# Estimating the Observation Area of a Stripmap SAR via an ISAR Image Sequence
## 3 ISAR Imaging and Method for Estimating the Stripmap SAR Observation Area
### Method for Estimating the Stripmap SAR Observation Area  
A spaceborne SAR is generally configured with a parabolic antenna or a panel phased array antenna, both of which have very distinctive edges. Cross-polarization ISAR images are very sensitive to the edges and corners of the target [24; 25]. Therefore, based on the above characteristics, the scatterers on the space target components with edge features are easy to associate and extract from the ISAR image sequence. Without a loss of generality, in this article, a spaceborne SAR with a parabolic antenna is used as a typical research object, and its 3D model is shown in Figure 6. In addition, it is assumed that the main body of the spaceborne SAR is three-axis-stabilized and Earth-oriented, which is a relatively common satellite attitude control method [26; 27]. As shown in Figure 7, for a spaceborne SAR operating in boresight stripmap mode, the component coordinate system is rotated around the Y-axis by a Euler angle \(\varphi\) to obtain the satellite body coordinate system. When the spaceborne SAR observes the left and right sides of the flight path, the corresponding Euler rotation angles are  
\[\varphi=-\gamma \tag{22}\]  
Figure 6: Three-dimensional model of the spaceborne SAR. \[\varphi=\gamma \tag{23}\]  
where \(\gamma\) is the look angle of the antenna when the spaceborne SAR operating in boresight stripmap mode observes the Earth. For any scatterer \(P_{q}\) on the edge of the spaceborne SAR antenna, its position vector in the component body coordinate system is denoted as \(\mathbf{P}_{q}\). In (21), (22), and (23), the range and Doppler in the SAR image at time \(t\) are expressed as  
\[\left[ \begin{array}{c}R_{q}(t)\\ D_{q}(t)\end{array} \right]=\left[ \begin{array}{c}\mathbf{e}_{l}(t)\\ \mathbf{e}_{d}(t)\end{array} \right]\mathbf{R}_{y}(\varphi)\mathbf{P}_{q}. \tag{24}\]  
For convenience, the range and Doppler values of the ISAR image of the scatterer \(P_{q}\) at time \(t_{m}\) are denoted as \(R_{qm}\) and \(D_{qm}\), respectively, and the projection vector in the range and Doppler dimension are denoted as \(\mathbf{e}_{lm}\) and \(\mathbf{e}_{dm}\), respectively. Therefore, the observation equation system constructed by the range and Doppler values of the ISAR image sequence of the scatterer \(P_{q}\) at time \(t_{1},t_{2},\cdots,t_{m}\) is expressed as  
\[\mathbf{W}_{q}=\mathbf{K}_{q}\mathbf{P}_{q} \tag{25}\]  
\[\text{where }\mathbf{W}_{q}=\left[ \begin{array}{c}R_{q1}-D_{q1}\\ R_{q2}-D_{q2}\\ \vdots\\ R_{qm}-D_{qm}\end{array} \right],\mathbf{K}_{q}=\left[ \begin{array}{c}\mathbf{e}_{l 1}-\mathbf{e}_{d1}\\ \mathbf{e}_{l2}-\mathbf{e}_{d2}\\ \vdots\\ \mathbf{e}_{lm}-\mathbf{e}_{dm}\end{array} \right]\mathbf{R}_{y}(\varphi)\text{ and }\mathbf{P}_{q}=\left[ \begin{array}{c}P_{ qx},P_{qy},P_{qz}\end{array} \right]^{T}.\]  
When the number of ISAR imaging moments \(m>3\), (25) is an overdetermined equation system, and its least square solution is expressed as  
\[\hat{\mathbf{P}}_{q}=\left(\mathbf{K}_{q}{}^{T}\mathbf{K}_{q}\right)^{-1}\mathbf{K}_{q}{}^{T} \mathbf{W}_{q}. \tag{26}\]  
Assume that the range and Doppler values of \(n\) scatterers on the parabolic antenna edge at \(m\) moments are extracted from the ISAR image sequence. Therefore, the position vectors of these \(n\) scatterers in the component body coordinate system can be obtained from (25), which are expressed as  
\[\hat{\mathbf{P}}_{j}=\left(\mathbf{K}_{j}{}^{T}\mathbf{K}_{j}\right)^{-1}\mathbf{K}_{j}{}^{T} \mathbf{W}_{j},j=1,2,\cdots,n. \tag{27}\]  
Figure 7: Component coordinate system and satellite body coordinate system. For a spaceborne SAR with a parabolic antenna, the scatterers on the parabolic antenna edge are all located in the same plane. The unit normal vector of this plane in the component body coordinate system is expressed as  
\[\mathbf{n}_{0}=(1,0,0)^{\mathrm{T}} \tag{28}\]  
For convenience, the number of combinations of \(M\) elements selected from \(N\) elements is denoted as \(\mathrm{C}_{N}^{M}\).