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝑐 𝑁𝑆𝑟,1 = 𝑁𝑆𝑟,1 + (𝑑𝑆𝐼𝐹12 ― 𝑏𝑆1 + 𝑏𝑆𝑟,1 ― 𝑑𝑆𝑟,1 + 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝜆𝑆1
(10)
The estimated unknown parameter vector is: 𝑿 = [𝒙 𝑑𝑡𝑟 𝑍𝑊 𝑁𝑆𝑟,1]
(11)
The model includes a code observation equation which provides code reference, and solves the rank deficiency problem exiting in the GRAPHIC model.
2.3 Ionosphere-constrained Model The ionosphere-constrained model adds virtual ionospheric observations from the external model with their corresponding constraints, which can be expressed as follows. 𝜌𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝐼𝑆𝑟,1 + 𝜀𝑆𝑟,1 𝑙𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 ― 𝐼𝑆𝑟,1 + 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 + 𝜉𝑆𝑟,1 𝐼𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑 = 𝐼𝑆𝑟,1
(12)
{
(13)
{ In the equation,
𝑑𝑡𝑟 = 𝑑𝑡𝑟 + 𝑑𝑆𝑟,1/𝑐 𝑆 𝑁𝑆𝑟,1 = 𝑁𝑆𝑟,1 + (𝑑𝑆𝐼𝐹12 ― 𝑏𝑆1 + 𝑏𝑆𝑟,1 ― 𝑑𝑆𝑟,1 ― 3𝛽12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝜆𝑆1 𝑆
𝐼𝑆𝑟,1 = 𝐼𝑆𝑟,1 ―𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑃1𝑃2 4
The estimated unknown parameter vector is: 𝑿 = [𝒙 𝑑𝑡𝑟
𝐼𝑆𝑟,1
𝑍𝑊 𝑁𝑆𝑟,1]
(14)
The ionosphere-constrained model involves constraining the ionospheric model correction value as a virtual observation value and then estimating the ionospheric delay parameters. Compared with model correction, reasonable weighting of the virtual observation value can accelerate the SF PPP initialization time. The DCB error must be considered in SF PPP. The DCB products can be provided by the analysis center of Center for Orbit Determination in Europe (CODE), the German Aerospace Center (DLR), and the Institute of Geodesy and Geophysics (IGG). Wang et al. verified that the above three products have the same accuracy (Wang et al. 2016). In addition, the broadcast ephemeris can broadcast the timing group delay (TGD) of the satellite clock in real time, which can be converted to DCB (Ge et al. 2017). The TGD parameters can be obtained in real time; however, the DCB parameters are not available. The two can be converted to each other for use depending on the condition. For the analysis, the three SF PPP models with different ionospheric correction approaches, namely the GRAPHIC, GRAPHIC with a code observation, and ionosphere-constrained model, are represented by SF1, SF2, and SF3, respectively.
3. Experimental Data and Processing Strategies A two-week dataset (DOY 013–026, 2019) from 10 stations of the MGEX network (Montenbruck et al. 2011) was selected for validation. Figure 1 and Table 1 present the distribution and details of the stations. The sampling interval of the observation is 30 s. Precise satellite clock product with 30 s interval and precise satellite orbit products with 5 min interval were employed from GFZ (gbm*.*). In this study, the SF PPP data were tested using the first frequency signal observations of GPS/GLONASS/BDS/Galileo/QZSS simplified as G/R/C/E/J. The satellite cut-off elevation angle was set depending on the test objective. In the SF1/SF2/SF3 models, the estimated parameters include the coordinate parameters, the receiver clock error parameter, ZWD parameters, ionospheric delay parameter, and ambiguity parameter. For the coordinate parameters, the constant estimation method was used in static PPP, and a random walk model was used in the dynamic PPP, in which the process noise was set to 100𝑚2. For the tropospheric delay parameters, the Saastamoinen model (Saastamoinen, 1972) and the GMF projection function (Roken et al. 1991) were used to calculate the tropospheric dry-component delay error and partial wet-component delay error, and the residual of the ZTW component delay was estimated as a piece-wise constant. The ambiguity parameter was considered a constant when satellite cycle slip did not occur and was re-initialized when it did. The receiver clock error was estimated as white noise. The phase center offset (PCO) and phase center variations (PCV) of the satellite and receiver antennae were corrected using igs14.atx_2032 (Rebischung et al. 2016) file provided by the International GNSS Service (IGS) (Dow et al. 2009). As the antenna file rarely updates PCO and PCV parameters of the receivers of the BDS, Galileo, and 5
QZSS, we used the correction value of GPS to correct them. The DCB error was corrected using the DCB products released by the IGG. The GIM products issued by CODE were used to correct the ionospheric errors in the SF2 model and as virtual ionospheric observations in the SF3 model. The GIM model has an accuracy of 2–8 total electron content unit (TECU), which is equivalent to a ranging error of 0.32–1.28 m for the GPS L1 frequency signal. The formulae for calculating the ionospheric delay errors using GIM products have been previously introduced (Cai et al. 2017) and are not detailed herein. The stochastic models of the observations represented by a sinusoidal function and based on the elevation angle of the satellite were utilized to reduce the effect of elevation angle-related errors (Witchayangkoon. 2000). The GPS, BDS, GLONASS, Galileo, and QZSS phase observation precisions were set to 0.003, 0.006, 0.004, 0.003, and 0.006 m, respectively, and the code observation precisions were set to 0.3, 1.2, 1.2, 0.3, and 1.2 m. Because of the poor accuracy of the orbit and clock products of the BDS GEO satellite (Guo et al. 2017), the weight of the BDS GEO satellite observations was reduced by 10 fold (Li et al. 2017). In the SF3 model, the stochastic model of the virtual observations can affect the positioning results. Because of the limited accuracy of the virtual observations, a higher weight was given to improve the convergence speed of PPP at the beginning of the estimation, and the weight was then gradually reduced to improve the positioning accuracy. The variance in the virtual observations was defined as follows. 𝜎2𝐼 (𝑖) = (1 + 0.02 ∙ 𝑖) ∙ 𝜎2𝑖𝑜𝑛(𝑖)/0.03
(15)
Here, i refers to the number of epochs, and 𝜎2𝑖𝑜𝑛 refers to the priori variance of the ionospheric delay, which was determined by the ionospheric model. The GIM model was used in the test, and the temporal and spatial correlation functions can be expressed as (Gao et al. 2017). σ2ion =
{
σ2ion,0
,
sin2E 2 𝜎𝑖𝑜𝑛,0
𝑡 < 8 𝑜𝑟 𝑡 > 20 𝑜𝑟 𝑡 > 𝜋/3 +
𝜎2𝑖𝑜𝑛,1
∙ cos(B) ∙ cos
(
),
𝑡 ― 14 12 ∙ 𝜋
other
(16)
Here, t is the local time at the ionospheric pierce point (IPP) in h, B refers to the satellite elevation angle, and 𝜎2𝑖𝑜𝑛,0 and 𝜎2𝑖𝑜𝑛,1 denote the variance in the zenith ionospheric delay and the zenith ionospheric delay variation, respectively, which are set to 0.09 𝑚2. The coordinates of the station in the file (*.snx) issued by the IGS were taken as reference values for the accuracy analysis, and the convergence condition was defined such that the positioning errors of the east (E), north (N), and up (U) components are lower than 0.3, 0.3, and 0.5 m, respectively.
6
Figure 1. Distribution of the selected MGEX stations Table 1. Detailed information of the receivers. Manufacturer LEICA TRIMBLE
Receiver type GR25 NETR9 POLARX5 POLARX4 TRE_G3TH DELTA
SEPTENTRIO JAVAD
Station name ALIC CCJ2/GMSD/KOUC/STK2/TSK2 CEDU/DARW TWTF JOG2
4. Results and Discussion 4.1 Contribution of QZSS in PDOP and Satellites Number As a regional satellite navigation system, the QZSS mainly serves the Asia-Pacific and its surrounding
areas.
