## 4 Analysis
### Simulation Experiments for the Optimal Satellite Orbit in the Common-View Mode between Two Stations
Additionally, the observation elevation angles of the two SLR stations observing the satellite are nearly identical. In this paper, a heuristic method that employs the simulated annealing algorithm was introduced to tackle the problem of designing optimal satellite orbits for multi-SLR stations in a common-view model. The improved simulated annealing algorithm is refined through a designed annealing strategy to break away from local optima and gradually converge towards the global optimum during the convergence process. The optimization results of the common-view time among two SLR field stations using three different annealing strategies are presented in Figure 6. Figure 6 illustrates the optimization results for the current received solution value. When employing an exponential distribution-based annealing strategy, rapid convergence is achievable, but there is a higher probability of getting trapped in local optima. Solely utilizing a geometric distribution-based annealing strategy helps diminish the likelihood of getting stuck in local optima by introducing disturbances to the optimal solution, yet it may still lead to local optima. However, with the annealing strategy outlined in Section 3.3, the simulated annealing algorithm minimizes the incidence of local optima while narrowing the search space around the current optimal solution, striving for the global optimum. This method satisfies the algorithm's functionality. Figure 5: Outcomes derived from the optimal common-view satellite orbit model between line-of-sight stations include the following: (**a**) common-view duration concerning varying right ascension of the ascending node at identical orbital inclination; (**b**) common-view duration concerning diverse orbital inclinations at the same right ascension of the ascending node; (**c**) a schematic depiction of the optimal satellite orbit (when the site connection is perpendicular to the satellite orbit); and (**d**) a schematic illustrating the scenario where the connection between sites runs parallel to the satellite’s orbit. The improved simulated annealing algorithm and particle swarm optimization algorithm were utilized to obtain results, as shown in Figure 7. The figure indicates that both algorithms achieved similar optimal satellite orbits, converging towards the global optimum. However, the computational performance of the particle swarm optimization algorithm is significantly affected by the number of iterations in the swarm. Due to the complexity of optimizing multiple-station satellite orbits in the common-view perspective, achieving comparable precision requires the particle swarm optimization algorithm to iterate 20 to 30 times more than the simulated annealing algorithm. The improved simulated annealing algorithm demonstrates reduced computational costs and shorter processing times compared to the particle swarm optimization algorithm, highlighting its superior computational efficiency. Figure 6: The convergence process and the optimization results of the common-view time among two SLR field stations using three different annealing strategies. Figure 7: Comparison chart depicting the optimization process and iteration counts between the improved simulated annealing algorithm and the particle swarm optimization algorithm. Thirty simulations were conducted using an improved simulated annealing algorithm and particle swarm algorithm to determine the optimal satellite orbit for two SLR field stations operating under the common-view model, as shown in Figure 8. The results of the optimization process using the improved simulated annealing algorithm and particle swarm algorithm are depicted in Figure 8a, where fluctuations in outcomes can be observed. However, the common-view time's fluctuation range is limited to only 3 s. Figure 8b displays histograms for both algorithms, illustrating the distribution of optimal solutions achieved when addressing the optimization problem of determining the best satellite orbits for multiple stations under the common-view model.