## 3 Methodology
### Configurations with the Lowest GDOP
\tag{26}\]  
If \(R=\frac{n_{1}}{n_{2}}\), Equation (26) can be transformed into  
\[\left\{ \begin{array}{l}x_{k}^{2}+y_{k}^{2}+z_{k}^{2}=1,k=1,2,...,n\\ n\frac{R}{1+R}z_{\min}^{2}+n\frac{1}{1+R}z_{up}^{2}=n/3\\ \frac{R}{1+R}nz_{\min}+\frac{1}{1+R}nz_{up}\to nz_{\min}\end{array} \right. \tag{27}\]  
By rounding \(n\) on both sides of Equation (27), we can obtain  
\[\left\{ \begin{array}{l}x_{k}^{2}+y_{k}^{2}+z_{k}^{2}=1,k=1,2,...,n\\ \frac{R}{1+R}z_{\min}^{2}+\frac{1}{1+R}z_{up}^{2}=1/3\\ \frac{R}{1+R}z_{\min}+\frac{1}{1+R}z_{up}\to z_{\min}\end{array} \right. \tag{28}\]  
\[\frac{R}{1+R}z_{\min}^{2}+\frac{1}{1+R}z_{up}^{2}=1/3 \tag{29}\]  
\[z1:3 ratio for satellite allocation based on optimal configuration analysis involving four satellites. In a similar vein, Ref. [12] conducted an analysis of optimal satellite configuration for a limited number of satellites (less than eight) by employing computer simulations that encompassed each data point. Theoretical analysis was conducted considering the case of restricted satellite positioning, leading to the derivation of a relationship between the allocation ratio of high and low satellites and the concept of a mask angle. Based on this theoretical foundation, a fast satellite selection algorithm for positioning with the LEO constellation was developed.  
A Satellite Selection Algorithm Incorporating a Genetic Algorithm to Overcome the Effect of Obstruction  
For observation-constrained satellites, the typical two-dimensional solution--i.e., satellites uniformly distributed on the circumference--is obviously not achievable. A reasonable way of dealing with this is to hold the following equations: \(x_{1}+x_{2}+\cdots+x_{n}=0\), \(y_{1}+y_{2}+\cdots+y_{n}=0\), and \(x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}=0\). If the minimum value cannot be selected directly, we can calculate it by minimizing the following objective function:  
\[\left(\sum_{i=1}^{n}x_{i}\right)^{4}+\left(\sum_{i=1}^{n}y_{i}\right)^{4}+ \left(\sum_{i=1}^{n}x_{i}y_{i}\right)^{2} \tag{32}\]  
This transforms the original problem into a parametric optimization problem for nonlinear problems, and a genetic algorithm can be used to solve it. In this paper, we use Equation (32) as the fitness function of the algorithm. For \(m\) visible satellites, selecting \(n\) satellites from them produces \(C_{m}^{n}\) schemes. Each particle represents a scheme. For large-scale satellite constellations, the combination of satellite selection results by an exhaustive search is not feasible and requires too much computation. A genetic selection algorithm can reduce the effect of obstruction on the geometric position satellite selection method and deliver a selection with a smaller GDOP. Meanwhile, by setting a ratio for high and low satellites, the number of satellites for selection can be effectively reduced, improving the algorithm's speed and performance. The basic process is shown in Figure 1, and the specific steps are as follows:  
* Step 1: Calculate the overall visible \(m\) satellites' elevation \(\theta\) and azimuth \(\omega\) by dividing all the satellites in view into two regions: \(m_{1}\): (0\({}^{\circ}\)-30\({}^{\circ}\)) and \(m_{2}\) (30\({}^{\circ}\)-90\({}^{\circ}\)), known as the low elevation area and the high elevation area. * Step 2: Selecting \(n_{1}=\lfloor n/(1+R)\rfloor\) satellites with elevation angles ranked from high to low from the high elevation area according to the ratio \(R=2\). * Step 3: Encoding \(m_{2}=m-m_{1}\) visible satellites and generating an \(n_{2}=n-n_{1}\) initial population
* Step 4: Calculate the fitness value for each chromosome. * Step 5: If the algorithm has converged, the algorithm will select a final satellite combination, stop searching, and proceed to Step 7. * Step 6: Apply selection, crossover, and mutation, and repeat Step 5. * Step 7: Export the result from the satellite selection process.