## 2 Concepts and Methods
### Multiplicative Speckle Noise Behavioral Formulation and Modeling  
The two basic principle features of an SAR system are the resolution of the image and the spatial resolution or the system resolution. The system resolution directly affects the image resolution, while the image resolution is completely dependent on the quality of the received backscattered data and its pertinent information. A serious difficulty in exploiting and analyzing the raw data is the presence of coherent multiplicative noise that inherently exists in these images, and is called speckle. According to Equations (4) and (16), in the case of electromagnetic waves backscattering from the surface to the SAR, both the phase \(\varphi(x,y)\) and the amplitude \(\mathrm{A(x,y)}\) of the signal are changed. These changes are due to the different physical properties of the CTT, as well as the deviations of the trajectory from the nominal or the attitude and forward velocity variations, which mostly happen during the observation time, as shown in Figure 1. In other words, these echo data are not coherent signal anymore because of atmospheric turbulence and radiometric errors that can easily affect the phase history of the raw data within the plane of the antenna \((\mathrm{e_{1}e_{2}})\)[24; 25]. In practice, phase errors are residue phases that degrade the standard level of the received signal components and make them not coherent anymore for the purpose of processing. Therefore, any time the SAR collects the complex value of \(\mathrm{Ae^{i\varphi}}\) from the distributed point scatterers within the terrain, the 2-D signals add coherently as constructive or destructive, depending on the relative phase; this is usually referred to as multiplicative speckle noise. In other words, speckles reduce the image quality like noise, but not randomly. This comes from the backscattering of many point scatterers within the CTT, and presents as image pixel anomalies. According to Equation (17), every pixel in an SAR image results from the backscattered wave of a large number of point scatterers located in one resolution cell, and this can be modeled as a complex value of Equation (4):  
\[\mathrm{Ae^{i\varphi}}=\sum_{i=1}^{N}\mathrm{A_{i}e^{i\varphi_{i}}} \tag{24}\]  
where \(\mathrm{A_{i}}\) and \(\varphi_{i}\) are the individual amplitude and phase of the backscattering, while N is the total number of scatterers within the resolution cell. The summation is over the number of point scatterers located in one cell illuminated by the electromagnetic wave, while different weights in the form of PSF and Equation (23) will be implemented. \(\mathrm{A_{i}}\) and \(\varphi_{i}\) are unobservable because the individual scatters are much smaller than the resolution cell. Hence, the speckle can be formulated as a multiplicative model of pixel anomaly:  
\[\mathrm{O}=\mathrm{I}\mathrm{Y} \tag{25}\]  
where \(\mathrm{O}\) is the observation result, \(\mathrm{I}\) is the average CTT backscattered intensity, and \(\mathrm{Y}\) is the noise intensity. \(\mathrm{I}\) and \(\mathrm{Y}\) are considered as two independent random variables. In the case of multiplicative noise demodulation modeling, the observed in-phase and quadrature-phase components of the speckle noise \(\mathrm{y_{1}}=\mathrm{Acos}(\varphi)\) and \(\mathrm{y_{2}}=\mathrm{Asin}(\varphi)\) are assumed to be identical distributed Gaussian random complex variables with zero means and a variance of \(\frac{1}{2}\)[3]. Consequently, the joint probability density function (PDF) can be written as:  
\[\mathrm{P}_{Y_{1},Y_{2}}\left(Y_{1},Y_{2}\right)=\frac{1}{\pi}\exp\left(-\left(Y _{1}^{2}+Y_{2}^{2}\right)\right) \tag{26}\]  
where the phase \(\varphi\) is assumed to be uniformly distributed. On the other side, the amplitude A has a Rayleigh distribution, as:  
\[\mathrm{P}_{\mathrm{A}}(\mathrm{y})=2\mathrm{y}\exp\left(-\mathrm{y}^{2} \right)\mathrm{y}\geq 0 \tag{27}\]  
with the mean value of \(\sqrt{\pi}/2\) and standard deviation of \(\sqrt{1-\pi/4}\).