# An Efficient Ground Moving Target Imaging Method for Airborne Circular Stripmap SAR
## 2 Proposed Range Model for CSSAR
### Proposed Range Model  
In order to accomplish an accurate imaging of the target and to allow for an accurate analytical 2-D spectrum to benefit the design of imaging method, the following range equation model is proposed  
\[R_{p}(t_{a})=\sqrt{R_{c}^{2}+v_{e}^{2}t_{a}^{2}+\alpha t_{a}}+\beta t_{a} \tag{10}\]  
As can be seen from (10), there are three variables (i.e., \(v_{e}\), \(\alpha\), and \(\beta\)). In order to obtain the expression for these parameters, a third-order Taylor series expansion is applied to (10) and then we obtain  
\[R_{p}(t_{a})\approx R_{c}+\bigg{(}\beta+\frac{\alpha}{2R_{c}}\bigg{)}(t_{a}- t_{ac})+\bigg{(}\frac{v_{e}^{2}}{2R_{c}}-\frac{\alpha^{2}}{8R_{c}^{3}}\bigg{)}(t_{a}- t_{ac})^{2}+\bigg{(}-\frac{\alpha v_{e}^{2}}{4R_{c}^{3}}+\frac{\alpha^{3}}{16R_{c}^{ 5}}\bigg{)}(t_{a}-t_{ac})^{3} \tag{11}\]Letting the coefficients of the first, second, and third terms of (11) be equal to those of (2), respectively, one can obtain the expressions for \(v_{e}\), \(\alpha\), and \(\beta\) as follows  
\[v_{e}=\sqrt{\left(\frac{R_{c}l_{3}}{l_{2}}\right)^{2}+2R_{c}l_{2}} \tag{12}\]  
\[\alpha=-\frac{2R_{c}^{2}l_{3}}{l_{2}} \tag{13}\]  
\[\beta=l_{1}+\frac{R_{c}l_{3}}{l_{2}} \tag{14}\]  
From (10) it can be seen that the proposed range model consists of a hyperbola and a linear term. The proposed range model has two advantages: (1) it allows for an accurate analytical 2-D spectrum which will benefit the design of imaging algorithm (the target's 2-D spectrum will be presented in the next section); and (2) its accuracy is significantly higher than those of the commonly used second-order and third-order Taylor-approximated range equations. In the following, the accuracy superiority of the proposed range model over the second-order and third-order Taylor approximated range models are demonstrated via numerical experiments. The phase error is utilized to indicate the accuracies of the proposed range equation model as well as the second-order and third-order Taylor-approximated range equations, and it is calculated as follows  
\[\Omega=\max_{t_{a}\in[-T_{a}/2,T_{a}/2]}\left\{4\pi\cdot\frac{|R(t_{a})-R^{ \prime}(t_{a})|}{\lambda}\right\} \tag{15}\]  
where \(R(t_{a})\) is the target's actual range equation, \(R^{\prime}(t_{a})\) indicates the approximated range models (i.e., the proposed range model, the second-order and third-order Taylor approximated range models), \(\lambda\) is the wavelength. \(T_{a}\) is the synthetic aperture time, and it can be approximately expressed as [16]  
\[ \begin{array}{ll}T_{a}&\approx\frac{\theta_{bw}R_{c}}{v_{g}}\\ &\approx 0.886\cdot\frac{\lambda R_{c}}{2v_{g}\rho_{a}}\end{array}  \tag{16}\]  
where \(\theta_{bw}\) is the 3 dB beamwidth of the radar, \(\rho_{a}\approx 0.886\lambda v_{g}/\left(2\theta_{bw}v_{a}\right)\) is the azimuth resolution, \(v_{g}=\omega\cdot r_{c}\) is the velocity of the beam footprint along the ground, and \(v_{a}=\omega\cdot r_{a}\) is the velocity of the radar platform. In Figures 2 and 3, the phase errors under different velocities and accelerations are illustrated, respectively. The color indicates the value of the phase error with the unit being radian. Comparing Figure 2a with Figure 2b and Figure 3a with Figure 3b, it can be seen that the phase errors of the proposed range model are much smaller than that of the second-order [PERSON] approximated range equation. Moreover, by comparing Figure 2a with Figure 2c and Figure 3a with Figure 3c, one can see that the phase errors of the proposed range equation model are even smaller than that of the third-order [PERSON] approximated range equation. These results validate the accuracy superiority of the proposed range equation model over the second-order and third-order [PERSON] approximated range equations. The parameters of the CSSAR for the simulations are presented in Table 1.  
Figure 2: Phase errors under different velocities. (**a**) Proposed range equation, (**b**) second-order [PERSON] approximated range equation, and (**c**) third-order [PERSON] approximated range equation.