## 5 Results and Discussion  
Given the objective function and the orbital considerations defined in Sections 2 and 3, we are seeking to verify a direct relationship between the numerical methodology characterized and the simulation setup in Section 4 that has been used in the literature to simulate GRACE-like constellations [7; 11; 23; 24; 25; 26; 27; 28]. This will prove that the GA with its objective function can be used to numerically explore the search space for a wide variety of constellation configurations. This tool aims not to find a single solution but to narrow down the search \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & **Truth** & **Nominal** & **Source** \\ \hline Static gravity field & EIGEN-GL04C & EIGEN-GL04C & [49] \\ Ocean tide & FES2004 & GOT00 & [50; 51] \\ Atmospheric & ECMWF & NCEP & [52; 53] \\ Ocean & OMCT & MOG2D & [54; 55] \\ Hydrological & GLDAS & - & [56] \\ Ice & ESA & - & [57] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation and model definitions. space and, at the same time, provide the mission designer with a family of constellation solutions that can fulfill the given requirements. This offers the constellation designer a variety of solutions that can be narrowed down based on other criteria (e.g., launch sites). Figure 8 shows how the GA uses the objective function effectively to explore the search space, given here by the spatial and temporal objective function J. Within 20 generations, it has narrowed down the search space and identified a set of optimal solutions. The constellations generated at every generation are sorted on several Pareto curves using the fitness values evaluated by the objective functions. The closer the Pareto curves are to the optimality point, the higher the inherent quality of the constellations is in regard to the fitness values. To show how the information contained in the search space is in direct relation to the solutions of the simulations in Section 4, we analyze the constellations in the first generation and how its Pareto curves relate to simulation results given by the NASA/GSFC tools. The constellations developed in the first generation have been chosen due to having more significant variability in the computed fitness values resulting in a more extensive span in the quality of the analyzed constellations. Of all available Pareto curves in the first generation, Figure 9 shows four selected Pareto curves that encompass the entirety of the population in the first generation, namely, Pareto curves 1, 10, 48, and 98. Figure 8: Constellation population for generations 1, 3, and 20. Each dot represents a different six-pair constellation. The values for the objective functions \(J_{so}\) and \(J_{to}\) are unitless. Figure 9: Pareto curves 1, 10, 48, and 98 span the search space both in spatial and temporal objective functions. The values for the objective functions \(J_{so}\) and \(J_{so}\) are unitless. Using the simulation setup described in Section 4, we can verify with the degree variance curves in Figure 10 that there is a direct relationship between the location of the Pareto curves in the search space and the average quality of their solution represented in the degree variance format. Geoid degree error plots are constructed using the following equation:  
\[\Delta N_{n}=r_{\oplus}\sqrt{\sum_{m=0}^{n}\left(\Delta\overline{C}_{nm}^{2}+ \Delta\overline{S}_{nm}^{2}\right)} \tag{11}\]  
where  
\[\Delta\overline{C}_{nm}=\left(C_{nm}\right)_{\text{estimate}}-\left(C_{nm} \right)_{\text{truth}} \tag{12}\]  
\[\Delta\overline{S}_{nm}=\left(S_{nm}\right)_{\text{estimate}}-\left(S_{nm} \right)_{\text{truth}}\]  
and \(r_{\oplus}\) is the radius of the Earth. This formulation quantifies the geoid error for each degree in units of length as the difference between the defined truth geophysical signal and its best estimate. This expression is a valid assumption since we are working in a strict simulation environment where the truth of the geophysical signals is known.