## 5 Results and Discussion
Here, we see that the closer the Pareto curves are to the optimality point, the better low and high degrees can be resolved since their corresponding error, in colored lines, is below the time-varying gravity signal, represented by the black line. The degree variance results for Pareto curve 98 could not be shown since the resulting constellations did not converge onto a usable solution. Having verified the spatial relation described by our methodology, we will now analyze the final set of GRACE-like constellations. After 20 generations, the GA converged onto a family of ten different constellations, each with six GRACE-like pairs. Figure 11 shows that according to the objective function, these constellations have a slight variation in both spatial as well as temporal coverage. Figure 10 has shown the spatial relation between the \(J_{so}\) and the resulting degree variance from the simulations. To verify the temporal correlation, we perform a temporal analysis of the constellations shown in Figure 11 for a simulation period of one day. For this, we compare the constellations that perform better according to the objective function \(J_{to}\) to the ones that do not. From Figure 12, we can see a one-to-one correlation between the quality of the resolved geophysical field in the degree variance curves and the corresponding temporal objective function value. Figure 10: Average degree variances for Pareto curves 1, 10, and 48. Additionally, we analyze the daily variation during the month. For this, we show the daily changes of constellation c06 in Figure 13. Here, we see that the daily degree variance stays within a small variation over one month. This concludes the verification of the transfer function between the designed methodology in Section 2 with the orbital configuration in Section 3 to the simulation setup of Section 4. Figure 11: Family of ten six-pair constellations defined by the Pareto front after twenty generations. The values for the objective functions \(J_{so}\) and \(J_{so}\) are unitless. Figure 12: 1-day average degree variances for constellations c01-c10. The 29-day average degree variance curves in Figure 14 show that the spatial resolutions of the selected family of ten constellations from Figure 11 perform equally well over a period of 29 days since there is no significant discrimination in the architectures. Using the temporal objective function to select preferred constellations is most beneficial for shorter time periods (i.e., daily). The sawtooth pattern, especially at low degrees, could result from how the data are combined. Currently, each pair is given equal weight in the simulations. We hypothesize that performing spectral weighting when combining the satellite pairs may alleviate this issue. Satellites with very low inclinations have poor observability in specific spectral bands. It may be helpful to down-weight these observations in these bands. Alternatively, mascons or local basis functions could suppress this pattern. This is subject to future study. The Keplerian orbital elements of the constellations shown in Figures 11-14 are shown in Figure 8:0 Here, the inclination \(i_{n}\), right ascension of the ascending node \(\Omega_{n}\), and mean anomaly \(M_{n}\) are given for each of the six pairs that make up the family of ten final constellations. Figure 8:1 29-day average degree variances for constellations c01 to c10. Figure 8:2 1-day degree variances for constellation c06 for the month of January 2003. As we can see, each constellation has at least one polar or near-polar pair, with the remaining constellation uniformly covering a large variety of latitude areas with additional pairs in high-latitude regions. For the following results, constellation six from Table 4 has been used. All constellations perform equally well.