## 2 Methodology
### Objective Function
#### 2.1.1 Spatial Objective Function \(J_{so}\)
First, we transform the velocities of all observations at each spatial grid cell to a local frame with Equation (4), where the vectors \(\mathsf{REN}\) represent the radial, east, and north direction. Subsequently, we superposition the local velocities for all spatial cells to form matrix \(\mathbf{A}\) by disregarding the radial velocity component since it does not add any  
Figure 4: The behavior of the Gini coefficient, as described in [44], adapted for a spatial grid cell analysis. A uniform distribution is defined when an equal percentage of the spatial grid observes the same percentage of the observations throughout the spatial domain (G1). The least desirable case (G4) occurs when a small percentage of the spatial grid observes a high percentage of the total observations. Cases G2 and G3 are intermediary stages. Figure 5: Relation of the local reference frame to the Earth-centered, Earth-fixed (ECEF) coordinate system used to describe the quality of the North–South and East–West information produced by the constellation ground track. The ECEF coordinates are shown in relation to the longitude (\(\lambda\)) and latitude (\(\varphi\)). information to the velocity vector's East-West and North-South local components. The size of matrix \(\mathbf{A}\) is \(2\times n\), where \(n\) identifies the number of observations at each spatial cell. \[\bar{\mathbf{v}}_{local}= \begin{bmatrix}\mathbf{\dot{R}}\mathbf{\dot{R}}\mathbf{ \dot{R}}\end{bmatrix} \bar{\mathbf{v}}_{ECEF}\qquad\mathbf{A}= \begin{bmatrix} \hat{\mathbf{v}}_{1}\\ \vdots\\ \hat{\mathbf{v}}_{n}\end{bmatrix}  \tag{4}\]  
The diagonal elements of the \(\mathbf{B}\) matrix define how much information the system has in the East-West compared with the North-South direction at each spatial cell. \[1-\frac{\mathbf{A}^{T}\mathbf{A}}{n}= \begin{bmatrix}B_{ew}&B_{cross}\\ B_{cross}&B_{ns}\end{bmatrix}  \tag{5}\]  
A visual representation of how the \(B_{ew}\) value varies depending on which satellite constellation is analyzed can be seen in Figure 6.  
Here, a two-pair satellite constellation has been used to illustrate the effect of the East-West information on the objective function. The first pair has been chosen to be polar, whereas the second pair has a varying inclination between \(50^{\circ}\) and \(80^{\circ}\). The two-pair solution with the second pair at \(50^{\circ}\) has very good East-West information in the equatorial region; still, it is missing all the information in the polar region. The case with the second pair at \(80^{\circ}\) is the direct opposite, where it has excellent East-West information coverage over the high latitudes but is missing the equatorial region. An ideal middle ground is the solutions at \(60^{\circ}\) and \(70^{\circ}\), which cover just enough of the equatorial region as well as part of the high-latitude regions. From previous research, we know that a possible mathematical optimum when analyzing the East-West information is at \(63^{\circ}\) of inclination [46]. As we can see through visual inspection of Figure 6, the increase in inclination of the second pair is correlated to a better overall value for the \(B_{ew}\) information up to an inclination between \(60^{\circ}\) and \(70^{\circ}\). By adding additional pairs at various inclinations, there is a superposition of the collected cross-track information of all pairs that make up a constellation into a single combined solution. Figure 6: \(B_{ew}\) information for a two-satellite-pair example with a polar pair and an inclined pair of \(50^{\circ}\) (**top left**), \(60^{\circ}\) (**top right**), \(70^{\circ}\) (**bottom left**), and \(80^{\circ}\) (**bottom right**). A lower value of \(B_{ew}\) directly relates to beneficial information in the East-West direction due to the ground track directionality. The global value is optimum between the inclinations of \(60^{\circ}\) and \(70^{\circ}\).