## 2 Methodology
### Objective Function
#### 2.1.1 Spatial Objective Function \(J_{so}\)  
For the analysis of the spatial domain, we make use of a 3\({}^{\circ}\) equal-area spatial grid cell since it corresponds to the \(\sim\)300 km\({}^{2}\) native spatial resolution of current mass change estimates [42]. This subdivision divides the surface of Earth into 4551 equal-area cells, as shown in Figure 3, which will be represented herein by m. If the spacecrafts' ground track, which is sampled at a 5 s interval, is over a specific spatial grid cell, this one is considered to be observed in its entirety once. For the purposes of describing the spatial fitness, the spatial objective function in Equation (1) is composed of four distinct subobjectives--\(J_{ob}\), \(J_{ro}\), \(J_{ev}\), and \(J_{ns}\)--which are calibrated using their own individual weighting factors \(W_{ob}\), \(W_{ro}\), \(W_{ew}\), and \(W_{ns}\), respectively. More to the selection of these values in Section 2.1.2. \[J_{so}=W_{ob}\cdot J_{ob}+W_{ro}\cdot J_{ro}+W_{ew}\cdot J_{ev}+W_{ns}\cdot J_{ ns} \tag{1}\]  
Figure 2: Pareto front characterization relative to the search space defined by the objective functions \(J_{so}\) and \(J_{to}\). The first spatial subobjective function, \(J_{ob}\), describes how the global coverage function \(J_{so}\) is related to the overall observability of the system. Here, we compare the sum of observed spatial grid cells \(N\) to the total number of cells available \(m\), where \(i\) stands for the subscript that identifies each spatial cell. A cell is observed if at least one satellite pair that makes up a constellation measures the corresponding spatial cell. This relation can be seen in Equation (2). \[J_{ob}=1-\frac{\sum_{i=1}^{m}N_{i}}{m} \tag{2}\]  
For the second parameter of Equation (1), we focus on the uniformity of repeat observations. Here, we make use of the Gini coefficient. [PERSON] originally developed this method in 1912 to describe income inequality for economic systems [44]. The application of this coefficient using repeat observations is shown in Equation (3). Here, \(M\) stands for the number of times a spatial grid cell has been observed more than once and \(i\) and \(j\) stand for the subscript that identifies the corresponding spatial cell. The resulting Gini coefficient in Equation (3) has a value that ranges between zero and one and represents how uniformly all spatial cells have been observed more than once, where zero stands for a perfectly uniform system in terms of repeated observations. \[J_{ro}=\frac{\sum_{i=1}^{m}\sum_{j=1}^{m}|M_{i}-M_{j}|}{2m\sum_{i=1}^{m}M_{i}} \tag{3}\]  
A visual representation of the behavior of the Gini coefficient adapted for this constellation design can be seen in Figure 4. The Gini coefficient drives the objective function to create constellations in which the number of revisits of each spatial grid cell is as uniformly distributed as possible. This directs the optimization function to seek globally uniform solutions concerning how often the constellation has revisited each spatial cell. A zero value equals an ideal constellation where a percentage of the spatial cells have observed the same percentage of observations. The opposite would be the case for a Gini coefficient of one, where a small percentage of spatial cells have observed a high percentage of observations. Figure 3: Three degrees equal-area spatial grid cells [43]. At least one GRACE-like pair within a constellation must be in a polar orbit to ensure global coverage. As we know from GRACE simulations, this gives us a good estimation of North-South information but lacks East-West information, leading to correlated errors in GRACE data manifesting as striping in unfiltered GRACE-derived gravity maps [45]. The last two subobjective functions of Equation (1) analyze the directionality of the ground track at each spatial grid cell, shown in Figure 5, as a description for the quality of East-West and North-South information that is present in each satellite constellation.