## 2 Orbit Selection Approach
In this way, we found the step size \(\Delta a=8.35\) m for semimajor axes in the range of 6718-6888 km, which corresponds to an altitude range of 340-510 km, and \(K\approx 800\) (50 days). On the left, Figure 4 presents all homogeneity values \(h_{l}<3\) in that altitude range within a time span of 1-50 days. This graph is particularly useful for selecting the altitude of the individual satellite pairs of the constellation. We only have to draw a horizontal line into the graph to identify whether an altitude provides small homogeneity values, i.e., a dense spatial sampling, for the desired time spans and obtain an overview of the achievable subcycles offered by the selected altitude. Generally, the graph in Figure 4 reveals that for shorter time spans small homogeneity values stretch over much larger altitude ranges than for longer time spans. At three days, for example, homogeneity values \(h_{l}<1.5\) stretch from 406 to 433 km, i.e., over an altitude range of 27 km. This altitude range offers many other small homogeneity values at longer time spans, which gives flexibility for optimizing the spatial sampling at multiple temporal resolutions. We propose using this flexibility to optimize the constellation in the following way: First, we note that the ground track will shift by  
\[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]8  
after \(l\) orbital revolutions. Figure 5 shows \(\Delta\lambda_{shift,l}\) for the homogeneity values \(h_{l}<3\), which are illustrated in \[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]9 For homogeneity values equal to 1, which correspond to repeat orbits, the ground track shift is obviously zero. Generally, homogeneity values close to \(1\) result in small ground track shifts. The first step of the constellation optimization is the selection of the shortest period that is longer than one day and shall be resolved by the gravity field modeling. Since the spatial sampling is obviously less dense for shorter time spans, we select altitudes of the satellite pairs such that both their ground tracks exhibit the same \(\Delta\lambda_{shift,l}\) within that short period. Then, the crossover points of the satellite pairs' ground tracks will shift in longitude by \(\Delta\lambda_{shift,l}\) after \(l\) orbital revolutions, whereas their latitude will not change. In a sense, the interleaving of these ground tracks for a single pair of satellites will nearly repeat every \(l\) orbital revolutions, such that the density of the spatial sampling within that short  
\[\Delta\lambda_{n}=\Big{(}\hat{\Omega}-\omega\Big{)}nT, \tag{2}\]0 Ground track analysis graphs for circular orbits with an inclination of 70\({}^{\circ}\): ground track homogeneity ranges for altitudes between 350 and 500 km. \[\Delta\lambda_{n}=\Big{(}\hat{\Omega}-\omega\Big{)}nT, \tag{2}\]1 Ground track analysis graphs for circular orbits with an inclination of 70\({}^{\circ}\): ground track shift in longitude per subcycle period represented on the x-axis. period does not change over time. This feature ensures a constant quality of gravity field models for each chosen estimation period, which is an important benefit for emergency and operational applications and services. Since we only require that \(\Delta\lambda_{shift,l,l}\) is the same for both satellite pairs, we still have the flexibility to select the altitudes such that the orbits offer near-homogeneity values at longer periods. In practice, we create plots of the homogeneity and ground track shift as shown in Figures 4 and 5 for both satellite pairs. Then, we search for the altitudes that offer near-homogeneity values for a number of periods, which relate to the desired temporal resolutions (for example, 1 week and 1 month), and fulfill the constraint that the ground track shift of the shortest period longer than one day is the same for both satellite pairs. Thus, the combination of the two plots as illustrated in Figures 4 and 5 enables us to optimize the sampling of each individual pair as well as the constellation.