*2.2. Pseudorange Multipath*

The combination of the pseudorange and carrier phase observation is used to calculate pseudorange multipath, which can eliminate the effect of tropospheric and ionospheric delays [22], and it can be expressed as follows:

$$MP\_k = P\_k - L\_i - \beta \left(L\_i - L\_j\right) = P\_k + \alpha L\_i + \beta L\_j + \varepsilon \cdot \alpha \tag{1}$$

$$
\alpha = f\_1^2 / f\_2^2 \tag{2}
$$

$$\beta = -\frac{\left(f\_{\vec{j}}^2 + f\_k^2\right)f\_{\vec{j}}^2}{\left(f\_i^2 + f\_{\vec{j}}^2\right)f\_k^2} \tag{3}$$

where *k*, *i*, *j* are frequency, *MPk* denotes the pseudorange multipath; *Pk* represents the pseudorange observation; *Li* and *Lj* are the carrier phase observation on frequency *i* and *j*, respectively. *fi* and *fj* are frequencies; *ε* is the noise.

The characteristics of pesudorange multipath for GPS, GLONASS, BDS-2, BDS-3, Galileo, and QZSS satellites are investigated in this study using 10 stations from the Asia-Pacific region. The pesudorange multipath values versus elevation angle for the BDS-2 C11, C16, BDS-3 C30, and C39 on DOY 283 for the JFNG station was shown in Figure 3. It can be seen that there is an opposite relationship between pseudorange multipath and elevation angle, the larger the pseudorange multipath, the smaller the elevation angle, and vice versa. The pseudorange multipath is significantly large when the elevation is extremely small in some cases, which may be caused by the observation noise. For the BDS-2 satellites, the pseudorange multipath of the B3I signal is better than those of B1I and B2I. In terms of the BDS-3 satellites, the B1I frequency band of the MEO satellite presents the best performance, its pseudorange multipath is the smallest. Overall, the BDS-3 pseudorange multipath is around 0.28 m, while it is about 0.3 m for the BDS-2 satellites. The time series of pseudorange multipath with respect to elevation on the JFNG station for GPS, GLONASS, Galileo, and QZSS is shown in Figure 4. Similar to BDS-2 and BDS-3 satellites, the pseudorange multipath shows an opposite relationship with the elevation angle. Among them, the pseudorange multipath value of Galileo is the smallest, and it is around 0.2 m, the GPS and QZSS systems show comparable performance, and its value is around 0.3 m, while the value is about 0.4 m for GLONASS satellites. Since the pseudorange multipath effect is an important error source at the receiver side, it has a negative impact on GNSS precise data processing. In GNSS precise data processing, the following measures can be adopted to eliminate or weaken the effect of it: Firstly, reducing or eliminating the weight of observation with low elevation; secondly, modeling the pseudorange multipath errors.

**Figure 3.** The MP series of BDS-2 and BDS-3 for DOY 283 on JFNG station. (The red wave indicates the sequence of elevation angle).

**Figure 4.** The MP series of GPS/QZSS/GLONASS/Galileo for DOY 283 on JFNG station. (The red wave indicates the sequence of elevation angle).

