# LEO Constellation-Augmented Multi-GNSS for 3D Water Vapor Tomography
## 2 Materials and Methods
### Tomographic Theory and Approach
Pressure-level data include relative humidity and temperature, which can be used to calculate water vapor density \(x_{ERA5}\) as follows [34]:  
\[x_{ERAS}=\frac{0.008\cdot e\cdot RH}{1+\frac{T}{273.15}} \tag{4}\]  
where \(RH\) is relative humidity expressed as percentage; \(T\) is temperature (K); and \(e\) denotes the saturated water vapor pressure (Pa), which can be calculated by Wexler method [35; 36], as follows:  
\[\ln e=\sum_{i=0}^{3}c_{i}T^{i-1}+c_{4}\ln T(273.15\text{K}\leq T<373.15\text{K}) \tag{5}\]  
\[\ln e=\sum_{i=0}^{4}c_{i}T^{i-1}+c_{5}\ln T(173.15\text{K}\leq T<273.15\text{K}) \tag{6}\]  
where \(c_{0}=-6.0436117\times 10^{3}\), \(c_{1}=1.89318833\times 10^{1}\), \(c_{2}=-2.8238594\times 10^{-2}\), \(c_{3}=1.7241129\times 10^{-5}\), and \(c_{4}=2.858487\) in Equation (5), and \(c_{0}=-5.8653696\times 10^{3}\), \(c_{1}=2.22410300\times 10^{1}\), \(c_{2}=1.3749042\times 10^{-2}\), \(c_{3}=-3.4031775\times 10^{-5}\), \(c_{4}=2.6967687\times 10^{-8}\), and \(c_{5}=6.918651\times 10^{-1}\) in Equation (6). Input value \(y_{simu}\) for the tomographic simulation can be calculated by kernel matrix \(A\) and water vapor density values \(x_{ERAS5}\), as follows:  
\[y_{simu}=A\cdot x_{ERA5} \tag{7}\]  
In order to use uniform data for research, the SWV observations of all GNSS inversions were also obtained by ERA5 data simulation as described above. The value of \(x_{ERA5}\) was regarded as the true water vapor density in voxels without errors. Therefore, \(y_{simu}\) did not \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Description** \\ \hline Horizontal resolution & Reanalysis: 0.25\({}^{\circ}\)\(\times\) 0.25\({}^{\circ}\) \\ Vertical coverage & 1000 to 1 hPa \\ Vertical resolution & 37 pressure levels \\ Temporal resolution & Hourly \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ERA5 hourly data parameters stratified by barometric pressure. contain any errors. All observation errors and those in the functional model are reflected in the SWV residuals. If a suitable residual value is constructed, the simulated observations are likely to be close to the true values. In this study, the following method is employed to calculate the residual values of SWV. [PERSON] et al. [37] investigated the observation with GPS receivers and radiometers at Platteville and Table Mountain, Colorado on 21-23 May, 1996. [PERSON] et al. [38] studied the SWV difference between the water vapor radiometer and GNSS in Xi'an and Nanjing, China, and concluded that the root mean square (RMS) of the difference is less than 4 mm. [PERSON] et al. [39] also compared the data of water vapor radiometer and GNSS in Germany. The SWV differences in different regions are all of the same order of magnitude. Moreover, [PERSON] et al. [37] provided the SWV differential scattering maps between GPS and water vapor radiometer at different elevation angles. Therefore, we quoted the data in the scatter plot by [PERSON] et al. [37]. By vectorizing this scatter plot, we extracted the maximal-difference data corresponding to different elevation angles. Extracted data are shown in Figure 4; the horizontal coordinate is the elevation angle, and the vertical coordinate is the maximal difference corresponding to each elevation angle. On the basis of the above extracted data, a fitting equation was constructed using linear interpolation. The function was employed to fit the relationship between elevation angle and the maximal difference as follows:  
\[\Delta_{l,\max}=f(ele_{l}) \tag{8}\]  
where \(\Delta_{l,\max}\) is the maximal difference in the \(l\)th ray; \(ele_{l}\) is the elevation angle of the \(l\)th ray; then the random error \(\varepsilon_{l}\) of the \(l\)th ray could be obtained, and \(\varepsilon_{l}\)\(\sim\) N(0, \(\Delta_{l,\max}^{2}\)), \(\varepsilon_{l}\) will be used as the residual term of the \(l\)th SWV. Lastly, we gave a distribution map of the SWV residual scatter of GNSS and 288 LEO satellite constellations in 30 min.