The precise point positioning (PPP) technique is a useful method in scientific research and civilian applications, which can achieve centimeter-level positioning accuracy at a single station with precise corrections for satellite orbit and clock biases [1
]. However, its widespread application is limited by the drawback of long convergence times [2
]. Thanks to PPP ambiguity resolution (AR) technology, it is possible to achieve higher positioning accuracy and shorter convergence times [5
]. In previous research, the positioning accuracy of the PPP AR method in the eastward, northward, and upward directions are higher than 0.4, 0.4, and 1.2 cm, respectively, and the total improvement is realized as 27%; the convergence time is also shortened from 19.5 to 12 min with a 38% improvement [8
Moreover, multiple global navigation satellite system (GNSS) PPP with increasing observations from multiple constellations is another efficient method to achieve higher positioning accuracy and shorter convergence times [5
]. In recent years, multi-GNSS has undergone rapid development, and its performance has been demonstrated in many studies. Due to the availability of GNSS satellites, the combined GPS and GLONASS PPP was investigated first [9
]. The convergence time was significantly shortened due to the availability of a greater number of satellites, and the positioning accuracy and reliability were also improved, especially when only satellites with a high cutoff elevation were applied in the processing. Further improvement has been confirmed for the combination of GPS and GLONASS [10
]. The performance of BDS-2 PPP was also investigated, and centimeter-level accuracy was obtained with the continuous construction of BDS-2 [13
]. An improvement for positioning accuracy and rapid convergence time was efficiently realized in the combined GPS/BDS-2 PPP [14
]. In addition, the Galileo satellites were also added to improve the combined multi-GNSS PPP performance [15
]. The combined GPS/BDS-2/GLONASS/Galileo PPP was implemented to present the advantages of multi-GNSS in single- or dual-frequency positioning [16
Recently, multi-GNSS PPP AR has been recommended to achieve the best positioning performance. The performance associated with the positioning bias convergence in PPP AR for GPS/BDS-2 and GPS/GLONASS was analyzed [11
]. Li (2017) demonstrated the performance of PPP AR with FCBs for GPS/BDS-2 in both static and dynamic modes [21
]. The improvements on the static positioning accuracy and its bias convergence were also obtained for GPS/BDS-2 PPP AR with STDs of 0.7, 0.5, and 1.9 cm for the eastward, northward, and upward directions, respectively [22
]. Liu (2017) investigated the GPS FCB estimation of GPS/GLONASS and indicated that the application of GLONASS observations significantly improved the GPS ambiguity fixing percentage over a short time span [24
]. When more GNSS constellations are added into the PPP combinations, more optimal results can be obtained [25
]. Recently, the multi-GNSS FCBs estimated with the ionospheric-free PPP model for GPS/BDS-2/Galileo have been routinely published at Wuhan University [27
]. PPP AR with GPS and Galileo observations can also be achieved using the FCBs from the Centre National d’Etudes Spatiales (CNES) or the center for orbit determination in Europe (CODE) rapid orbit/clock solutions [28
However, the previous studies are based on the combined PPP method using ionospheric-free observations. In ionospheric-free combinations, the measurement noise is amplified three times. Practically, the accuracy of the wide-lane (WL) ambiguities in the Melbourne–Wübbena (MW) combination is degraded, because this combination is sensitive to systematic pseudorange errors, such as multipath errors. Meanwhile, the uncombined PPP model, which uses raw observations, directly estimates the original ambiguities of each frequency, avoiding the abovementioned drawbacks. In the uncombined PPP model, the individual signal of each frequency is treated as an independent observable, which is flexible and can process multi-frequency signals from different GNSSs. Its positioning performance in terms of accuracy and bias convergence has been demonstrated for both single- and dual-frequency PPP [17
], as well as real-time PPP with atmosphere augmentation corrections [30
]. For the PPP-RTK method, instantaneous AR solutions using uncombined observations are achieved with atmospheric augmented corrections in the regional reference network [31
]. In the Integer Recovery Clock (IRC) [7
] and the Decoupled Satellite Clock (DSC) models [6
], raw observations after correcting phase bias are also adopted for PPP AR [36
]. Moreover, this approach has been used in ionospheric modeling [38
], differential code bias (DCB) estimation [40
], and FCB estimation [8
]. Li (2013) used the uncombined PPP method to shorten the convergence time with ambiguity-fixed solutions [41
]. Gu (2015) and Li (2018) estimated FCB products from raw ambiguities using the uncombined PPP method with BDS-2 triple-frequency observations [42
]. Xiao (2019) proposed a common uncombined model to achieve multi-frequency PPP AR using the triple-frequency signals from Galileo and BDS-2 [44
]. The equivalence of the FCB estimation between observations combined PPP model and uncombined PPP models has been confirmed and verified [8
]. The advantages of flexibly and efficiently estimating FCBs using the uncombined PPP model have also been analyzed [8
]. With the rapid development of GNSS, the potential of the uncombined PPP method for shortening convergence time and for improving positioning accuracy with ambiguity resolution should be further delineated.
