Precise Point Positioning Using Triple GNSS Constellations in Various Modes

This paper introduces a new dual-frequency precise point positioning (PPP) model, which combines the observations from three different global navigation satellite system (GNSS) constellations, namely GPS, Galileo, and BeiDou. Combining measurements from different GNSS systems introduces additional biases, including inter-system bias and hardware delays, which require rigorous modelling. Our model is based on the un-differenced and between-satellite single-difference (BSSD) linear combinations. BSSD linear combination cancels out some receiver-related biases, including receiver clock error and non-zero initial phase bias of the receiver oscillator. Forming the BSSD linear combination requires a reference satellite, which can be selected from any of the GPS, Galileo, and BeiDou systems. In this paper three BSSD scenarios are tested; each considers a reference satellite from a different GNSS constellation. Natural Resources Canada’s GPSPace PPP software is modified to enable a combined GPS, Galileo, and BeiDou PPP solution and to handle the newly introduced biases. A total of four data sets collected at four different IGS stations are processed to verify the developed PPP model. Precise satellite orbit and clock products from the International GNSS Service Multi-GNSS Experiment (IGS-MGEX) network are used to correct the GPS, Galileo, and BeiDou measurements in the post-processing PPP mode. A real-time PPP solution is also obtained, which is referred to as RT-PPP in the sequel, through the use of the IGS real-time service (RTS) for satellite orbit and clock corrections. However, only GPS and Galileo observations are used for the RT-PPP solution, as the RTS-IGS satellite products are not presently available for BeiDou system. All post-processed and real-time PPP solutions are compared with the traditional un-differenced GPS-only counterparts. It is shown that combining the GPS, Galileo, and BeiDou observations in the post-processing mode improves the PPP convergence time by 25% compared with the GPS-only counterpart, regardless of the linear combination used. The use of BSSD linear combination improves the precision of the estimated positioning parameters by about 25% in comparison with the GPS-only PPP solution. Additionally, the solution convergence time is reduced to 10 minutes for the BSSD model, which represents about 50% reduction, in comparison with the GPS-only PPP solution. The GNSS RT-PPP solution, on the other hand, shows a similar convergence time and precision to the GPS-only counterpart.


Introduction
Global navigation satellite systems (GNSS) precise point positioning (PPP) has proven to be capable of providing positioning accuracy at the sub-decimeter and decimeter levels in static and kinematic modes, respectively. PPP differs from differential positioning methods in that differential techniques require access to GNSS observations from one or more reference stations with precisely known coordinates. This provides an advantage for PPP over differential methods, as only a single receiver is required at the user's end. Consequently, the spatial operating range limit of differential techniques is overcome through PPP. However, a major disadvantage of PPP in comparison with

