## 3 Analytical Non-Stationary S2A Channel Model
### Channel Impulse Response
\[h_{\text{SPE}}\left(t,\tau\right)=\Gamma\left(t\right)\cdot\alpha_{\text{SPE} }\left(t\right)\cdot g_{\text{SPE}}\left(t\right)\cdot\delta\left(\tau-\tau_{ \text{SPE}}\left(t\right)\right) \tag{13}\]  
\[\alpha_{\text{SPE}}\left(t\right)=\frac{\lambda\sqrt{G_{\text{T}}^{\text{SPE }}\left(t\right)G_{\text{R}}^{\text{SPE}}\left(t\right)}}{4\pi d_{\text{SPE}} \left(t\right)} \tag{14}\]  
\[g_{\text{SPE}}\left(t\right)=\exp\left(\text{j}k_{0}d_{\text{SPE}}\left(t \right)+\text{j}\frac{2\pi}{\lambda}\int_{0}^{t}\left(\mathbf{n_{TO}^{\text{ T}}\left(t\right)\cdot v_{T}\left(t\right)+n_{RO}^{\text{T}}\left(t\right)\cdot v_{R} \left(t\right)}\right)\text{d}t\right) \tag{15}\]  
In Equations (14) and (15), \(\mathbf{n_{\text{TO}}}\) and \(\mathbf{n_{\text{RO}}}\) are the unit directional vector towards O observed from T and R, respectively. And \(\Gamma\left(t\right)\) can be calculated as Fresnel Equations (16) and (17), with vertical and horizontal polarization, respectively,where \(\epsilon_{r}^{e}\left(t\right)\) is the relative dielectric permittivity of the reflection surface, which is a random process since the reflector and the scatterers may be changed with the movements of the transmitter and the receiver, the reflecting and scattering areas are for example, the sea, earth surface, etc. [25; 26; 27]. \[\Gamma_{\bot}\left(t\right)=\frac{\cos\theta\left(t\right)-\sqrt{\epsilon_{r}^ {e}\left(t\right)-\sin^{2}\!\theta\!\left(t\right)}}{\cos\theta\!\left(t\right) +\sqrt{\epsilon_{r}^{e}\left(t\right)-\sin^{2}\!\theta\!\left(t\right)}} \tag{16}\]  
\[\Gamma_{//}\left(t\right)=\frac{\epsilon_{r}^{e}\left(t\right)\cos\theta\left( t\right)-\sqrt{\epsilon_{r}^{e}\left(t\right)-\sin^{2}\!\theta\left(t\right)}}{ \epsilon_{r}^{e}\left(t\right)\cos\theta\left(t\right)+\sqrt{\epsilon_{r}^{e }\left(t\right)-\sin^{2}\!\theta\left(t\right)}} \tag{17}\]  
In the simulation of this paper, \(\epsilon_{r}^{e}\left(t\right)\) is calculated as in [27] with the assumption that the scattering surface is sea surface. Suppose that the number of the scatterers is \(L\), then the impulse response of the scattering path is as Equation (18). \[h_{\text{DIF}}\left(t,\tau\right)=\sum_{l=1}^{L}\alpha_{\text{DIF},l}\left(t \right)g_{\text{DIF},l}\left(t\right)\delta\left(\tau-\tau_{\text{DIF},l}\left( t\right)\right) \tag{18}\]  
When the radar cross section (RCS) \(\sigma\) is constant for each scatterer, the amplitude attenuation of the \(l^{\text{th}}\) subpath \(\alpha_{l}\left(t\right)\) is calculated by Equation (19) in accord with [2; 28]. In Equation (19), \(d_{\text{TD},l}\left(t\right)\) and \(d_{\text{DR},l}\left(t\right)\) are the distance between the transmitter and the scatterer, and the distance between the scatterer and the receiver, respectively. \[\alpha_{\text{DIF},l}\left(t\right)=\frac{\lambda\sqrt{G_{\text{T},l}^{\text{ DIF}}\left(t\right)G_{\text{R},l}^{\text{DIF}}\left(t\right)}}{4\pi d_{\text{TD},l} \left(t\right)d_{\text{DR},l}\left(t\right)}\sqrt{\sigma} \tag{19}\]  
According to the radar equation and the theory presented in [2], when RCS is not constant in the diffuse scattering area, we can denote it as \(\sigma\left(x,y\right)\). Thus \(\alpha_{\text{DIF},l}\left(t\right)\) is calculated as in (20) with the assumption that the shape of each scatterer is a circle \(C_{l}\) with radius \(R_{l}\). \[ \begin{split}\alpha_{\text{DIF},l}\left(t\right)&= \sqrt{\frac{\lambda^{2}}{\left[4\pi d_{\text{LOS}}\left(t\right)\right]^{2}} \cdot\iint_{x,y\in C_{l}}\frac{\sigma\left(x,y\right)d_{\text{TR}}^{2}\left(t \right)}{4\pi d_{\text{TD},l}\left(t\right)d_{\text{DR},l}^{2}\left(t\right)} G_{\text{T}}\left(x,y\right)G_{\text{R}}\left(x,y\right)\text{d}x\text{d}y}\\ &=\frac{\lambda}{8\pi d_{\text{TD},l}\left(t\right)d_{\text{DR},l} \left(t\right)\sqrt{\pi}}\sqrt{\iint_{x,y\in C_{l}}\sigma\left(x,y\right)G_{ \text{T}}\left(x,y\right)G_{\text{R}}\left(x,y\right)\text{d}x\text{d}y}\end{split}  \tag{20}\]\(\sigma\left(x,y\right)\) can be calculated as Equations (21) and (22) if the diffuse surface is sea surface[29; 30].