## 3 Methodology  
### Configurations with the Lowest GDOP  
For a single-point positioning configuration that consists of \(K\) satellites, the GDOP optimal problem can be described as an unconstrained extremum problem. Their extremum points satisfy a set of matrix condition equations. \[\mathbf{G}^{\mathrm{T}}\mathbf{G}= \begin{bmatrix}\frac{K}{m}\mathbf{I}\in\Re^{m\times m}& \mathbf{0}\\ \mathbf{0}&K\end{bmatrix}  \tag{23}\]  
\(\Re^{m\times m}\) means that the satellite is visible in the \(m\times m\) dimension space. 1. Since the matrix Equation (23) is a nonlinear equation, it may have a finite number of solutions, no solutions, or infinite solutions. 2. GDOP is a continuous function; that is, in the vicinity of the \"ideal extreme point\", its GDOP will increase continuously without any sudden change. 3. GDOP minimization requires a symmetry condition \(\mathbf{\mathrm{F}}\mathbf{\mathrm{k}}_{\mathrm{K}}=0\), which is difficult to satisfy for most 3D applications. This means that, when the control points are distributed on a hemisphere or the same plane, GDOP cannot reach a theoretical minimum value [20; 22]. For the satellite positioning system, as users can only receive signals whose elevation angles are above the mask angle, the configuration that can be selected has to satisfy the condition \(z\geq z_{\mathrm{min}}\) (\(z_{\mathrm{min}}\) refers to the minimum normalized height coordinate at which the elevation angle is equal to the mask angle). In the discussion, the configuration parameter is given by  
\[\mathbf{F}= \begin{bmatrix}x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\\ \vdots&\vdots&\vdots\\ x_{n}&y_{n}&z_{n}\end{bmatrix}  \tag{24}\]  
Under the observation-constrained condition \(z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}=n/3\) and \(z_{1}+z_{2}+\cdots+z_{n}=nz_{\mathrm{min}}\), it is not possible to achieve both conditions simultaneously. Consequently, the configuration can only converge to \(z_{1}+z_{2}+\cdots+z_{n}\to nz_{\mathrm{min}}\) as closely as possible. The convergence to \(nz_{\mathrm{min}}\) represents the best approximation that can be attained within \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Population size & Number of bottom satellites to be selected \\ Max Generations & 100 \\ Function Tolerance & \(10^{-6}\) \\ Crossover probability & 0.8 \\ Mutation probability & 0.01 \\ perturbation function & Gaussian perturbation function \(x_{i}=x_{i}+(1-\frac{g}{\xi})\cdot N(0,1)\). \(x_{i}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operating parameters of the basic genetic algorithm. the given constraints.The nonlinear system of equations to analyze the minimum value of GDOP reads as follows:  
\[\left\{ \begin{array}{l}x_{k}^{2}+y_{k}^{2}+z_{k}^{2}=1,k=1,2,...,n\\ x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}=0\\ x_{1}z_{1}+x_{2}z_{2}+\cdots+x_{n}z_{n}=0\\ y_{1}z_{1}+y_{2}z_{2}+\cdots+y_{n}z_{n}=0\\ x_{1}^{2}+y_{1}^{2}+z_{2}^{2}+\cdots+x_{n}^{2}+y_{n}^{2}=2n/3\\ z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}=n/3\\ x_{1}+x_{2}+\cdots+x_{n}=0\\ y_{1}+y_{2}+\cdots+y_{n}=0\\ z_{1}+z_{2}+\cdots+z_{n}=nz_{\min}\end{array} \right. \tag{25}\]  
When \(n_{1}\) points are selected to be uniformly distributed on the circumference of \(z=z_{\min}\), and the remaining \(n_{2}\) points are located on the circumference of \(z=z_{up}\), \(x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}=0\), \(x_{1}z_{1}+x_{2}z_{2}+\cdots+x_{n}z_{n}=0\), \(y_{1}z_{1}+y_{2}z_{2}+\cdots+y_{n}z_{n}=0\), \(x_{1}+x_{2}+\cdots+x_{n}=0\), and \(y_{1}+y_{2}+\cdots+y_{n}=0\) are satisfied simultaneously. \(z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}=n/3\) must be true if \(x_{1}^{2}+y_{1}^{2}+z_{1}^{2}\cdots+x_{n}^{2}+y_{n}^{2}+z_{n}^{2}=n\) and \(x_{1}^{2}+y_{1}^{2}+x_{2}^{2}+y_{2}^{2}+\cdots+x_{n}^{2}+y_{n}^{2}=2n/3\). Equation (25) is transformed into  
\[\left\{ \begin{array}{l}x_{k}^{2}+y_{k}^{2}+z_{k}^{2}=1,k=1,2,...,n\\ z_{1}^{2}+z_{2}^{2}+\cdots+z_{n-2}^{2}+z_{n-1}^{2}+z_{n}^{2}=n/3\\ z_{1}+z_{2}+\cdots+z_{n}\to nz_{\min}\end{array} \right.