To
analyze
the
contribution
of
the
QZSS
to
the
SF
PPP
of
GPS/BDS/GLONASS/Galileo, the influences of the QZSS on the number of satellites and position dilution of precision (PDOP) values of each system in the service area were first analyzed. Figure 2 and
Figure
3
show
the
average
satellite
number
and
PDOP
distribution
of
the
GPS/BDS/GLONASS/Galileo single system and GPS/BDS/GLONASS/Galileo and QZSS dual system in 2019 DOY 024 days. The global region was divided into grids of 2.5º × 5º. Subsequently, the PDOP values of each grid were calculated using the broadcast ephemeris (brdm0200.19p). The cut-off elevation angle was set to 7.5º, and the time resolution was 10 min. For every grid, one calculation result of the PDOP was obtained at each epoch, which generated a PDOP sequence. Thereafter, we could determine the mean value from this sequence for every grid. Figure 2 shows that the QZSS increases the number of visible satellites of GPS/BDS/ GLONASS/Galileo single system in the Asia-Pacific and southern Indian Ocean regions, particularly in southeastern Asia and the western Pacific. As shown in Figure 3, the QZSS system significantly improved the PDOP values of the GLONASS and Galileo single systems in the Asia-Pacific region. GPS has full navigation service capability worldwide and good graphic intensity of satellites. Therefore, the QZSS increases the number of GPS satellites in the Asia-Pacific region but has little influence on GPS PDOP. In the Asia-Pacific region, the PDOP of BDS is smaller and the QZSS has little impact on its. 7
Figure 2. Satellite distribution of GPS, BDS, GLONASS, and GALILEO single system, and QZSS and GPS, BDS, GLONASS, GALILEO dual system (DOY 016 on DOY 024, 2019)
Figure 3. PDOP distribution of GPS, BDS, GLONASS, and Galileo single system, and QZSS and GPS, BDS, GLONASS, Galileo dual system (DOY 016 on DOY 024, 2019)
4.2 MP Combinations Analysis of Code Measurements for BDS The MP effect, which is mainly due to signal reflection, is an intractable error source for GNSS observations, particularly for code observations. MP combinations after factored out with an average term contain MP errors and combined observation noise of the code and phase measurements, which is calculated by combining one-frequency code observation and dual-frequency phase observation (Wanninger et al. 2015; Zhang et al. 2019). Given that the MP error and noise of phase are smaller than that of the code, the former can be neglected. The BDS constellation includes the geostationary orbit (GEO), inclined geostationary earth orbit (IGSO), and medium earth orbit (MEO). The MP combinations time series of BDS IGSO and MEO are elevation-dependent and usually neglected for other GNSS systems. Taking GPS as an example, Figure 4 shows the MP combinations time series and elevation angles of BDS GEO/IGSO/MEO and GPS satellites. The MP combinations value of the BDS IGSO/MEO satellite decreases with the increase in the satellite elevation angle, while those of the GPS have no obvious correlation with the satellite elevation angle. As the BDS GEO satellites are visible at almost constant elevation angles, the MP combination time series has no obvious variation. Therefore, we limited our analysis to BDS MEO and IGSO satellites. 8
Figure 4. MP combinations and elevation angles time series of GPS and BDS at the frequency band L1and B1 respectively The BDS-2 satellites provide three frequency bands, i.e., B1I, B2I and B3I, centered at 1561.098, 1207.140 and 1268.520 MHz, respectively. Figure 5 shows the relationship between the MP combinations values of the three frequency bands and elevation angles. For different types of satellites, the MP combinations value of MEO is larger than that of IGSO at the B1I and B2I frequency bands. The MP combinations effect at the frequency band of B1I is the highest and that at the B3I frequency is the lowest. Many experts and scholars have modeled and corrected the MP combinations effect of the BDS-2 MEO/IGSO satellite at different frequencies based on the relationship between the satellite elevation angles and the MP combinations value. In this study, we used the MP combinations correction model proposed by Wanninger et al. (Wanninger et al. 2015). Figure 6 shows the MP combinations time series of the uncorrected and corrected measurements. After correction, the MP combinations value decrease, and the correlation with the elevation angle decreases. Because the MP combinations effect at the B1I frequency band is the biggest, the MP combinations effect on the SF PPP was tested using the observation at the B1I frequency band. Figure 7 shows the positioning bias in the static model before and after correction. The U component has a systematic bias of approximately 0.4–1.0 m when the MP combinations effect is not considered. The positioning error of the U component is lower than 0.2 m, and the positioning accuracies of the E and N components are improved accordingly after correction.
9
Figure 5. MP combinations and elevation angles time series of BDS IGSO and MEO at B1/B2/B3 frequency bands. The black line represents the elevation angles, and the yellow, red, and green lines represent the MP combinations time series at the B1I, B2I, and B3I frequency bands, respectively
Figure 6. MP combinations time-series of BDS IGSO and MEO for uncorrected and corrected B1I, B2I, and B3I measurements
Figure 7. Comparison of E/N/U components of DARW Station before and after correction of BDS MP combinations at B1I frequency in DOY 013-DOY 019
10
The QZSS has three frequency bands for Navigation and Timing Service (PNT) of L1, L2 and L5. To analyze the relation between MP combinations and elevation angles of the QZSS, the MP combinations time series and elevation angles of the QZSS GSO and GEO satellites are shown in Figure 8. Although the elevation angles of the QZSS GSO satellite vary significantly, the MP combination value does not obviously change. The MP combination time series of the QZSS GEO satellites have no obvious variation at almost constant elevation angles. Therefore, the MP combinations of the QZSS have no significant correlation with elevation angles.
Figure 8. MP combinations and elevation angles time series of QZSS GSO/GEO satellites at L1 frequency band. The black line represents the elevation angles, and the red line represents the MP combinations time series of L1
4.3 Comparison and Analysis of Positioning Results of Three SF PPP Models To compare and analyze the positioning results of the three models, a test was performed using GPS observations of 10 stations in 2019 DOY 013-026 days. The satellite cut-off elevation angle was set to 7º. Figure 9 shows the final positioning results of the one-day solution for all days of the three models. The positioning result for most stations obtained using the three models reaches the decimeter level to centimeter level in the E/N/U components. The mean RMS values of the single-day solution bias of SF1, SF2, and SF3 models are (0.059, 0.044, 0.074) m, (0.059, 0.044, 0.075) m, and (0.062, 0.043, 0.074) m, which are largely equal.
11
Figure 9. Bias distribution of single-day static solution of the three models for DOY 013 -DOY 026 days in all stations in 2019
Table 2 summarizes the average convergence time, the STD, and RMS values of one-day solutions for all stations after convergence in static PPP. From the point of convergence time, the convergence speed of the SF3 model with the GIM ionospheric constraint is higher than those of the SF1 and SF2 models, and the convergence speeds of SF3 and SF2 are largely equal. From the perspective of positioning accuracy, the positioning results of the three models are largely similar, and the RMS value of the SF1 model in the U component is slightly lower than those of the SF1 and SF2 models. For the SF3 model, the higher weight of the virtual observation at the beginning helps improve the convergence speed. The weight is lower after the convergence; therefore, it has no positive influence on the positioning accuracy after convergence. Unlike the SF1 model, the SF2 model includes a code observation equation, which provides code reference and solves the rank deficiency problem observed in the SF1 model. However, the convergence speed is not improved, and the positioning accuracy of the U component is somewhat reduced. The reason for the low precision may be not only the low precision of the code, but also the limited accuracy of the ionospheric model products. Therefore, when there is no external ionosphere product, the SF1 model can get similar 12
positioning accuracy like the other two models with external ionosphere product.
Table 2. Average values of the convergence time, STD, and RMS of static SF PPP models for all the stations
Model SF1 SF2 SF3
Convergence time (min) 68.84 67.84 52.19
E 0.037 0.039 0.036
STD (m) N 0.017 0.018 0.018
U 0.045 0.048 0.047
E 0.063 0.064 0.062
RMS (m) N U 0.029 0.069 0.029 0.075 0.029 0.074
To further study the characteristics of the positioning results of the three models, Figure 10 shows the residuals of the GRAPHIC observation of the SF1 and SF2 models at the CEDU station. As the observation equations of the two methods are consistent, the residual magnitude and RMS are equal. Figure 11 shows the residuals of the code observations for the SF2 and SF3 models. The residuals of the SF3 code observations are lower than the SF2 residuals, showing that the ionospheric parameter estimation method can better eliminate the ionospheric error than the model correction method.
Figure 10. Residual of GRAPHIC observation of SF1 and SF2 models at CEDU station on DOY 015 in 2019
Figure 11. Residual of code of SF2 and SF3 models at CEDU station on DOY 015 in 2019 13
Figure 12 shows the estimated ZWD parameters, receiver clock error parameters, and their differences for the three models. The ZWD estimated using the three models are largely equal. The receiver clock error parameters estimated by the SF2 and SF3 models are similar and stable; however, the receiver clock error parameters estimated by the SF1 model differ from those estimated by SF2 and SF3 models with a system bias. The reason may be that the receiver clock error parameters estimated by the SF2 and SF3 models absorbed a part of the ionospheric errors. However, the STD values of the difference in the receiver clock errors between the three models are low, indicating the presence of a error with good consistency.
