# A Method to Improve the Distribution of Observations in GNSS Water Vapor Tomography
## 2 Tomography Theory and Method
### Method for Improving Distribution of Observations  
Only the GNSS signal rays passing through from the top boundary of the tomographic region can be used to construct observation equations. Thus, most of the rays in the low elevation angle, which contains abound water vapor information in low layer voxels and boundary voxels, are eliminated. In addition, most of the signal rays used to construct the observation equation only penetrate the vertical voxels above the GNSS stations. Therefore, the distribution of observations in the tomography equations is relatively centralized, resulting in many voxels without signal rays passing through and in overabundance of zero elements in the tomographic matrix. In this paper, virtual observations in specific elevation and azimuth angle (yellow paths shown in Figure 1) are introduced  
Figure 1: 3-D distribution of slant water vapor and virtual slant water vapor. (**a–d**) represent the GNSS receiver located in four different tomographic regions, namely northwest, northeast, southwest, and southeast.  
to improve the distribution of observations. Boundary points are selected, coordinates are transformed, and the wet mapping function and the wet delay gradient in the north-south and east-west directions are used in this method. The corresponding procedure is as follows:  
1. The direction of the GNSS station in the tomographic area (northwest, southwest, southeast, and northeast) is determined.
2. The GNSS stations are selected as the starting points by considering the boundary points at the top of the tomographic region as the terminal points of the virtual observations.
3. The geodetic coordinate is converted to a site-centric coordinate. \[BLH\to NEU\] (7)
4. The elevation \(ele^{\prime}\) and azimuth \(azi^{\prime}\) of the virtual observations are calculated as follows: \[ele^{\prime}=\arctan\left(\frac{U_{t}}{\sqrt{{N_{t}}^{2}+{E_{t}}^{2}}}\right)\] (8) \[azi^{\prime}=\arctan\left(\frac{E_{t}}{N_{t}}\right)\] (9) where \(N_{t},E_{t},U_{t}\) represent the coordinate of the terminal point in the north, east, and zenith directions, respectively, of the site-centric coordinate, which can be obtained from the information of tomography boundary.
5. The virtual slant water vapor SWV\({}^{\prime}\) is calculated, as follows: \[\text{SWV}^{\prime}=f\big{(}ele^{\prime}\big{)}\times\text{PWV}+\Pi\times f \big{(}ele^{\prime}\big{)}\times\cot\big{(}ele^{\prime}\big{)}\times\big{(}G_{ NS}^{w}\times\cos\big{(}azi^{\prime}\big{)}+G_{WE}^{w}\times\sin\big{(}azi^{ \prime}\big{)}\big{)}\] (10)
6. The virtual observation equation is constructed to improve the distribution of observations, as follows: \[\text{SWV}^{\prime}=\sum_{i=1}^{n}d_{i}^{\prime}\times x_{i}\] (11) where \(d_{i}^{\prime}\) is the distance of the virtual slant water vapor inside voxel \(i\), which can be calculated using these coordinates.
7. The new functional model of water vapor tomography is constructed and the water vapor density in each voxel is solved using the least square method: \[\left[\begin{array}{c}y_{swv}\\ y_{h}\\ y_{v}\\ y_{swv^{\prime}}\end{array}\right]=\left[\begin{array}{c}A_{swv}\\ A_{h}\\ A_{v}\\ A_{swv^{\prime}}\end{array}\right]\times X+\left[\begin{array}{c}\Delta_{swv }\\ \Delta_{h}\\ \Delta_{v}\\ \Delta_{swv^{\prime}}\end{array}\right]\] (12) where \(y_{swv},y_{swv^{\prime}}\) are the column vectors with a set of SWV and virtual SWV, respectively; \(y_{h}\), \(y_{v}\) are the column vectors with the value of 0 for horizontal and vertical constraint equations, respectively; \(A_{SWV}\), \(A_{h}\), \(A_{v}\), \(A_{SWV^{\prime}}\) represent the coefficient matrices of observation equation, horizontal constraint equation, vertical constraint equation, and virtual observation equation, respectively;\(X=[x_{1},x_{2}\cdots x_{n}]^{T}\) is a vector of water vapor to be estimated; and \(\Delta_{swv},\Delta_{h},\Delta_{v},\Delta_{swv^{\prime}}\) denote the noises of observation equation, horizontal constraint equation, vertical constraint equation, and virtual observation equation, respectively.