# LEO Constellation-Augmented Multi-GNSS for 3D Water Vapor Tomography
## 2 Materials and Methods
### Tomographic Theory and Approach  
Since the orbital altitude of LEO satellites is 1000 km, the signal of LEO and GNSS satellites passes through the troposphere when it reaches the ground-based receiver. Hence, tropospheric water vapor information can be derived from LEO and GNSS signals. Slant water vapor (SWV), as an input observation for the tomographic model, could be obtained by the method proposed in [3]. The observations of the tropospheric water vapor tomography are the integration of water vapor in the direction of the ray-path SWV which can be expressed as:  
\[\text{SWV}=\!\!\int\rho(s)ds \tag{1}\]  
where \(\rho\) is water vapor density, \(s\) denotes the distance of the line of sight in the troposphere, and \(ds\) represents the distance of a satellite ray path element. For the tomographic observation model, the studied region can be divided into several voxels in the horizontal and vertical directions, as shown in Figure 3. The observation rays can be determined by the positions of station and satellites that pass through the discrete voxels. The water vapor density value parameter is considered to be a constant for the selected time period. The slant path is replaced by a straight line. [PERSON] et al. [17] pointed out that when the elevation angle is less than 30 degrees, there will be an error in treating the observation ray as a straight line, but this error is relatively small. Therefore, the observation ray will be replaced by a straight line in this paper. Thus, the total SWV value could be expressed as the sum of the delayed fractions in each voxel that were discrete along the satellite ray path, as follows:  
\[\text{SWV}=\sum_{i=1}^{m}\sum_{j=1}^{n}\sum_{k=1}^{p}\left(a_{ijk}\cdot x_{ijk}\right) \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Satellite** & **Constellation** & **Orbit Type** &
\begin{tabular}{c} **Orbit Inclination** \\ **(deg)** \\ \end{tabular} & **Altitude (km)** \\ \hline
60 & 6 planes & Polar & 90 & 1000 \\
288 & 12 planes & Polar & 90 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed orbital configurations of designed LEO constellations. Figure 3: Satellite observation rays passing through the tomographic region. where \(m,n,p\) are the total number of voxels in the longitude, latitude, and vertical directions, respectively; \(a_{ijk}\) is the distance of satellite rays; and \(x_{ijk}\) is the unknown water vapor density in voxel \((i,j,k)\). The linear equation of SWV can be described as follows:  
\[A\cdot x=y \tag{3}\]  
where \(y\) is the derived SWV; \(x\) is the unknown state vector that is to be estimated in the voxel; and \(A\) is the kernel matrix containing the sub paths of each slant path in each voxel, which could be obtained by calculating the relative positions from the satellite, ground receiver, and tomographic voxels. The satellite position includes the positions of the GNSS and LEO satellites, of which the latter could be obtained from simulated LEO satellite orbital information. However, the observed values of \(y\) in the course of this experiment were simulated from reanalysis datasets. Fifth-generation accurate global atmospheric reanalysis (ERA5) data released by the European Centre for Medium-Range Weather Forecasts (ECMWF) were employed for SWV simulation. In this study, ERA5 hourly data of pressure levels from 1979 to the present were employed [33]. The parameters of ERA5 data are shown in Table 2.  
Water vapor density is calculated for each ERA5, which is 3D grid data, and water vapor density values for each tomographic voxel centroid could be obtained by inverse distance weighted interpolation.