# Estimating the Observation Area of a Stripmap SAR via an ISAR Image Sequence
## 2 Geometry of Stripmap SAR Observation
### Model of the Stripmap SAR Observation Area  
As shown in Figure 2, the spaceborne SAR observes the Earth with a look angle of \(\gamma\). T\({}_{1}\) is the intersection point of the normal line between the antenna and the ground, T\({}_{11}\) and T\({}_{12}\) are the intersection points of the beam edge and the ground in the zero Doppler plane, D is the real-time position of the spaceborne SAR, C is the sub-satellite point, and E is the center of Earth. T\({}_{1}\)T\({}^{\prime}\)\(\perp\)DE is drawn through T\({}_{1}\) and the vertical foot is T\({}^{\prime}\). \(r\) and \(d\) are the radius of the spherical circle with T\({}^{\prime}\) as the center and passing through T\({}_{1}\) and the distance from T\({}^{\prime}\) to the sphere's center, respectively. \(h\) is the orbital height and \(\Theta\) is the central angle \(\angle\)TED. Therefore, \(r\) and \(d\) are expressed as  
\[d=\cos\left[\text{asin}\left(\frac{r_{e}+h}{r_{e}}\sin\gamma\right)-\gamma \right]r_{e} \tag{1}\]  
and  
\[r=\sin\left[\text{asin}\left(\frac{r_{e}+h}{r_{e}}\sin\gamma\right)-\gamma \right]r_{e}. \tag{2}\]  
As shown in Figure 3, D is the real-time position of the spaceborne SAR and N is the ascending node. T\({}^{\prime}\)T\({}^{\prime\prime}\) is drawn perpendicular to the equatorial plane through T\({}^{\prime}\) and the vertical foot is T\({}^{\prime\prime}\). T\({}^{\prime\prime}\)F\(\perp\)EN is drawn through T\({}^{\prime\prime}\) and the vertical foot is F. \(\angle\)T\({}^{\prime\prime}\)FT\({}^{\prime}\) is the orbital inclination angle, which is denoted as \(i\). T\({}^{\prime}\), T\({}_{1}\), and T\({}_{2}\) are defined in Figure 2. Therefore, T\({}_{1}\) and T\({}_{2}\) are the positions observed by the spaceborne SAR operating in boresight stripmap mode with the look angle \(\gamma\). Figure 1: Boresight imaging geometry of spaceborne SAR operating in stripmap mode. Figure 2: Observation geometry of spaceborne SAR operating in stripmap mode. Assume that the spaceborne SAR moves around the Earth from west to east. For convenience, the geodetic coordinates of sub-satellite point C, that is, the longitude and latitude, are denoted as \((\alpha_{0},\beta_{0})\). The geodetic coordinates of the area observed by the spaceborne SAR operating in stripmap mode are denoted as \((\alpha,\beta)\). Since \(\mathrm{T_{1}T_{2}}\subset\mathrm{planeT^{\prime}FT^{\prime\prime}}\), the angles of \(\mathrm{T^{\prime}T_{2}}\) and \(\mathrm{T^{\prime}T_{1}}\) with respect to the positive Z-axis direction are \(i\) and \(\pi-i\), respectively. Therefore, the following equation is obtained and expressed as  
\[\sin\beta=d\sin\beta_{0}\pm r\cos i. \tag{3}\]  
Since the research scenario is located in the northern hemisphere, the latitude range of the observation area is \([0,\pi/2]\). Therefore, the following equation is obtained and expressed as  
\[\cos\beta=\sqrt{1-(d\sin\beta_{0}\pm r\cos i)^{2}}. \tag{4}\]  
Because of \(\mathrm{T_{1}T_{2}\bot T^{\prime}E}\), the following equation is obtained and expressed as  
\[\cos(\beta_{0})\cos(\alpha_{0})\cos(\beta)\cos(\alpha)+\cos(\beta_{0})\sin( \alpha_{0})\cos(\beta)\sin(\alpha)+\sin(\beta_{0})\sin(\beta)=d. \tag{5}\]  
For convenience, when the spaceborne SAR observes the left area of the flight path, that is, the high latitude area, the geodetic coordinates of the observation area are denoted as \((\alpha_{H},\beta_{H})\), and when the spaceborne SAR observes the right area of the flight path, that is, the low latitude area, the geodetic coordinates of the observation area are denoted as \((\alpha_{L},\beta_{L})\). In addition, the relative longitude from the sub-satellite point to the ascending node is denoted as \(\Delta_{Lng}\). Substituting (3) and (4) into (5), when the spaceborne SAR observes the left area of the flight path, the geodetic coordinates of the observation area are expressed as  
\[