Figure 12. Estimated ZWD and receiver clock errors of the SF1, SF2, and SF3 models, and the differences in ZWD and receiver clock errors between the SF1, SF2, and SF3 models
4.4 Contribution of QZSS in SF PPP As the QZSS satellite has a high satellite elevation angle, we selected cut-off elevation angles of 15, 20, and 25º to study the influence of the QZSS on the SF PPP of GPS/BDS/GLONASS/Galileo. The SF3 model with less convergence time was employed for the test. Considering the number of satellite systems in the stations, the CCJ2/CEDU/DARW/GMSD/STK2/TSK2 /TWTF station, CCJ2/CEDU/JOG2/KOUC/STK2/TSK2/TWTF/ALIC station, CCJ2/CEDU/DARW/GMSD/JOG2/ STK2/TSK2 station, CEDU/DARW station and CEDU/DARW/GMSD/KOUC station (in 2019 DOY 013-019 days) were selected to test the contribution of QZSS to the SF PPP of GPS-only, GLONASS-only, Galileo-only, BDS-only and GPS/BDS/GLONASS/Galileo respectively. Figure 13 shows the positioning bias comparison between GPS/BDS/GLONASS/Galileo single system and GPS/BDS/GLONASS/Galileo + QZSS dual-system, GPS/BDS/GLONASS/Galileo and GPS/BDS/GLONASS/Galileo + QZSS at cut-off altitude angles of 15, 20, and 25º, respectively, in the static PPP model. Table 3 lists the average values of the convergence time, STD, and RMS of the 14
one-day solution after convergence. Figure 13 (a) shows the positioning bias comparison between GPS-only and GPS + QZSS. Fig. 13 (c) shows the positioning bias comparison between Galileo-only and Galileo + QZSS. At cut-off elevation angles of 15, 20, and 25º，the QZSS does not significantly improve the accuracy and convergence time of GPS-only or Galileo-only . Figure 13 (b) shows the positioning bias comparison between GLONASS and GLONASS + QZSS. At cut-off elevation angles of 15, 20, and 25º, the positioning accuracy and precision of GLONASS + QZSS is similar to that of GLONASS-only, and the convergence times are reduced by 4.3, 30.8, and 12.7%, respectively. Figure 13 (d) shows the positioning bias comparison between BDS-only and BDS + QZSS. At the different cut-off elevation angles, adding the QZSS system helped reduce the convergence time and improve the positioning accuracy, particularly in the U direction. When the cut-off elevation angle of the satellite is 25º, the convergence time of the BDS-only is considerable or even non-convergent in some cases. Therefore, the positioning results were not included in Table 3. At cut-off elevation angles of 15 and 20º, the positioning precision of BDS + QZSS is similar to that of BDS-only, the convergence times are reduced by 37.6 and 39.2%, and the positioning accuracies in horizontal component are improved by 18.6 and 14.1% and the positioning accuracies in vertical component components are increased by 13.9 and 21.4%, respectively. Figure 13 (e) shows the positioning bias comparison between GPS/BDS/GLONASS/Galileo and GPS/BDS/GLONASS/Galileo + QZSS. At cut-off elevation angles of 15, 20, and 25º, the QZSS has little impact on the positioning accuracy and precision of GPS/BDS/ GLONASS/Galileo, while the convergence times are reduced by 7.4 and 4.3% at cut-off elevation angles of 20 and 25º, respectively. Similar to GPS-only and Galileo-only, the GPS/GLONASS/BDS/Galileo has longer initialization time under the low cut-off after adding the QZSS. This indicates the lower accuracy of orbit and clock of the QZSS. However, the QZSS helps get shorter initialization time of GPS-only, Galileo-only and GPS/BDS/GLONASS/Galileo with the increase of cut-off elevation angles.
(a)
(b)
15
(c)
(d)
(e) Figure 13. SF static positioning bias comparison in DOY 016 days in 2019 under cut-off elevation angles of 15º/20º/25º: (a) Positioning bias comparison between GPS-only and GPS + QZSS, (b) Positioning bias comparison between GLONASS-only and GLONASS + QZSS, (c) Positioning bias comparison between Galileo-only and Galileo + QZSS, (d) Positioning bias comparison between BDS and BDS-only + QZSS, (e) Positioning bias comparison between GPS/BDS/GLONASS/Galileo and GPS/BDS/ GLONASS/Galileo + QZSS Table 3. Average values of the convergence time, STD, and RMS for static SF PPP of all the stations under cut-off elevation angles of 15º/20º/25º Cut-off elevation angles (°)
15
Satellite type
Convergenc e time (min)
G G+J R R+J C C+J E
47.26 51.74 100.54 96.24 290.75 181.43 61.53
STD (m)
RMS (m)
E
N
U
E
N
U
0.033 0.033 0.037 0.035 0.051 0.043 0.033
0.018 0.017 0.020 0.019 0.016 0.019 0.019
0.058 0.065 0.064 0.066 0.095 0.090 0.054
0.057 0.056 0.045 0.045 0.076 0.056 0.059
0.033 0.033 0.031 0.029 0.061 0.056 0.035
0.078 0.087 0.094 0.093 0.151 0.130 0.078
16
20
25
E+J
59.42
0.033
0.019
0.052
0.058
0.035
0.073
GRCE
44.66
0.029
0.013
0.052
0.054
0.037
0.072
GRCE+J
0.029
0.013
0.054
0.054
0.038
0.075
G G+J R R+J C C+J E E+J
48.18 57.15 57.67 120.44 83.36 302.75 183.96 70.26 73.12
0.034 0.034 0.041 0.033 0.046 0.036 0.030 0.028
0.018 0.017 0.019 0.017 0.014 0.017 0.017 0.016
0.069 0.069 0.072 0.072 0.123 0.125 0.059 0.057
0.058 0.057 0.053 0.054 0.064 0.044 0.057 0.055
0.034 0.033 0.031 0.021 0.056 0.059 0.034 0.034
0.102 0.101 0.112 0.122 0.191 0.150 0.094 0.084
GRCE
48.48
0.029
0.014
0.056
0.054
0.039
0.091
GRCE+J
0.029
0.014
0.056
0.054
0.039
0.089
G G+J R R+J C C+J E E+J
44.88 74.92 76.04 148.78 129.91 91.42 90.16
0.032 0.031 0.040 0.038 0.031 0.031
0.017 0.017 0.020 0.021 0.018 0.017
0.083 0.082 0.078 0.081 0.067 0.066
0.055 0.054 0.058 0.054 0.058 0.057
0.034 0.033 0.032 0.031 0.034 0.034
0.128 0.127 0.107 0.126 0.126 0.118
GRCE
67.79
0.028
0.014
0.059
0.052
0.039
0.111
GRCE+J
64.86
0.028
0.014
0.059
0.053
0.039
0.110
Figure 14 shows the positioning biases comparison between GPS/BDS/GLONASS/Galileo single system and GPS/BDS/GLONASS/Galileo + QZSS, GPS/BDS/GLONASS/Galileo and GPS/BDS/ GLONASS/Galileo + QZSS at cut-off altitude angles of 15, 20, and 25º, respectively, in imitating dynamic SF PPP. Similar to the static positioning results, the QZSS has no obvious effect on the SF PPP of GPS-only, GALILEO-only and GPS/BDS/ GLONASS/Galileo at these cut-off elevation angles; nevertheless, the QZSS significantly improved the positioning accuracy of GLONASS-only and BDS-only, particularly of BDS-only. Compared to GPS-only and Galileo-only, GLONASS-only and BDS-only show poor performance in SF PPP, which are mainly caused by the satellite-specific inter-frequency bias (IFB) of GLONASS and code bias of BDS except the reason of Geometric Dilution of Precision (GDOP) (Lou et al. 2016). The IFB of GLONASS at the receiver is strongly related to the ionosphere and then influences on the SF positioning result. The elevation-dependent code bias of BDS has negative impact on SF PPP results, especially in the vertical component. Based on the experimental results, the QZSS greatly improves the positioning results of BDS-only and GLONASS-only, and GLONASS and QZSS are coupled together to provide better positioning service as well as BDS and QZSS in static and imitating dynamic SF PPP.
17
(a)
(b)
(c)
(d)
(e) Figure 14. SF imitating dynamic positioning bias comparison on DOY 016 days in 2019 under cut-off elevation angles of 15º/20º/25º: (a) Positioning bias comparison between GPS-only and GPS + QZSS, (b) Positioning bias comparison between GLONASS-only and GLONASS + QZSS, (c) Positioning bias comparison between Galileo-only and Galileo + QZSS, (d) Positioning bias comparison between BDS and BDS-only + QZSS, (e) Positioning bias comparison between GPS/BDS/GLONASS/Galileo 18
and GPS/BDS/ GLONASS/Galileo + QZSS
5. Conclusions In this study, the observations provided by MGEX and the precise product of IGS were used to test and analyze the positioning results of the contribution of the QZSS to GPS/BDS/ GLONASS/Galileo, three SF PPP models and the code bias od BDS. The QZSS increases the number of visible satellites of GPS/BDS/ GLONASS/Galileo single system particularly in southeastern Asia and the western Pacific, and improves PDOP value of GLONASS and Galileo single system. Moreover, the QZSS greatly improves the SF PPP performance of BDS-only and GLONASS-only. Compared to GPS-only, Galileo-only and GPS/GLONASS/BDS/Galileo, the QZSS helps get longer initialization time under the low cut-off elevation angles due to the lower accuracy of orbit and clock of the QZSS,
but get shorter initialization time with the increase of the cut-off altitude angles. The
code bias of BDS IGSO and MEO elevation-dependent has negative impact on the SF PPP accuracy especially in the vertical component. In static SF PPP, the U component had a systematic bias of approximately 0.4–1.0 m. After correcting the MP value, the positioning error of the U component was found to be lower than 0.2 m, and the positioning accuracy of the E and N components improved accordingly. In addition, the MP combinations of the QZSS have no significant correlation with elevation angles. The positioning result of the three models reached the decimeter level to centimeter level for the E/N/U components. The ionosphere-constrained model exhibited the fastest convergence, whereas the positioning accuracies of the three models were largely equal. Therefore, the GRAPHIC model can get similar positioning accuracy like the other two models in the absence of external ionosphere products, but its convergence speed is slower.
