## 2 Problem Description
### Variables  
An observation demand is regarded as an observation mission, and the set of missions is defined as \(T=\{t_{1},\cdots,t_{N}\}\), where \(N\) is the sum of the number of static and dynamic missions. Each mission \(t_{i}=\{pos_{i},p_{i},d_{i},arrT_{i},ETW_{i}\}\in T\) is composed of five tuples:  
* \(pos_{i}\): the location information of the mission including the latitude, longitude, and altitude. * \(p_{i}\): the profit level, as defined by the Earth observation system, serves to reflect the importance of the mission. The profit level of the dynamic mission is usually higher than that of the static mission. * \(d_{i}\): the required observation duration, which determines the size of the observed data. * \(arrT_{i}\): the arrival time of the mission. The arrival time of the dynamic mission is randomly distributed over the scheduling horizon, while the arrival time of the static mission is considered the start time of the scheduling horizon. * \(ETW_{i}=\{eq_{i},lot_{i}\}\): the effective observation time window (ETW). \(est_{i}\) and \(let_{i}\) are the earliest effective observation time and the latest effective observation time of the mission. The observation for each mission must be completed within the ETW. Compared to the static missions, the ETW length of the dynamic missions is usually shorter because the dynamic missions have an urgent observation deadline. The set of EOSs is denoted as \(S=\{s_{1},\cdots,s_{M}\}\), where \(s_{i}\) is the i-th EOS and \(M\) is the number of EOSs. The predefined scheduling horizon is divided into several orbits based on satellite orbit parameters, and the set of orbits for EOS \(s_{i}\) is represented as \(O_{i}=\{o_{i1},\cdots,o_{in_{i}}\}\), where \(o_{ij}\) is the j-th orbit for EOS \(s_{i}\) and \(m_{i}\) is the number of orbits for EOS \(s_{i}\). Once the mission set \(T\) and EOS set \(S\) are determined, all visible time windows (VTWs) between missions and EOSs can be calculated. We denote the set of VTWs as \(W=\{W_{i_{P}},\cdots,W_{N}\}\), and \(W_{i}=\{w_{1i},\cdots,w_{in_{i}}\}\) is the set of VTWs of the mission \(t_{i}\), where  
Figure 1: An example of dynamic observation mission scheduling. \(w_{ij}\) is the j-th VTW of mission \(t_{i}\) and \(n_{i}\) is the number of VTWs of mission \(t_{i}\). Each VTW can be defined as \(w_{ij}=\{s_{ij},e_{ij},SID_{ij},OID_{ij},\gamma_{ij},\pi_{ij},\pi_{ij},\psi_{ji}\}\):  
* \(st_{ij}\) and \(et_{ij}\); the start time and end time of VTW \(w_{ij}\). * \(SID_{ij}\) and \(OID_{ij}\): the identifiers of the EOS and orbit for VTW \(w_{ij}\)
* \(\gamma_{ij}\), \(\pi_{ij}\), and \(\psi_{ji}\): the look angles for the roll, pitch, and yaw in VTW \(w_{ij}\)  
The transition time between two adjacent observation missions is necessary for EOSs to adjust the observation angle, and the transition time between mission \(t_{i}\) and \(t_{f}\) is calculated by:  
\[ \begin{split}\text{z}_{ii^{\prime}}=\begin{cases}a_{1}+\theta_{ ii^{\prime}}/v_{1}&\theta_{ii^{\prime}}\leq 15\\ a_{2}+\theta_{ii^{\prime}}/v_{2}&15<\theta_{ii^{\prime}}\leq 40\\ a_{3}+\theta_{ii^{\prime}}/v_{3}&40<\theta_{ii^{\prime}}\leq 90\\ a_{4}+\theta_{ii^{\prime}}/v_{4}&\theta_{ii^{\prime}}>90\end{cases}\end{split}  \tag{1}\]  
where \(\theta_{ii^{\prime}}\) is the total of the transition angle between the observation for two adjacent missions \(t_{i}\) and \(t_{f^{\prime}}\), \(v_{1},v_{2},v_{3},v_{4}\) are the angular velocities of attitude rotation, and \(a_{1},a_{2},a_{3},a_{4}\) are constant terms corresponding to different angular velocities indicating the rotation capacity of the EOS's camera. The perturbation \(\delta\) is defined to measure the distance between a new observation plan and the initial observation plan. Similar to [38], the distances are summarized into the following three types of variance:  
* 1. Variance of the observation time within the same VTW. * 2. Variance of the observation time among different VTWs. * 3. Rejection.