1. INTRODUCTION
The concept of Precise Point Positioning (PPP) was first proposed in the mid-1990s (Héroux and Kouba, Reference Héroux and Kouba1995; Zumberge et al., Reference Zumberge, Heflin, Jefferson, Watkins and Webb1997). The PPP technique is usually carried out with the use of dual-frequency code and carrier phase observations as well as the correction data from the precise satellite orbit and clock products (Kouba and Héroux, Reference Kouba and Héroux2001). Over the past two decades, significant progress has been achieved in dual-frequency PPP, which is currently able to provide millimetre-level accuracy in static mode and centimetre-level accuracy in kinematic mode (Li et al., Reference Li, Zhang and Ge2011; Reference Li, Ge, Zhang and Wickert2013a). However, single-frequency Global Navigation Satellite System (GNSS) receivers are still widely used in most positioning and navigation applications due to the low costs (Øvstedal, Reference Øvstedal2002; Montenbruck, Reference Montenbruck2003). Therefore the development of a single-frequency PPP technique has aroused great interest in the GNSS community (Cai et al., Reference Cai, Liu and Luo2013; Li et al., Reference Li, Ge, Zhang, Nischan and Wickert2013b; Sterle et al., Reference Sterle, Stopar and Prešeren2015).
In single-frequency PPP, proper handling of the ionospheric delay is a key issue. There are presently two main ways to solve this issue. One way is to apply ionospheric models, such as the Global Ionosphere Maps (GIM) ionospheric model and the Klobuchar model, to mitigate or eliminate the ionospheric effect (Øvstedal, Reference Øvstedal2002; Le and Tiberius, Reference Le and Tiberius2007; Li et al., Reference Li, Ge, Zhang, Nischan and Wickert2013b). Another way is to use the code and carrier phase observations on a single frequency to form an ionosphere-free observable, which is known as the GRoup And PHase Ionospheric Correction (GRAPHIC) (Yunck, Reference Yunck, Parkinson and Spilker1996; Montenbruck, Reference Montenbruck2003). Numerical results show that single-frequency PPP based on GRAPHIC can achieve centimetre-level accuracy in static mode and decimetre-level accuracy in kinematic mode, which is several times better than that of ionospheric mitigation models (Sterle et al., Reference Sterle, Stopar and Prešeren2015). In this study, the GRAPHIC ionosphere-free linear combination is adopted to remove the ionospheric effect.
According to the above discussions, most of the research work about single-frequency PPP has focused on the correction of ionospheric delay and was carried out relying on the GPS system. Efforts were also made to improve the single-frequency PPP performance by combining GPS with GLONASS (Cai et al., Reference Cai, Liu and Luo2013). The results demonstrated that both the positioning accuracy and convergence time were improved by approximately 30%. Moreover, the improvement in precision and convergence speed for dual-frequency PPP has been widely confirmed when observations from other GNSS systems are available in addition to GPS (Li et al., Reference Li, Zhang, Ren, Fritsche, Wickert and Schuh2015a; Li et al., Reference Li, Ge, Dai, Ren, Fritsche, Wickert and Schuh2015b; Cai et al., Reference Cai, Gao, Pan and Zhu2015; Pan et al., Reference Pan, Cai, Santerre and Zhu2014). The joint use of multi-constellation signals has become a trend in GNSS development (Li et al., Reference Li, Zhang, Ren, Fritsche, Wickert and Schuh2015a). Multi-constellation integrated single-frequency PPP has the potential to significantly improve positioning accuracy and reduce the position convergence time due to the increased number of visible satellites and the improved satellite sky distribution, especially when positioning is performed in areas with GNSS signal blockages.
GLONASS, BeiDou and Galileo systems have boomed in recent years. A full GLONASS constellation consisting of 24 operational satellites has been completely revitalised since 2012. BeiDou was declared to provide navigation and position services over the Asia-Pacific region with a constellation of 14 operational satellites on 27 December 2012. Four new generation BeiDou satellites were successfully launched on 30 March, 25 July and 30 September 2015, respectively, which marks the start of the BeiDou system expansion from regional to global scale. Currently the four new BeiDou satellites are still being commissioned. The Galileo constellation has had four In-Orbit Validation (IOV) satellites and four Full Operational Capability (FOC) satellites since 27 March 2015. GNSS users are able to simultaneously use four-constellation signals under the current GNSS constellations, which provides an opportunity to investigate the performance of Four-Constellation integrated Single-Frequency PPP (FCSF-PPP) with GPS, GLONASS, BeiDou and Galileo measurements. In this study, a FCSF-PPP model based on the GRAPHIC ionosphere-free linear combination is presented and its performance is investigated using the datasets collected at 47 globally distributed four-system Multi-GNSS Experiment (MGEX) stations on seven consecutive days and a kinematic experimental dataset.