Un-Differenced Post-Processing PPP Models
Traditionally, PPP has been carried out using dual-frequency ionosphere-free linear combinations of carrier-phase and pseudorange GPS measurements. Equations (1) to (6) show the ionosphere-free linear combinations of GPS, Galileo, and BeiDou observations [29,30].
where the subscripts G, E, and B refer to the GPS, Galileo, and BeiDou satellite systems, respectively; P G IF , P E IF , and P B IF are the ionosphere-free pseudoranges in meters for GPS, Galileo, and BeiDou systems, respectively; Φ G IF , Φ E IF , and Φ B IF are the ionosphere-free carrier phase measurements in meters for GPS, Galileo, and BeiDou systems, respectively; GGTO is the GPS to Galileo time offset; GB is the GPS to BeiDou time offset; ρ is the true geometric range from receiver at reception time to satellite at transmission time in meter; dt r , dt s are the clock errors in seconds for the receiver at signal reception time and the satellite at signal transmission time, respectively; are ionosphere-free linear combinations of frequency-dependent initial fractional phase biases in the receiver and satellite channels for both GPS, Galileo, and BeiDou in meters, respectively; c is the speed of light in vacuum in meter per second; are the ionosphere-free linear combinations of the relevant noise and un-modeled errors in meter; α G , β G , α E , β E , α B , β B are the ionosphere-free linear combination coefficients for GPS, Galileo, and BeiDou which are given, respectively, by: α G " , β E " . where f 1 and f 2 are GPS L 1 and L 2 signals frequencies; f E1 and f E5a are Galileo E 1 and E 5a signals frequencies; f B1 and f B2 are BeiDou B 1 and B 2 signals frequencies.
where λ 1 and λ 2 are the GPS L1 and L2 signals wavelengths in meters; λ E1 and λ E5a are the Galileo E1 and E5a signals wavelengths in meters; λ B1 and λ B2 are the BeiDou B1 and B2 signals wavelengths in meters; N 1 , N 2 are the integer ambiguity parameters of GPS signals L1 and L2, respectively; N E1 , N E5a are the integer ambiguity parameters of Galileo signals E1 and E5a, respectively; N B1 , N B2 are the integer ambiguity parameters of BeiDou signals B1 and B2, respectively. Precise orbit and satellite clock corrections of IGS-MGEX networks are produced for GPS, Galileo and BeiDou observations and are referred to GPS time. IGS precise GPS satellite clock correction includes the effect of the ionosphere-free linear combination of the satellite hardware delays of L1/L2 P(Y) code, while the Galileo counterpart includes the effect of the ionosphere-free linear combination of the satellite hardware delays of the Galileo E1/E5a pilot code. In addition, BeiDou satellite clock correction includes the effect of the ionosphere-free linear combination of the satellite hardware delays of B1/B2 code [11]. By applying the precise clock products for GPS, Galileo, and BeiDou observations, Equations (1)-(6) will take the following form: For simplicity, the receiver and satellite hardware delays are written as: In the combined GPS, Galileo and BeiDou un-differenced post-processed PPP solution, the GPS receiver clock error is lumped to the GPS receiver differential code biases. To maintain consistency in the estimation of a common receiver clock offset, this convention is used when combining the ionosphere-free linear combination of GPS L1/L2, Galileo E1/E5a, and BeiDou B1/B2 observations in the post-processed PPP solution. This, however, introduces an additional bias in the Galileo ionosphere-free PPP mathematical model, which represents the difference in the receiver differential code biases of both systems. Such an additional bias is commonly known as the inter-system bias, which is referred to as ISB in this paper. In our PPP model, the Hopfield tropospheric correction model along with the Vienna mapping function are used to account for the hydrostatic component of the tropospheric delay [27,31]. Other corrections are also applied, including the effect of ocean loading [32,33], Earth tide [28], carrier-phase windup [8,34], Sagnac [35], relativity [7], and satellite and receiver antenna phase-center variations [36]. The noise terms are modeled stochastically using an exponential model, as described in [37]. With the above consideration, the GPS/Galileo/BeiDou ionosphere-free linear combinations for the pseudorange and carrier-phase measurements can be written as: where r dt rG represents the sum of the receiver clock error and receiver hardware delay r dt rG " cdt rGb r P ; ISB is the inter system bias as follows ISB GE " b r E´b r P ; ISB GB " b r B´b r P ; r N G IF , r N E IF and r N B IF are given by:

BSSD Post-Processing PPP Models
When combining the GPS, Galileo, and BeiDou observations in an un-differenced PPP model, the ambiguity parameters lose their integer nature as they are contaminated by the receiver and satellite hardware delays. It should be pointed out that the number of unknown parameters in the combined PPP solution equals the number of visible satellites from any system plus seven parameters, while the number of equations equals double the number of the visible satellites. This means that the redundancy equals n G + n E + n B´7 . In other words, at least seven mixed satellites are needed for the solution to exist. In comparison with the GPS-only un-differenced scenario, which requires a minimum of five satellites for the solution to exist, the addition of Galileo or BeiDou satellites increases the redundancy by n E + n B´2 . In other words, we need a minimum of three satellites from both Galileo and BeiDou systems in order to contribute to the solution.
As indicated earlier, the reference satellite can be selected from any of the three satellite constellations [37]. If a GPS satellite is selected as a reference for all GNSS observables, using Equations (16)- (21), BSSD mathematical models can be written as: Similarly, when a Galileo satellite is selected as a reference, using Equations (16)-(21), BSSD mathematical models can be written as: , r N lk E IF , and r N lh BE IF are the BSSD non-integer ambiguity parameters lumped to the receiver and satellite hardware delays, which are given by: When selecting a BeiDou satellite as a reference, using Equations (16)-(21), BSSD mathematical models can be written as: , r N hk EB IF , and r N hu B IF are the BSSD non-integer ambiguity parameters lumped to the receiver and satellite hardware delays, which are given by: Under the assumption that the observations are uncorrelated and the errors are normally distributed with zero mean, the covariance matrix of the un-differenced observations takes the form of a diagonal matrix. The elements along the diagonal line represent the variances of the code and carrier phase measurements. In our solution, we consider the ratio between the standard deviation of the code and carrier-phase measurements to be 100. When forming BSSD, however, the differenced observations become mathematically correlated. This leads to a fully populated covariance matrix at any particular epoch.

