## 4 Experiments and Results
### Comparison with the Existing Correction Model
As far as the quantitative comparison of WVD is concerned, the accuracy of the tomographic solutions from Scheme III (mean RMSE are 0.69 and 0.95 g/m\({}^{3}\) on both days) is superior to those from Scheme I (mean RMSE are 1.40 and 1.30 g/m\({}^{3}\) on both days) and Scheme II (mean RMSE are 0.73 and 1.20 g/m\({}^{3}\) on both days), which reveals that the proposed model has better reconstruction capabilities than the Yao model and traditional model in both weather conditions. In addition, to further show the superiority of the proposed method in the WVD comparison, the RMSE of tomographic results from the three schemes was calculated, and the results are shown in Figure 16, including the 31-day period from DOY 213 to DOY 243, 2017, at 00:00 and 12:00 UTC daily. It is clear that, among the three schemes, Scheme III has the smallest RMSE compared to Scheme I and Scheme II in the majority of time periods. The average improvement of Scheme III with respect to Scheme I and Scheme II was 0.51 and 0.20 g/m\({}^{3}\) in terms of the RMSE, respectively. As a further assessment, Table 2 lists the statistics of the three tomography solutions, including the maximum, minimum, and mean values of RMSE, STD, and bias. Figure 16: Comparisons of the RMSE of the tomography results derived from different schemes during the 31-day period from DOY 213 to DOY 243, 2017, at **(a)** 00:00 UTC and **(b)** 12:00 UTC daily. Figure 15: Comparison of the tomographic water vapor profiles for three schemes under different weather conditions using the radiosonde data (black lines) as a reference. **(a)** 00:00 UTC, DOY 220; **(b)** 12:00 UTC, DOY 220; **(c)** 00:00 UTC, DOY 234; and **(d)** 12:00 UTC, DOY 234. It should be noted from Table 6 that all of the means of the three statistics in Scheme II and III are superior to those of Scheme I, which suggests that there is an improvement of the GNSS tomography solution when side observations are included in the tomography modeling. As far as Scheme II and III are concerned, the average RMSE and STD for the former are 1.28 and 1.25 g/m\({}^{3}\) and the latter are 1.08 and 1.07 g/m\({}^{3}\), respectiely, and the mean accuracy improvement rates are 15.62% and 14.40% for both statistics, respectively, which demonstrates that the proposed HFM provides more accurate side SWV observations and three-dimensional water vapor fields. Following this, to quantitatively compare the accuracy of each layer's tomographic results, the reference value of WVD, along the vertical columns, was interpolated using the radiosonde data, and the mean difference between the tomographic results obtained from the three schemes and the radiosonde data during the tomographic period was calculated and is reported in Figure 17. It is evident that the absolute value of the average differences dramatically decreases with increasing height layers. Besides, it should be noted that in the earth surface layer from 0 to 2 km, the difference of most voxels is less than 0 in the three schemes, i.e., the WVD retrieved from the GNSS tomography solution is lower than that obtained from the radiosonde. However, it is noticeable that there is an impressive improvement of the WVD in these layers in Scheme II and Scheme III, although the voxels in this range are not penetrated by side signals according to Figure 10. This can be explained by the fact that when combining both integral signals and inside ones for GNSS tomography, the number of effective observations is obviously increased and the spatial geometric defect of the three-dimensional tomographic model and the ill condition of the tomographic observation equations are gradually remedied, which is beneficial for improving the accuracy and stability of the tomography results. Furthermore, the conclusion that the differences of Scheme III are smaller than those of Scheme II in each layer can be drawn from Figure 17, which highlights the advantages of the proposed HFM.