2. POSITIONING MODEL WITH FOUR CONSTELLATIONS
The pseudorange and carrier phase observations on L1/G1/B1/E1 frequencies are used, and they can be expressed as:
$$P_r^s =\rho _r^s +c\lpar dt_r -dt^s\rpar +d_{orb}^s +I_r^s +d_{r\comma trop}^s +c\lpar b_r -b^s\rpar +\varepsilon _{r\comma P}^s $$
$$L_r^s =\rho _r^s +c\lpar dt_r -dt^s\rpar +d_{orb}^s -I_r^s +d_{r\comma trop}^s +\lambda N_r^s +\lambda \lpar d_r -d^s\rpar +\varepsilon _{r\comma L}^s $$
where the indices s and r refer to the satellite and receiver, respectively. P is the measured pseudorange in metres, L is the measured carrier phase in metres, ρ is the geometric range between the phase centres of the satellite and receiver antennas in metres, c is the speed of light in a vacuum in metres per second, dt r is the receiver clock offset in seconds, dt s is the satellite clock offset in seconds, d orb is the satellite orbit error in metres, I is the ionospheric delay in metres, d trop is the tropospheric delay in metres, λ is the wavelength in metres, N is the phase ambiguity term in cycles, b r and b s are the code biases of the receiver and the satellite in seconds, d r and d s are the receiver- and satellite-dependent Uncalibrated Phase Delays (UPDs) in cycles, and ε is the observation noises including multipath in metres. The tropospheric delays are usually split into a hydrostatic (dry) and a non-hydrostatic (wet) part (Davis et al., Reference Davis, Herring, Shapiro, Rogers and Elgered1985). The Saastamoinen (Reference Saastamoinen1972) tropospheric model is used to correct the dry part from the measurements whereas the wet part is estimated from the measurements. The Global Mapping Functions (GMF) (Boehm et al., Reference Boehm, Niell, Tregoning and Schuh2006) are used to project the slant delays to the zenith delay.
Based on the GRAPHIC linear combination, the ionosphere-free combined observable Φ can be obtained by combining Equations (1) and (2). Φ can be described below:
$$\eqalign{\Phi_r^s = 0 \cdot 5\lpar P_r^s +L_r^s \rpar & =\rho _r^s +c\lpar dt_r -dt^s\rpar +d_{orb}^s +m_r^s \cdot Z_r +0 \cdot 5\lambda N_r^s \cr & \quad +0 \cdot 5\lambda \lpar d_r -d^s\rpar +0 \cdot 5c\lpar b_r -b^s\rpar +\varepsilon _{r\comma \Phi }^s} $$
where m is the wet mapping function, and Z r is the tropospheric Zenith Wet Delay (ZWD) at the station r in metres.
In order to investigate the emerging new navigation systems such as BeiDou and Galileo, MGEX has been established by the International GNSS Service (IGS) to serve as a platform (Rizos et al., Reference Rizos, Montenbruck, Weber, Weber, Neilan and Hugentobler2013). Four-constellation mixed precise satellite orbit file “gbmwwwwd.sp3” and clock file “gbmwwwwd.clk” are provided daily by GFZ, which is one of the MGEX analysis centres, are adopted for FCSF-PPP processing (Li et al., Reference Li, Ge, Dai, Ren, Fritsche, Wickert and Schuh2015b). The ionosphere-free linear combination of pseudorange P1 and P2 is usually used for the generation of precise satellite clock products, but the code biases are ignored during the estimation process. As such, the satellite clock correction
$dt_{I}^{s}$
provided in the precise satellite clock products contains a specific linear combination of satellite code biases in P1 and P2 (Guo et al., Reference Guo, Zhang and Wang2015), which can be shown as follows:
$$dt_I^s = dt^s+cb_{IF}^s $$
$$b_{IF}^s = \displaystyle{{f_1^2 \cdot b_1^s -f_2^2 \cdot b_2^s }\over{f_1^2 -f_2^2}}$$
where f
1 and f
2 are two carrier-phase frequencies in Hertz.
$b_{1}^{s}$
and
$b_{2}^{s}$
are the code biases on the satellite end in P1 and P2 in seconds, respectively.