#### **3. Broadcast Ephemeris Performance**

#### *3.1. Broadcast Ephemeris Clock Offset Performance*

The broadcast ephemeris clock offset accuracy of GPS, BDS-2, BDS-3, GLONASS, Galileo, and QZSS satellites from DOY 283 to 289 in 2021 are depicted in Figure 5, and the mean clock offset accuracy is listed in Table 3. The clock offset accuracy assessment method is referred to Huang et al. [23]. For GPS, the clock offset accuracy of the G08 satellite is 2.57 ns, which shows the poorest performance among all GPS satellites since the cesium atomic clock was installed on it. Furthermore, except for the G03, G17, G28, and G29 satellite clock, the clock offset accuracy of other satellites is better than 1 ns. The mean accuracy is 0.65 ns. The broadcast ephemeris clock offset accuracy is 0.98 ns, 1.72 ns, and 2.11 ns for BDS-2 GEO, IGSO, and MEO satellites, respectively. The clock offset accuracy of GEO satellites outperforms that of IGSO and MEO satellites, the reason is that the stations applied to estimate broadcast ephemeris clock offset are mainly located in China. Compared to the IGSO and MEO satellites, the observation arc of GEO satellites is longer, and the data used for predicting clock offset are more, resulting in the clock offset accuracy being higher, whereas the observation data of MEO satellites are few, leading to inferior clock offset accuracy. Moreover, the frequency stability of BDS-2 onboard satellite clocks is poorer, which has a negative impact on broadcast ephemeris clock offset accuracy. The broadcast ephemeris clock offset accuracy of the BDS-3 satellites is about 1 ns apart from the C38, C39, and C40 satellites, the operation period of these three satellites is shorter, and the stations can receive the signal of these three satellites are few. In a word, the mean accuracy is 1.04 ns, its accuracy is improved compared to that of BDS-2 satellites, which can be attributed to the following reasons: Firstly, the improved rubidium atomic clocks and highperformance Passive Hydrogen Masers (PHM) are equipped on BDS-3 satellites, the frequency stability is extremely improved compared to BDS-2 satellites; secondly, since the Inter-Satellite Link (ISL) technology is employed to estimate BDS-3 broadcast ephemeris clock offset [24], the broadcast ephemeris clock offset accuracy can be significantly improved.

**Table 3.** Clock offset accuracy of each system (units: ns).


**Figure 5.** Clock offset accuracy of GPS/BDS-2/BDS-3/Galileo/GLONASS/QZSS satellites from broadcast ephemeris.

The cesium atomic clocks are installed on GLONASS satellites, previous studies have demonstrated that the performance of cesium atomic clocks is poorer than that of rubidium atomic clocks and PHM [25]. Compared to other systems, the broadcast ephemeris clock offset accuracy of GLONASS is worse, and the broadcast ephemeris clock offset accuracy of R13, R16, and R22 satellites is poorer than 4 ns. The mean clock offset accuracy is 3.10 ns. In terms of Galileo satellites, the broadcast ephemeris clock offset accuracy of each satellite is better than 1 ns, and the mean is 0.61 ns, which shows superior performance; the reason may be that the high-precision rubidium atomic clocks and PHM are employed on Galileo satellites [26]. Moreover, the update frequency of Galileo satellites is high, it is updated every 10 min, which also guarantees the high-precision performance of broadcast ephemeris clock offset. Each QZSS satellite clock offset accuracy is better than 1 ns, and the mean is 0.75 ns, the atomic clocks equipped on QZSS satellites are the same as that of GPS, while the clock offset accuracy is poorer than that of GPS, which may be related to the satellite orbit type. The QZSS broadcast ephemeris clock offset difference from DOY 283 to 289 is presented in Figure 6. It can be clearly seen that the clock offset difference time series is rather stable, and the value is between −2 ns and 2 ns, the mean is 0.75 ns, 0.71 ns, 0.88 ns, and 1.64 ns for J01, J02, J03, and J07 satellites, respectively.

**Figure 6.** Clock offset difference time series of QZSS satellites from broadcast ephemeris.

#### *3.2. Broadcast Ephemeris Orbit Performance*

The broadcast ephemeris orbit accuracy of GPS, BDS-2, BDS-3, GLONASS, Galileo, and QZSS satellites from DOY 283 to 289 in 2021 are presented in Figure 7, and the mean orbit accuracy is listed in Table 4. For GPS satellites, the radial accuracy of G04, G14, G18, and G23 satellite orbit is worse than that of other satellites, which may be that these four satellites belong to Block III, and the operation period is shorter, and their broadcast ephemeris orbit accuracy still needs to be improved; the orbit accuracy in the radial direction of other satellites is better than 0.25 m. For the along and cross components, the orbit accuracy is between 0.25 m and 1.5 m, and 0.25 m and 1 m, respectively. The mean 3D RMS is 0.60 m. The orbit accuracy is significantly poor for BDS-2 satellites, especially for GEO satellites, which mainly be since the GEO satellites are static relative to the ground stations, resulting in a strong correlation among the observations at different epochs. Compared to BDS-2 satellites, the broadcast ephemeris orbit accuracy of BDS-3 satellites is considerably improved, the satellite orbit accuracy in radial, along, cross, and 3D RMS is 0.11 m, 0.25 m, 0.25 m, and 0.21 m, respectively, and the improvements are 94.24%, 81.06%, 90.23%, 89.29%, respectively. The reasons can be attributed to the following: Firstly, the inter-satellites link technology is employed to determine the satellite orbit. Secondly, the number of BDS-3 satellites is much more than that of BDS-2, and the satellite type is mainly MEO satellites, the redundancy of observation data has been improved. The satellite orbit accuracy after the PRN 38 is slightly inferior, and the operation period of these satellites is shorter, the stations that can receive the signal of these satellites are few. With the increase of stations, satellite orbit accuracy can be improved in the near future.

