## 7 Ground-Segment Optimization Aspects
### Optimization Metrics Related to the Ground Segment  
The study in [149] looked at the coverage as the optimization metric in the context of Beidou GNSS and aimed at having the same satellite in view for a predetermined interval of time by at least four ground stations (thus, they called this metric a 'quadruple coverage' metric). Although the study compared only two deployments based on 18 ground stations each (thus, there was no complex optimization problem involved), it served as a good example of utilizing sky coverage percentage as a metric for the general optimization goal of maximizing coverage in GS planning. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Convergence** & **Global Convergence** & **Large-Scale Applicability** \\ \hline Theoretical & & & \\ Modeling & & & \\ \hline GA & & & \\ \hline PSO & & & \\ \hline SA & & & \\ \hline DNN & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparative Summary of Optimization Methods for LEO Space-Segment Optimization. The sky coverage is a metric such that it requires a mathematical model to operate with, which is typically constructed slightly differently depending on the objective of the analysis. In [149], the authors defined it as _'the percentage of observing time to a certain satellite by at least \(M\) stations within a specified time period'_ and formulated it from the GS visible curve to the satellite using simplified geometric calculations. The problem of finding the best location of a GS was addressed in [150] in the context of MEO Satellite-Based Augmentation System (SBAS). The target metrics for the geographical location optimization were the scintillation fade depth, the GDOP, the number of visible satellites, the ionospheric delay, and the rainfall attenuation, which were all varying metrics indicating the signal-availability level. While the problem was formulated as a multi-modal optimization problem, no final optimal solution was given, but rather a trade-off criterion by criterion was presented. The number of satellites and GDOP are related metrics; as the number of satellites in view increases, the value of GDOP decreases, which ensures good accuracy. In [150], the authors followed the GDOP calculations from [151] and used GDOP as the optimization metric. The GDOP is defined in Equation (18) based on the least square solution seen in Equation (17). \[\Delta y=\left(H^{\mathrm{T}}H\right)^{-1}\!H^{\mathrm{T}}\Delta x \tag{17}\]  
\[\mathrm{GDOP}=\sqrt{tr{\left(H^{\mathrm{T}}H\right)}^{-1}} \tag{18}\]  
Above, \(\Delta y\) is the position offset as obtained from the least squares solution of the linearized pseudorange measurement model of the system, \(H\) is the satellite geometry matrix, and \(\Delta x\) is the net error in the pseudorange value. \(tr{\left(.\right)}\) refers to the trace operation. Note that, in order to formulate the set of equations whose least squares solution provides the GDOP, a modeling process is required that expresses the geometry between the users (for our case, this is the geometry between the GS and the satellite). In addition, modified versions of GDOP such as the weighted GDOP [151] also exist, which include a weight matrix \(W\) to the model, as in Equation (19). \[WGDOP=\sqrt{tr{\left(H^{\mathrm{T}}WH\right)}^{-1}} \tag{19}\]  
Following the optimization metric definitions from [150], the scintillation fade depth \(FD_{\mathrm{s}}(p)\) at level \(\rho\) is calculated in dB, as summarized in Equation (20), where \(\sigma_{\mathrm{s}}\) is the standard deviation of the signal for a desired period, \(t(\rho)\) is the time percentage factor, and \(\rho\) is the percentage availability within the range \(0.01\%<\rho<50\%\).