# A Method to Improve the Distribution of Observations in GNSS Water Vapor Tomography
## 2 Tomography Theory and Method  
### Water Vapor Tomography Modeling  
Zenith total delay (ZTD) is an average parameter in spatial aspects and can be obtained from GNSS observations. The zenith wet delay (ZWD) is the wet component of ZTD affected by water vapor content along the signal travel path and can be retrieved by extracting the zenith hydrostatic delay (ZHD) from ZTD using the formula ZWD = ZTD \(-\) ZHD. The empirical model used to calculate the corresponding ZHD is as follows [22]:  
\[\text{ZHD}=\frac{0.002277\times P_{s}}{1-0.00266\times\cos(2\varphi)-0.00028 \times H} \tag{1}\]  
where \(P_{s}\) (unit: hPa) is the surface pressure, which can be obtained from measured meteorological data or calculated by the Global Pressure and Temperature Model (GPT). \(\varphi\) and \(H\) (unit: km) denote the latitude and the geodetic height of the station, respectively. ZWD can be translated into PWV on the basis of a linear relationship by using the following formula [23]:  
\[\text{PWV}=\Pi\times\text{ZWD}=\frac{10^{6}}{\rho_{w}\times\frac{R}{m_{w}} \Big{(}\frac{k_{3}}{T_{m}}+k_{2}-\frac{m_{w}}{m_{d}}\times k_{1}\Big{)}}\times \text{ZWD} \tag{2}\]  
where \(\Pi\) is the conversion factor. \(\rho_{w}\) represents the liquid water density (unit:\(g\cdot m^{-3}\)); \(R\) indicates the universal gas constant (\(R=8314\ \text{Pa}\cdot\text{m}^{3}\cdot\text{K}^{-1}\cdot\text{kmol}^{-1}\)); \(m_{w}\) and \(m_{d}\) are the molar mass of water and the dry atmosphere (\(m_{w}=18.02\ \text{kg}\cdot\text{kmol}^{-1}\), \(m_{d}=28.96\ \text{kg}\cdot\text{kmol}^{-1}\)), respectively. \(k_{1}\), \(k_{2}\), and \(k_{3}\) are empirical physical constants (\(k_{1}=77.604\ \text{K}\cdot\text{hPa}^{-1}\), \(k_{2}=70.4\ \text{K}\cdot\text{hPa}^{-1}\), \(k_{3}=3.775\times 10^{5}\ \text{K}^{2}\cdot\text{hPa}^{-1}\)); and \(T_{m}\) (unit: K) denotes the weighted mean temperature, which is a function of vapor pressure and temperature at different altitude. In practice, it can be approximated using a measurement of the surface temperature \(T_{s}\) in K (\(T_{m}=85.63+0.668T_{s}\)) [24; 25]. The slant water vapor is the integrated water vapor along the slant path between the ground receiver and GNSS satellite; it is expressed as follows:  
\[\text{SWV}=\int\limits_{s}\rho_{v}ds=\text{PWV}\times f(ele)+\Pi\times f(ele) \times\cot(ele)\times\big{(}G_{NS}^{w}\times\cos(azi)+G_{WE}^{w}\times\sin(azi )\big{)}+R \tag{3}\]  
where \(s\) is the path of a satellite signal ray, \(\rho_{v}\) represents the water vapor density (unit: \(g\cdot m^{-3}\)), \(f\) is the wet mapping coefficient. \(ele\) and \(azi\) are the satellite elevation and azimuth, respectively. \(G_{NS}^{w}\) and \(G_{WE}^{w}\) are the wet delay gradient parameters in the north-east and east-west directions, respectively. \(R\) refers to the non-homogeneous variation of water vapor, it is calculated by multiplying the post-fit residual by the conversion factor [26; 27]. The observation equation, which is based on the unknown water vapor density within each tomographic voxel and the distances of GNSS signal rays crossing the divided voxel (red paths shown in Figure 1), is established as follows:  
\[\text{SWV}=\sum\limits_{i=1}^{n}d_{i}\times x_{i} \tag{4}\]  
where SWV denotes the slant water vapor that can be acquired by the tropospheric estimation. \(d_{i}\) denotes the distance of signal rays inside voxel \(i\), which can be achieved by the station and satellite coordinates, and \(x_{i}\) is the unknown parameter, which denotes water vapor density of voxel \(i\). A spatial relation exits between water vapor in a specific voxel and its surrounding ones. In the horizontal direction, [PERSON] et al. [28] pointed out that the distribution of water vapor density is relatively stable within a small region. Thus, water vapor density within a voxel is related to the weighted average of its adjacent voxels in the same grid layer.