# A High-Resolution SAR Focusing Experiment Based on GF-3 Staring Data
## 2 Focusing Experiment Based on GF-3 Staring Data
### Curved Orbit and Cubic Phase Error
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{r}\) **(m)** & \(\mathbf{v}_{e}\) **(m/s)** & \(\mathbf{\cos\varphi}_{e}\) & \(f_{D}\) **(Hz)** & \(f_{R}\) **(Hz/s)** \\ \hline
934,633.16 & 7125.15 & \(-0.000083\) & \(-21.21\) & \(-1956.82\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Some parameters of acquisition geometry. Figure 2: Schemes of hyperbolic range equivalent model (HREM): **(a)** The geometric figure of the HREM; **(b)** The model error between the actual range history and the equivalent range history in the ST experiment. \[c_{3}=r_{st}\bigg{(}\frac{1}{2}x_{3}+\frac{1}{16}x_{1}^{3}-\frac{1}{4}x_{1}x_{2} \bigg{)} \tag{8}\]  
\[c_{4}=r_{st}\bigg{(}-\frac{1}{4}x_{1}x_{3}-\frac{1}{8}x_{2}^{2}+\frac{1}{2}x_{4} -\frac{5}{128}x_{1}^{4}+\frac{3}{16}x_{1}^{2}x_{2}\bigg{)} \tag{9}\]  
and \(x_{1},x_{2},x_{3},x_{4}\) can be found in [18] (Equations (10)-(13)). \(f_{D}\) and \(f_{R}\) of the HREM can be expressed as  
\[\left\{ \begin{array}{l}f_{D}=2c_{1}/\lambda\\ f_{R}=4c_{2}/\lambda\end{array} \right. \tag{10}\]  
Then, the deviation can be expressed as  
\[\Delta R_{st}(t)=R_{e}(t)-R_{st}(t) \tag{11}\]  
The actual SAR echo signal can be expressed as  
\[s_{0}(\tau,\;t)=A_{0}\cdot w_{r}\bigg{(}\tau-\frac{2R_{st}(t)}{c}\bigg{)}\cdot w _{a}(t)\cdot\exp\bigg{(}-\frac{4\pi R_{st}(t)}{\lambda}\bigg{)}\cdot\exp\Bigg{\{} jK_{r}\bigg{[}\tau-\frac{2R_{st}(t)}{c}\bigg{]}^{2}\Bigg{\}} \tag{12}\]  
where \(\tau\) is the range time, \(A_{0}\) is a complex constant, \(c\) is the speed of light, \(K_{r}\) is the chirp rate, \(w_{r}(\cdot)\) is the window in range, and \(w_{a}(\cdot)\) is the window in azimuth. Ignoring \(A_{0}\) and the windows, the above equation can be expressed as  
\[s_{0}(\tau,\;t)\approx\exp\bigg{(}-j\frac{4\pi R_{e}(t)}{\lambda}\bigg{)}\cdot \exp\Bigg{\{}jK_{r}\bigg{[}\tau-\frac{2R_{e}(t)}{c}\bigg{]}^{2}\Bigg{\}}\times \exp\bigg{\{}j4\pi\Delta R_{st}(t)\bigg{[}\frac{K_{r}\tau}{c}+\frac{1}{\lambda }\bigg{]}\bigg{\}} \tag{13}\]  
By application of the principle of stationary phase (POSP) [27], the range Fourier transform of the above equation can be written as  
\[S_{0}(f_{\tau},\;t)=\mathcal{F}_{\tau}\Bigg{\{}\exp\bigg{(}-j\frac{4\pi R_{e} (t)}{\lambda}\bigg{)}\cdot\exp\Bigg{\{}jK_{r}\bigg{[}\tau-\frac{2R_{e}(t)}{c} \bigg{]}^{2}\Bigg{\}}\Bigg{\}}\exp\bigg{\{}j4\pi\Delta R_{\text{st}}(t)\cdot \bigg{(}\frac{f}{c}+\frac{1}{\lambda}\bigg{)}\bigg{\}} \tag{14}\]  
where \(\mathcal{F}_{\tau}\{\cdot\}\) represents the range Fourier transform and \(f_{\tau}\) is the range frequency. Then, the actual range history becomes an ideal hyperbola as described by Equation (1) after compensating  
\[\Phi_{1\text{st}}=\exp\bigg{\{}j4\pi\cdot\Delta R_{\text{st}}(t)\cdot\bigg{(} \frac{f}{c}+\frac{1}{\lambda}\bigg{)}\bigg{\}} \tag{15}\]  
in azimuth-time domain. This operation is just like the first-order motion compensation mentioned in [28], which can correct both phase and position. Even though Equation (15) is just valid for the reference target located in the center of imaged scene, the beam width is very small (about 0.287\({}^{\circ}\) in ST mode) and the correction is used on the whole imaged scene in this experiment. After compensating the above derivation, a pure hyperbolic range history is forced, so that the classical imaging algorithms can be used without modification. But in the case of high-resolution, higher-order items, Equation (4) will play an important role in focusing. In this experiment, only third-order items were considered, and \(c_{3}\) in Equation (4) can be obtained by fitting \(R_{st}(t)\) and \(t\). In general, \(c_{3}\ll c_{2}\), so the quadratic stationary point calculated by the POSP can be approximated as cubic stationary point.