After applying the precise satellite orbit and clock corrections, the ionosphere-free observable Φ can be rewritten as:
$$\Phi_r^s =\rho _r^s +cdt_r +m_r^s \cdot Z_r +0 \cdot 5\lambda N_r^s +B_r^s +\varepsilon_{r\comma \Phi }^s $$
$$B_r^s =0 \cdot 5\lambda \lpar d_r -d^s\rpar +0 \cdot 5c\lpar b_r +2b_{IF}^s -b_1^s \rpar $$
where B is a linear combination of code biases and phase delays on the receiver and satellite ends, and B can be decomposed into an average term B α and a satellite-dependent bias term δB as given below:
$$B_r^s =B_{r\comma \alpha }^s +\delta B_r^s $$
The average term B α and the satellite-dependent bias term δB will be absorbed into the receiver clock and phase ambiguity items, respectively, in the parameter estimation process. Equation (6) can then be rewritten as:
$$\Phi_r^s =\rho_r^s +cd\widetilde{t}_r +m_r^s \cdot Z_r +0 \cdot 5\lambda \widetilde{N}_r^s +\varepsilon _{r\comma \Phi }^s $$
$$d\widetilde{t}_r = dt_r +B_{r\comma \alpha }^s /c $$
$$\widetilde{N}_r^s = N_r^s +2\delta B_r^s /\lambda $$
To ensure accuracy and reliability, the Differential Code Biases (DCB) products can also be used for code bias corrections. Currently both the receiver-specific code biases for MGEX stations and the satellite-specific code biases for GPS, GLONASS, BeiDou and Galileo are provided by MGEX in the annual files “MGEXyyyy.bsx” and “MGEXyyyy_all.bsx” (Guo et al., Reference Guo, Zhang and Wang2015).
The FCSF-PPP observation model can be depicted as:
$$\Phi _r^G =\rho _r^G +cd\widetilde{t}_{rG} +m_r^G \cdot Z_r +0 \cdot 5\lambda _G \widetilde{N}_r^G +\varepsilon _{r\comma \Phi }^G $$
$$\Phi _r^R =\rho _r^R +cd\widetilde{t}_{rR} +m_r^R \cdot Z_r +0 \cdot 5\lambda _R \widetilde{N}_r^R +\varepsilon _{r\comma \Phi }^R $$
$$\Phi _r^C =\rho _r^C +cd\widetilde{t}_{rC} +m_r^C \cdot Z_r +0 \cdot 5\lambda _C \widetilde{N}_r^C +\varepsilon _{r\comma \Phi }^C +b_{sys}^C $$
$$\Phi _r^E =\rho _r^E +cd\widetilde{t}_{rE} +m_r^E \cdot Z_r +0 \cdot 5\lambda _E \widetilde{N}_r^E +\varepsilon _{r\comma \Phi }^E $$
where the superscripts G, R, C and E refer to GPS, GLONASS, BeiDou and Galileo satellites, respectively. The four receiver clock offset parameters in Equations (12)–(15) are different because the code biases and phase delays partially absorbed by them are different for different satellite systems (Li et al., Reference Li, Zhang, Ren, Fritsche, Wickert and Schuh2015a). In the precise products provided by GFZ, the time scales of the four GNSS systems are unified to the GPS time. The different receiver clock offsets should not be attributed to the system time offsets between GNSS systems. Hence, the unknown parameters include three receiver coordinates, four receiver clock offsets, one tropospheric ZWD, and real-valued ambiguity parameters equal to the number of the observed GPS, GLONASS, BeiDou and Galileo satellites. In Equation (14), b sys is the unique systematic bias in BeiDou code observations. The code observations from BeiDou Geostationary Orbit (GEO) satellites are down-weighted to weaken the effects of the systematic biases in this study. As to BeiDou Inclined Geosynchronous Orbit (IGSO) and Medium Earth Orbit (MEO) satellites, the elevation-dependent correction model for code systematic biases is adopted (Wanninger and Beer, Reference Wanninger and Beer2015).
The file “igs08_wwww.atx” generated and released by IGS is used to correct the GPS and GLONASS satellite Phase Centre Offsets (PCOs) and Variations (PCVs). The antenna offsets recommended by MGEX are used to correct the PCOs of BeiDou and Galileo satellites, while the BeiDou and Galileo PCV corrections are still unavailable (Rizos et al., Reference Rizos, Montenbruck, Weber, Weber, Neilan and Hugentobler2013). As other error mitigations have been well described in work such as Kouba and Héroux (Reference Kouba and Héroux2001), they will not be covered again.