**Figure 7.** Broadcast orbit accuracy of GNSS five systems.

**Table 4.** Average accuracy of DOY 283–289 broadcast orbit for Multi-GNSS satellites (unit: m).


The satellite orbit accuracy in radial and cross is small for GLONASS satellites, most satellites are better than 1.5 m, whereas the broadcast ephemeris orbit accuracy in along component shows poorer performance, the mean orbit accuracy is 0.46 m, 0.99 m, 1.92 m, and 1.27 m for radial, cross, along, and 3D direction, respectively. The orbit difference of each Galileo satellite is small, and the orbit accuracy for most satellites is better than 1.0 m in along, cross, and radial directions, and the mean 3D RMS is 0.62 m, which is comparable to that of GPS. The orbit accuracy of QZSS satellites is similar to that of BDS-2, the orbit accuracy of three IGSO satellites is tremendously better than that of GEO satellites. The mean 3D RMS is 0.83 m, which is poorer than that of BDS-3, GPS, and Galileo satellites.

#### *3.3. Signal-in-Space Ranging Errors*

To comprehensively assess the accuracy of GNSS broadcast ephemeris orbits and clock offset, the SISRE of all systems was investigated [27]. The SISRE for GPS/BDS-2/ BDS-3/GLONASS/Galileo/QZSS satellites from 283 to 289 days in 2021 is given in Figure 8. The mean and RMS of SISRE are given in Table 5. It can be found that the SISRE of GPS and Galileo show the best performance among the five systems, and its fluctuation is between 0 m and 2 m. However, the SISRE of Galileo is larger at some epochs. The rank of SISRE from best to poorest is QZSS, BDS-3, GLONASS, and BDS-2. For the BDS MEO satellites, the SISRE value of the BDS-2 MEO satellites (C11, C12) and the BDS-3 MEO satellites (C25, C30) are from 0 m to 7 m and 0 m to 4 m, respectively. In terms of BDS IGSO satellites, the fluctuation of BDS-2 IGSO satellites (C13, C16) and BDS-3 IGSO satellites (C39, C40) are between 0 m and 6 m, and 0 m and 4 m, respectively. The SISRE of BDS-3 is better than BDS-2 for both MEO and IGSO satellites. Since SISRE can reflect the combined error of orbit and clock offset, and the accuracy of broadcast ephemeris orbit and clock offset are calculated using the observations from ground stations, there is a certain relationship between the observation data quality and SISRE. By comparing the observation data quality and SISRE, it can be found that when the observation data quality is better, the broadcast ephemeris SISRE is also better, and vice versa.

