# Node-Based Optimization of GNSS Tomography with a Minimum Bounding Box Algorithm
## 1 Introduction
Second, the voxels and their nodes (as the cell of parameterization) in the empty spaces are unnecessary for the tomographic model and become a computational burden. Third, the fixed region for all epochs has larger redundant space (i.e., the region must ensure that various distributions of signals at each epoch are covered), which does not fit the distribution of GNSS signals at a defined specific epoch. To address the above problems, a node-based innovative approach to dynamically determine the dimensions of the tomographic region at each epoch by the minimum bounding box algorithm is \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**[PERSON]** & **[PERSON] et al. [1]** & **[PERSON] [10]** & **[PERSON] et al. [11]** & **[PERSON] et al.** [12]** \\ \hline Tomographic region (km\({}^{3}\)) & \(20\times 20\times 15\) & \(27\times 23\times 10\) & \(30\times 30\times 9\) & \(20\times 20\times 10\) \\ \hline [PERSON] & [PERSON] et al. [13] & [PERSON] et al. [14] & [PERSON] et al. [15] & [PERSON] [16] \\ \hline Tomographic region (km\({}^{3}\)) & \(300\times 150\times 15\) & \(30\times 20\times 10\) & \(50\times 40\times 10\) & \(266\times 166\times 15\) \\ \hline [PERSON] & [PERSON] et al. [17]; [PERSON] et al. [18]; [PERSON] et al. [19]; [PERSON] [20] \\ \hline Tomographic region (km\({}^{3}\)) & The Hong Kong Satellite Positioning Reference Station Network (SatRef) roughly covers & & & \\  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The dimensions of the Global Navigation Satellite Systems (GNSS) tomographic region. proposed. In each epoch, the new approach defines the two-dimensional tomographic boundaries at different height of tomographic planes, which comprise the tomographic region in the three-dimensional space. Two steps determine the tomographic boundaries: (1) the first approximation model using the convex hull algorithm to define the boundary (i.e., convex polygon) of the collection of rays; (2) the secondary approximation model using the minimum bounding box algorithm to define the final tomographic boundary (i.e., minimum-area rectangle) that encases this convex polygon. By using the secondary approximation modeling method, the position of the nodes can be easily assigned in the tomographic region. The expected intervals, defined in Section 3.2, were used to adjust the nodes' distribution at each tomographic plane. When the nodes have been determined, the discretization of the tomographic region is finished. Moreover, the new approach is not limited to the test in this study but can be utilized for all GNSS networks. In this paper, we focus on the improvement of the common node-based GNSS tomographic approach. The common approach was used for comparative analysis and validation of this new approach. This paper mainly discusses the new approach that improves node-based GNSS tomography with a minimum bounding box algorithm. This paper is structured as follows. Section 2 introduces the GNSS tomography technique. This section details the observations used in the tomographic model and the common tomographic method. Section 3 proposes a new approach, which can dynamically define the tomographic model. Section 4 presents the test results and analysis. At the end, a few conclusions are drawn in Section 5.