Since a Kalman filter is applied for FCSF-PPP, appropriate stochastic models for the measurements and dynamic models for the parameters need to be provided. As can be seen from Equation (3), the GRAPHIC ionosphere-free combined observations are a linear combination of the original code and phase measurements. Assuming that there is no correlation between the two types of measurements, the initial variances of the ionosphere-free observations can be obtained by applying the law of random error propagation as follows:
$$\sigma_{\Phi}^{2} =0 \cdot 25\sigma_P^2 +0 \cdot 25\sigma_L^2 \approx 0 \cdot 25\sigma_P^2 $$
where
${\sigma}_P^{2}$
and
${\sigma}_L^{2}$
are the variances of code and phase observations, respectively. The variance of phase observations is far smaller than that of code observations, and thus can be neglected. The actual variances are defined as a function of the initial variances and the satellite elevation angles (Gerdan, Reference Gerdan1995).
As to the dynamic models for the unknown parameters, the ambiguity parameters and static receiver coordinates are modelled as constants, while the receiver clock offsets and kinematic receiver coordinates may be modelled as a Random Walk (RW) or a first-order Gauss–Markov process. The ZWD can be modelled as a RW process (Cai et al., Reference Cai, Gao, Pan and Zhu2015).
In the specific implementation of FCSF-PPP dynamic and stochastic models, the spectral density values for the receiver clock offsets and ZWD are empirically set to 105 and 10−9 m2/s, respectively. In kinematic mode, a value of 102 m2/s is set as the spectral density value for receiver coordinates. The precision of the GPS code observations is set to 0·3 m (Pan et al., Reference Pan, Cai, Santerre and Zhu2014). The precision of the GLONASS code observations is set to 0·6 m partly because of its higher code noise level. The higher code measurement noises of GLONASS are due to its lower code chipping rate equal to half that of GPS code observations (Hauschild et al., Reference Hauschild, Montenbruck, Sleewaegen, Huisman and Teunissen2012; Montenbruck, Reference Montenbruck2003). In addition, the satellite-specific receiver code biases may also contribute to the down-weighting of GLONASS code observations. Since the accuracy of BeiDou and Galileo satellite orbits and clocks are relatively lower than GPS (Zhao et al., Reference Zhao, Guo, Li, Qu, Hu, Shi and Liu2013; Steigenberger et al., Reference Steigenberger, Hugentobler, Loyer, Perosanz, Prange, Dach, Uhlemann, Gendt and Montenbruck2015), their measurements are down-weighted. The precision of code observations is set to 0·6 m for Galileo, BeiDou IGSO and MEO satellites, while the code observation precision of BeiDou GEO satellites is set to 0·9 m.
3. RESULTS AND ANALYSIS
3.1. Data description
Datasets collected at 47 globally distributed four-system MGEX stations on 1–7 April 2015 are used to assess the performance of FCSF-PPP. The geographical distribution of the selected stations is shown in Figure 1. All selected stations were equipped with multi-GNSS receivers which can produce GPS, GLONASS, BeiDou and Galileo observations. All observations were post-processed and recorded at a sampling interval of 30 s. The satellite elevation mask angle was set to 10°. Since the precise coordinates of eight selected MGEX stations, which are marked in red in Figure 1, are not available in file “igs15Pwwwwd.snx” provided by IGS, their coordinate values are computed through an Online Positioning User Service (OPUS), which is capable of providing centimetre-level or even millimetre-level positioning accuracy (Ghoddousi-Fard and Dare, Reference Ghoddousi-Fard and Dare2006) and available at: http://www.ngs.noaa.gov/OPUS. As the repeatability of the OPUS processing results is at a millimetre-level from day to day, the average differential positioning results over the seven days are used to assess the single-frequency PPP solutions at the eight stations. In order to assess the OPUS processing results, the station coordinates that are available in SINEX solutions are re-computed using the OPUS. The difference between the SINEX solutions and the average OPUS solutions over seven days is 1–3 mm.
Geographical distribution of 47 four-system MGEX stations.