Real-Time PPP Satellite Clock Corrections
IGS launched its real-time service (RTS) in April 2013, which provides the users with real-time satellite orbit and clock corrections. At present, the IGS RTS uses a network of 130 globally distributed real-time tracking stations (IGS, 2016). Generally, the IGS satellite orbit and clock corrections are available to users with a delay based on the stated accuracies of the corrections, which are intended to be used in the post-processed positioning mode, e.g., the final IGS orbit and clock corrections have a delay of about 14 days. The IGS produced "ultra rapid" precise satellite correction products, which can be used in near real-time and real-time positioning; however, the prediction part of these corrections are based on earlier observations and are significantly less accurate than the other IGS products [26,38].
The IGS RTS produces and publishes real-time GNSS orbit and clock corrections, which are streamed to users in the Radio Technical Commission for Maritime services (RTCM) format. The RTCM State Space Representation (SSR) format is capable of supporting sub-decimeter RT-PPP anywhere in the world. Currently, the RTS products are offered for the GPS, Galileo, and GLONASS constellations. Table 1 outlines the IGS RTS products, their formats and frequency [25,26]. In order to access the RTS-IGS data streams that contain the satellite orbit and clock corrections, an NTRIP client application must be used. The Bundesamt für Kartographie und Geodäsie (BKG) NTRIP Client (BNC) version 2.11.1 is used to access these data streams. BKG Ntrip Client (BNC) is an open source application that support a variety of GNSS positioning applications [38].
IGS01/IGC01 precise satellite orbit and clock corrections are computed using a single epoch GPS combination. The solution epochs in this product are completely independent of each other, which has the advantage that the full accuracy is available as soon as product generation starts. While the IGS02 precise satellite orbit and clock corrections is extracted from one of the incoming ultra-rapid solutions. Both of the IGS02 and IGS03 use Kalman filtering and require a few minutes to converge to their full accuracy. The major difference between IGS03 and IGS02 is that the former includes GNSS corrections in addition to GPS [39].