Acknowledgments: The work is partly supported by the program of National Key Research and Development Plan of China (Grant No: 2016YFB0501804), National Natural Science Foundation of China (Grant No: 41674034, 41974032, 11903040) and Chinese Academy of Sciences (CAS) programs of “Pioneer Hundred Talents” (Grant No: Y923YC1701), and “The Frontier Science Research Project” (Grant No: QYZDB-SSW-DQC028). References
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S0273-1177(20)30006-5 https://doi.org/10.1016/j.asr.2020.01.003 JASR 14600
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
8 October 2019 3 January 2020 6 January 2020
Please cite this article as: Hong, J., Tu, R., Zhang, R., Fan, L., Zhang, P., Han, J., Contribution Analysis of QZSS to Single-frequency PPP of GPS/ BDS/ GLONASS/ Galileo, Advances in Space Research (2020), doi: https:// doi.org/10.1016/j.asr.2020.01.003
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Contribution Analysis of QZSS to Single-frequency PPP of GPS/ BDS/ GLONASS/ Galileo Ju Hong1,2, Rui Tu1,2,3 *, Rui Zhang1,3, Lihong Fan1,3, Pengfei Zhang1,3, Junqiang Han1,3 1 National Time Service Center, Chinese Academy of Sciences, Shu Yuan Road, 710600, Xi'an, China 2 Key Laboratory of precision navigation and timing technology, Chinese Academy of Sciences, Xi’an, 710600, China 3 University of Chinese Academy of Sciences, Yu Quan Road, Beijing, 100049, China *Correspondence: [email protected]; Tel.: +86-029-8389-0326
Abstract:The Quasi-Zenith Satellite System (QZSS) established by the Japan Aerospace Exploration Agency mainly serves the Asia-Pacific region and its surrounding areas. Currently, four in-orbit satellites provide services. Most users of GNSS in the mass market use single-frequency (SF) receivers owing to the low cost. Therefore, it is meaningful to analyze and evaluate the contribution of the QZSS to SF precise point positioning (PPP) of GPS/BDS/GLONASS/Galileo systems with the emergence of GNSS and QZSS. This study compares the performances of three SF PPP models, namely the GRoup and PHase Ionospheric Correction (GRAPHIC) model, GRAPHIC with code observation model, and an ionosphere-constrained model, and evaluated the contribution of the QZSS to the SF PPP of GPS/BDS/GLONASS/Galileo systems. Moreover, the influence of code bias on the SF PPP of the BDS system is also analyzed. A two-week dataset (DOY 013–026, 2019) from 10 stations of the MGEX network is selected for validation, and the results show that: (1) For cut-off elevation angles of 15, 20, and 25º, the convergence times for the static SF PPP of GLONASS + QZSS are reduced by 4.3, 30.8, and 12.7%, respectively, and the positioning accuracy is similar compared with that of the GLONASS system. Compared with the BDS single system, the convergence times for the static SF PPP of BDS + QZSS under 15 and 25º are reduced by 37.6 and 39.2%, the horizontal positioning accuracies are improved by 18.6 and 14.1%, and the vertical components are improved by 13.9 and 21.4%, respectively. At cut-off elevation angles of 15, 20, and 25º, the positioning accuracy and precision of GPS/BDS/GLONASS/Galileo + QZSS is similar to that of GPS/BDS/GLONASS/Galileo. And the convergence times are reduced by 7.4 and 4.3% at cut-off elevation angles of 20 and 25º, respectively. In imitating dynamic PPP, the QZSS significantly improves the positioning accuracy of BDS and GLONASS. However, QZSS has little effect on the GPS-only, Galileo-only and GPS/BDS/ GLONASS/Galileo. (3) The code bias of BDS IGSO and MEO cannot be ignored in SF PPP. In static SF PPP, taking the frequency band of B1I whose multipath combination is the largest among the frequency bands as an example，the vertical component has a systematic bias of approximately 0.4–1.0 m. After correcting the code bias, the positioning error in the vertical component is lower than 0.2 m, and the positioning accuracy in the horizontal component are improved accordingly. (3) The SF PPP model with ionosphere constraints has a better convergence speed, while the positioning accuracy of the three models is nearly equal. Therefore the GRAPHIC model can be used to get good positioning accuracy in the absence of external ionosphere products, but its convergence speed is slower. Keywords Quasi-Zenith satellite system; Single-frequency precise point positioning; GNSS; Analysis of multipath; Ionospheric correction
1
1. Introduction The Quasi-Zenith Satellite System (QZSS) is a regional satellite navigation and augmentation system developed and implemented by the Japan Aerospace Exploration Agency (JAXA) (JAXA. 2016; Zhao et al. 2017). The QZSS was originally designed to improve the service capability of GPS in the Asia-Pacific region, particularly for Japan’s mountainous landscape and urban high-rise buildings with severe signal occlusion. The track is a highly inclined ellipse, known as the quasi-zenith orbit (QZO). With similar figure-eight shaped ground tracks, the QZSS satellites spend an extended time at high elevations above Japan with a separation of 8 h passing the same region (Teunissen et al. 2017). In Asia-Oceania region, all the four QZSS satellites are located above an elevation angle of 10º (Safoora et al. 2018). Since the launch of the first satellite of the QZSS in 2010, three IGSO and one GEO satellites have been launched and officially served in November 2018. Table 1 lists the detailed information of the in-orbit satellites. The QZSS is planned to be extended to a seven-satellite system by 2024. Many domestic and foreign scholars have explored and evaluated the QZSS and the contribution of QZSS to other navigation systems with the development of the QZSS. Hauschild et al. analyzed the multipath (MP), yaw attitude, and orbit of QZSS-1 using tracking data obtained from the CONGO monitoring network (Hauschild et al. 2012). Guo (2017) verified the quality of precise orbit and clock products of the QZSS by comparing with those obtained using individual analysis centers (ACs), satellite laser ranging (SLR) residuals, and precise point positioning (PPP) (Zumberge et al. 1997) tests. The orbit and clock of the QZSS have good consistency among ACs, showing agreements of 0.2–0.4 m and 0.4–0.8 ns, respectively, and the accuracy of the QZSS orbit is 0.2 m lower than those of Galileo and BDS orbits. The QZSS can improve the single-point positioning (SPP) accuracy of the GPS/BDS single system, the fixed degree of ambiguity, and the PPP accuracy of BDS; however, the accuracy of GPS PPP has not been improved (Zhang et al. 2018). Most users of the GNSS in the mass market use single-frequency (SF) receivers owing to the low cost (Ola. 2002). The mitigation of the ionospheric delay as the big challenge for SF PPP has attracted wide attention at home and abroad, and several approaches have been developed for correcting the ionospheric delay. Model correction and algorithm correction methods are commonly used. Model correction has gradually developed from the broadcast ephemeris correction model with a low precision, to the grid ionosphere map (GIM) ionospheric model with a higher precision, while GRoup and PHase Ionospheric Correction (GRAPHIC), ionospheric delay parameter estimation, and ionosphere-constrained model are commonly used among algorithm correction methods. There are advantages and disadvantages to each approach which are worthy of further verification and exploration. Meanwhile, for BDS, the multipath (MP) of the code bias is also an important factor affecting the accuracy of SF PPP (Wanninger et al. 2015), which is both elevation- and frequency-dependent bias (Lou et al. 2017; Lin et al. 2017). It must be corrected in SF PPP. With the development of multi-GNSS, extensive researches have focused on using multi-GNSS observation data to accelerate convergence speed and improve positioning accuracy of SF PPP. Lou et al. exploited the contribution of multi-GNSS to SF PPP solution based on raw observations. With more satellites available, the SF PPP multi-GNSS PPP positioning accuracy has shown improvements compared to the GPS results, while combination of BDS did not give better results due to the low GEO orbit accuracy and the negative impacts of code bias (Lou et al. 2016). However, a few scholars evaluated the contribution of the QZSS to the SF PPP of other satellite navigation systems. Based on this background, three SF PPP models commonly used are first given for the mitigation of ionospheric delay to compare different methods, including the GRAPHIC model, the GRAPHIC with code 2
observation model and ionosphere-constrained model. Then, the experimental data and processing strategies are introduced. Finally, contributions of the QZSS to GNSS in PDOP, satellites number and SF PPP are analyzed and compared. The SF PPP results of the three models and impacts of code bias of BDS are also assessed and explored.
