## 2 Concepts and Methods
### SAR Received Backscattered Signal Modeling
If \(|z-x|\gg\hat{L}q\) and A are assumed to be slowly varying, hence:  
\[\mathrm{U}_{y}(\tau_{r},z)\,\sim\frac{\mathrm{P}(\tau_{r}-|z-x|/c_{0})}{4\pi|z -x|}\mathrm{e}^{\mathrm{i}k\hat{L}q} \tag{10}\]  
Therefore, the far-field from the antenna is given by Equation (11):  
\[ \begin{split}&\mathrm{U}_{x}^{\mathrm{in}}(\tau_{r},z)\\ &=\int_{-L/2}^{L/2}\int_{-D/2}^{D/2}\mathrm{U}_{x+s_{1}\hat{e}_{1 }+s_{2}\hat{e}_{2}}(\tau_{r},z)\mathrm{d}s_{1}\mathrm{d}s_{2}\,\sim\int_{-L/2 }^{L/2}\int_{-D/2}^{D/2}\frac{\mathrm{P}(\tau_{r}-|z-x|/c_{0})}{4\pi|z-x|} \mathrm{e}^{\mathrm{i}k\hat{L}\cdot(s_{1}\hat{e}_{1}+s_{2}\hat{e}_{2})} \mathrm{d}s_{1}\mathrm{d}s_{2}\\ &=\frac{\mathrm{P}(\tau_{r}-|z-x|/c_{0})}{4\pi|z-x|}\omega(\hat{L} )\end{split}  \tag{11}\]  
where \(\omega\big{(}\hat{L}\big{)}\) is the antenna radiation pattern and is directed perpendicular to the antenna plane (\(e_{1}e_{2}\)) as Equation (12):  
\[\omega\big{(}\hat{L}\big{)}=2D\,\sin c\big{(}k\hat{L}.e_{1}D/2\big{)}\,2L\,\sin c \big{(}k\hat{L}.e_{2}L/2\big{)} \tag{12}\]  
The sinc function has its main peak at 0 and its zero at \(\pi\). Therefore, the first zero of \(\sin c\big{(}k\hat{L}.e_{1}D/2\big{)}\) occurs when \(k\hat{L}.e_{1}D/2=\pi\). Using the fact that \(2\pi/k\) is precisely the wavelength \(\lambda\), it can be written as \(\hat{L}.e_{2}=\lambda/L\). In this case, the main lobe of the antenna beam pattern has an angular width of \(2\lambda/D\)[1; 3]. On the other hand, with the help of scattering theory, the linearized scattering model of Equation (1) can be considered as below [1]:\[\Psi(\tau_{r},x)=\Psi^{in}(\tau_{r},x)+\Psi^{sc}(\tau_{r},x) \tag{13}\]  
where:  
\[\Psi^{sc}(\tau_{r},x)=\iint G_{0}(\tau_{r}-t,x-z)\Bigg{(}\frac{1}{c_{0}^{2}}- \frac{1}{c^{2}(z)}\Bigg{)}\partial_{t}^{2}\Psi^{sc}(t,z)dtd^{3}z \tag{14}\]  
According to single scattering approximation, \(\Psi\) on the right side of Equation (13) is replaced by incident field \(\Psi^{in}\) as Equation (15):  
\[\Psi^{sc}(\tau_{r},x)\approx\Psi^{B} (\tau_{r},x)\] \[=\iint G_{0}(\tau_{r}-t,x-z)\Bigg{(}\frac{1}{c_{0}^{2}}-\frac{1}{ c^{2}(z)}\Bigg{)}\partial_{t}^{2}\Psi^{in}(\tau_{r},x)dtd^{3}z \tag{15}\] \[=\int\ \frac{V(z)}{4\pi|x-z|}\partial_{\tau_{r}}^{2}\Psi^{in}( \tau_{r}-|x-z|/c_{0},z)d^{3}z\]  
In the SAR remote sensing application, the antenna radiates a series of pulses in the form of Equation (11) as it moves along the flight track. Hence, if the platform is assumed to be located at position \(x^{n}\) at time nT, then the incident field is:  
\[\Psi^{in}(t,z)=\sum_{n}\Psi^{in}_{n}(t,z) \tag{16}\]  
\[\Psi^{in}(t,z)=U^{in}_{x^{n}}(t-nT,z)\] \[=\iint\limits_{antenna}U_{x^{n}+q}(t-nT,z)d^{2}q\sim\frac{P(t-nT -|z-x^{n}|/c_{0})}{4\pi|z-x^{n}|}\omega\Bigg{(}\widehat{z-x^{n}}\Bigg{)} \tag{17}\]  
where \(\Psi^{in}(t,z)\) is the nth emission. With the help of Equation (17) and Equation (15), the scattered field due to the nth emission at the center of the SAR antenna will be derived as Equation (18):  
\[S_{n}(\tau_{r})=\Psi^{B}_{n}(\tau_{r}-nT,x^{n})\approx-\int\frac{\omega_{0}^{ 2}P(\tau_{r}-nT-2|z-x^{n}|/c_{0})}{4\pi|z-x^{n}|}\omega\Bigg{(}\widehat{z-x^{ n}}\Bigg{)}d^{3}z \tag{18}\]  
where \(2|z-x^{n}|/c_{0}\) is the two-way travel time from the center of the antenna to the point z and factor \(4\pi|z-x^{n}|\) in the denominator corresponding to the geometrical spreading of the spherical wave from the antenna and from point z. It should be noted, in practice the factor of \(|z-x^{n}|\) will be replaced by the distance from z to the platform as R\({}_{0}\), or the slantrange distance. For SAR image reconstruction, a match filter is applied to the received backscattered signal to from the 2-D raw data \(S_{n}(\tau_{r})\) as Equation (19):  
\[\Psi(\tau_{r},x)=\Psi^{in}(\tau_{r},x)+\Psi^{sc}(\tau_{r},x) \tag{13}\]0  
where the * denotes the complex conjugate.