## 2 Methodology
### Emergency Observation Mission Scheduling Model
#### 2.3.1 Constraint Analysis of Satellite Scheduling  
The mission defined in this paper refers to the point target that can be observed by the satellites in a field of view. Considering multi-resource constraints, we established a mathematical model to describe the scheduling problem. In the observation scheduling stage, the main constraints are as follows. The parameter and label definitions in the constraints are listed in Table 2, and function definitions are shown in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notation** & **Definition** & **Notation** & **Definition** \\ \hline \multirow{4}{*}{\(S\)} & \(S=\{s_{1},s_{2},\ldots\ s_{n}\}\), where \(S\) is the set of satellites and & \multirow{4}{*}{\(N_{ij}\)} & \(N_{ij}\) represents the number of visibility time \\  & \(n\) represents the number of satellites. & & \\  & \(T=\{t_{1},t_{2},\ldots\ t_{m}\}\), where \(T\) is the set of missions & & \\  & and \(m\) represents the number of missions. \(\{tsh_{i},tse_{i}\}\) & \multirow{4}{*}{\(p_{i}\)} & \(p_{i}\) is the priority level of mission \(t_{i}\). \\  & represents the validity time window of mission \(t_{i}\). & & \\  & \(GSW_{j}\) is the set of the ground station time & & \\  & windows, representing the time window set of & & \\  & satellite \(s_{j}\) for all ground stations. & & \\  & \(SA_{ij}^{l}\) is the angle of the side swing, representing the & & \\  & side swing angle of the sensor of satellite \(s_{j}\) to the & \(l\)-th visible time window of mission \(t_{i}\). & \\  & \(OS_{ij}^{l}=\left[\text{bqs}_{j}^{l},\text{cos}_{ij}^{l}\right]\) where \(OS_{ij}^{l}\) represents the \(l\)-th & & \\  & time window of the sensor of satellite \(s_{j}\) to mission & & \\  & \(t_{i}\), \(\text{bqs}_{i}^{l}\) is the start time of time window, and \(\text{cos}_{ij}^{l}\) is & & \\  & the end time of time window. & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter and label definitions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Function** & **Definition** \\ \hline \multirow{4}{*}{\(CapLen(t_{i})\)} & The shooting duration, which is related to the mission requirements and is obtained \\  & before scheduling. \\  & The start-up time of satellite \(s_{j}\), which can be calculated from the specific execution time \\  & after scheduling. \\  & The shutdown time of satellite \(s_{j}\), which can be calculated from the specific execution time \\  & after scheduling. \\  & The solar elevation angle, which can be calculated from the geographical position of \\  & mission \(t_{i}\) and the time \(tm\). \\  & The observation angle, which can be calculated from the geographical position of mission \\  & \(t_{i}\), the path of satellite \(s_{j}\) and the time \(tm\). \\  & The storage capacity of satellite \(s_{j}\), which is occupied by the observation mission \(t_{i}\) \\  & arranged between any two data transmission windows \(w_{1}\) and \(w_{2}\). It can be calculated \\  & from the planned observation mission \(ti\) and the speed of reading and writing \(v_{j}\) of \\  & satellite \(s_{j}\). \\  & The number of on-off switches for satellite \(s_{j}\), which can be calculated from the start-up \\  & time and shutdown time of satellite \(s_{j}\). \\  & The working wave of satellite \(s_{j}\) which is related to mission \(t_{i}\). \\  & An indicator represents whether mission \(t_{i}\) is an urgent mission. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Function definitions in the constraints. 1. Visible window constraint. Satellite payloads and observation targets must be visible: \[ \begin{array}{l}\forall tm\in ST_{ij},\exists l:p_{ijl}(tm),\\ bst_{ij}\geq bos^{s}_{ij^{\prime}},est_{ij}\leq eos^{s}_{ij^{\prime}},\\ est_{ij}-bst_{ij}\geq CapLen(t_{i})\end{array} \] (4)
2. Start-up time constraint. The start-up time must be between the minimum start-up time \(MinT_{j}\) and the maximum start-up time \(MaxT_{j}\) of sensor \(s^{\prime}_{j}\) of satellite \(s_{j}\). The time interval between two switches cannot exceed the preset minimum time interval \(MinPeriod_{j}\). \[