## 6 Conclusions  
Due to an increasing trend in the miniaturization of hardware and accelerations in the quality of small form factor satellites, we consider that a future mass change mission could have the shape of an \(n_{pair}\) GRACE-like satellite constellation. Such a constellation aims to reduce temporal aliasing by recovering high-frequency temporal variations in the gravitational field. This is achieved through the inherent higher sampling frequency due to the chosen orbit geometry that can be achieved with such an \(n_{pair}\) satellite constellation.  
The goal of this paper was to develop a tool to optimize the orbits of an \(n_{pair}\) GRACE-like satellite constellation to provide increased spatial and temporal resolution in determining hydrology, ice mass variations, and ocean bottom pressure signals globally. The search space for this type of problem is infinite. To efficiently explore the search space, we balance exploration and exploitation with a search algorithm by fixing the altitude at \(500\pm 5\) km and the number of satellite pairs to six. Although smaller constellations were possible, six satellite pairs were the smallest constellation size to provide a satisfactory recovered gravity field. The constellations have been optimized simultaneously for a monthly and daily time period. We considered only circular orbit in a repeat period of 29 days, which guaranteed us an equal performance on a month-to-month basis. The intersatellite separation amongst satellite pairs has been assumed to be 100 km. Using a fast and elitist multiobjective genetic algorithm as our search algorithm to explore the search space, we developed objective functions that seek to optimize a satellite constellation design regarding uniformity in spatial coverage, ground track directionality, and temporal sampling. These parameters have been calibrated using prior design points and weighted by information on the importance of geophysical signals to equally capture hydrology, ocean, and ice mass variations.  
State-of-the-art geodetic parameter estimation and precision orbit determination software provided by NASA's GSFC have been used to verify the geophysical signal recovered by constellations developed by the genetic algorithm. The resulting set of orbital elements describes a family of solutions that, regarding spatial resolution, do not have significant discrimination between the architectures of the different constellations. Due to the temporal objective function, we can discriminate among more suited architectures for shorter (i.e., daily) geophysical recovery. Due to the varying inclinations present in the designed constellations, no postprocessing through destriping nor smoothing was necessary.  
The proposed scheme addresses the problem of a future \(n_{pair}\) GRACE-like satellite constellation design given by an infinite search space by providing a family with a wide variety of candidate solutions that can be applied for a future mass change study.  
Conceptualization, C.M.A.D., R.S.N., and D.N.W.; methodology, C.M.A.D.; software, C.M.A.D.; validation, C.M.A.D.; formal analysis, C.M.A.D.; investigation, C.M.A.D.; resources, C.M.A.D., R.S.N., and D.N.W.; data curation, C.M.A.D.; writing-original draft preparation, C.M.A.D.; writing--review and editing, C.M.A.D., R.S.N., and D.N.W.; visualization, C.M.A.D.; supervision, R.S.N.; project administration, R.S.N. and D.N.W.; funding acquisition, C.M.A.D., R.S.N., and D.N.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the NASA Earth and Space Science Fellowship grant number 18-EARTH18F-0380 and NASA ROSES Earth's Surface and Interior call grant number NNH18 ZDA001N-ESI.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
We would like to thank [PERSON] and [PERSON] from NASA Goddard Space Flight Center for their support in all aspects related to the GEODYN software package. Your help is immensely appreciated.  
The authors declare no conflict of interest.