3.2. FCSF-PPP result analysis
In order to investigate the performance improvement attributing to multi-constellation integration, the dataset from station JFNG on 7 April 2015 is processed in different constellation combinations, i.e. GPS-only, GPS/GLONASS and GPS/GLONASS/BeiDou/Galileo. The station JFNG is located in Wuhan, China and covered by the BeiDou service of the Asia-Pacific area. Figure 2 shows the epoch-wise positioning errors for the three different combination cases. Compared with the GPS-only case, GPS/GLONASS single-frequency PPP achieves better convergence performance, especially in the vertical coordinate component. The convergence time is further reduced after adding BeiDou and Galileo observations. As to positioning accuracy, the Root Mean Square (RMS) values of position errors over the last 15 minutes are 0·3, 2·0 and 1·5 cm for four-constellation case in the east, north and up directions, respectively. The above accuracy is better than that of GPS-only and GPS/GLONASS cases, which are 0·4, 2·8 and 3·1 cm, and 0·4, 2·2 and 1·7 cm in the three directions, respectively. Figure 3 presents the number of visible satellites and Position Dilution of Precision (PDOP) for each processing case. It is clear that multi-constellation integration significantly increases the number of visible satellites and simultaneously decreases the PDOP values. The average satellite numbers and PDOP values for the above three processing cases are 8·1, 14·6 and 25·7, and 2·0, 1·5 and 1·2, respectively. In conjunction with the position solutions, it is concluded that the performance of multi-constellation integrated single-frequency PPP benefits from the increased number of satellites and improved satellite sky distribution.
Single-frequency PPP positioning errors for three different constellation combinations at JFNG on 7 April 2015. The abbreviations GLO, BDS and GAL represent GLONASS, BeiDou and Galileo, respectively.
Satellite number and PDOP at JFNG on 7 April 2015.
Since observation residuals contain measurement noises and other unmodelled errors, they can also be used as an important index to evaluate the FCSF-PPP model. Figure 4 shows the observation residuals in the four-constellation integrated processing case. Different colours represent different satellites. As can be seen, most GPS, GLONASS and BeiDou observation residuals vary within a similar range of −0·5 m ~ 0·5 m, whereas the Galileo observation residuals vary in a significantly smaller range. There are almost no observation residuals for Galileo when only one Galileo satellite is available. This is because two more unknown parameters, namely Galileo receiver clock offset and phase ambiguity, need to be estimated after adding one Galileo measurement. The RMS statistics are also displayed in each panel. The statistical results clearly demonstrate that GLONASS has the largest observation residuals with 0·161 m, while Galileo has the smallest observation residuals with 0·084 m. GPS and BeiDou observation residuals are comparable with values of 0·130 m and 0·136 m, respectively. Since observation residuals show a comprehensive effect of remaining satellite orbit and clock error, ranging error, ionospheric delay error and other unmodelled errors, they are different for different constellations. The smallest observation residuals from Galileo are partly due to the lower measurement redundancy and the higher satellite elevation angles. Overall, the RMS residuals are smaller than 0·17 m for all four GNSS systems and no systematic errors can be found in the residuals, suggesting that various errors and biases in GRAPHIC ionosphere-free observations from different constellations have been properly handled in the FCSF-PPP model.
Observation residuals for FCSF-PPP at JFNG on 7 April 2015.
To compare the performance of single- and dual-frequency PPP, Figure 5 shows a comparison between FCSF-PPP and Four-Constellation integrated Dual-Frequency PPP (FCDF-PPP) solutions using observations at JFNG. The FCDF-PPP model developed by Cai et al. (Reference Cai, Gao, Pan and Zhu2015) is adopted. It is clear that FCSF-PPP takes much longer than FCDF-PPP before its position solutions converge. If it is considered to have converged when the positioning errors reach 0·1 m and keep within 0·1 m, the position filter for FCSF-PPP needs 64·5, 73·5 and 85·0 minutes to converge in the east, north and up directions, respectively, while it needs only 21·5, 15·0 and 32·5 minutes for FCDF-PPP in the three directions. The RMS statistical values of position errors over the last 15 minutes are 0·3, 2·0 and 1·5 cm for FCSF-PPP and 0·3, 1·0 and 0·9 cm for FCDF-PPP in the three directions, respectively. It is indicated that FCSF-PPP still achieves a worse positioning accuracy than FCDF-PPP even after a long convergence time.
Single-frequency vs. dual-frequency PPP position errors using combined GPS, GLONASS, BeiDou and Galileo observations at JFNG on 7 April 2015.
