# 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  
Initially, the traditional imaging algorithms were based on airborne SAR acquisition geometry that assumed that the target on the ground was still and the platform flew at a constant velocity. But in spaceborne cases, the platform's flight path as well as the Earth's surface are curved, and the Earth rotates continuously. In order to make the most of existing algorithms, the HREM, which used equivalent velocity and squint angle to describe the range history was proposed. This model (see Figure 2a) assumes that the sensor flies at a constant velocity \(v_{e}\), the sensor locates on point O when the antenna points to the target T on the ground, and the azimuth time \(t=0\). The sensor locates on point P at an arbitrary azimuth time \(t\), A is the nadir point, the distance between O and T is \(r\), the distance between P and T is \(R_{e}(t)\), \(\theta_{L}\) is the looking angle, \(\theta_{sq}\) is the squint angle, and \(\varphi_{e}\) is the complement angle of \(\theta_{sq}\). Then, the range history can be expressed as:  
\[R_{e}(t)=\sqrt{r^{2}+\left(v_{e}t\right)^{2}-2 rv_{e}t\cdot\cos\varphi_{e}} \tag{1}\]  
\[\left\{ \begin{array}{c}v_{e}=\sqrt{\frac{\lambda rf_{R}}{2}+\left(\frac{ \lambda f_{R}}{2}\right)^{2}}\\ \varphi_{e}=\arccos\left(-\frac{\lambda f_{D}}{2v_{e}}\right)\end{array} \right. \tag{2}\]  
where \(\lambda\) is the carrier wavelength; \(f_{D}\) is the Doppler centroid frequency, and \(f_{R}\) is the Doppler frequency modulation rate, and they can be obtained by calculating the first- and second-order derivative of the actual range history between the target and the sensor. In this experiment, some of these parameters are sketched in Table 2.  
In most cases, the integration time is usually short (no more than 2 s), and the trajectory can be described by the HREM accurately. In SL or ST mode, however, targets on the ground that are illuminated by a complete beam have a longer integration time as compared to stripmap mode. As shown in Figure 2b, the integration time of the target in the middle of the imaged scene is about8.58 s. Because of the long integration time and the spaceborne acquisition geometry, the actual range history is not a pure hyperbola and the phased error caused by the range deviation becomes larger as the integration time extends. In addition, the third- and higher-order items of Equation (1) cannot be ignored in the long integration time or high-resolution. In operational cases, with the help of satellite attitude and orbit parameters provided by Global Positioning System (GPS), \(f_{D}\) and \(f_{R}\) can be calculated. Then, the equivalent range history can be obtained according to Equations (1) and (2), and we compensate the model error in two aspects--range deviation and cubic phase error. In order to use the HREM to deal with raw data, the deviation between the actual and the equivalent range history must be compensated. The range history vector \(\mathbf{R}_{st}(t)\) of a target can be obtained with the help of attitude and orbit parameters, and it can be expanded by the Taylor formula:  
\[\mathbf{R}_{st}(t)=\mathbf{R}_{st}+\mathbf{V}_{st}t+\frac{1}{2}\mathbf{A}_{st}t^{2}+\frac{1}{6 }\mathbf{B}_{st}t^{3}+\frac{1}{24}\mathbf{C}_{st}t^{4}+\cdots \tag{3}\]  
where the subscript s means satellite, the subscript t means target; \(\mathbf{R}_{st}\), \(\mathbf{V}_{st}\), \(\mathbf{A}_{st}\), \(\mathbf{B}_{st}\), \(\mathbf{C}_{st}\) denote the distance, velocity, acceleration, rate of acceleration, and rate of rate of acceleration vector between the satellite and the target when \(t=0\). Then, the distance between the satellite and the target can be expressed as  
\[R_{st}(t)=|\mathbf{R}_{st}(t)|=r_{st}\cdot\sqrt{1+\frac{\mathbf{R}_{st}(t)\mathbf{R}_{st}( t)-r_{st}^{2}}{r_{st}^{2}}}=r_{st}+c_{1}t+c_{2}t^{2}+c_{3}t^{3}+c_{4}t^{4}+\cdots \tag{4}\]  
where  
\[r_{st}=|\mathbf{R}_{st}| \tag{5}\]  
\[c_{1}=\frac{1}{2}r_{st}x_{1} \tag{6}\]  
\[c_{2}=r_{st}\bigg{(}\frac{1}{2}x_{2}-\frac{1}{8}x_{1}^{2}\bigg{)} \tag{7}\]