## 5 Conclusions  
A novel tropospheric tomography methodology based on the HFM, combining integral and side signals, has been introduced and demonstrated in this paper. The SWD of GNSS rays, which is a crucial observation for the GNSS tomography system, has been separated into two parts--isotropic and anisotropic components--corresponding to the isotropic and anisotropic height factors in the HFM. With these two factors, the SWD of the side rays can be estimated accurately. It should be noted that the signals with an elevation angle of less than 15\({}^{\circ}\) were eliminated due to bending of the ray path. In addition, the dynamic tomography top boundary was determined based on 30-year radiosonde data, which preserved highly accurate water vapor profiles at different altitudes.  
The proposed method was validated and assessed by four special schemes, in which the tomography area was adjusted to force the RS-voxels located in different tomographic boundaries. When the input of the tomography system was only integral signals, the average utilization of GNSS signals was 47.12%, and the mean RMSE and STD were 1.63 and 1.61 g/m\({}^{3}\), respectively. When side rays were combined, the former was increased to 80.54% and was stable for the four schemes. The latter was improved to 1.16 and 1.15 g/m\({}^{3}\), respectively. In the four experimental comparisons, whether in terms of RMSE or STD, it was observed that there was a significant improvement of the tomography results with the side signals. Furthermore, a special comparison of the proposed method and one existing correction approach was performed based on the three other schemes. Compared to the currently optimized model, the tomography results of the proposed model showed mean accuracy improvement rates of 15.62% and 14.40% in terms of the RMSE and STD, separately, which highlights the advantages of the proposed HFM in estimating the side SWV observations and retrieving the three-dimensional water vapor field.  
In addition, the proposed HFM method provides a new way to address the tropospheric delays. From the perspective of isotropic and anisotropic components, some issues resulting from atmospheric water vapor may be skillfully handled, e.g., tropospheric delay correction in InSAR and VLBI, which may inspire new concepts for relevant researchers. Future work will focus on assimilating multi-source remote sensing data into tropospheric tomography, e.g., MODIS, MERIS, and InSAR, providing high-resolution PWV maps, which may be useful for GNSS tomography.  
Conceptualization, W.Z. and S.Z.; methodology, W.Z.; software, W.Z.; validation, W.Z., N.D., and Q.Z.; formal analysis, W.Z. and S.Z.; writing--original draft preparation, W.Z., N.D., and Q.Z.; writing--review and editing, Q.Z. and S.Z.; funding acquisition, S.Z. and N.D. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 41774026; the National Natural Science Foundation of China, grant number 41904013; and the Natural Science Basic Research Project of Shaanxi, grant number 2020 JQ-738.  
The authors acknowledge the support of the Survey and Mapping Office (SMO) of Lands Department, Hong Kong, for the provision of the SatRef GNSS data and the ground-based meteorological data. The King's Park Observatory is also acknowledged for providing the multi-years radiosonde data. The GAMIT/GLOBK software was provided by the Department of Earth Atmospheric and Planetary Sciences, MIT. The authors would like to thank anonymous reviewers for the reviews and suggestions for improving this paper.  
The authors declare no conflict of interest.