3.3. Performance of FCSF-PPP in constrained visibility environments
An accuracy improvement of only 1–2 mm was found after the integration with BeiDou and Galileo under an open sky, namely under the observing conditions with no signal blockage around the receiver, as shown in Figure 2. However, very often single-frequency PPP users are in environments with limited satellite visibility such as in urban canyons, mountainous areas and open-pit mines, etc. In order to investigate the performance of FCSF-PPP in these constrained visibility environments, the cut-off elevation angles are set from 20° to 50° in steps of 10° to simply simulate the real harsh environments. Figure 6 illustrates the positioning errors under different elevation mask angles using the dataset from JFNG. It is clearly seen that the vertical positioning errors increase rapidly with the increment of elevation mask angles whereas it is not the case for horizontal components. For the GPS-only case, the vertical accuracy starts to significantly degrade when the elevation mask angle increases to 20°. Similar situations are also found in the GPS/GLONASS case at a mask angle of 30°. By contrast, FCSF-PPP can still achieve better positioning performance at a mask angle of 40°. At a cut-off elevation angle of 50°, there are fairly large positioning errors in the vertical direction for all three processing cases, but FCSF-PPP still achieves the best positioning performance. The above phenomenon that the vertical accuracy decreases rapidly as the cut-off elevation angle increases may be explained by satellite sky distribution. The average Dilution of Precision (DOP) values in the horizontal and vertical components, namely HDOP and VDOP, are calculated. The average HDOP values for FCSF-PPP are 0·7, 0·9, 1·4 and 2·4 at cut-off angles from 20° to 50°, respectively, whereas the corresponding VDOP values are 1·4, 2·2, 4·6 and 8·9. The RMS statistics of positioning errors over the last 15 minutes as well as the average number of satellites under different elevation mask angles are given in Table 1. According to the statistical results, the three-dimensional (3D) accuracy improvement for the GPS/GLONASS case over the GPS-only case is 40%, 41%, 67% and 26%, and for the four-constellation case over the GPS/GLONASS case is 80%, 77%, 23% and 45% when the satellite elevation mask angle is set to 20°, 30°, 40° and 50°, respectively.
Single-frequency PPP positioning errors for three different constellation combinations under different elevation mask angles at JFNG on 7 April 2015.
RMS statistics of positioning errors and average number of satellites for three different constellation combinations under different elevation mask angles at JFNG on 7 April 2015.
3.4. Global FCSF-PPP accuracy assessment
In order to assess the positioning accuracy of the FCSF-PPP in both static and kinematic modes, the FCSF-PPP solutions are compared with cases of GPS-only and GPS/GLONASS using datasets collected at 47 globally distributed four-system MGEX stations on seven consecutive days. Table 2 provides the statistical results of the static single-frequency PPP solutions for three different processing cases with different session lengths of 15 and 30 min, 1, 2, 4, 6 and 12 h. The RMS values are calculated from all static single-frequency PPP solutions over all the selected stations and days. It can be seen that the positioning accuracy is continuously improved along with the increase of observational length. With the same session length, the four-constellation integrated case achieves the highest positioning accuracy, and the GPS/GLONASS case follows. Usually, the positioning accuracy in the north coordinate component is the best, while the vertical component is the most inaccurate component. The GPS-only single-frequency PPP requires more than two hours to achieve a positioning accuracy better than 10 cm for all three coordinate components. As to the other two cases, an accuracy better than 10 cm for all three components is available within two hours. In addition, the FCSF-PPP can also achieve a 3D positioning accuracy better than 10 cm within two hours. After undergoing a long convergence time of 12 hours, the 3D positioning accuracy for GPS-only, GPS/GLONASS and four-constellation cases is 4·2, 3·1 and 2·2 cm, respectively. The correlation between coefficient matrices of adjacent epochs is very high. Therefore, the repeat observation in a short period of time will do little to help the convergence of parameters or the improvement of accuracy because of the small changes in satellite sky distribution. In contrast, a long observation time of up to 12 hours can be very helpful, which benefits from the availability of numerous measurements and the strong satellite geometry. For comparison, the RMS values are also calculated from all static FCDF-PPP solutions. The 3D positioning accuracy for four-constellation integrated dual-frequency case with seven different session lengths is 37·7, 15·6, 6·5, 3·9, 2·3, 1·6 and 0·7 cm, respectively.
RMS values of static single-frequency PPP solutions for three different constellation combinations with different session lengths of 15 min, 30 min, 1 h, 2 h, 4 h, 6 h and 12 h.
