**4. PPP Analysis of Regional Clock Product**

We selected IGS stations in the Asia-Pacific region and national stations which had not participated in the calculation in each region to verify the PPP performance of the regional satellite clock product, and the selected stations can receive BDS-2 and BDS-3 signals and use the B1B3 signal for PPP processing. The distribution of IGS stations and national stations is as Figure 13.

**Figure 13.** PPP station distribution.

The satellite clocks calculated by the global, Chinese, Northern, and Southern China regional stations are used to conduct PPP experiments for the selected stations on days 214–220 of 2021. The first epoch of 20 consecutive epochs when the error is less than 10 cm is used as the convergence time to count the post-convergence accuracy. The seven-day averages of Root Mean Square Error (RMS) after convergence at different IGS stations are counted in Figure 14; the RMS in E, N, and U directions are combined, i.e.,

$$
\delta = \sqrt{\delta\_{\rm E}^{2} + \delta\_{N}^{2} + \delta\_{\rm U}^{2}} \tag{13}
$$

In Equation (13), *δ* represents the three-dimensional (3D) RMS in the space; *δE*, *δN*, and *δ<sup>U</sup>* represent the RMS error in the direction of E, N, and U. It should be noted that in the RMS calculated by the Equation (13), due to the large value of RMS in the U direction, small values in the E and N directions will be submerged, making the overall RMS statistical data seems large.

**Figure 14.** Effects of Different Regional Clocks on Different IGS Stations.

From Figure 14, it can be seen that the PPP results of the stations near China solved by the global SCB product have the same accuracy, and the 3D accuracy is at the level of 10 cm. The RMS error of the PPP calculated by the small size regional clock product is higher than that of the global product, which indicates that the reduction of the calculation range of the regional clock has a negative impact on the PPP. The difference between the accuracy of the Chinese regional clock solution and the global solution is insignificant, and they are both at the same level. This indicates that the accuracy of PPP experiments is very close to that of the global clock products if only the SCB products calculated by the national stations in the Chinese region are used. When comparing within the Chinese region, the PPP accuracy of CUSV using the regional clocks in South China in the figure is much higher than that in North China. This phenomenon combined with Figure 8 shows that when performing PPP, the accuracy level of the regional stations calculated far from the clock product increases rapidly with the number of observable satellites. However, the stations closer to the center of the study area, such as PTGG, ULAB, and WUH2, have a larger base of observation satellites, and the continued increase in the number of observation satellites will reduce the contribution to the improvement of PPP accuracy. However, the low-latitude regional SCB product has poor accuracy, as shown in Figure 9, and a large deviation from its regional influence, which prevents further improvement of the accuracy. It will even reduce the PPP accuracy of some stations such as JFNG.

The combination of Figures 14 and 15 shows that the accuracy has reached the level of global clock accuracy when the corresponding satellite clocks are used for PPP in the regional range calculated by the same regional SCB. The stations QHZD, XJAK, YNML, and NXLW in the western region have larger errors, whereas the errors in the southern region are smaller. This phenomenon is similar to the CUSV in Figure 14. This is because the number of satellite observations in the northern region is smaller and the number of satellite observations of the stations far from the satellite clock solution area decrease, and the PPP result accuracy becomes worse. The 3D accuracy of other stations in the North and South China clock calculation regions is at the same level as the global clock, indicating that the SCB products in the smaller mid-latitude range are not suitable for extending the positioning area excessively due to the low number of satellite clock calculations.

**Figure 15.** Effects of Different Regional Clocks on Different National Stations.

Figures 16 and 17 show the time series of the jump in the PPP results. From the regional clocks of China and North China in Figure 16, in kinematic mode, it can be seen that the PPP results by the regional SCB of China and the global SCB are at the same level, i.e., centimeter level. As shown in kinematic mode, the PPP accuracy of the regional clocks in South China and North China is at the centimeter level in most of the epochs, and there are jumps in the kinematic PPP result sequences of CUSV and URM stations, which shows that some periods after convergence diverge again, and the regional clocks in mid-latitude regions have more jumps and large fluctuations in PPP of stations in low-latitude regions. The regional clock in South China has the same effect on the PPP results of FJZA and CUSV stations, both of which are in the southeast corner of the Chinese region, indicating that the effect here comes from the REB, whereas the jump in the same region using the regional clock in North China has the same obvious consistency as that of the regional clock in South China, and the REB of the regional clock in North China has no obvious effect on the PPP. In the North China region, the low number of observations leads to a decrease in the number of satellite observations in the region of some stations far from the calculation region affecting the accuracy, which leads to the jump in the positioning results of the North China regional clock and affects the positioning accuracy of the regional clock products. In the kinematic mode, the PPP results of the previous epoch are not transferred to the next epoch for iteration to improve the convergence results, whereas in the static PPP mode, the process noise in the state transfer matrix is 0. From the results of static mode positioning, it can be seen that the positioning accuracy of the regional clock reaches

the level of the global clock, but the convergence time is significantly longer. From the comparison between Figures 16 and 17, it can be seen that the static mode can effectively eliminate the jump caused by the steep drop in the number of observation satellites in one day, but from the positioning results of the South China clock for FJZA on CUSV, the effect of REB cannot be eliminated completely. Moreover, the convergence time generally reaches more than 2h, which is obviously inferior to the convergence time of the global clock.

**Figure 16.** Kinematic PPP Time Series Plots of Different Regional Stations on Day 216 of 2021.

**Figure 17.** Static PPP Time Series Plots of Different Regional Stations on Day 216 of 2021.

#### **5. Discussion**

We have mentioned the problems of accuracy and regional influence caused by station area and latitude in Sections 3 and 4, and we introduce the concept of regional effect bias (REB). It is found in Section 4 that the stations distributed in the same area have the same fluctuations in the kinematic mode. We believe that the influence of these regional station selections on the BeiDou SCB products comes from the spatial correlation errors such as atmospheric parameters absorbed over the small regional stations. If the research scope is larger, as shown in Figure 2, the difference in spatial correlation error between global and region is small, so the deviation of regional influence is reduced. In Section 3.1, we know that the regional clock has many breakpoints. The calculation after each breakpoint needs to be initialized with the pseudo-range. Therefore, in theory, the regional clock must have a lot of errors due to initialization, and the positioning results should not be as good as the global SCB. However, the positioning accuracy of the regional clock is higher than that of the global SCB; this phenomenon is due to the effect of the SCB parameters absorbing regional system deviation. When the regional SCB calculating, the corresponding range fluctuates when the atmospheric environment changes abruptly and will be absorbed by the regional SCB. These mutations are the main factor leading to the loss of precision in PPP. The atmospheric parameters absorbed by the regional SCB products can offset the effects of these sudden changes during PPP, thereby improving positioning accuracy. Otherwise, some papers also provide another hypothesis that when using globally distributed stations, satellites

are always directly above the visible station [33]. Therefore, the clock estimated from the global network cannot compensate for the orbital error of the tangential component, which may also lead to the degradation of the positioning performance of the global network.

Another important question is whether REB will affect the precision statistics of SCB; because Equation (9) of the regional clock should include the regional effect bias REB, it may affect the statistics and analysis of the SCB accuracy after the double-difference. However, according to the conclusions obtained from Figures 10–12 in Section 3.3, the fluctuations of the REBs of each satellite clock calculated in the same area are the same, and the trend term of REB will be greatly weakened after the difference between satellites. The STD level of the final obtained SCB statistics in Figure 9 is consistent with the normal situation, so the SCB products calculated by the inter-satellite difference method adopted in this paper are effective.
