## 3 Methodology
### Objective and Methodology for Global Deployment  
As satellite position is considered only for the observation model, at least 3 GSs should be visible to each satellite at any time. Therefore, it is necessary to achieve 3-HC. The objective is to maximize the observing rate and minimize the number of GSs. Anothercriterion is added as a sub-objective (third objective), which is the maximization of the distance between the two nearest GSs to avoid having a high density of GSs in the same area. Therefore, the objective fitness function becomes  
\[\left\{ \begin{array}{l}C_{1}=\min_{k}X_{k}(1)=\min_{k}(n_{GS}^{k})\\ \widetilde{C}_{2}=\min_{k\_Heavy\_Coverage}(R_{Non-obs})\\ c_{3}=\min(1/\min(d_{i}))\end{array} \right.. \tag{16}\]  
The coastline of the whole world, including continental land masses and ocean islands, in addition to Antarctica, was obtained from shapefiles (*.shp) from the National Oceanic and Atmospheric Administration (NOAA)'s [34] Global Self-consistent, Hierarchical, High-resolution Geography (GSHHG) database. The method used for GS deployment in this study is based on the integration of three evolutionary algorithms, i.e., non-dominated sorting genetic algorithm-III (NSGA-III), strength Pareto evolutionary algorithm-II (SPEA-II) and multi-objective particle swarm optimization (MOPSO). A flow chart of this method and its detailed steps is shown in Figure 11 in Appendix A. Figure 3 shows the flow chart of the GS deployment optimization for global distribution. The method used for GS deployment based on evolutionary algorithms is outlined below (Algorithm 1). ```
Step 0:  Initialize the parameters: evolutionary algorithm parameters, number of populations \(N\) and maximum iteration (\(I\)). Step 1:  For a given k-heavy coverage of \(k=3\), generate \(N\) random integer values (according to the size of population), each in the interval \(\{n_{GSmin},\ldots,n_{GSmax}\}\). Step 2:  For each random integer \(m\), generate \(m\) random coordinates, and check whether each set corresponds to land. Step 3:  For each population, evaluate the following objectives: the number of GSs and the observing rate according to the \(k\)-heavy coverage, in addition to the evaluation of the distance between the two nearest GSs in the population. Step 4:  Sort the population according to a non-dominated solution. Step 5:  Apply the main loop of the evolutionary algorithm (Appendix A) with all operations of crossover, mutation and selection. Step 6:  For each individual \(P\) in the population, check whether the value of the number of GSs (\(X_{P}\)(1) refers to the variable of the number of the GSs used as expressed in Equation (1) or whether it is changed (due to crossover and mutation operators) relative to the total number of GSs (\(|GS|\)). If the number of GSs (\(n_{GS}\)) in a population has changed due to the crossover and mutation operations, generate a new deployment of GSs according to the new \(n_{GS}\) (the same as for the generation of the initial populations). Step 7:  Due to crossover, mutation operations could change the variables; check the coordinates of GSs to see whether they correspond to land. Step 8:  Find the nearest land, and move the GSs here. Step 9:  Evaluate each individual population through the objective function. Step 10:  Check the number of iterations, and if iteration \(It<=I_{max}\) (maximum iteration), go to Step 4; else, end of optimization. ```  
**Algorithm 1**Optimal deployment method for global distribution of GSs  
In order to perform the optimization process of GS global deployment, the range of variables (i.e., the number of GSs and geographic coordinates) of each case is defined, as presented in Table 1.  
The limits for the number of GSs are set according to the heavy observing coverage. In order to reach triple HC for ideal distribution (when oceans and seas are ignored), no less than 60 GSs are used. In addition, 150 is set as the upper limit for the number of GSs used based on several tests. Therefore, the lower and upper limits are fixed to 60 to 150, respectively.