## 2 Preliminary Work  
### Theoretical Foundation  
The theoretical foundations for VTW calculation for Earth observation payloads involve spacecraft orbital dynamics, space geometry, and related principles. The calculation process entails inputting data such as space station orbital information, payload parameters, and observation target details to output the VTW [28]. Orbital data describe the spatial position of space station over a period, typically obtained by providing Two-Line Element (TLE) data as initial parameters and utilizing precise orbit propagation models such as SGP4 and SDP [29]. This yields accurate Earth-centered inertial coordinates (\(x\), \(y\), \(z\)) and velocities (\(v_{x}\), \(v_{y}\), \(v_{z}\)) at a given time. The SGP4 model is commonly used for near-Earth orbits due to its high predictive accuracy [30]. The visibility range of the payload is influenced by the payload's field of view type, commonly characterized by conical or rectangular fields of view [31]. The observation target region is generally represented as a closed polygon with a broad coordinate span that cannot be covered by a single instantaneous field of view or scanning strip of the payload. A schematic diagram of a VTW is shown in Figure 1, where the circular region M represents the circular field of view projection of the payload, the blue rectangle N represents the ground regional target, and the dashed line represents the sub-point track. Based on the given information, the geometric relationship between the current moment's projection region M and the regional target N is determined. Starting from the initial time, it is assumed that the regions M and N are disjointed. When at time \(a\), they intersect, marking the start of timing, and this continues until time \(b\) when they separate, marking the end of timing. The duration between these two critical times \(a\) and \(b\) represents the VTW of the payload for target N [32]. The specific approach for calculating the VTW involves using the payload's orbital position information and the latitude and longitude information of the ground region. It calculates the angle (referred to as \(\beta\) in Figure 2) between the line connecting the sub-point and the payload and the line connecting the target and the payload, along with the payload's half field of view angle (referred to as \(\alpha\) in Figure 2). This is illustrated in Figure 2, adopting the spherical Earth model with O as the Earth's center, S as the space station platform (payload), \(L\) as the target position, and \(T\) as the intersection point between the payload-to-Earth's-center line and the ground. The three orange lines, LO, SO, and SL, form a triangle. Here, ST represents the height of the space station orbit \(h\), TO represents the  
Figure 1: Schematic diagram of VTW. radius of the Earth \(r\), and LO represents the distance from the ground target to the Earth's center. \(Q\) is the intersection point between the payload's field of view and the ground. Assuming the accuracy of the space station's orbital position obtained based on the TLE and SGP4 models, firstly, based on the space station's position, the position of the sub-point \(T\) is calculated. The central angle \(\varepsilon\) between point \(T\) and the ground point \(L\) (\(lon_{L}\), \(lat_{L}\)) is then determined. Subsequently, using this angle and the lengths of the two sides, the length of side SL \(b\) is calculated using the law of cosines. Finally, the angle \(\beta\) between sides SO and SL is determined using the law of arscines. By comparing this angle \(\beta\) with the payload's half field of view angle \(\alpha\), if \(\beta\) is less than or equal to \(\alpha\), then the target is within the visible range. The position of point \(T\) is recorded at this moment. The corresponding time is calculated as the start time of the VTW, denoted as '\(T_{a}\)', and it continues until \(\beta\) starts to exceed \(\alpha\). Similarly, the position of point \(T\) is recorded at this moment and denoted as '\(T_{b}\)'.