Results and Discussion
To verify the developed combined PPP model, three-constellation GNSS (GPS, Galileo, and BeiDou) observations at four globally distributed stations were selected from the IGS tracking network (Figure 1) [36]. Those stations are occupied by GNSS receivers, which are capable of simultaneously tracking the GNSS constellations. Only one hour of observations with maximum possible number of Galileo and BeiDou satellites at each station is considered in our analysis. All data sets have an interval of 30 s.
The positioning results for station Delf1 located at Delft University of Technology, The Netherlands, are presented below. Similar results are obtained for the other stations. However, a summary of the convergence times and the three-dimensional PPP solution standard deviations are presented below for all stations. Natural Resources Canada's GPSPace PPP software is modified to handle data from GPS, Galileo, and BeiDou systems, which enables a combined PPP solution as detailed above. BNC version 2.11.1 software is used to combine GPS and Galileo observations to obtain a real-time PPP solution. In addition to the combined PPP solution, we also obtained the PPP solutions of the un-differenced ionosphere-free GPS-only, which is used to assess the performance of the newly developed PPP model. Figure 2 summarizes the satellite availability during the one-hour observation time for each constellation at DLF1 station. As shown in Figure 2 Figure 3 summarizes the convergence times for all un-differenced post-processing PPP models with different GNSS constellation combinations. As can be seen, the un-differenced GPS-only post-processed PPP solution indicates that the model is capable of obtaining a sub-decimetre level accuracy. However, the solution takes about 20 min to converge to decimetre level precision. As shown in Figure 3, the convergence time of the combined GNSS post-processed PPP solutions takes about 15 min to reach the decimeter level precision, which represent a 25% improvement in comparison with the GPS-only post-processed PPP solution. To further assess the performance of the various un-differenced post-processing PPP models, the solution output is sampled every 10 min and the standard deviation of the computed station coordinates is calculated for each sample. Figure 4 shows the position standard deviations in the East, North, and Up directions, respectively. As can be seen, the precision of the combined un-differenced post-processed PPP solutions is comparable to that of the GPS-only post-processed PPP solution. As shown in Figure 2, eight to nine GPS satellites were visible during the one-hour observation time span. The addition of Galileo and BeiDou systems increase the number of visible satellites to 19-20.  Figure 3 summarizes the convergence times for all un-differenced post-processing PPP models with different GNSS constellation combinations. As can be seen, the un-differenced GPS-only post-processed PPP solution indicates that the model is capable of obtaining a sub-decimetre level accuracy. However, the solution takes about 20 min to converge to decimetre level precision. As shown in Figure 3, the convergence time of the combined GNSS post-processed PPP solutions takes about 15 min to reach the decimeter level precision, which represent a 25% improvement in comparison with the GPS-only post-processed PPP solution. To further assess the performance of the various un-differenced post-processing PPP models, the solution output is sampled every 10 min and the standard deviation of the computed station coordinates is calculated for each sample. Figure 4 shows the position standard deviations in the East, North, and Up directions, respectively. As can be seen, the precision of the combined un-differenced post-processed PPP solutions is comparable to that of the GPS-only post-processed PPP solution.     Figure 5 summarizes the convergence times for the GNSS BSSD post-processed PPP solutions using different reference satellites. As shown in Figure 5, using BSSD post-processing PPP model reduces the convergence time to 10 min, which represents a 50% improvement compared to the GPS-only post-processed PPP solution. Similar to the un-differenced solution, the BSSD solution output is sampled every 10 min and the standard deviation of the estimated station coordinates is calculated for each sample.    Figure 5 summarizes the convergence times for the GNSS BSSD post-processed PPP solutions using different reference satellites. As shown in Figure 5, using BSSD post-processing PPP model reduces the convergence time to 10 min, which represents a 50% improvement compared to the  Figure 6 shows a summary of the standard deviations of the station coordinates in the East, North, and Up directions, respectively. As shown in Figure 6, the standard deviations of the GNSS BSSD post-processed PPP solutions are improved compared to the un-differenced post-processed PPP solutions. In addition, as the number of epochs, and consequently the number of measurements, increases the performance of the various models tends to be comparable. In order to assess the RT-PPP solution, all of the IGS RTS products are used to produce various real-time PPP solutions. GPS-only post-processed PPP solution. Similar to the un-differenced solution, the BSSD solution output is sampled every 10 min and the standard deviation of the estimated station coordinates is calculated for each sample. Figure 6 shows a summary of the standard deviations of the station coordinates in the East, North, and Up directions, respectively. As shown in Figure 6, the standard deviations of the GNSS BSSD post-processed PPP solutions are improved compared to the un-differenced post-processed PPP solutions. In addition, as the number of epochs, and consequently the number of measurements, increases the performance of the various models tends to be comparable. In order to assess the RT-PPP solution, all of the IGS RTS products are used to produce various real-time PPP solutions.       based on multi-constellation GNSS solution. The other RTS-IGS products provided longer PPP solution convergence times. Figure 8 summarizes the convergence times of the RT-PPP solutions for the various test stations when the IGS03 satellite corrections are used. As can be seen, the convergence time for the RT-PPP solution is similar to the convergence time of the GPS-only RT-PPP solution. As mentioned earlier the RTS-IGS satellite clock corrections are not available for the BeiDou system. Similar to previous cases, the RT-PPP solution output is sampled every 10 min and the standard deviation of the estimated station coordinates is calculated for each sample.  Figure 9 shows the standard deviations of the station coordinates in the East, North, and Up directions, respectively. As can be seen, the use of IGS03 satellite corrections improves the RT-PPP solution precision, in comparison with other real-time satellite correction products.  Figure 9 shows the standard deviations of the station coordinates in the East, North, and Up directions, respectively. As can be seen, the use of IGS03 satellite corrections improves the RT-PPP solution precision, in comparison with other real-time satellite correction products.  Figure 9 shows the standard deviations of the station coordinates in the East, North, and Up directions, respectively. As can be seen, the use of IGS03 satellite corrections improves the RT-PPP solution precision, in comparison with other real-time satellite correction products.

Conclusions
This paper developed a triple-constellation GNSS PPP mode in various positioning modes. Post-processed PPP solutions are obtained in both of the un-differenced and BSSD modes, while a RT-PPP solution is obtained the un-differenced mode only. All post-processed PPP and RT-PPP solutions are compared with the traditional un-differenced GPS-only counterparts. Three scenarios are considered when forming BSSD, each uses a reference satellite from a different GNSS constellation. All of the RTS-IGS satellite corrections currently produced by the IGS are used to produce the RT-PPP solutions.
It has been shown that combining the GPS, Galileo, and BeiDou observations in an un-differenced post-processing PPP model improves the convergence time by about 25% in comparison with the GPS-only counterpart. However, no noticeable improvement is obtained in the PPP solution precision. The use of BSSD linear combination improves the PPP solution convergence time by about 50% and the precision of the estimated PPP parameters by about 25%, in comparison with the GPS-only post-processed PPP solution. RTS IGS03 satellite corrections provided the shortest RT-PPP convergence time, which was about 25 min. In addition, the precision of the RT-PPP solution was better when IGS03 was used, in comparison with the cases when IGS01, IGC01, and IGS02 satellite corrections were used.