**Figure 8.** SISIRE of GPS/BDS-2/BDS-3/GLONASS/Galileo/QZSS.


**Table 5.** The average accuracy and RMS of DOY 283–289 SISRE for Multi-GNSS satellites (unit: m).

#### **4. PPP Accuracy Evaluation**

To investigate the positioning performance of multi-GNSS in the Asia-Pacific region, 10 MGEX stations were selected to conduct static and kinematic PPP experiments. For the data processing strategies, the sampling interval of observation data is 30 s, and the period is from DOY 283 to 289 in 2021. At present, several MGEX analysis centers can provide precise satellite orbit and clock offset products for the five systems, and the consistency between WUM orbit and clock offset products and other MGEX analysis centers is about 3–10 cm and 0.1–0.3 ns, respectively, showing better consistency with the products from other analysis centers. The positioning performance in the Asia-Pacific region can be reflected using the WUM orbit and clock offset products [28,29]. Therefore, the multi-GNSS final satellite orbit, clock offset, and earth rotation parameter (ERP) products generated from Wuhan University are applied in this study [30]. The satellite antenna phase center variation (PCV) and phase center deviation (PCO) values are used from igs14.atx [31]. The dual-frequency ionosphere-free is employed to eliminate the effect of the first-order ionosphere and the higher-order ionosphere is ignored [32]. The zenith hydrostatic delay of the troposphere is corrected using the Saastamoinen model [33], while the zenith wet delay is estimated as the parameter. The carrier phase ambiguities are estimated as float solutions [34]. The station coordinates of the static PPP are estimated as a constant, while it is estimated as white noise in the kinematic model. The receiver clock offset is estimated as white noise. In addition, the relativistic effects, satellite antenna phase wind-up, and station tides are weakened or eliminated using existing models [35–37]. To compare the positioning performance differences between single-system and multi-GNSS combinations in the Asia-Pacific region, six mode combinations in the static and kinematic PPP experiments were conducted, which is: GPS(G), GPS/QZSS combination (GJ), BDS (C), BDS/QZSS combination (CJ), GLONASS (R), and Galileo (E), respectively. The convergence time and positioning accuracy are used to evaluate the positioning performance. The convergence time is that the current epoch with 20 consecutive epochs is better than 10 cm, and the positioning accuracy is the RMS of the positioning error after convergence [38]. It is noted that the positioning error is the positioning difference between the PPP solutions and IGS weekly solution [39].

#### *4.1. Static PPP Performance*

The mean convergence time for static PPP of six combinations for each station in the east, north, and up directions is presented in Figure 9, and the mean convergence time of each combination is listed in Table 6. It can be seen from Figure 9 and Table 6 that the convergence time of GPS is the shortest among six combinations, which are 6.01 min, 5.53 min, and 16.52 min for the east, north, and up directions, respectively. For the east and north directions, the convergence time of Galileo is faster than that of BDS, and the GLONASS show the longest convergence time, which may be that all selected stations are located in the region with low latitudes, and better positioning performance can be achieved at high latitudes than low latitudes for GLONASS [38]. In terms of up component, the convergence time of Galileo is shorter than that of GLONASS, while the BDS presents the poorest convergence performance in the up component among the six combinations, and its convergence time is 32.17 min. Furthermore, compared to the GPS-only solution, the convergence time of the GPS/QZSS combination can be shorted, and the improvements are 10.37%, 0.90%, and 1.15% in the east, north, and up directions, respectively. While compared to the BDS-only solution, the BDS/QZSS solutions only short the convergence time in the up direction, the improvement is 1.65%. The reason may be that the number of BDS satellites in Asia-Pacific is larger than 20, when conducting PPP, by adding QZSS satellites, the improvements in convergence time for static PPP are limited.

**Figure 9.** Mean convergence time for static PPP in each station.


**Table 6.** Convergence time and positioning accuracy of different combination static PPP.

The mean positioning accuracy for static PPP of six combinations for each station in the east, north, and up directions are presented in Figure 10, and the mean positioning accuracy of each combination is listed in Table 6. It can be clearly seen that the positioning accuracy is better than 3 cm in the east direction except for PIMO and SIN1 stations. Apart from the USUD station, the positioning accuracy of other stations is better than 2 cm in the north component. The positioning accuracy is outperformed 5 cm in the up direction except for YARR station. The positioning accuracy of GPS is the best, and it is 1.09 cm, 0.78 cm, and 1.68 cm in east, north, and up directions, respectively. For three components, the positioning accuracy using BDS is better than that of GLONASS, while Galileo shows the worst performance. Compared to the GPS-only solution, the positioning accuracy of GPS/QZSS solutions can be improved, and the improvement is 0.92%, 1.28%, and 1.19% in the east, north, and up directions, respectively. The few improvements may be caused by the limited number of QZSS satellites. The improvement of the BDS/QZSS solution in terms of

positioning accuracy is 1.43%, 1.94%, and 1.92% in east, north, and up directions compared to the single BDS-only solution. Totally, the positioning accuracy of six combinations is better than 3 cm, 2 cm, and 4 cm in the east, north, and up directions, respectively.

