## 3 Method
### Algorithm Steps  
#### 3.2.1 Orbit Filtering  
This section mainly involves orbit segmentation, region expansion, and calculation of the intersection time interval between the expanded rectangle and the sub-point, to achieve  
Figure 6: Flowchart of VTW calculation process.  
orbit filtering and exclusion of large-scale invalid times, and to obtain the approximate time interval of the VTW. The main steps include orbit segmentation, region expansion, and position calculation.  
Step 1: Orbit Segmentation. First, the orbits within the global time interval \(\Delta t_{1}\) are segmented into circles. The circles are numbered based on two consecutive passes of the sub-point over the equator, and they are sequentially ordered by time. As shown in Figure 7, Circle 1 starts from the beginning of the interval (\(t_{1\text{s}}\)) until the sub-point passes the equator for the first time, indicated in blue. Circle 17 extends from the last time the sub-point passes the equator to the end of the interval (\(t_{1\text{e}}\)), indicated in red. The intermediate circles are defined by consecutive two passes of the sub-point over the equator [\(t_{si}\), \(t_{\alpha}\)], and different circles are marked with different colors [25].  
Step 2: Region Expansion. Based on the location information of the ground target and the orbit calculation model, the four limit points of the target are obtained, i.e., \(V(lon_{v}\), \(lat_{v})\), \(W(lon_{w}\), \(lat_{w})\), \(Y(lon_{y}\), \(lat_{y})\), and \(U(lon_{u}\), \(lat_{u})\). Next, the four latitude and longitude values are determined, and a rectangular region C is expanded, as shown in Figure 8. The coordinates of its four vertices are \(E(lon_{1}\), \(lat_{1})\), \(F(lon_{2}\), \(lat_{1})\), \(G(lon_{2}\), \(lat_{2})\), and \(H(lon_{1}\), \(lat_{1})\). The coordinate calculation formula is given by Formula (5).  
\[\begin{array}{l}lon_{1}=lon_{u}-\varepsilon\\ lon_{2}=lon_{w}+\varepsilon\\ lat_{1}=lat_{p}-\varepsilon\\ lat_{2}=lat_{y}+\varepsilon\\ \end{array} \tag{5}\]  
Step 3: Position Calculation. The relationship between the sub-point S (\(lon_{s}\), \(lat_{s}\)) and the expanded rectangle C is determined. This involves iterating through the longitude and  
Figure 8: Relationship discrimination diagram between sub-point and extended region.  
Figure 7: Schematic diagram of orbit segmentation.  
latitude of the sub-point at each time step to check if they fall within the coordinate range of the extended region. For instance, if the longitude \(lon_{s1}\) of the sub-point S\({}_{1}\) is within [\(lon_{1}\), \(lon_{2}\)], but its latitude \(lat_{s1}\) is not within [\(lat_{1}\), \(lat_{2}\)], then it is not inside the extended region. Conversely, for point S\({}_{2}\), if its longitude \(lon_{s2}\) is within [\(lon_{1}\), \(lon_{2}\)] and latitude \(lat_{s2}\) is within [\(lat_{1}\), \(lat_{2}\)], then it is inside the extended region. Formula (6) represents the position judgment formula [37]. By iterating through each time step, a collection of sub-points located within the extended region can be obtained, and their corresponding time intervals are recorded to derive the \(\Delta t_{2}\) during which they are inside. This process effectively eliminates sub-points outside the extended region range, thereby achieving the orbit filtering process.  
\[\begin{array}{c}lon_{1}\leq lon_{s}\leq lon_{2}\\ lat_{1}\leq lat_{s}\leq lat_{2}\end{array} \tag{6}\]  
Through the above two steps, it is possible to filter out the large time interval when the two regions are separated, reducing the global computation time \(\Delta t_{1}\) to a relatively small interval \(\Delta t_{2}\).