## 2 Concepts and Methods  
### SAR Received Backscattered Signal Modeling  
SAR is an active coherent microwave remote sensing device, the purpose of which is to form an image from the local interaction of a terrain illuminated by incidental electromagnetic waves. According to Figure 1, as the SAR platform flies along its designated path, known as the slow-time direction, the antenna radiates pulses of electromagnetic waves in a directed beam perpendicular to the flight direction, known as the fast-time. These waves scatter off the CTT and will be received and detected by the same antenna. The only processing performed on the received backscattered data is the striping of time-frame formatted signals, and re-assembling them into the synchronized unprocessed contiguous radar range lines, which are known as SAR raw data. The received backscattered signal depends on the fast-time (\(\tau_{r}\)) and the position along the slow-time direction, x, which results in a function of two variables based on [PERSON]'s scalar wave equation [1; 2; 3]:  
\[\left(\
abla^{2}-\frac{1}{c_{0}{}^{2}}\partial_{\tau_{r}}{}^{2}\right)\text{ U}(\tau_{r},\text{x})=0 \tag{1}\]  
where, \(c_{0}\) is the speed of light. In the free space, the field \(G_{0}(t-\tau_{r},\text{x}-\text{y})\) at position x and time \(\tau_{r}\) due to the delta function point scatterer at position y and time t will be driven as Equation (2):  
\[G_{0}(\tau_{r}-t,\text{x}-\text{y})=\frac{\delta(\tau_{r}-t-\left|\text{x}- \text{y}\right|/c_{0})}{4\pi\left|\text{x}-\text{y}\right|} \tag{2}\]  
This fields satisfies Equation (3):  
\[\left(\
abla^{2}-\frac{1}{c_{0}{}^{2}}\partial_{\tau_{r}}{}^{2}\right)G_{0}( \tau_{r}-t,\text{x}-\text{y})=-\delta(\tau_{r}-t)\delta(\text{x}-\text{y}) \tag{3}\]  
If the point scatterer at position y has the time history as Equation (4):  
\[\text{P}(\tau_{r})=\text{A}(\tau_{r})\text{e}^{\text{i}\omega_{0}\tau_{r}} \tag{4}\]  
then the resulting field \(\text{U}_{y}(\tau_{r},\text{z}-\text{y})\) will satisfy Equation (5):  
\[\left(\
abla^{2}-\frac{1}{c_{0}{}^{2}}\partial_{\tau_{r}}{}^{2}\right)\text{ U}_{y}(\tau_{r},\text{z}-\text{y})=-\text{P}(\tau_{r})\delta(\text{z}-\text{y}) \tag{5}\]  
Figure 1: Geometry of the illumination from the antenna toward the CTT. and is thus given by Equation (6):  
\[ \begin{split}\mathrm{U}_{y}(\tau_{r},z)=(\mathrm{G}_{0}*\mathrm{P})( \tau_{r},z-y)\\ =&\int\frac{\delta\Big{(}\tau_{r}-t-\big{|}z-y\big{|}/c_ {0}\Big{)}}{4\pi|z-y|}\mathrm{P}(t)\mathrm{dt}=\frac{\mathrm{P}\Big{(}\tau_{r}- \big{|}z-y\big{|}/c_{0}\Big{)}}{4\pi|z-y|}\\ =&\frac{\mathrm{A}\Big{(}\tau_{r}-\big{|}z-y\big{|}/c_ {0}\Big{)}}{4\pi|z-y|}\mathrm{e}^{\mathrm{i}\omega_{0}(\tau_{r}-|z-y|/c_{0})} \end{split}  \tag{6}\]  
If the length and width of the SAR antenna dimensions are denoted by L and D, and the center of the antenna is denoted by x, respectively, a point on the antenna can be written \(y=x+q\) where q is a vector from the center of the antenna to a point on the antenna, which is also introduced by Equation (7):  
\[q=s_{1}\hat{e}_{1}+s_{2}\hat{e}_{2} \tag{7}\]  
where, \(\hat{e}_{1}\) and \(\hat{e}_{2}\) are unit vectors along the width and length of the antenna. The vector \(\hat{e}_{2}\) points along the direction of the slow-time and \(\hat{e}_{1}\) is tilted with respect to the radiation direction or the fast-time. Considering point \(z\) far from the antenna, for which \(\big{|}q\big{|}\ll|z-x|\):  
\[\big{|}z-y\big{|}=|z-x|-\widehat{(z-x)}q+O\big{(}L^{2}/|z-x|\big{)} \tag{8}\]  
where the hat denotes a unit vector and \(\widehat{(z-x)}\) is supposed as (\(\hat{L}\)). With the help of Equation (6):  
\[\mathrm{U}_{y}(\tau_{r},z)\,\sim\frac{\mathrm{A}\Big{(}\tau_{r}-|z-x|/c_{0}+( \hat{L})q+\dots\Big{)}}{4\pi|z-x|}\mathrm{e}^{\mathrm{i}\omega_{0}(\tau_{r}-|z -x|/c_{0})}\mathrm{e}^{\mathrm{i}k\hat{L}q} \tag{9}\]  
where \(k=\omega_{0}/c_{0}\).