**Figure 10.** Positioning accuracy of static PPP solution in each station.

#### *4.2. Kinematic PPP Performance*

The mean convergence time for kinematic PPP of six combinations for each station in the east, north, and up directions is presented in Figure 11, and the mean convergence time of each combination is listed in Table 7. It can be seen that the convergence time of the Galileo solution is the shortest in the east direction, it is 23.82 min, while the shortest convergence time is the GPS-only solution in the north and up directions, with convergence times being 8.49 min and 24.4 min, respectively. For the east direction, the convergence time of the GPS-only solution is faster than that of the GLONASS-only solution, and the BDS-only solution is the longest, nearly one hour is still needed to obtain the centimeter-level position accuracy in kinematic PPP mode, whereas it is about 20 min for the GPS/QZSS solutions. Compared to the GPS-only and BDS-only solutions, the improvement of convergence time for the GPS/QZSS and BDS/QZSS solutions is 24.82%, 7.66%, 10.90%, and 11.06%, 19.94%, and 6.66% in the east, north, and up components, respectively. Compared to the static PPP, the improvement rate of convergence time for the GPS/QZSS and BDS/QZSS solutions is larger, which may be that the increased number of satellites and better geometry distribution of satellites are beneficial to the convergence of kinematic PPP.

**Table 7.** Convergence time and positioning accuracy of different combination kinematic PPP.


**Figure 11.** Mean convergence time for kinematic PPP in each station.

The kinematic PPP mean positioning accuracy of six combinations for each station in the east, north, and up directions are presented in Figure 12, and the mean positioning accuracy of each combination is listed in Table 7. One can see that the GPS-only, GPS/QZSS, BDS-only, and BDS/QZSS solutions show similar positioning accuracy in three directions except for the LAUT and MIZU station, which is better than 3 cm in the three directions. The positioning performance of GLONASS and Galileo in the Asia-Pacific region is relatively poorer. The positioning accuracy of BDS presents the best performance in the east and north component, and the positioning accuracy is 1.77 cm and 1.59 cm, respectively. This is due to the launch of BDS-3 satellites, the number of BDS satellites is more than other satellite systems in the Asia-Pacific region. The positioning accuracy in the up direction of GPS shows the best performance, and it is 4.65 cm. For the east and north directions, the positioning accuracy of GPS is better than that of Galileo, whereas the GLONASS is the worst. In terms of the up direction, the positioning accuracy of BDS is better than that of Galileo, while the GLONASS is still the worst. Moreover, compared to the GPS-only solution and BDS-only solutions, the positioning accuracy of the GPS/QZSS and BDS/QZSS solutions can be improved from 2.80 cm, 2.03 cm and 4.65 cm to 2.43 cm, 1.91 cm and 4.15 cm, with the improvement being 13.21%, 5.91%, and 10.75%, from 1.77 cm, 1.59 cm, and 4.69 cm to 1.76 cm, 1.57 cm, and 4.50 cm for the east, north, and up directions, with the improvement being 0.56%, 1.26%, and 4.05%, respectively. Except for the GLONASS-only and Galileo-only solutions, the positioning accuracy of 3 cm, 3 cm, and 5 cm in the east, north, and up components for kinematic PPP can be achieved.

The consistency of performance improvement among systems based on multi-GNSS data quality, broadcast ephemeris accuracy, and precision positioning performance is analyzed. It can be found that the observation data quality of GPS is improved by 28.11%, and the improvement for SISRE, static PPP, and kinematic PPP is 75%, 44.31%, and 69.24% compared to GLONASS, respectively. Compared to GLONASS, the improvement of data quality, SISRE, static PPP, and kinematic PPP of Galileo is improved by 32.71%, 75%, 5.52%, and 28.37%, respectively. The improvement of BDS-3 is 33% and 32.78% compared to BDS-2 in terms of observation data quality and SISRE, respectively. These results show that there is a consistency between data quality, SISRE, and PPP.

**Figure 12.** Positioning accuracy of kinematic PPP for each station.
