*3.2. Experimental Analysis*

The number of satellite signal rays available for the four satellite systems in each tomographic solution is counted and their averages during the 31 days from DOY 121 to 151, 2021 are shown in Figure 2. It can be seen that the BDS has the largest number (704) of available signal rays, followed by GPS, Galileo, and GLONASS with the average values of 507, 329, and 351, respectively. The percentages of available signal rays in BDS that exceed GLONAA and Galileo are more than 100%, achieving 114% and 101%, respectively. Compared with GPS, the value also reaches 39%. The number of signal rays used in GPS is the most stable during the experimental period, the difference between the maximum and minimum value is less than 100 with the standard deviation (STD) being only 23. While the other three satellite systems have obvious fluctuations in the number of available signal rays, with the differences between the maximum and minimum value far greater than 100 and the STDs reach 51, 48, and 34 for BDS, GLONASS, and Galileo, respectively. Note that the average number of signal rays used in Galileo is greater than that of GLONASS, but there are still days when GLONASS has more available signal rays than Galileo. When the satellite systems are combined, only the available signal rays of RE have just reached the level of BDS and the other combinations are all obviously improved compared with these single systems, especially since the average number of signal rays used in the combination of four systems could be close to 2000.

**Figure 2.** Average number of signal rays used in each solution based on different satellite systems during the experimental period.

The number of voxels passed through by signal rays for the four satellite systems in each tomographic solution is also counted, and their average values are shown in Figure 3. It was observed that the GPS has the largest number of voxels crossed by signal rays, followed by BDS, GLONASS, and Galileo with average values of 425, 424, 392, and 377, respectively. Corresponding to the 560 voxels in the entire tomographic region, the coverage rate of the four satellite systems reaches 75.9%, 75.4%, 70%, and 67.3%, respectively. Note that GPS and GLONASS with fewer available signal rays have more penetrated voxels than BDS and Galileo, respectively, and in fact, their differences are relatively small. In addition, the number of voxels crossed by signal rays for the four satellite systems all show a certain fluctuation during the experimental period.

**Figure 3.** Average number of voxels penetrated by signal rays in each solution based on different satellite systems during the experimental period.

When combining the satellite systems, the number of crossed voxels and their coverage rate is counted and shown in the form of a histogram in Figure 4. It can be seen that the number and coverage rate of voxels are increased after the combinations compared with single satellite system. In addition, the performances of the three-satellite systems combination are better than those of the two-satellite systems combination, and four satellite systems combination outperforms the three-satellite systems combination. Specifically, combination of GCRE achieved the best performance with the number and coverage rate of voxels of 468 and 83.6%, respectively.

In the tomographic experiment, we found the existence of voxels that were only penetrated by a few signal rays, thus the concept of voxels crossed by sufficient signal rays was introduced from the relevant literature [7]. Based on the fact that a ray crossed a minimum number of voxels when the signal ray passed vertically through the tomographic region, the minimum probability that a voxel will be penetrated by a ray could be calculated. In this experiment, the value is 10/560, namely 1.79%. Then, the value of minimum probability multiplied by total SWV used is regarded as the criteria to determine whether a voxel is crossed by sufficient signal rays. Therefore, the number of voxels passed through by sufficient signal rays for the four single satellite systems and their combinations are counted and listed in Table 1 during the experimental period. It was observed that GPS had the largest number of voxels penetrated by sufficient signal rays among the four single systems, and only 7 voxels more than Galileo with the least effective voxels. After the combinations, the number of voxels increased but very little and the value of the

combination of four system was only 278. Regarding the coverage rate, the difference of those 15 values in Table 1 is even smaller.

**Figure 4.** Average number of voxels penetrated by signal rays in each solution based on different combination during the experimental period.


**Table 1.** Average number of voxels penetrated by sufficient signal rays based on different satellite systems and different combinations during the experimental period.

Further, the situation that each voxel passed through by signal rays in a certain tomographic solution (UTC 11:45–12:15, DOY 137, 2021) is shown in detail in Figure 5, in which the black and white rectangles represent the voxels crossed by sufficient and in sufficient signal rays, respectively. Note that only the four single satellite systems and the combination of the four systems are illustrated in this figure. It is observed in the figure that the distribution of the black and white rectangles for different systems is very similar, especially in the lower and middle layers. From this point, for water vapor tomography in Hong Kong, the selection of a single satellite system or multi-GNSS combination has little effect on the structure of the tomographic model.