For the purpose of evaluating the positioning accuracy of kinematic FCSF-PPP solutions, the station coordinates are estimated epoch-by-epoch without imposing any constraints between the epochs. Figure 7 depicts the distribution of kinematic positioning errors, number of satellites and PDOP for GPS-only, GPS/GLONASS and four-constellation processing cases. All data are processed in kinematic mode, and each error value in Figure 7 refers to the positioning error at an epoch. As the position solutions in the first two hours are still in the converging stage, they are not used for the kinematic accuracy statistics. It is clearly seen that the positioning errors in all three directions are approximately normally distributed. The four-constellation case accounts for the largest percent of smaller errors, and the GPS/GLONASS case follows. The statistical results in terms of mean values, Standard Deviation (STD) values and RMS values are also given in Figure 7. According to the RMS values, the improvement of the GPS/GLONASS case on the positioning accuracy is 35%, 26% and 21% over the GPS-only case in the east, north and up directions, respectively. Furthermore, the accuracy improvement of FCSF-PPP over GPS/GLONASS single-frequency PPP is 18%, 15% and 21% in the three directions, respectively. The kinematic positioning accuracy for the FCSF-PPP is 6·7, 6·7 and 12·6 cm in the three directions, respectively. The average number of satellites and PDOP values for the above three processing cases are 8·3, 15·3 and 20·5, and 2·1, 1·4 and 1·3, respectively.
Distribution of epoch-wise kinematic positioning errors, number of satellites and PDOP for three different constellation combinations.
The positioning errors of single-frequency PPP in kinematic mode are larger than those in static mode. For the purpose of the convergence analysis, the precision threshold is set to 0·5 m for the east and north directions and 1 m for the up direction, respectively. The average convergence time for all single-day kinematic solutions of single-frequency PPP is presented in Table 3. The improvement of the FCSF-PPP on the convergence time is 41%, 37% and 55% over the GPS-only case and 13%, 11% and 23% over the GPS/GLONASS case in the east, north and up directions, respectively.
Convergence time for single-frequency PPP kinematic solutions (min).
Since the current BeiDou system is in the stage of regional deployment, the BeiDou satellites are still not globally evenly distributed. In addition, only a few Galileo satellites are available under the current Galileo constellation, leading to the quite different visibility of Galileo satellites in the global scale. In order to truly evaluate the contributions of BeiDou and Galileo systems, Figure 8 shows the 3D accuracy improvement against the PDOP improvement after adding BeiDou and Galileo observations to the GPS/GLONASS single-frequency PPP processing in kinematic mode. Only the results at a certain epoch that includes at least two BeiDou satellites or two Galileo satellites are involved in the statistics. The RMS statistics of 3D positioning errors and average PDOP values over all the available epochs at each selected station on each selected day are calculated respectively, and thus there is a total of 329 blue points in Figure 8. The red line shown in Figure 8 is the linear fitting of the blue points, which reveals a trend that the 3D accuracy improvement rates increase as the PDOP improvement rates increase. This demonstrates that the improvement of positioning accuracy is dependent on the improvement of satellite sky distribution.
Dependence of 3D accuracy improvement on the PDOP improvement after adding BeiDou and Galileo observations to the GPS/GLONASS single-frequency PPP processing in kinematic mode.
3.5. Kinematic results and analysis
The kinematic accuracy evaluation shown in the previous section is implemented by processing the static measurements with the strategies adopted in kinematic mode. In order to assess the FCSF-PPP performance in the real kinematic mode, a kinematic experiment was conducted within the new campus of the Central South University in Changsha, China, on 16 August 2014. The experiment started at the local time 18:00 (GPS time 10:00) and lasted for two hours. The experiment time was planned in advance so that all three operational Galileo IOV satellites at that time could be tracked in the experimental area. Figure 9 shows the field observation equipment and environment for both base and rover stations. At the rover station, a “Trimble NetR9” GNSS receiver with a “Trimble Zephyr Model 2” geodetic antenna was carried by an electric bicycle to collect kinematic data from four constellations. The moving vehicle was driven at a speed of around 10 km/h. At the base station, the same type of receiver with a “TRM55971.00” antenna and a radome was set up on the roof of the Mining Building of the Central South University to help determine the reference coordinates of the rover station at cm level accuracy using a double-difference Real-Time Kinematic (RTK) approach. The distance between the base and rover stations was less than 2·5 km. The kinematic data was collected at a sampling interval of 1 s. The satellite elevation mask angle was set to 10°.
Field observation equipment and environment for FCSF-PPP kinematic experiment on 16 August 2014.
