# Scheduling Sparse LEO Satellite Transmissions for Remote Water Level Monitoring
## 2 Materials and Methods
### Simulation Model for Online Learning Evaluation  
Setting up a number of sensors in the representative environment is expensive in time and money. Instead, the algorithm is tested with a simulated environment, similar to the methodology chosen in previous work on indirect-to-satellite scheduling [11]. Using simulations first can demonstrate the ability of the algorithm to learn underlying unknown patterns about satellite pass qualityand tune the discount factor \(\lambda\) parameter. To simulate the algorithm, two key components are needed: (1) virtual transmitters with an underlying probability model for which pass qualities are likely to result in transmission, and (2) randomly generated satellite pass characteristics and RF noise data. For virtual transmitters, three conceptual preference models were created, Table 5, to see how different transmitting obstacles would affect the algorithm. Note that the preferences in Table 5 refer to the conditions required for a high likelihood of success. For example, the first preference model requires high angles, long durations, and low noise for a high likelihood of success. To create the virtual transmitter models, a function is constructed that outputs a transmission success probability by multiplying three stretched-and-shifted sigmoid (theshold activation) curves, one for each of three preference variables from Table 5. For example, the sigmoid to represent a preference for high angles would produce a value close to 1 for high angles (e.g., 70 degrees and higher) but a value close to 0 for low angles (e.g., 30 degrees and lower). The general form of the preference models is shown in Equation (9). \[P(success)=\sigma(k_{\theta}(\theta-\theta_{0}))\times\sigma(k_{d}(d-d_{0})) \times\sigma(k_{\gamma}(\gamma-\gamma_{0})) \tag{9}\]  
where \(\sigma(x)\) represents the sigmoid function, \(\theta\) represents the max elevation angle, \(d\) represents the pass duration, and \(\gamma\) represents the RF background noise. Note that \(k_{\theta}\), \(k_{d}\), \(k_{\gamma}\), \(\theta_{0}\), \(d_{0}\), and \(\gamma_{0}\) represent configurable stretching and threshold shifting constants to represent the different conceptual preference models. The values of these constants used to create the three preference models by Equation 9 are shown in Table 6.  
Simulated satellite passes are presented to virtual transmitters by agents imbued with a preference model and a value function approximator. The generated pass characteristics are randomly generated: each agent is exposed to random RF background noise, a vector \(\mathbf{a}\) of satellite passes with corresponding random midpoint times \(\mathbf{t}\), and random pass characteristics \(\mathbf{s}\) (except each \(s_{i}\in\mathbf{s}\) also includes the RF noise value). The randomly generated pass characteristics are drawn from a uniform distribution, and the RF background noise values are drawn from two differing distributions:  
1. Uniform across all buckets (\(-107\) to \(-93\) dBm). 2. Uniform within one bucket (\(-107\) to \(-105\) dBm). to express that a given sensor may experience either a full range of RF noise, or (as expected in a remote location) a narrow sub-range of RF noise. \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Preference Model** & \(\mathbf{k_{\theta}}\) & \(\mathbf{\theta_{0}}\) & \(\mathbf{k_{d}}\) & \(\mathbf{d_{0}}\) & \(\mathbf{k_{\gamma}}\) & \(\mathbf{\gamma_{0}}\) \\ \hline
1 & 0.5 & 70 & 0.5 & 35 & \(-1\) & \(-102\) \\
2 & 0.5 & 50 & 0.5 & 20 & \(-1\) & \(-99\) \\
3 & 0.5 & 30 & 0.5 & 10 & \(-1\) & \(-96\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Constants for the three preference models. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Model** & **Elevation Angle** & **Pass Duration** & **RF Background Noise** \\ \hline
1 & High angles & Long time & Low noise \\
2 & Mid to high angles & Mid to long time & Low to mid noise \\
3 & Low to high angles & Short to long time & Low to high noise \\ \hline \hline \end{tabular}
\end{table}