## 4 Sentinel-1: SAR simulation  
In this section, we introduce the basic concepts of SAR raw signal simulation, which is here used to simulate both Sentinel-1 and COSMO-SkyMed data and to discuss the expected performances of Sentinel-1 in the development of the WARM-SAR applications described in Section 2. Let \(x\) and \(r\) be the independent space variables, standing respectively for azimuth and range. By using primed coordinates for the independent variables of the SAR raw signal, \(s(x^{\prime},r^{\prime})\), this can be expressed as [30]:  
\[s(x^{\prime},r^{\prime})=\iint\gamma(x,r)g(x^{\prime}-x,r^{\prime}-r;r)dxdr, \tag{1}\]  
where \(\gamma(x,r)\) is the reflectivity pattern of the scene, and \(g(x^{\prime}-x,r^{\prime}-r;r)\) the unit impulse response of the SAR system [30]. Therefore, in order to obtain \(s(x^{\prime},r^{\prime})\) we need to evaluate both the reflectivity function and the impulse response of the system. The evaluation of the reflectivity function requires the use of adequate electromagnetic scattering models, able to provide the solution of interest as a function of the considered sensor and surface parameters. Hence, we need a description of the macroscopic aspects of the surface at the scale of the sensor resolution: this is accomplished by providing as input to the simulator a DEM. The behavior of the DEM is then approximated using a two-scale model [30], _i.e._, using plane facets, over which a microscopic random roughness is superimposed. Then, the mean square value of the field backscattered from each facet can be evaluated providing an adequate stochastic description of the microscopic roughness. The roughness can be described using parameters resulting from the introduction of different models for the shape of the surface [31]: in the present paper, the roughness is described through a fractal _fractional Brownian motion_ (fBm) process [31], _i.e._, using only two independent parameters, the _Hurst coefficient_\(H\) and the _topothesy_\(T\) [m]. Finally, to complete the description of the surface, the relative dielectric constant \(\epsilon\) and the conductivity \(\sigma\) [S/m] of the observed surface must be provided as input to the simulator. The small perturbation model, with the appropriate fractal power law spectrum, can be then used for the evaluation of the reflectivity function of the surface [31]. Note that the reflectivity function is evaluated in a ground range-azimuth reference system and is necessary to project it in the sensor-centered slant range-azimuth reference system. After this transformation, the obtained reflectivity function is filtered according to the impulse response of the SAR system, providing as output the raw signal, as shown in Equation (1). In order to compute the impulse response, the radar and orbital parameters of the sensor are needed: satellite height \(h\) and velocity \(v\), the sensor look angle \(\theta\) and frequency \(f_{0}\), the chirp duration \(\tau\) and bandwidth \(B\), the sampling rate \(f_{s}\), and the pulse repetition frequency PRF. Finally, after standard focusing, the obtained raw signal provides the final simulated complex SAR image. The block diagram of the algorithm employed to evaluate the raw signal is depicted in Figure 2.  
Another important aspect of the simulation procedure is the appropriate inclusion of the speckle effect [32]. Its presence is accounted for thanks to the aforementioned two-scale model. In fact, thanks to this approach, the spatial scales both smaller and larger than the resolution can be treated differently. In particular, the signal macroscopic behavior is described evaluating the mean square value of the field scattered from the plane facets locally approximating the considered surface, assuming knowledge of the microscopic roughness parameters (\(H\) and \(T\)) and the electromagnetic parameters (\(\epsilon\) and \(\sigma\)) of thesurface [31].