Figure 10 shows the kinematic positioning errors of GPS-only, GPS/GLONASS and four-constellation single-frequency PPP solutions with respect to the reference coordinate values in the east, north and up directions. Figure 11 presents the corresponding number of satellites and PDOP. The average number of satellites and PDOP values for the above three processing cases are 8·4, 14·2 and 24·2, and 1·9, 1·4 and 1·2, respectively. It is obvious that the positioning errors are significantly reduced for the GPS/GLONASS case over the GPS-only case, especially in the vertical direction. The positioning accuracy is further improved after a further integration with BeiDou and Galileo. The RMS statistical values are computed using the positioning errors in the last one hour in which the position solutions in all three directions have reached stable values. GPS-only single-frequency PPP has the worst positioning accuracy with RMSs of 18·6, 13·7 and 33·2 cm in the east, north and up directions, respectively. With the combination of GPS and GLONASS, the positioning accuracy is improved by 30%, 20% and 28% over the GPS-only case to 13·0, 10·9 and 23·9 cm in the three directions, respectively. The FCSF-PPP improves the positioning accuracy by 11%, 28% and 13% over the GPS/GLONASS single-frequency PPP to 11·6, 7·8 and 20·9 cm in the three directions, respectively. When using dual-frequency observations, the four-constellation case can achieve an accuracy of 4·0, 2·1 and 9·6 cm in the three directions, respectively. Compared with the results shown in Figure 7, it is indicated that the positioning accuracy in the real kinematic mode is worse than that in the simulated kinematic mode.
Single-frequency PPP kinematic positioning errors for three different constellation combinations in the kinematic experiment.
Number of satellites and PDOP for three different constellation combinations in the kinematic experiment.
4. CONCLUSIONS
For most positioning and navigation applications, single-frequency GNSS receivers are widely used due to the low costs. Therefore the high-precision single-frequency PPP technique has attracted great interest in the GNSS community. Multi-constellation integration has the potential to significantly improve the positioning accuracy and reduce the convergence time due to the increased number of visible satellites and the improved satellite sky distribution. Since BeiDou and Galileo have already begun to transmit signals for position determination, the four-constellation integrated single-frequency PPP has become feasible. In this study, a FCSF-PPP model based on the GRAPHIC ionosphere-free combination to remove the ionospheric effect is presented to simultaneously process the pseudorange and carrier phase observations on a single frequency from GPS, GLONASS, BeiDou and Galileo. Datasets from 47 globally distributed four-system MGEX stations on seven consecutive days as well as a kinematic experimental dataset are employed to fully assess the performance of FCSF-PPP. The performance of FCSF-PPP is compared to GPS-only and combined GPS/GLONASS single-frequency PPP.
Using the MGEX datasets, the statistical results indicate that the 3D positioning accuracy for the FCSF-PPP is 154·5, 51·5, 23·1, 9·9, 5·7, 3·3 and 2·2 cm with different session lengths of 15 and 30 min, 1, 2, 4, 6 and 12 h in the static mode, respectively. In the kinematic mode, the FCSF-PPP can achieve an accuracy of 6·7, 6·7 and 12·6 cm in the east, north and up directions, respectively. In addition, the positioning performance of the four-constellation case is better than that of the GPS-only and GPS/GLONASS cases. Both the BeiDou and Galileo constellations will be fully deployed in the next few years, and thus greater benefits from the four-constellation integration can be expected. It is noted that the presented results are obtained using multi-frequency geodetic-type receivers rather than single-frequency navigation-type receivers. If the latter ones are used, the positioning accuracy could degrade due to the relatively poorer data quality. Nevertheless, the exhibited results are a good indication of the FCSF-PPP performance under the current GNSS constellations.
Considering that an accuracy better than 2 cm after an observation time of 12 hours in static mode and 13 cm in kinematic mode can be achieved in all three coordinate components, FCSF-PPP has large implications for development of low-cost and high-precision applications, such as precision agriculture, vegetation boundaries, hydrography and navigation applications in the automotive markets. Future work includes tests of Real-Time Kinematic PPP (PPP-RTK) ambiguity resolution for a truly single-frequency low-cost receiver with the use of precise corrections comprising satellite clocks, satellite phase biases and ionospheric delays from a regional Continuously Operating Reference Station (CORS) network.
ACKNOWLEDGMENTS
The contribution of data from IGS and GFZ is appreciated. This study was supported by National Natural Science Foundation of China (Grant No. 41474025).
