# Fast Screening Algorithm for Satellites Based on Multi-Constellation System
## 3 Fast Satellite Screening Algorithm Based on Multi-Threshold Optimization
### Satellite Elevation Angle and Carrier-to-Noise Ratio Modeling  
In the signal transmission process from the satellite to the receiver, the measurement pseudo-range of low-elevation-angle satellites has a large error caused by, for example, ionospheric error, tropospheric error, and multipath effect, so its ranging accuracy is also lower, and is prone to pseudo-range outliers [11]. Therefore, it is necessary to exclude such satellites by setting a satellite screening threshold based on the elevation angle.  
Ionospheric refraction is closely related to factors such as signal frequency, time of observation, and location. If ionospheric errors are not corrected or eliminated, the accuracy of the observation results will be seriously affected. The approximate expression for phase propagation in the ionosphere is  
\[n_{G}=1+\frac{c_{2}}{f^{2}} \tag{7}\]  
where \(f\) is the signal frequency, \(c_{2}\) is an estimated coefficient, whose estimated value is \(c_{2}=40.28N_{e}\text{Hz}^{2}\), and \(N_{e}\) is the electron density. Then, Equation (8) is corrected as  
\[n_{G}=1+\frac{40.28N_{e}}{f^{2}} \tag{8}\]  
Pseudo-range propagation velocity in the ionosphere is  
\[V_{G}=\frac{c}{n_{G}}=c(1-40.28\frac{N_{e}}{f^{2}}) \tag{9}\]  
The ionospheric delay correction can then be expressed as  
\[\left(V_{ion}\right)_{G}=-\frac{40.28}{f^{2}}\underset{s}{\int}N_{e}\text{ds}= \frac{40.28}{f^{2}}TEC \tag{10}\]  
where \(TEC\) is the total electron content and its value is \(\underset{s}{\int}N_{e}\text{ds}\), According to Equation (11), the ionospheric delay is determined by \(TEC\). And the smaller the elevation angle of the satellite, the longer the propagation path of the satellite signal in the ionosphere and the larger the value of \(TEC\).  
For tropospheric delays, the Saastamoinen model is usually used for correction or exclusion. According to this model, the tropospheric delay can be divided into two parts, which can be expressed as  
\[T_{h}=\frac{0.0022768p}{1.0-0.00266\cos(2\phi)-0.00028h\times 10^{-3}}\times \frac{1}{\cos z} \tag{11}\]  
\[T_{w}=0.0022768(\frac{1255}{T}+0.05)e\times\frac{1}{\cos z} \tag{12}\]where \(T_{h}\) stands for hydrostatic delay and \(T_{w}\) stands for wet delay. In the equation, \(p\) is the atmospheric pressure, \(h\) is the altitude, \(T\) is the atmospheric humidity, \(e\) is the water vapor pressure in the atmosphere, and \(z\) is the zenith angle. In the case of constant altitude, only \(z\) is a variable in Equations (12) and (13), and \(z\) can be expressed as  
\[z=\frac{\pi}{2}-el \tag{13}\]  
where \(el\) is the satellite elevation angle.  
From the above, it can be seen that for low-elevation-angle satellites, due to the long propagation path distance of their satellite signals in the troposphere and ionosphere, it is easy to produce large measurement errors, and these satellites are regarded as ineffective satellites and need to be excluded. This paper adopted the mainstream masking angle standard, and selects the minimum masking angle of \(5^{\circ}\) for satellites.  
Second, a carrier-to-noise ratio-based satellite screening threshold was set. Reference [12] pointed out that satellites in the elevation angle interval of 30-60\({}^{\circ}\) have a higher probability of being rejected by the GDOP value selection method, and there are more satellites with similar azimuth values in this interval. Therefore, the satellites with elevation angles in the range of 30-60\({}^{\circ}\) can be screened, and the satellites with excessive carrier-to-noise ratios can be eliminated without too much influence on the volume of the polyhedron composed of satellites and users, which ensures the positioning accuracy of the satellites after being screened out.