# A New GNSS-Derived Water Vapor Tomography Method Based on Optimized Voxel for Large GNSS Network
## 2 Data and Methods
### Optimized Solution for WV Tomography  
Regarding the conventional tomographic approach, the additional constraints are some virtual observations generated from the empirical function, which may lead to solutions against the truth. Therefore, the optimal WV tomography approach should reduce the dependence on constraints as much as possible. Another problem is the underlying causes of the rank deficiency of matrix \(A\). Besides the null-signal-passed voxels, massive linear dependence equations exist in Equation (7), which also leads to the rank deficiency of matrix \(A\). The origin of the existence of the too large number of linear dependence equations is visualized in Figure 2. We assume that the satellite elevation cut-off is \(10^{\circ}\) (minimum elevation cut-off angle allowed in GNSS data processing). A signal emerges from the satellite, intersects with the 8-km-high tropopause at point (c), and finally reaches the receiver (a). The segment (a-b) is the projection of the signal line segment (a-c) on the ground with a length of approximately 45 km. This is the maximum span of a signal segment, in theory. In practice, only a few signals have elevation angles close to \(10^{\circ}\), which indicates that most signal ground projections (such as (a-b) in Figure 2) are considerably shorter than 45 km. Hence, in the tomographic region, far station spacing or sparse grids can lead to a significant number of signals only passing through voxels in the same column. The tomographic equations consisting of these rays can exhibit obvious linear dependence. In other words, after the elementary row transformation, only a few nonzero lines of the coefficient matrix \(A\) in Equation (7) remain. Figure 2: Ground projection distance (**a–b**) of the segment (**a–c**) between the receiver (**a**) and point (**c**). The point (**c**) is the intersection of the ray and tropopause. To solve the second problem, a smaller grid spacing can be designed that will significantly increase the number of voxels. More voxels imply more null-ray-passed voxels, which leads to a worse ill-posed problem in the tomographic equation. Hence, we need to impose more constraints to overcome the above problem. However, these constraints may bias the tomographic results. Thus, we may not simultaneously overcome the two shortcomings in the conventional tomographic approach via a single approach; if the second problem is solved, the first problem would be more complex, and vice versa. In this study, we propose a novel tomographic approach, based on the principle of the elimination of null-ray-passed voxels. It can simultaneously reduce the contradictory disadvantages encountered in the traditional approach. We present the optimized tomographic approach in Figure 3.  
Figure 3a shows a 3D cube divided into a \(3\times 3\times 3\) grid, which represents a tomographic area composed of 27 self-enclosed 3D voxels. Figure 3b shows the tomographic zone and several visualized rays from three sites. This simulates the penetration of the voxel by rays. Most voxels are penetrated by rays, but a few voxels, particularly those at the bottom, are not penetrated. In this regard, the traditional tomographic method imposes constraints on the first layer in the tomographic zone, which act as virtual signals, so that almost all voxels are passed-through by signals. This increases the rank score of the tomographic coefficient matrix but introduces errors to the results. Hence, we propose a new WV tomographic technique, i.e., voxel redistribution, which does not depend on constraints originating from empirical functions. As shown in Figure 3c, the modified tomographic zone eliminates all voxels without signal penetration, which, to some extent, reduces the number of estimates, but significantly mitigates the rank deficiency problem of the tomographic coefficient matrix. Additionally, based on the above principle, we use Equations (12)-(15) to auxiliarily describe the voxel redistribution.