# 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
### ISAR Imaging Principle and Model
#### 3.2.2 ISAR Imaging Model for Spaceborne SAR  
In this section, based on the defined coordinate systems and the principle of ISAR imaging, and under the assumption of the two-body orbit model, the ISAR imaging model  
Figure 4: Illustration of the ISAR imaging projection principle.  
for spaceborne SAR is established to derive the instantaneous ISAR imaging plane and the ISAR imaging projection matrix.  
For convenience, we made a convention for the positive rotation direction. When rotating around a certain coordinate axis according to the right-hand rule, that is, with counterclockwise rotation, it is noted as the positive direction; otherwise, it is the negative direction. The coordinate transfer matrix is denoted as \(\mathbf{R}_{m}(\theta)\), where \(m=x,y,z\) is the coordinate axis and \(\theta\) is the rotation angle. The transfer matrix expressions are consistent with [22].  
As shown in Figure 5, S is the location of the radar station with geodetic coordinates \((R_{Lat},R_{Lng},R_{All})\); D is the real-time position of the spaceborne SAR; D\({}_{0}\) is the zenith when the ground-based radar is observing the spaceborne SAR; J is the perigee; \(\theta_{G}(t)\) is the Greenwich hour angle; \(\Omega\) is the right ascension of the ascending node; \(\omega\) is the argument of the perigee; \(f_{0}\) is the true anomaly at zero time, that is, the time when the ground-based radar observes the spaceborne SAR passing the zenith; \(f=f_{0}+n_{st}\) is true anomaly at time \(t\); and \(n_{s}\) is the angular velocity of the spaceborne SAR.  
For convenience, assume that Earth is simplified as a standard sphere. Therefore, the coordinates of the radar station in the Earth-centered, Earth-fixed coordinates are expressed as  
\[S_{ECEF}=\left[\begin{array}{c}(r_{e}+R_{Alt})\cos R_{Lat}\cos R_{Lng}\\ (r_{e}+R_{Alt})\cos R_{Lat}\sin R_{Lng}\\ (r_{e}+R_{Alt})\sin R_{Lat}\end{array}\right]. \tag{14}\]  
The transformation matrix from E_XECEFYECEFZECEF to E_XECYECIZECI is expressed as  
\[\mathbf{M}_{1}(t)=\mathbf{R}_{z}(-\theta_{G}(t)). \tag{15}\]  
The transformation matrix from E_XECIYECIZECI to E_XREFYREFZREF is expressed as  
\[\mathbf{M}_{2}=\mathbf{R}_{z}(u_{0})\mathbf{R}_{x}(i)\mathbf{R}_{z}(\Omega). \tag{16}\]  
The transformation matrix and shift vector from E_XREFYREFZREF to O_XSBYSBZSB are expressed as  
\[\mathbf{M}_{3}(t)=\mathbf{R}_{z}(f-f_{0}) \tag{17}\]  
and  
\[\mathbf{A}(t)=[R(t),0,0]^{T} \tag{18}\]  
where \(R(t)=a\big{(}1-e^{2}\big{)}/(1+e\cos f)\), \(a\) is semi-major axis, and \(e\) is the eccentricity. Since a spaceborne SAR is generally a low-Earth satellite with a nearly circular orbit, \(R(t)\) is approximately equal to \(a\).  
Figure 5: Radar station and orbital elements of spaceborne SAR.  
In (14)-(18), the radar line-of-sight vector is expressed as  
\[\mathbf{p}(t)=\mathbf{A}(t)-\mathbf{M}_{3}(t)\mathbf{M}_{2}\mathbf{M}_{1}(t)\mathbf{ S}_{ECEF}. \tag{19}\]  
In (10), the tangential velocity of radar line-of-sight vector is expressed as  
\[\mathbf{v}_{c}(t)=\frac{\mathbf{p}(t)\times\dot{\mathbf{p}}(t)}{\left|\mathbf{p}(t)\right|^{2}} \times\mathbf{p}(t). \tag{20}\]  
In (13), (19), and (20), the real-time 2D ISAR imaging projection matrix is expressed as  
\[\mathbf{H}(t)=\left[\begin{array}{c}\mathbf{e}_{l}(t)\\ \mathbf{e}_{d}(t)\end{array}\right] \tag{21}\]  
where \(\mathbf{e}_{l}(t)=\mathbf{p}(t)/\left|\mathbf{p}(t)\right|\) and \(\mathbf{e}_{d}(t)=\mathbf{v}_{c}(t)/\left|\mathbf{v}_{c}(t)\right|\). Therefore, the location of the radar station and the orbital elements of the spaceborne SAR determine the 2D ISAR imaging plane, which describes the projection from the satellite body coordinate system to the ISAR imaging coordinate system.