## 3 Methodology
### Objectives and Methodology for Regional Deployment  
In the regional scheme of GS deployment, there is no way to reach any k-HC, which makes it difficult to determine an optimal configuration based only on the optimal SPDOP value. Therefore, the number of GSs is fixed for algorithm simplification. In the case where the number of GSs is a variable for the optimization problem, the objective (SPDOP) of visible satellites in the regional area tends to be enhanced in proportion to the increase in the number of GSs. Moreover, there is an obvious concentration of GSs as long as SPDOP is evaluated only for visible satellites. The number of GSs for global deployment is considered as a decision variable in order to obtain the minimum number needed to reach three-heavy observing coverage. Fixing the number of GSs in global deployment means an excessive number of scenarios need to be studied. For each fixed number of GSs, the optimal deployment must be found to reach the highest observing rate. Therefore, it is suitable to use the number of GSs as a decision variable for global deployment, unlike for regional cases, even though the algorithm complexity increases. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameter** & **Value** \\ \hline Non-dominated sorting genetic & Crossover probability & 0.8 \\ algorithm-III & Mutation probability & 0.2 \\ (NSGA-III) & Mutation rate & 0.05 \\ \hline Multi-objective particle swarm & Inertia weight & 0.4 \\ optimization & Damping rate & 0.5 \\ (MOPSO) & Inflation rate & 0.5 \\  & Mutation rate & 0.1 \\  & Personal learning & 1 \\  & Global learning & 2 \\ \hline Strength Pareto evolutionary & Crossover probability & 0.5 \\ algorithm-II & Mutation probability & 0.5 \\ (SPEA-II) & Number archive to population ratio & 1/3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Algorithm parameters of the hybrid method for GS deployment optimization. For the purpose of simplifying the optimization problem, the number of GSs is fixed for differing scenarios. The variables of this problem are the coordinates of the GSs. \[X=[\lambda_{1},\varphi_{1},\ldots,\lambda_{n_{GS}},\varphi_{n_{GS}}], \tag{17}\]  
where \(\lambda_{i}\) and \(\varphi_{i}\) are the longitude and latitude of the \(i\)th ground station, respectively, and \(i=[1,2,\ldots,n_{GS}]\), where \(n_{GS}\) is the number of GSs used for optimal deployment. There is a single objective of optimization in these cases (only the mean SPDOP value). In addition, a sub-objective is added to ensure the good dispersion of GSs throughout the regional area to obtain a maximum number of observed satellites each time. This objective function is the maximization of the distance between the two nearest GSs (as the global distribution scheme). Therefore, the objective functions are expressed as  
\[\left\{ \begin{array}{l}C=\underset{t\in\text{Observation}}{mean}\left( \underset{P\in\text{lat,Long}}{mean}\left(\underset{S\in\text{Visible}\_Sat}{SDOP}(t, \text{P})\right)\right.\\ \widetilde{c}=\min(1/\min(d_{i}))\end{array} \right., \tag{18}\]  
where \(t\) is the observing epoch (epoch of 1 min within 24 h of observation), P denotes the satellite's position, \(S\) denotes all the visible satellites at epoch \(t\), and \(d_{i}\) is the distance between each station and the nearest GS in the same deployment (individual in the population). The flow chart of the optimization method for GS regional deployment with the objective of SPDOP value minimization is shown in Figure 4. The principal step of the method used to determine the optimal deployment of GSs for local distribution is presented below (Algorithm 2). ```
Step 0: Initialize the parameters of the algorithm and population size of \(N\) individuals. Step 1: Define the number \(m\) of GSs that are used for optimal deployment. Step 2: For each population, generate \(m\) random GSs in the selected regional area.