# A Method to Improve the Distribution of Observations in GNSS Water Vapor Tomography
## 4 Discussion  
The proposed method allowed the signal rays to pass through more voxels, thereby enriching the voxels' information in the tomographic region and improving the distribution of observations. The different tomography results were compared with the radiosonde data and SWV, and it was found that the proposed method could effectively improve the accuracy of results. In order to analyze whether the ill-conditioned nature of the tomography model is weakened when the distribution of the observation is improved, the concept of matrix condition number is introduced. The condition number measures the degree of dispersion of the eigenvalues of the coefficient matrix, which can be used to  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**DOY**} & \multicolumn{2}{c}{**Scheme \#1**} & \multicolumn{2}{c}{**Scheme \#2**} & \multicolumn{2}{c}{**Improvement**} \\ \cline{2-7}  & **MAE** & **RMS** & **MAE** & **RMS** & **MAE** & **RMS** \\ \hline
182 & 1.22 & 1.34 & 0.86 & 1.07 & 0.36 & 0.27 \\
183 & 1.37 & 1.76 & 0.89 & 1.21 & 0.48 & 0.55 \\
184 & 1.81 & 2.59 & 0.93 & 1.79 & 0.88 & 0.80 \\
185 & 1.72 & 2.46 & 1.17 & 1.82 & 0.45 & 0.64 \\
186 & 1.55 & 2.19 & 1.04 & 1.26 & 0.51 & 0.93 \\
187 & 1.54 & 2.49 & 1.12 & 2.21 & 0.42 & 0.28 \\
188 & 1.40 & 2.01 & 0.83 & 1.48 & 0.57 & 0.53 \\ \hline
**Average** & 1.51 & 2.09 & 0.98 & 1.51 & 0.53 & 0.57 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Statistical results of the water vapor density profile comparison between radiosonde and different schemes at UTC 12:00 from DOY 182 to 188, 2017.  
Figure 9: Linear regression of water vapor density from radiosonde and two schemes. (**a**) and (**b**) represent the regression results of Scheme #1 and #2, respectively.  
diagnose the ill-posed problem of matrix. When the condition number of a matrix is high, then the ill-condition becomes serious [36]. It is defined as follows:  
\[\kappa(N)=\|N^{-1}\|\times\|N\| \tag{13}\]  
where \(N=A^{T}A\), \(\kappa(N)\) denotes the condition number of matrix \(N\), and \(\|N\|\) is the norm of matrix \(N\).  
With the addition of various types of equations, the change in the condition number of the coefficient matrix in each tomography solution was considered. Table 7 lists the situation of tomography solution at 12:00 UTC of DOY 182, 2017. We can see that the condition number of coefficient matrix was INF only when the observation equation existed in the tomography model, implying that the matrix has a serious ill-condition. With the addition of horizontal and vertical constraint equations, the condition numbers decreased gradually, indicating that the constraint equations could alleviate the ill-posed issue of the tomography matrix. The condition number increased slightly when the virtual observations of the proposed method were added. The same trend was shown in other tomography solutions due to the existence of multicollinearity between the virtual observation equation and the observation equation. However, the singular value decomposition (SVD) method used in tomography calculation can overcome the problem of unstable results, and the accuracy of the tomography results of the proposed method was still higher than that of the conventional method. Therefore, the method of improving the observation distribution could not optimize the ill-conditioned tomographic matrix, but it caused abundant water vapor information of voxels, which could greatly affect the accuracy of the tomography result. In follow-up research, the ill-condition issue of the coefficient matrix in the tomography equation and the problem of insufficient observations should be considered and distinguished from each other.