## 2 Signal Model  
In the two-dimensional slant plane, the observation geometry between the side-looking airborne SAR platform with velocity \(v\) (\(x\)-axis direction) and a ground moving target is shown in Figure 1. At the azimuth time \(t_{a}=0\), there is a target moving with uniform acceleration at the same azimuth position of the platform, and the coordinate position of the moving target is \(P_{0}=\left(0,R_{0}\right)^{T}\), where \(T\) stands for vector transposition. This paper assumes that the target is accelerated uniformly. Its initial velocity and acceleration vectors along-track and cross-track are \(V_{t}=\left[v_{x},v_{r}\right]^{T}\) and \(a_{t}=\left[a_{x},a_{r}\right]^{T}\), respectively. After a period of time, the target moves to \(P_{t}\), and the instantaneous coordinates of the moving target can be expressed as  
\[P_{t}=P_{0}+V_{t}t_{a}+\frac{1}{2}a_{t}t_{a}^{2}=\left[v_{x}t_{a}+\frac{1}{2}a _{t}t_{a}^{2},R_{0}+v_{r}t_{a}+\frac{1}{2}a_{r}t_{a}^{2}\right]^{T} \tag{1}\]  
According to the geometric relationship between the radar and the moving target, the instantaneous range from the antenna phase center to the moving target can be written as [2]  
\[R(t_{a})=\sqrt{\left(vt_{a}-v_{x0}t_{a}-\frac{1}{2}a_{x}t_{a}^{2}\right)^{2}+ \left(R_{0}-v_{r}t_{a}-\frac{1}{2}a_{r}t_{a}^{2}\right)^{2}} \tag{2}\]  
For high-resolution SAR systems, due to the existence of target acceleration, the instantaneous range \(R(t_{a})\) can be expanded to the third-order term based on the Taylor series expansion. Ignoring the fourth and above higher orders, we can obtain [8; 14; 18; 19]  
\[R(t_{a})\approx R_{0}+a_{1}t_{a}+a_{2}t_{a}^{2}+a_{3}t_{a}^{3} \tag{3}\]  
where \(a_{1}=-v_{r}\), \(a_{2}=\frac{\left(v-v_{r}\right)^{2}}{2R_{0}}-\frac{a_{r}}{2}\), \(a_{3}=\frac{v_{r}\left(v-v_{r}\right)^{2}}{2R_{0}^{2}}+\frac{a_{r}\left(v_{r }-v\right)}{2R_{0}}\) are the unknown parameters that need to be estimated. The system transmits a linear frequency modulation signal, and, after pulse compression in range direction, the received baseband signal can be expressed as [20]  
\[S_{1}(t_{r},t_{a})=Aw_{a}(t_{a})sinc\left[B\left(t_{r}-\frac{2R(t_{a})}{c} \right)\right]\exp\left\{-j\frac{4\pi f_{c}R(t_{a})}{c}\right\} \tag{4}\]  
\(A\) is the product of the complex reflectivity and the range compression gain of the target, \(w_{a}(t_{a})=rect\left(\frac{t_{a}}{t_{a}}\right)\) represents the azimuth window function, \(sinc(\cdot)\) represents the  
Figure 1: The observation geometry between airborne SAR platform and a moving target. point spread function, and \(B\), \(c\) and \(f_{c}\) represent the bandwidth of the transmitted signal, the speed of light and the signal carrier frequency, respectively. By calculating the first and second derivatives of the phase of Equation (4), the Doppler shift and the instantaneous azimuth modulation frequency of the moving target can be written down, respectively [10]:  
\[f_{dc}=\frac{2v_{r}}{\lambda}\,\text{Ka}=\frac{4(a_{2}+3a_{3}t_{a})}{\lambda} \tag{5}\]  
where \(\lambda=c/f_{c}\) represents the wavelength of the signal. The received signal in the range-frequency domain can be expressed as [9]  
\[ \begin{split} S_{1}(f_{r},t_{a})&=Aw_{a}(t_{a})rect \left(\frac{f_{r}}{B}\right)\exp\left\{-j\frac{4\pi(f_{r}+f_{c})}{c}R(t_{a}) \right\}\\ &=Aw_{a}(t_{a})rect\left(\frac{f_{r}}{B}\right)\exp\left\{-j\frac {4\pi(f_{r}+f_{c})}{c}\left(R_{0}+a_{1}t_{a}+a_{2}t_{a}^{2}+a_{3}t_{a}^{3} \right)\right\}\end{split}  \tag{6}\]  
It can be seen from Equation (5) that the Doppler frequency rate of the moving target is not only affected by the target motion, but is also time-varying. Therefore, it is difficult to obtain a good focusing effect only using second-order parameter estimation. For a moving target, \(a_{1}\) causes the range walk and Doppler center shift, parameters \(a_{2}\) and \(a_{3}\) produce range curvature and Doppler spectral broadening. In a synthetic aperture time \(T_{s}\), the range migration spans multiple range cells.