2. Single-Frequency PPP Model The original observation equations for the code and carrier phase of satellites and receivers can be expressed as follows (Zhou et al. 2018): 𝑃𝑆𝑟,𝑔 = 𝜌𝑠𝑟 +𝑐 ∙ 𝑑𝑡𝑟 ―𝑐 ∙ 𝑑𝑡𝑆 +𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝛾𝑠𝑔 ∙ 𝐼𝑆𝑟,1 + 𝑑𝑆𝑟,𝑔 ― 𝑑𝑆𝑔 + 𝜀𝑆𝑟,𝑔 Φ𝑆𝑟,𝑔
=
𝜌𝑆𝑟
+𝑐 ∙ 𝑑𝑡𝑟 ―𝑐 ∙ 𝑑𝑡 +𝑚( 𝑆
𝑒𝑙𝑒𝑆𝑟
) ∙ 𝑍𝑤 ―
𝛾𝑆𝑔
∙
𝐼𝑆𝑟,1
+
𝜆𝑆𝑔
∙
𝑁𝑆𝑟,𝑔
+
𝑏𝑆𝑟,𝑔
―
(1) 𝑏𝑆𝑔
+
Here, 𝑔, r and S represent the frequency, receiver, and satellite system, respectively, 𝜙𝑆𝑟,𝑔
are the code and carrier phase measurements, respectively,
𝜌𝑆𝑟
𝜉𝑆𝑟,𝑔 𝑃𝑆𝑟,𝑔
(2) and
is the geometric distance between
the phase centers of the satellite and receiver antennae (at the signal transmission and reception times), 𝑑𝑡𝑟 is the receiver clock error, c is the speed of light, and 𝑑𝑡𝑆 is the satellite clock error. 𝑚(𝑒𝑙𝑒𝑆𝑟) is the projection function from the tropospheric zenith direction of the station to the inclined direction of the station and satellite, 𝑒𝑙𝑒𝑆𝑟 is the elevation angle of the satellites, 𝑍𝑤 is the tropospheric wet delay error of the zenith component (ZWD), 𝐼𝑆𝑟,1 is the slant ionospheric delay at the frequency band of 𝑓𝑆1, 2
𝛾𝑆𝑔 is the frequency-dependent multiplier factor (𝛾𝑆𝑔 = (𝑓𝑆1/𝑓𝑆𝑔) ), 𝑑𝑆𝑟,𝑔 and 𝑑𝑆𝑔 are uncalibrated code delays (UCDs) at the satellite and receiver ends, respectively. 𝑏𝑆𝑟,𝑔 and 𝑏𝑆𝑔 are the uncalibrated phase delays (UPDs) at the satellite and receiver ends, respectively, 𝜆𝑆𝑔 is the wavelength of the carrier phase, 𝑁𝑆𝑟,𝑔 is the integer ambiguity, which is observed at the frequency band of 𝑓𝑆𝑔. 𝜀𝑆𝑟,𝑔 and 𝜉𝑆𝑟,𝑔 denote the code and carrier phase measurement noises, respectively. The phase center offset and variation, relativistic effects, tidal loading, ocean tides, earth rotation effects, and phase wind-up can be corrected using existing models (Dach et al. 2007) and are therefore not explicitly included in Equations (1) and (2). The satellite clock error is corrected using precise clock error products, which contain the satellite UCD of the ionosphere-free code observations. Thus, we have, 𝑐 ∙ 𝑑𝑡𝑆𝐼𝐹12 = 𝑐 ∙ 𝑑𝑡𝑆 + (𝛼𝑆12 ∙ 𝑑𝑆1 + 𝛽𝑆12 ∙ 𝑑𝑆2) = 𝑐 ∙ 𝑑𝑡𝑆 + 𝑑𝑆𝐼𝐹12
(3)
In this study, the observation value at the frequency band of 𝑓𝑆1 was used. Equation (3) is substituted into Equations (1) and (2) and linearized to obtain the following after applying precise clock error products:
{
𝜌𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝐼𝑆𝑟,1 + 𝑑𝑆𝑟,1 ― 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2 + 𝜀𝑆𝑟,1 𝑆
𝑙𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 ― 𝐼𝑆𝑟,1 + 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 + 𝑏𝑟,1 ― 𝑏𝑆1 + 𝑑𝑆𝐼𝐹12 + 𝜉𝑆𝑟,1
(4)
Here, 𝜌𝑆𝑟,1 and 𝑙𝑆𝑟,1 denote the observed minus computed values of the code and carrier phase observables, respectively, 𝝁𝑆𝑟 is the unit vector of the component from the receiver to the satellite, and x is the vector of the receiver position increments relative to the a priori position. The symbols in the equation can be expressed as:
{
2
2
2
2
2
2
𝛼𝑆𝑚𝑛 = (𝑓𝑆𝑚) /((𝑓𝑆𝑚) ― (𝑓𝑆𝑛) ),𝛽𝑆𝑚𝑛 = ― (𝑓𝑆𝑛) /((𝑓𝑆𝑚) ― (𝑓𝑆𝑛) ) 𝑑𝑆𝐼𝐹𝑚𝑛 = 𝛼𝑆𝑚𝑛 ∙ 𝑑𝑆𝑚 + 𝛽𝑆𝑚𝑛 ∙ 𝑑𝑆𝑛 𝐷𝐶𝐵𝑆𝑃1𝑃2 = 𝑑𝑆1 ― 𝑑𝑆2
(5)
2.1 GRAPHIC Model The GRAPHIC model eliminates the ionospheric delay error through the linear combination of 3
code and carrier observations, based on the fact that the ionospheric delay errors in the code and carrier observations are equal in magnitude and opposite in sign (Gold et al. 1994). According to Equation (4), the linearized observation equation obtained by re-parameterization is as follows: 1 𝐺𝑆𝑟,1 = ∙ (𝜌𝑆𝑟,1 + 𝑙𝑆𝑟,1) = 𝝁𝑆𝑟 ∙ 𝒙 +𝑐 ∙ 𝑑𝑡𝑆𝑟 +𝑚 2
(𝑒𝑙𝑒𝑆𝑟) ∙ 𝑍𝑤 +1/2 ∙ 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 +1/2 ∙ (𝜀𝑆𝑟,1 + 𝜉𝑆𝑟,1)
(6)
In the equation,
{
𝑑𝑡𝑟 = 𝑑𝑡𝑟 + [1/2 ∙ (𝑏𝑆𝑟,1 + 𝑑𝑆𝑟,1)]/𝑐 𝑁𝑆𝑟,1 = 𝑁𝑆𝑟,1 + (𝑑𝑆𝐼𝐹12 ― 𝑏𝑆1 ― 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝜆𝑆1
(7)
The estimated unknown parameter vector is: 𝑿 = [𝒙 𝑑𝑡𝑟 𝑍𝑊 𝑁𝑆𝑟,1]
(8)
The GRAPHIC combined observables are dominated by the noise of the code measurements and the GRAPHIC combination from single-frequency code and phase observations produces a rank-deficient mathematical problem, which needs a long convergence time (Montenbruck. 2003;
Shi et al. 2012).
2.2 GRAPHIC with Code Observation Model Compared with GRAPHIC PPP, a code observation equation is added to the model. The equation includes the ionospheric delay error, which can be corrected by the external products. After correcting the ionospheric error, the linearized observation equation is obtained as follows: 𝜌𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝜀𝑆𝑟,1 𝐺𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 1/2 ∙ 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 + 1/2 ∙ (𝜀𝑆𝑟,1 + 𝜉𝑆𝑟,1)
{
(9)
In the equation,
{
𝑑𝑡𝑟 = 𝑑𝑡𝑟 + (𝑑𝑆𝑟,1 ― 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝑐 𝑁𝑆𝑟,1 = 𝑁𝑆𝑟,1 + (𝑑𝑆𝐼𝐹12 ― 𝑏𝑆1 + 𝑏𝑆𝑟,1 ― 𝑑𝑆𝑟,1 + 𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝜆𝑆1
(10)
The estimated unknown parameter vector is: 𝑿 = [𝒙 𝑑𝑡𝑟 𝑍𝑊 𝑁𝑆𝑟,1]
(11)
The model includes a code observation equation which provides code reference, and solves the rank deficiency problem exiting in the GRAPHIC model.
