# An Efficient Ground Moving Target Imaging Method for Airborne Circular Stripmap SAR
## 2 Proposed Range Model for CSSAR  
### Geometry  
The geometric relationship between an airborne CSSAR and a ground moving target is shown in Figure 1. The aircraft travels along a horizonal circle with height \(h\) and radius \(r_{a}\) at a constant angular velocity \(\omega\), with its antenna pointing perpendicular to the aircraft direction fixedly. At the azimuth slow time \(t_{a}=0\), the radar and the target are assumed to be located at (\(r_{a}\), 0, \(h\)) and (\(r_{0}\) cos\(\theta_{0}\), \(r_{0}\) sin\(\theta_{0}\), 0), respectively, where \(r_{0}\) is the distance between the target and the coordinate origin, and \(\theta_{0}\) is the target's azimuth angle. In addition, it is assumed that the target moves with velocities (\(v_{x}\), \(v_{y}\)) and constant accelerations (\(a_{x}\), \(a_{y}\)) along the x-and y-axes.  
According to the geometric relationship shown in Figure 1, the instantaneous distance from the radar to the target (i.e., the target's range equation) can be expressed as  
\[R(t_{a})=\sqrt{\left(r_{0}\cos\theta_{0}+v_{x}t_{a}+\frac{1}{2}a_{x}{t_{a}}^{2 }-r_{a}\cos(\omega t_{a})\right)^{2}+\left(r_{0}\sin\theta_{0}+v_{y}t_{a}+ \frac{1}{2}a_{y}{t_{a}}^{2}-r_{a}\sin(\omega t_{a})\right)^{2}+h^{2}}} \tag{1}\]Applying Taylor series expansion to (1) around the target's beam center crossing time, one can obtain  
\[R(t_{a})=R_{c}+l_{1}(t_{a}-t_{ac})+l_{2}(t_{a}-t_{ac})^{2}++l_{3}(t_{a}-t_{ac})^{3 }+\cdots \tag{2}\]  
With  
\[l_{1}=\frac{v_{tc}(r_{c}-r_{a})}{R_{c}} \tag{3}\]  
\[l_{2}=\frac{v_{ta}^{2}-2r_{a}\omega\cdot v_{ta}+r_{a}r_{c}\omega^{2}}{2R_{c}}+ \frac{a_{tc}(r_{c}-r_{a})}{2R_{c}}+\frac{h^{2}v_{tc}^{2}}{2R_{c}^{3}} \tag{4}\]  
\[l_{3} =\frac{v_{ta}\big{(}a_{tc}{+}r_{a}\omega^{2}\big{)}{+}a_{ta}(v_{tc }{-}r_{a}\omega)}{2R_{c}}\ -\ \frac{v_{ta}(r_{c}{-}r_{a})\big{[}v_{tc}^{2}{-}2v_{tc}r_{a} \omega{+}v_{ta}^{2}{+}r_{a}r_{c}\omega^{2}{+}a_{tc}(r_{c}{-}r_{a})\big{]}}{2R_ {c}^{3}} \tag{5}\] \[+\frac{v_{ta}^{3}(r_{c}{-}r_{a})^{3}}{2R_{c}^{3}}\]  
\[v_{tc} =v_{x}\cos\theta_{c}+v_{y}\sin\theta_{c} \tag{6}\] \[v_{ta} =v_{y}\cos\theta_{c}-v_{x}\sin\theta_{c}\] (7) \[a_{tc} =a_{x}\cos\theta_{c}+a_{y}\sin\theta_{c}\] (8) \[a_{ta} =a_{y}\cos\theta_{c}-a_{x}\sin\theta_{c} \tag{9}\]  
where \(t_{ac}\) is the beam center crossing time, \(R_{c}\) is the range between the radar and the target at \(t_{a}=t_{ac}\), \(r_{c}\) is the range between the origin of coordinate and the target at \(t_{a}=t_{ac}\), \(v_{tc}\) and \(a_{tc}\) are the projections of the target's velocity and acceleration onto the vertical direction of the radar platform velocity at \(t_{a}=t_{ac}\), respectively, and \(v_{ta}\) and \(a_{ta}\) are the projections of the target's velocity and acceleration onto the direction of the radar platform velocity at \(t_{a}=t_{ac}\), respectively, and \(\theta_{c}\) is the target's azimuth angle at \(t_{a}=t_{ac}\).  
The accuracy of the Taylor-approximated range equation increases with the order of Taylor series expansion. However, when this order is more than two, the derivation of an accurate analytical 2-D spectrum and the design of imaging method will be challenging. Thus, a second-order Taylor expansion is usually applied to the target's range equation. Nevertheless, when the azimuth resolution goes high or the target moves with accelerations, the accuracy of the second-order Taylor-approximated range equation would be insufficient. Herein, a novel range equation model is proposed. The accuracy of the proposed range model is even higher than that of the third-order Taylor approximated range equation. Moreover, based on the proposed range model, an accurate analytical expression for the target's 2-D spectrum can be easily derived.