## 2 Methodology  
The main issue with generating an \(n_{pair}\) satellite constellation is the size of the search space. When considering constellations with one or two satellite pairs, it is possible to perform a Monte Carlo study [31]; however, with an increasing number of pairs, the size of the search space grows exponentially. To explore the search field efficiently, we use a genetic algorithm (GA). These stochastic search methods take their inspiration from [PERSON]'s theory of natural evolution [32]. In this study, the GA analyzes a set of constellations that define the population at each generation. Each constellation contains a set of GRACE-like \(n_{pair}\) satellites. The relevant characteristics of each constellation are encoded as a string of bits that can be either zeros or ones. We analyzed the impact of three orbital parameters for this investigation: the inclination i, right ascension of the ascending node \(\Omega\), and mean anomaly M. These orbital elements are encoded to span the whole viable search space of constellation configurations' characteristics. The altitude is fixed at \(500\pm 5\) km. The designed constellations fly in circular orbits to avoid changes in relative distance due to eccentric orbits. In this paper, the authors selected a constellation consisting of six GRACE-like pairs to verify the described methodology. Previous design points suggest it to be the smallest constellation size to recover reasonable time-varying gravity fields at daily time intervals [33]. The resulting populations are then evaluated by measuring their so-called fitness value, determined by an objective function, described in Section 2.1. This function is defined by the characteristic we intend to optimize (e.g., spatial and temporal resolution). Once these fitness values are obtained, they serve as a quality metric in the GA for each population. GA operators utilize them to improve the set of solutions for the next generation. The genetic algorithm is a formidable search algorithm that is capable of dealing with a very large discrete search space [34]. GAs use biological and computationally principled methods to explore a given number of variables. Using techniques to balance exploration against exploitation, the search space is efficiently explored for candidate \(n_{pair}\) satellite constellations. There is no proof of convergence, but with a large population and multiple runs, the probability of finding a global optimum can be maximized. A potent type of GA is the Multiobjective Genetic Algorithms (MOGA) that allow for more than one objective function to be optimized simultaneously. The main drawback of this method is that the complexity directly correlates to the number of objectives and the size of the selected population. To circumvent this issue, we make use of the Nondominated Sorting Genetic Algorithm II (NSGA-II) developed by [PERSON] [35]. A flowchart depicting the NSGA-II algorithm can be seen in Figure 1. This method uses a fast nondominated sorting approach and encourages diversity of solutions, with a quick convergence, and has an aptness for global optimization. Additionally, it decreases computational complexity. Starting from a set of constellations, called populations, picked at random, it computes the fitness of each constellation by an objective function, described in Section 2.1, which we have developed specifically for a GRACE-like multipair constellation. Figure 2 shows a sample figure of individual constellations, represented by dots, on a search space spanned  
Figure 1: Algorithm scheme of the nondominated sorting genetic algorithm II (NSGA-II). by two objective functions: \(J_{so}\) on the x-axis and \(J_{to}\) on the y-axis. These constellations are organized by the their fitness values onto Pareto curves, which act as a ranking metric. The Pareto curve closest to the axes origin, also called the optimality point, is ranked higher than each subsequent front.