2.3 Ionosphere-constrained Model The ionosphere-constrained model adds virtual ionospheric observations from the external model with their corresponding constraints, which can be expressed as follows. 𝜌𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 + 𝐼𝑆𝑟,1 + 𝜀𝑆𝑟,1 𝑙𝑆𝑟,1 = 𝝁𝑆𝑟 ∙ 𝒙 + 𝑐 ∙ 𝑑𝑡𝑟 + 𝑚(𝑒𝑙𝑒𝑠𝑟) ∙ 𝑍𝑤 ― 𝐼𝑆𝑟,1 + 𝜆𝑆1 ∙ 𝑁𝑆𝑟,1 + 𝜉𝑆𝑟,1 𝐼𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑒𝑑 = 𝐼𝑆𝑟,1
(12)
{
(13)
{ In the equation,
𝑑𝑡𝑟 = 𝑑𝑡𝑟 + 𝑑𝑆𝑟,1/𝑐 𝑆 𝑁𝑆𝑟,1 = 𝑁𝑆𝑟,1 + (𝑑𝑆𝐼𝐹12 ― 𝑏𝑆1 + 𝑏𝑆𝑟,1 ― 𝑑𝑆𝑟,1 ― 3𝛽12 ∙ 𝐷𝐶𝐵𝑆𝑃1𝑃2)/𝜆𝑆1 𝑆
𝐼𝑆𝑟,1 = 𝐼𝑆𝑟,1 ―𝛽𝑆12 ∙ 𝐷𝐶𝐵𝑃1𝑃2 4
The estimated unknown parameter vector is: 𝑿 = [𝒙 𝑑𝑡𝑟
𝐼𝑆𝑟,1
𝑍𝑊 𝑁𝑆𝑟,1]
(14)
The ionosphere-constrained model involves constraining the ionospheric model correction value as a virtual observation value and then estimating the ionospheric delay parameters. Compared with model correction, reasonable weighting of the virtual observation value can accelerate the SF PPP initialization time. The DCB error must be considered in SF PPP. The DCB products can be provided by the analysis center of Center for Orbit Determination in Europe (CODE), the German Aerospace Center (DLR), and the Institute of Geodesy and Geophysics (IGG). Wang et al. verified that the above three products have the same accuracy (Wang et al. 2016). In addition, the broadcast ephemeris can broadcast the timing group delay (TGD) of the satellite clock in real time, which can be converted to DCB (Ge et al. 2017). The TGD parameters can be obtained in real time; however, the DCB parameters are not available. The two can be converted to each other for use depending on the condition. For the analysis, the three SF PPP models with different ionospheric correction approaches, namely the GRAPHIC, GRAPHIC with a code observation, and ionosphere-constrained model, are represented by SF1, SF2, and SF3, respectively.
3. Experimental Data and Processing Strategies A two-week dataset (DOY 013–026, 2019) from 10 stations of the MGEX network (Montenbruck et al. 2011) was selected for validation. Figure 1 and Table 1 present the distribution and details of the stations. The sampling interval of the observation is 30 s. Precise satellite clock product with 30 s interval and precise satellite orbit products with 5 min interval were employed from GFZ (gbm*.*). In this study, the SF PPP data were tested using the first frequency signal observations of GPS/GLONASS/BDS/Galileo/QZSS simplified as G/R/C/E/J. The satellite cut-off elevation angle was set depending on the test objective. In the SF1/SF2/SF3 models, the estimated parameters include the coordinate parameters, the receiver clock error parameter, ZWD parameters, ionospheric delay parameter, and ambiguity parameter. For the coordinate parameters, the constant estimation method was used in static PPP, and a random walk model was used in the dynamic PPP, in which the process noise was set to 100𝑚2. For the tropospheric delay parameters, the Saastamoinen model (Saastamoinen, 1972) and the GMF projection function (Roken et al. 1991) were used to calculate the tropospheric dry-component delay error and partial wet-component delay error, and the residual of the ZTW component delay was estimated as a piece-wise constant. The ambiguity parameter was considered a constant when satellite cycle slip did not occur and was re-initialized when it did. The receiver clock error was estimated as white noise. The phase center offset (PCO) and phase center variations (PCV) of the satellite and receiver antennae were corrected using igs14.atx_2032 (Rebischung et al. 2016) file provided by the International GNSS Service (IGS) (Dow et al. 2009). As the antenna file rarely updates PCO and PCV parameters of the receivers of the BDS, Galileo, and 5
QZSS, we used the correction value of GPS to correct them. The DCB error was corrected using the DCB products released by the IGG. The GIM products issued by CODE were used to correct the ionospheric errors in the SF2 model and as virtual ionospheric observations in the SF3 model. The GIM model has an accuracy of 2–8 total electron content unit (TECU), which is equivalent to a ranging error of 0.32–1.28 m for the GPS L1 frequency signal. The formulae for calculating the ionospheric delay errors using GIM products have been previously introduced (Cai et al. 2017) and are not detailed herein. The stochastic models of the observations represented by a sinusoidal function and based on the elevation angle of the satellite were utilized to reduce the effect of elevation angle-related errors (Witchayangkoon. 2000). The GPS, BDS, GLONASS, Galileo, and QZSS phase observation precisions were set to 0.003, 0.006, 0.004, 0.003, and 0.006 m, respectively, and the code observation precisions were set to 0.3, 1.2, 1.2, 0.3, and 1.2 m. Because of the poor accuracy of the orbit and clock products of the BDS GEO satellite (Guo et al. 2017), the weight of the BDS GEO satellite observations was reduced by 10 fold (Li et al. 2017). In the SF3 model, the stochastic model of the virtual observations can affect the positioning results. Because of the limited accuracy of the virtual observations, a higher weight was given to improve the convergence speed of PPP at the beginning of the estimation, and the weight was then gradually reduced to improve the positioning accuracy. The variance in the virtual observations was defined as follows. 𝜎2𝐼 (𝑖) = (1 + 0.02 ∙ 𝑖) ∙ 𝜎2𝑖𝑜𝑛(𝑖)/0.03
(15)
Here, i refers to the number of epochs, and 𝜎2𝑖𝑜𝑛 refers to the priori variance of the ionospheric delay, which was determined by the ionospheric model. The GIM model was used in the test, and the temporal and spatial correlation functions can be expressed as (Gao et al. 2017). σ2ion =
{
σ2ion,0
,
sin2E 2 𝜎𝑖𝑜𝑛,0
𝑡 < 8 𝑜𝑟 𝑡 > 20 𝑜𝑟 𝑡 > 𝜋/3 +
𝜎2𝑖𝑜𝑛,1
∙ cos(B) ∙ cos
(
),
𝑡 ― 14 12 ∙ 𝜋
other
(16)
Here, t is the local time at the ionospheric pierce point (IPP) in h, B refers to the satellite elevation angle, and 𝜎2𝑖𝑜𝑛,0 and 𝜎2𝑖𝑜𝑛,1 denote the variance in the zenith ionospheric delay and the zenith ionospheric delay variation, respectively, which are set to 0.09 𝑚2. The coordinates of the station in the file (*.snx) issued by the IGS were taken as reference values for the accuracy analysis, and the convergence condition was defined such that the positioning errors of the east (E), north (N), and up (U) components are lower than 0.3, 0.3, and 0.5 m, respectively.
6
Figure 1. Distribution of the selected MGEX stations Table 1. Detailed information of the receivers. Manufacturer LEICA TRIMBLE
Receiver type GR25 NETR9 POLARX5 POLARX4 TRE_G3TH DELTA
SEPTENTRIO JAVAD
Station name ALIC CCJ2/GMSD/KOUC/STK2/TSK2 CEDU/DARW TWTF JOG2
4. Results and Discussion 4.1 Contribution of QZSS in PDOP and Satellites Number As a regional satellite navigation system, the QZSS mainly serves the Asia-Pacific and its surrounding
areas.