**Figure 5.** Distribution of voxel with sufficient signal rays at each layer for GPS, BDS, GLONASS, Galileo, and a combination of GCRE.

#### **4. Discussion**

To assess the performance of water vapor tomography using different satellite systems, SWV of the GNSS sites for validation were computed using these 15 tomographic results and the distances of signal ray in each voxel based on the observation equation established in Equation (5). The 15 tomography-computed SWV were then compared with the GAMITestimated SWV (as a reference). Figure 6 shows the change of tomography-computed vs. GAMIT-estimated SWV residuals with elevation angle during the experimental period for the four single systems. The change of the SWV residuals has the same trend in the four satellite systems, and they decrease as the elevation angle increases. It is observed that the residuals of four systems all ranged from −10 to 10 mm, and most of them concentrated between −2.0 and 2.0 mm. The percentage of absolute residuals smaller than 2.0 mm are 86.9%, 88.1%, 85.7%, and 85.3% for GPS, BDS, GLONASS, and Galileo, respectively. The largest absolute residual of the four satellite systems is 8.46, 9.63, 9.37, and 9.91 mm, respectively. We obtained the SWV residuals for various combinations of satellite systems, which also follow a decreasing trend with increasing elevation. These ranges and concentrated areas of the SWV residuals are unchanged compared with the four single satellite systems.

**Figure 6.** Scatter diagram of the change of SWV residuals with elevation angle for the four satellite systems.

To further assess their performance, SWV values were grouped into individual elevation bins of 5◦, i.e., all SWVs with an elevation angle between 15 and 20◦ were evaluated as a single unit. Thus, the RMSE of each elevation bin for these 15 tomographic results was calculated and is shown in Figure 7. It can be seen from the left panel that the GLONASS and Galileo performance is not as good as the BDS and GPS at low elevation angles. As the elevation angle increases, their differences become very small. BDS achieved the best RMSE with a value of 1.59 mm, followed by GPS, Galileo, and GLONASS. In fact, the differences between these RMSEs are small and the values do not exceed 0.2 mm. Considering the magnitude range of SWV, these differences can be negligible. After the combinations, the RMSE of SWV residuals in each elevation bin were shown in the middle and right panels, which are the combination of two systems and multi systems, respectively. The RMSE difference of the SWV residuals for various combinations is relatively small in each elevation bin. Specifically, the RMSEs of whole SWV residuals are 1.66, 1.59, 1.75, 1.74, 1.68, 1.64, 1.67, 1.62, 1.63, 1.60, 1.59, 1.64, 1.65, 1.65, and 1.63 mm for G, C, R, E, GC, GR, GE, CR, CE, RE, GCR, GCE, GRE, CRE, and GCRE, respectively. Considering the magnitude range of SWV values, the differences of RMSE mentioned above not more than 0.2 mm could be negligible. Therefore, it is concluded that the tomographic results of different satellite systems and different combinations have little difference in SWV validation.

Radiosonde data are well suited as a reference to validate the accuracy of the water vapor tomography results, since they can provide a water vapor density profile with high precision based on the atmospheric parameters obtained at different altitudes. Figure 8 illustrates the water vapor density comparisons between radiosonde data and these 15 tomographic results for different altitudes on UTC 11:45–12:15, DOY 137, 2021, which is consistent with the time of tomographic solution shown in Figure 5. It is clear from the profiles that the water vapor density decreased with increasing height. The water vapor density profiles reconstructed by these 15 tomographic results conform with those derived from radiosonde data. From Figure 8, it is difficult to observe the difference in the water vapor density reconstructed by different satellite combinations. Therefore, the radiosonde comparison of 31 days from DOY 121 to 151, 2021 was conducted and the statistical results were listed in Table 2 to further illustrate their performances. From the mean value of RMSE, the difference between the WVD results reconstructed by single system tomography is 0.05 gm−3, and BDS and GPS outperforms GLONASS and Galileo slightly. Compared with the single system, improvement can be observed from the WVD results reconstructed after the satellite system combination. The largest improvement appears from the Galileo with a RMSE of 1.46 gm−<sup>3</sup> to the combination of GCR with a RMSE of 1.30 gm−3. The number of satellite systems in the combination (two, three, or four satellite systems) did not present an obvious impact on the WVD results reconstructed by water vapor tomography.


**Table 2.** Statistical results of the water vapor density composition between radiosonde and tomographic results of different combinations during the experimental period.

**Figure 7.** Comparison of SWV residuals in each elevation bin for various combinations.

**Figure 8.** Water vapor density comparisons between radiosonde and 15 tomographic results.
