# Fast Screening Algorithm for Satellites Based on Multi-Constellation System
## 3 Fast Satellite Screening Algorithm Based on Multi-Threshold Optimization
### Fast Satellite Selection Algorithm Based on Spatial Zoning  
For a GNSS positioning system consisting of S constellations, in order to satisfy the needs of positioning solution, at least \(n=3+\text{S}\) visible satellites are required. The accuracy of the GNSS positioning system can be expressed as  
\[\sigma_{p}=\text{GDOP}\cdot\sigma_{\text{U}EE} \tag{16}\]  
where \(\sigma_{p}\) is the standard deviation of positioning accuracy and \(\sigma_{\text{U}EE}\) is the standard deviation of the user equivalent distance error. It can be seen that the smaller the GDOP value, the higher the positioning accuracy. Therefore, combinations with smaller GDOP values should be selected as much as possible in the positioning solution. For the single-constellation weighted least squares system, when the visible star is four, the GDOP can be expressed as  
\[\text{GDOP}=\sqrt{tr\left[\left(G^{T}WG\right)^{-1}\right]}=\frac{\sqrt{tr \left[\left(G^{T}WG\right)^{*}\right]}}{\|G\|}=\frac{A}{6V} \tag{17}\]  
where \(A=\sqrt{tr\left[\left(G^{T}WG\right)^{*}\right]}\) and \(V\) denotes the volume of the tetrahedron formed by the endpoints of the unit vector from the satellite to the user, and the situation is similar when multidimensionality is used for positioning. In a multi-constellation satellite positioning system, if the number of visible satellites N is very large, using the traditional optimal GDOP value selection algorithm, if the percentage of selected satellites is set to 70%, it is necessary to carry out \(\text{C}_{N}^{0,7N}\) times the GDOP value solution. The amount of computation is very large, so the positioning computational resources are very large, which is unacceptable for real-time users. Thus, a fast satellite selection algorithm based on elevation and azimuthal angle division of space is proposed. In this satellite screening algorithm, the larger the selected top star altitude is, the more uniform the azimuthal distribution of the remaining low-altitude angle satellites is, and the smaller the obtained GDOP value [15]. The proposed algorithm can effectively reduce the redundant information involved in positioning, while ensuring the number of satellites, which have a wide range of application:  
1. The elevation angles of visible stars are divided into three regions: 5-30\({}^{\circ}\), 30-60\({}^{\circ}\), and 60-90\({}^{\circ}\),which are called the low-elevation-angle region, medium-elevation-angle-region, and high-elevation-angle region, respectively, and the number of satellites in the high-elevation-angle region and the low-elevation-angle region are calculated as \(k_{1}\), \(k_{2}\), and their ratios, so that we can obtain the distribution information of the elevation angles of the visible stars as a whole. 2. We determined the number of satellites in the low-elevation region based on the ratio of low-elevation and high-elevation satellites in each quadrant and distributed the remaining satellites evenly in different elevation quadrants [16]. The remaining satellites were evenly distributed in different elevation quadrants, and the remaining satellites were selected by the permutation algorithm. * With the satellite receiver as the center of the circle, the sky is divided into four quadrants according to the satellite azimuth angles as (0\({}^{\circ}\),90\({}^{\circ}\)), (90\({}^{\circ}\),180\({}^{\circ}\)), (180\({}^{\circ}\),270\({}^{\circ}\)), and (270\({}^{\circ}\),360\({}^{\circ}\)), and the satellite with the largest spatial elevation angle in each quadrant is selected as the first of four satellites in the star selection strategy. For the remaining visible stars, sorting is performed according to the azimuth angle, and the azimuth difference between the two satellites before and after is calculated in turn. If the azimuth angle of two satellites is too close, one of them is excluded based on the overall observable satellite elevation angle, and the other satellite information can largely ensure that the volume of the polyhedron composed of all satellites and receivers is kept as constant as possible.