## 3 Satellite Selection Algorithm
### Weighted Quasi-Optimal Satellite Selection Algorithm  
Although the quasi-optimal satellite selection algorithm is close to the optimal selection algorithm and is more efficient, the satellite navigation system positioning accuracy is not only related to the GDOP but also affected by the uncertainty error of each pseudo-range measurement value, as shown in Equation (17). Pseudo-range measurements are affected by atmospheric ionospheric and tropospheric delays, multipath effects, and orbital errors [5]. The signal-to-noise ratio of the satellite signal will decrease with the increase in the noise of the measured value and increase with the increase in the satellite elevation angle, while the multipath effect and satellite orbit error will also reduce the signal-to-noise ratio. So, at the same time, considering the satellite elevation angle, signal-to-noise ratio, and multipath effect, a weighted quasi-optimal satellite selection algorithm (WQOSA) of the weight function can be constructed. This paper refers to the literature [27] to give a weight function construction method:\[W_{i}=\frac{E_{\max}}{E_{i}}+\alpha_{i}\cdot\frac{SNR_{\max}}{SNR_{i}} \tag{23}\]  
where \(E_{i}\) and \(SNR_{i}\) are the elevation angle and signal-to-noise ratio of satellite i, \(E_{\max}\) and \(SNR_{\max}\) are the maximum elevation angle and the maximum signal-to-noise ratio of the visible satellites in the same calendar element, and \(\alpha_{i}\) is the multipath scale factor. Under normal driving conditions, the multipath scale factor is usually set to 1, and when multipath effects are encountered, the multipath scale factor can be set to a value greater than 1. So, the multipath scale factor can be expressed as a function of the reflection coefficient \(R_{coef}\), which, in turn, is correlated with the carrier-to-noise density ratio (\(C/N_{0}\)) [28]:  
\[\begin{split}\alpha_{i}=1+\frac{\sqrt{R_{coef}}-1}{\sqrt{R_{coef }}+1}\\ R_{\text{coef}}=\frac{10^{\left(\frac{(C/N_{0})_{\max}}{2v} \right)}}{10^{\left(\frac{(C/N_{0})_{1}}{20}\right)}}\end{split} \tag{24}\]  
In the above equation, \((C/N_{0})_{i}\) is the carrier-to-noise ratio of satellite i, which indicates the strength of the received signal and is usually used to estimate the signal quality [29], and it can be seen that when the carrier-to-noise ratio is larger, \(\alpha_{i}\) is close to 1, which corresponds to a weaker satellite multipath effect.  
Since the cost function of the QOSA has a value range from \(-1\) to \(+1\), to avoid the impact of positive and negative attributes on the weight function, it is also necessary to transform the cost function of the QOSA before constructing the cost function of the WQOSA. In this paper, we present the new cost function:  
\[J_{ij}=e^{\cos 2\theta_{ij}} \tag{25}\]  
The cost function for a single satellite \(i\) is then expressed as  
\[J_{i}=\sum_{j}e^{\cos 2\theta_{ij}} \tag{26}\]  
The cost function for the WQOSA can then be constructed:  
\[J_{i}=W_{i}\sum_{j}e^{\cos 2\theta_{ij}} \tag{27}\]  
The satellite selection process of the WQOSA is similar to that of the QOSA, but it involves an additional weight calculation process. To provide a clearer illustration of the satellite selection process of the WQOSA, a flowchart of the algorithm execution is presented in Figure 2.