# Node-Based Optimization of GNSS Tomography with a Minimum Bounding Box Algorithm
## 3 New Node-Based Tomographic Approach
### Determination of the Position and Density of Nodes  
For the tomographic region, the convex hulls in Figure 3b are considered to be the first approximation and the minimum bounding boxes in Figure 4 are the secondary approximation. The second approximation keeps the balance between the proximity of the nodes to the distribution of GNSS signals and the complexity of the node-set. In this section, the nodes with different water vapor parameters are assigned based on the shape of the minimum bounding box.  
It should be noted that the areas of the minimum bounding boxes at different heights of the tomographic planes are different. For example, the rectangular area at the bottom of the tomographic region is 1450 km\({}^{2}\) and that of the top tomographic boundary is 4309 km\({}^{2}\). However, the number of GNSS signals that pierce through the different planes at each height of the tomographic region is the same. This implies that the density of the GNSS signals in the lower layers is higher than that in the higher layer. So, for different heights, the densities of the nodes should be adjusted to match the tightness of the GNSS signals. The density of nodes can be expressed by  
\[\mathrm{F}_{\mathrm{L}}^{\mathrm{i}}=\left[\ \mathrm{D}_{\mathrm{L}}^{ \mathrm{i}}/\hat{\mathrm{f}}_{\mathrm{L}}^{\mathrm{i}}\ \right] \tag{3}\]  
\[\mathrm{F}_{\mathrm{s}}^{\mathrm{i}}=\left[\mathrm{D}_{\mathrm{S}}^{\mathrm{i} }/\hat{\mathrm{f}}_{\mathrm{S}}^{\mathrm{i}}\right] \tag{4}\]  
where \(\mathrm{D}_{\mathrm{L}}^{\mathrm{i}}\) and \(\mathrm{D}_{\mathrm{S}}^{\mathrm{i}}\) are the length of the long side and short side of the minimum bounding box at \(i\)th tomographic plane; \(\hat{\mathrm{f}}_{\mathrm{L}}^{\mathrm{i}}\) and \(\hat{\mathrm{f}}_{\mathrm{S}}^{\mathrm{i}}\) are the expected intervals for the long side and short side; the brackets round \(\mathrm{D}_{\mathrm{L}}^{\mathrm{i}}/\hat{\mathrm{f}}_{\mathrm{L}}^{\mathrm{i}}\) and \(\mathrm{D}_{\mathrm{S}}^{\mathrm{i}}/\hat{\mathrm{f}}_{\mathrm{S}}^{\mathrm{i}}\) to the nearest integer; \(\mathrm{F}_{\mathrm{L}}^{\mathrm{i}}\) and \(\mathrm{F}_{\mathrm{S}}^{\mathrm{i}}\) are the segments for the long side and short side, respectively. As shown in Figure 6, when the \(\mathrm{F}_{\mathrm{L}}^{\mathrm{i}}\) and \(\mathrm{F}_{\mathrm{S}}^{\mathrm{i}}\) are defined, the nodes (black square) can be determined easily at the tomographic plane.  
As mentioned before, the densities of the nodes must be appropriate for the size of the minimum bounding boxes at a different height. Therefore, the variables \(\mathrm{f}_{\mathrm{L}}\) and \(\mathrm{f}_{\mathrm{S}}\) vary at different heights of the tomographic region (details seen in Table 2). When the nodes at each of the tomographic planes have been determined, the discretization of the region is finished. These nodes are herein named _tomographic nodes_.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Height (m)** & **Below 2400** & **2800–4500** & **5000–5500** & **6500–8500** & **11000** \\ \hline \(\hat{\mathrm{f}}_{\mathrm{L}}^{\mathrm{i}}\) and \(\hat{\mathrm{f}}_{\mathrm{S}}^{\mathrm{i}}\) (m) & 7000 & 8000 & 9000 & 9800 & 15000 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The size of \(\hat{\mathrm{f}}_{\mathrm{L}}^{\mathrm{i}}\) and \(\hat{\mathrm{f}}_{\mathrm{S}}^{\mathrm{i}}\) at different heights of the minimum bounding boxes.  
Figure 6: The even-distributed nodes (square dots) at the top of the tomographic region in Figure 4; the blue box is the boundary of the minimum-area encasing rectangle at the top of the tomographic region.