Furthermore, the inter-system bias (ISB) resulting from system-specific differences, such as different time and coordinate systems, needs to be properly modeled in multi-GNSS PPP. Theoretically, ISB mainly originates from the differences of the receiver/antenna hardware delays between different GNSSs. However, the datum differences between different GNSSs in satellite clock offset estimation are introduced into the multi-GNSS PPP model, which are reflected in the ISBs. Generally, three different strategies are introduced into multi-GNSS PPP models, where the ISB parameter is modeled either with a random constant, a random walk, or a white noise process, respectively [17
]. Zeng (2017) analyzed the ISB stability of BDS-2 relative to GPS with six types of receivers, and the great stable variation of the one day ISB series was shown in the experiment [50
]. Zang (2020) further investigated the daily stability of the ISBs among GNSSs and proposed to add constraints using the stable property of the ISB to improve the positioning performance [47
]. Jiang (2017) also proposed a model for GPS/BDS-2 PPP, constrained with the predicted ISBs [51
]. Zhou (2019) compared the positioning differences using three different ISB estimation models, which indicated that the ISB variation was also related to the clock datum differences among the different GNSSs in satellite clock correction estimation [46
]. Therefore, the ISB parameters should be carefully considered to precisely deal with satellite clock corrections in multi-GNSS PPP AR.
This work contributes to the literature by demonstrating an uncombined PPP model for estimating the FCB products and by analyzing the positioning performance using GPS, BDS-2, and Galileo observations. In addition, the ISBs developed with the GPS, BDS-2, and Galileo observations are also carefully evaluated. Herein, we firstly present the uncombined PPP model with multi-GNSS observations, and then, FCB products are estimated with raw ambiguities. Then, the data process strategy is elaborated in detail. The performance of the ISB stability and FCB estimations is verified with the GPS, BDS-2, and Galileo dual-frequency raw observations from the globally distributed Multi-GNSS Experiment (MGEX) stations. Furthermore, we evaluate the improved performance of the PPP ambiguity resolution with the combined model using GPS/BDS-2/Galileo observations.
We comprehensively studied the performance of uncombined multi-GNSS PPP processing, as well as FCB estimation with GPS/BDS-2/Galileo observations. The performance of the positioning accuracy, the convergence time, and the ambiguity success fixing rate was evaluated. Firstly, the differences between the BDS-2 and Galileo ISB series were presented to exhibit the fluctuations of ISBs. Although ISB is mainly affected by the receiver-dependent bias, the clock datum of the satellite clock offset estimation also accounts for the variations of the one day ISB series. Hence, we selected the white noise model for ISB estimation, considering the estimation of precise satellite clock offset products among different GNSS constellations. With the GFZ precise satellite clock products, the averaged STD of the BDS-2 ISB series was 1.32 ns, while the value associated with Galileo was 0.29 ns.
Furthermore, the FCBs of GPS, BDS-2, and Galileo were estimated and evaluated with the ambiguity a posteriori residual distributions and the STDs. The RMSs of the residuals for the NL and WL combinations among all systems were better than 0.1 cycles, indicating that relatively high-precision FCB is more likely to achieve PPP AR. For the RMSs of the residuals for the FCB estimation of the NL combination, the values were 0.067, 0.063, and 0.058 cycles for GPS, BDS-2, and Galileo, respectively, while for the WL combination, they were 0.069, 0.091, and 0.045 cycles.
Finally, rapid convergence and high positioning accuracy were achieved in the implementation of PPP AR using the multi-GNSS observations. For the ambiguity-fixed solutions, the positioning accuracy of all type of GNSS combinations was improved by over 30% compared to that of the float solutions. The RMS of the positioning accuracy for “GCE” PPP AR was 1.14 cm and it exhibited a 15% improvement compared to the result for “G”. Correspondingly, the convergence time was improved to 7.5 min for “GCE”, resulting in a 29% improvement. A higher ambiguity success fixing rate was also achieved for “GCE” within only 10 min of data, and the percentages were computed as 75.5%, 94.3%, 86.5%, and 99.0% for “G”, ”GC”, “GE”, and “GCE”, respectively.
With the rapid development of BDS-3, its orbits and positioning performance have been demonstrated with initial evaluations [60
]. FCB estimation and PPP AR performance using BDS-3 satellites will be studied in future work.