To
analyze
the
contribution
of
the
QZSS
to
the
SF
PPP
of
GPS/BDS/GLONASS/Galileo, the influences of the QZSS on the number of satellites and position dilution of precision (PDOP) values of each system in the service area were first analyzed. Figure 2 and
Figure
3
show
the
average
satellite
number
and
PDOP
distribution
of
the
GPS/BDS/GLONASS/Galileo single system and GPS/BDS/GLONASS/Galileo and QZSS dual system in 2019 DOY 024 days. The global region was divided into grids of 2.5º × 5º. Subsequently, the PDOP values of each grid were calculated using the broadcast ephemeris (brdm0200.19p). The cut-off elevation angle was set to 7.5º, and the time resolution was 10 min. For every grid, one calculation result of the PDOP was obtained at each epoch, which generated a PDOP sequence. Thereafter, we could determine the mean value from this sequence for every grid. Figure 2 shows that the QZSS increases the number of visible satellites of GPS/BDS/ GLONASS/Galileo single system in the Asia-Pacific and southern Indian Ocean regions, particularly in southeastern Asia and the western Pacific. As shown in Figure 3, the QZSS system significantly improved the PDOP values of the GLONASS and Galileo single systems in the Asia-Pacific region. GPS has full navigation service capability worldwide and good graphic intensity of satellites. Therefore, the QZSS increases the number of GPS satellites in the Asia-Pacific region but has little influence on GPS PDOP. In the Asia-Pacific region, the PDOP of BDS is smaller and the QZSS has little impact on its. 7
Figure 2. Satellite distribution of GPS, BDS, GLONASS, and GALILEO single system, and QZSS and GPS, BDS, GLONASS, GALILEO dual system (DOY 016 on DOY 024, 2019)
Figure 3. PDOP distribution of GPS, BDS, GLONASS, and Galileo single system, and QZSS and GPS, BDS, GLONASS, Galileo dual system (DOY 016 on DOY 024, 2019)
4.2 MP Combinations Analysis of Code Measurements for BDS The MP effect, which is mainly due to signal reflection, is an intractable error source for GNSS observations, particularly for code observations. MP combinations after factored out with an average term contain MP errors and combined observation noise of the code and phase measurements, which is calculated by combining one-frequency code observation and dual-frequency phase observation (Wanninger et al. 2015; Zhang et al. 2019). Given that the MP error and noise of phase are smaller than that of the code, the former can be neglected. The BDS constellation includes the geostationary orbit (GEO), inclined geostationary earth orbit (IGSO), and medium earth orbit (MEO). The MP combinations time series of BDS IGSO and MEO are elevation-dependent and usually neglected for other GNSS systems. Taking GPS as an example, Figure 4 shows the MP combinations time series and elevation angles of BDS GEO/IGSO/MEO and GPS satellites. The MP combinations value of the BDS IGSO/MEO satellite decreases with the increase in the satellite elevation angle, while those of the GPS have no obvious correlation with the satellite elevation angle. As the BDS GEO satellites are visible at almost constant elevation angles, the MP combination time series has no obvious variation. Therefore, we limited our analysis to BDS MEO and IGSO satellites. 8
Figure 4. MP combinations and elevation angles time series of GPS and BDS at the frequency band L1and B1 respectively The BDS-2 satellites provide three frequency bands, i.e., B1I, B2I and B3I, centered at 1561.098, 1207.140 and 1268.520 MHz, respectively. Figure 5 shows the relationship between the MP combinations values of the three frequency bands and elevation angles. For different types of satellites, the MP combinations value of MEO is larger than that of IGSO at the B1I and B2I frequency bands. The MP combinations effect at the frequency band of B1I is the highest and that at the B3I frequency is the lowest. Many experts and scholars have modeled and corrected the MP combinations effect of the BDS-2 MEO/IGSO satellite at different frequencies based on the relationship between the satellite elevation angles and the MP combinations value. In this study, we used the MP combinations correction model proposed by Wanninger et al. (Wanninger et al. 2015). Figure 6 shows the MP combinations time series of the uncorrected and corrected measurements. After correction, the MP combinations value decrease, and the correlation with the elevation angle decreases. Because the MP combinations effect at the B1I frequency band is the biggest, the MP combinations effect on the SF PPP was tested using the observation at the B1I frequency band. Figure 7 shows the positioning bias in the static model before and after correction. The U component has a systematic bias of approximately 0.4–1.0 m when the MP combinations effect is not considered. The positioning error of the U component is lower than 0.2 m, and the positioning accuracies of the E and N components are improved accordingly after correction.
9
Figure 5. MP combinations and elevation angles time series of BDS IGSO and MEO at B1/B2/B3 frequency bands. The black line represents the elevation angles, and the yellow, red, and green lines represent the MP combinations time series at the B1I, B2I, and B3I frequency bands, respectively
Figure 6. MP combinations time-series of BDS IGSO and MEO for uncorrected and corrected B1I, B2I, and B3I measurements
Figure 7. Comparison of E/N/U components of DARW Station before and after correction of BDS MP combinations at B1I frequency in DOY 013-DOY 019
10
The QZSS has three frequency bands for Navigation and Timing Service (PNT) of L1, L2 and L5. To analyze the relation between MP combinations and elevation angles of the QZSS, the MP combinations time series and elevation angles of the QZSS GSO and GEO satellites are shown in Figure 8. Although the elevation angles of the QZSS GSO satellite vary significantly, the MP combination value does not obviously change. The MP combination time series of the QZSS GEO satellites have no obvious variation at almost constant elevation angles. Therefore, the MP combinations of the QZSS have no significant correlation with elevation angles.
Figure 8. MP combinations and elevation angles time series of QZSS GSO/GEO satellites at L1 frequency band. The black line represents the elevation angles, and the red line represents the MP combinations time series of L1
4.3 Comparison and Analysis of Positioning Results of Three SF PPP Models To compare and analyze the positioning results of the three models, a test was performed using GPS observations of 10 stations in 2019 DOY 013-026 days. The satellite cut-off elevation angle was set to 7º. Figure 9 shows the final positioning results of the one-day solution for all days of the three models. The positioning result for most stations obtained using the three models reaches the decimeter level to centimeter level in the E/N/U components. The mean RMS values of the single-day solution bias of SF1, SF2, and SF3 models are (0.059, 0.044, 0.074) m, (0.059, 0.044, 0.075) m, and (0.062, 0.043, 0.074) m, which are largely equal.
11
Figure 9. Bias distribution of single-day static solution of the three models for DOY 013 -DOY 026 days in all stations in 2019
Table 2 summarizes the average convergence time, the STD, and RMS values of one-day solutions for all stations after convergence in static PPP. From the point of convergence time, the convergence speed of the SF3 model with the GIM ionospheric constraint is higher than those of the SF1 and SF2 models, and the convergence speeds of SF3 and SF2 are largely equal. From the perspective of positioning accuracy, the positioning results of the three models are largely similar, and the RMS value of the SF1 model in the U component is slightly lower than those of the SF1 and SF2 models. For the SF3 model, the higher weight of the virtual observation at the beginning helps improve the convergence speed. The weight is lower after the convergence; therefore, it has no positive influence on the positioning accuracy after convergence. Unlike the SF1 model, the SF2 model includes a code observation equation, which provides code reference and solves the rank deficiency problem observed in the SF1 model. However, the convergence speed is not improved, and the positioning accuracy of the U component is somewhat reduced. The reason for the low precision may be not only the low precision of the code, but also the limited accuracy of the ionospheric model products. Therefore, when there is no external ionosphere product, the SF1 model can get similar 12
positioning accuracy like the other two models with external ionosphere product.
Table 2. Average values of the convergence time, STD, and RMS of static SF PPP models for all the stations
Model SF1 SF2 SF3
Convergence time (min) 68.84 67.84 52.19
E 0.037 0.039 0.036
STD (m) N 0.017 0.018 0.018
U 0.045 0.048 0.047
E 0.063 0.064 0.062
RMS (m) N U 0.029 0.069 0.029 0.075 0.029 0.074
To further study the characteristics of the positioning results of the three models, Figure 10 shows the residuals of the GRAPHIC observation of the SF1 and SF2 models at the CEDU station. As the observation equations of the two methods are consistent, the residual magnitude and RMS are equal. Figure 11 shows the residuals of the code observations for the SF2 and SF3 models. The residuals of the SF3 code observations are lower than the SF2 residuals, showing that the ionospheric parameter estimation method can better eliminate the ionospheric error than the model correction method.
Figure 10. Residual of GRAPHIC observation of SF1 and SF2 models at CEDU station on DOY 015 in 2019
Figure 11. Residual of code of SF2 and SF3 models at CEDU station on DOY 015 in 2019 13
Figure 12 shows the estimated ZWD parameters, receiver clock error parameters, and their differences for the three models. The ZWD estimated using the three models are largely equal. The receiver clock error parameters estimated by the SF2 and SF3 models are similar and stable; however, the receiver clock error parameters estimated by the SF1 model differ from those estimated by SF2 and SF3 models with a system bias. The reason may be that the receiver clock error parameters estimated by the SF2 and SF3 models absorbed a part of the ionospheric errors. However, the STD values of the difference in the receiver clock errors between the three models are low, indicating the presence of a error with good consistency.
Figure 12. Estimated ZWD and receiver clock errors of the SF1, SF2, and SF3 models, and the differences in ZWD and receiver clock errors between the SF1, SF2, and SF3 models
4.4 Contribution of QZSS in SF PPP As the QZSS satellite has a high satellite elevation angle, we selected cut-off elevation angles of 15, 20, and 25º to study the influence of the QZSS on the SF PPP of GPS/BDS/GLONASS/Galileo. The SF3 model with less convergence time was employed for the test. Considering the number of satellite systems in the stations, the CCJ2/CEDU/DARW/GMSD/STK2/TSK2 /TWTF station, CCJ2/CEDU/JOG2/KOUC/STK2/TSK2/TWTF/ALIC station, CCJ2/CEDU/DARW/GMSD/JOG2/ STK2/TSK2 station, CEDU/DARW station and CEDU/DARW/GMSD/KOUC station (in 2019 DOY 013-019 days) were selected to test the contribution of QZSS to the SF PPP of GPS-only, GLONASS-only, Galileo-only, BDS-only and GPS/BDS/GLONASS/Galileo respectively. Figure 13 shows the positioning bias comparison between GPS/BDS/GLONASS/Galileo single system and GPS/BDS/GLONASS/Galileo + QZSS dual-system, GPS/BDS/GLONASS/Galileo and GPS/BDS/GLONASS/Galileo + QZSS at cut-off altitude angles of 15, 20, and 25º, respectively, in the static PPP model. Table 3 lists the average values of the convergence time, STD, and RMS of the 14
one-day solution after convergence. Figure 13 (a) shows the positioning bias comparison between GPS-only and GPS + QZSS. Fig. 13 (c) shows the positioning bias comparison between Galileo-only and Galileo + QZSS. At cut-off elevation angles of 15, 20, and 25º，the QZSS does not significantly improve the accuracy and convergence time of GPS-only or Galileo-only . Figure 13 (b) shows the positioning bias comparison between GLONASS and GLONASS + QZSS. At cut-off elevation angles of 15, 20, and 25º, the positioning accuracy and precision of GLONASS + QZSS is similar to that of GLONASS-only, and the convergence times are reduced by 4.3, 30.8, and 12.7%, respectively. Figure 13 (d) shows the positioning bias comparison between BDS-only and BDS + QZSS. At the different cut-off elevation angles, adding the QZSS system helped reduce the convergence time and improve the positioning accuracy, particularly in the U direction. When the cut-off elevation angle of the satellite is 25º, the convergence time of the BDS-only is considerable or even non-convergent in some cases. Therefore, the positioning results were not included in Table 3. At cut-off elevation angles of 15 and 20º, the positioning precision of BDS + QZSS is similar to that of BDS-only, the convergence times are reduced by 37.6 and 39.2%, and the positioning accuracies in horizontal component are improved by 18.6 and 14.1% and the positioning accuracies in vertical component components are increased by 13.9 and 21.4%, respectively. Figure 13 (e) shows the positioning bias comparison between GPS/BDS/GLONASS/Galileo and GPS/BDS/GLONASS/Galileo + QZSS. At cut-off elevation angles of 15, 20, and 25º, the QZSS has little impact on the positioning accuracy and precision of GPS/BDS/ GLONASS/Galileo, while the convergence times are reduced by 7.4 and 4.3% at cut-off elevation angles of 20 and 25º, respectively. Similar to GPS-only and Galileo-only, the GPS/GLONASS/BDS/Galileo has longer initialization time under the low cut-off after adding the QZSS. This indicates the lower accuracy of orbit and clock of the QZSS. However, the QZSS helps get shorter initialization time of GPS-only, Galileo-only and GPS/BDS/GLONASS/Galileo with the increase of cut-off elevation angles.
(a)
(b)
15
(c)
(d)
(e) Figure 13. SF static positioning bias comparison in DOY 016 days in 2019 under cut-off elevation angles of 15º/20º/25º: (a) Positioning bias comparison between GPS-only and GPS + QZSS, (b) Positioning bias comparison between GLONASS-only and GLONASS + QZSS, (c) Positioning bias comparison between Galileo-only and Galileo + QZSS, (d) Positioning bias comparison between BDS and BDS-only + QZSS, (e) Positioning bias comparison between GPS/BDS/GLONASS/Galileo and GPS/BDS/ GLONASS/Galileo + QZSS Table 3. Average values of the convergence time, STD, and RMS for static SF PPP of all the stations under cut-off elevation angles of 15º/20º/25º Cut-off elevation angles (°)
15
Satellite type
Convergenc e time (min)
G G+J R R+J C C+J E
47.26 51.74 100.54 96.24 290.75 181.43 61.53
STD (m)
RMS (m)
E
N
U
E
N
U
0.033 0.033 0.037 0.035 0.051 0.043 0.033
0.018 0.017 0.020 0.019 0.016 0.019 0.019
0.058 0.065 0.064 0.066 0.095 0.090 0.054
0.057 0.056 0.045 0.045 0.076 0.056 0.059
0.033 0.033 0.031 0.029 0.061 0.056 0.035
0.078 0.087 0.094 0.093 0.151 0.130 0.078
16
20
25
E+J
59.42
0.033
0.019
0.052
0.058
0.035
0.073
GRCE
44.66
0.029
0.013
0.052
0.054
0.037
0.072
GRCE+J
0.029
0.013
0.054
0.054
0.038
0.075
G G+J R R+J C C+J E E+J
48.18 57.15 57.67 120.44 83.36 302.75 183.96 70.26 73.12
0.034 0.034 0.041 0.033 0.046 0.036 0.030 0.028
0.018 0.017 0.019 0.017 0.014 0.017 0.017 0.016
0.069 0.069 0.072 0.072 0.123 0.125 0.059 0.057
0.058 0.057 0.053 0.054 0.064 0.044 0.057 0.055
0.034 0.033 0.031 0.021 0.056 0.059 0.034 0.034
0.102 0.101 0.112 0.122 0.191 0.150 0.094 0.084
GRCE
48.48
0.029
0.014
0.056
0.054
0.039
0.091
GRCE+J
0.029
0.014
0.056
0.054
0.039
0.089
G G+J R R+J C C+J E E+J
44.88 74.92 76.04 148.78 129.91 91.42 90.16
0.032 0.031 0.040 0.038 0.031 0.031
0.017 0.017 0.020 0.021 0.018 0.017
0.083 0.082 0.078 0.081 0.067 0.066
0.055 0.054 0.058 0.054 0.058 0.057
0.034 0.033 0.032 0.031 0.034 0.034
0.128 0.127 0.107 0.126 0.126 0.118
GRCE
67.79
0.028
0.014
0.059
0.052
0.039
0.111
GRCE+J
64.86
0.028
0.014
0.059
0.053
0.039
0.110
Figure 14 shows the positioning biases comparison between GPS/BDS/GLONASS/Galileo single system and GPS/BDS/GLONASS/Galileo + QZSS, GPS/BDS/GLONASS/Galileo and GPS/BDS/ GLONASS/Galileo + QZSS at cut-off altitude angles of 15, 20, and 25º, respectively, in imitating dynamic SF PPP. Similar to the static positioning results, the QZSS has no obvious effect on the SF PPP of GPS-only, GALILEO-only and GPS/BDS/ GLONASS/Galileo at these cut-off elevation angles; nevertheless, the QZSS significantly improved the positioning accuracy of GLONASS-only and BDS-only, particularly of BDS-only. Compared to GPS-only and Galileo-only, GLONASS-only and BDS-only show poor performance in SF PPP, which are mainly caused by the satellite-specific inter-frequency bias (IFB) of GLONASS and code bias of BDS except the reason of Geometric Dilution of Precision (GDOP) (Lou et al. 2016). The IFB of GLONASS at the receiver is strongly related to the ionosphere and then influences on the SF positioning result. The elevation-dependent code bias of BDS has negative impact on SF PPP results, especially in the vertical component. Based on the experimental results, the QZSS greatly improves the positioning results of BDS-only and GLONASS-only, and GLONASS and QZSS are coupled together to provide better positioning service as well as BDS and QZSS in static and imitating dynamic SF PPP.
17
(a)
(b)
(c)
(d)
(e) Figure 14. SF imitating dynamic positioning bias comparison on DOY 016 days in 2019 under cut-off elevation angles of 15º/20º/25º: (a) Positioning bias comparison between GPS-only and GPS + QZSS, (b) Positioning bias comparison between GLONASS-only and GLONASS + QZSS, (c) Positioning bias comparison between Galileo-only and Galileo + QZSS, (d) Positioning bias comparison between BDS and BDS-only + QZSS, (e) Positioning bias comparison between GPS/BDS/GLONASS/Galileo 18
and GPS/BDS/ GLONASS/Galileo + QZSS
5. Conclusions In this study, the observations provided by MGEX and the precise product of IGS were used to test and analyze the positioning results of the contribution of the QZSS to GPS/BDS/ GLONASS/Galileo, three SF PPP models and the code bias od BDS. The QZSS increases the number of visible satellites of GPS/BDS/ GLONASS/Galileo single system particularly in southeastern Asia and the western Pacific, and improves PDOP value of GLONASS and Galileo single system. Moreover, the QZSS greatly improves the SF PPP performance of BDS-only and GLONASS-only. Compared to GPS-only, Galileo-only and GPS/GLONASS/BDS/Galileo, the QZSS helps get longer initialization time under the low cut-off elevation angles due to the lower accuracy of orbit and clock of the QZSS,
but get shorter initialization time with the increase of the cut-off altitude angles. The
code bias of BDS IGSO and MEO elevation-dependent has negative impact on the SF PPP accuracy especially in the vertical component. In static SF PPP, the U component had a systematic bias of approximately 0.4–1.0 m. After correcting the MP value, the positioning error of the U component was found to be lower than 0.2 m, and the positioning accuracy of the E and N components improved accordingly. In addition, the MP combinations of the QZSS have no significant correlation with elevation angles. The positioning result of the three models reached the decimeter level to centimeter level for the E/N/U components. The ionosphere-constrained model exhibited the fastest convergence, whereas the positioning accuracies of the three models were largely equal. Therefore, the GRAPHIC model can get similar positioning accuracy like the other two models in the absence of external ionosphere products, but its convergence speed is slower.
Acknowledgments: The work is partly supported by the program of National Key Research and Development Plan of China (Grant No: 2016YFB0501804), National Natural Science Foundation of China (Grant No: 41674034, 41974032, 11903040) and Chinese Academy of Sciences (CAS) programs of “Pioneer Hundred Talents” (Grant No: Y923YC1701), and “The Frontier Science Research Project” (Grant No: QYZDB-SSW-DQC028). References
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