## 3 Modified SBR-FBSSA Method for Composite Scattering
### Modified SBR Algorithm  
The SBR algorithm describes the scattering fields by GO and PO methods. The GO method is utilized to trace the EM wave propagation paths. When the EM waves leave the composite model or finish bouncing, the PO method is used to compute the scattering field. The incident electric field \(\mathbf{E}_{i}^{0}\) by the plane wave is expressed by  
\[\mathbf{E}_{i}^{0}=A_{i}^{0}\hat{\mathbf{e}}_{i}^{0}e^{i\phi_{0}} \tag{7}\]  
where \(\phi_{0}\) denotes the initial phase of \(\mathbf{E}_{i}^{0}\), \(A_{i}^{0}\) is the incident wave electric field intensity, and \(\hat{\mathbf{e}}_{i}^{0}\) is the unit vector of vertical-polarization or horizontal-polarization, described by  
\[\hat{\mathbf{e}}_{i}^{0}=\left\{ \begin{array}{l}(\cos\theta_{i}\cos\varphi_{i}, \cos\theta_{i}\sin\varphi_{i},-\sin\theta_{i})\text{ V-polarization}\\ (-\sin\varphi_{i},\cos\varphi_{i},0)\quad\text{ H-polarization}\end{array} \right. \tag{8}\]  
where the incident pitch angle and azimuth angle are denoted by \(\theta_{i}\) and \(\varphi_{i}\), respectively. As shown in Figure 5, assuming the \(i\)-th reflection point \(P_{i}\) located at the facet \(S_{n}\) with the normal direction \(\hat{n}_{i}\), \(\theta_{i}\) and \(\theta_{r}\) are the incident- and reflect-pitch angles, respectively, \(\hat{\mathbf{R}}_{i}\) and \(\hat{\mathbf{R}}_{r}\) are the incident- and reflect-unit vector, respectively, and \(P_{i+1}\) denotes the \((i+1)\)-th reflection point. Based on the GO method,  
\[\left\{ \begin{array}{l}\theta_{r}=\theta_{i}\\ \hat{\mathbf{R}}_{r}=\hat{\mathbf{R}}_{i}-2(\hat{\mathbf{R}}_{i}\cdot\hat{\mathbf{n}}_{i})\\ \mathbf{E}_{i}^{\prime}(\mathbf{r}_{p_{i}})=E_{v}\hat{\mathbf{e}}_{v}+E_{h}\hat{\mathbf{e}}_{h} \\ \mathbf{E}_{r}^{\prime}(\mathbf{r}_{p_{i}})=\left(DF\right)_{i}(R_{v}E_{v}\hat{\mathbf{e}}_ {v}+R_{h}E_{h}\hat{\mathbf{e}}_{h})\end{array} \right. \tag{9}\]  
where \(\mathbf{E}_{i}^{\prime}(\mathbf{r}_{p_{i}})\) and \(\mathbf{E}_{r}^{\prime}(\mathbf{r}_{p_{i}})\) denote the incident- and reflect-electric field, respectively. The subscripts v and h denote the vertical- and horizontal-polarization mode, \(\left(DF\right)_{i}\) represents the diffusion coefficient [34] at the point \(P_{i}\). Figure 5: Sketch map of the \(i\)-th reflection on the facet \(S_{n}\). In traditional SBR method, the \(R_{v}\) and \(R_{h}\) are Snell reflection coefficients meaning the facets of the rough surface are considered smooth planes. However, the facets of rough surfaces are rough in practice. Hence, this situation is considered in our modified SBR method by introducing a high-order reflection model \(R_{p}\) as follows [35]  
\[R_{p}=\rho_{S}\Big{(}R_{p00}+m^{2}\big{(}R_{p02}+R_{p20}\big{)}\Big{)} \tag{10}\]  
where subscript \(p\) denotes the polarization mode, is v- or h-; m is the root mean square slope [36], and \(\rho_{S}\) is described as  
\[\rho_{S}= \begin{cases}\exp\Big{[}(-2\pi\tau)^{2}\Big{]}&0\leq\tau\leq 0.1\\ 0.812537\Big{/}\Big{[}1,+,2,(2\pi\tau)^{2}\Big{]}&\tau\geq 0.1\end{cases}  \tag{11}\]  
where \(\tau=h_{rms}\cos\theta_{i}/\lambda\), \(\lambda\) is the wavelength. Continually, \(R_{p00}\), \(R_{p02}\) and \(R_{p20}\) are Taylor coefficients of the local reflection coefficient. For v-polarization, these Taylor coefficients can be obtained by:  
\[\left\{ \begin{array}{l}R_{v00}=\frac{\eta_{1}\cos\theta_{i}-\eta_{2}\cos \theta_{i}}{\eta_{1}\cos\theta_{i}+\eta_{2}\cos\theta_{i}}\\ R_{v10}=\frac{\left(\eta_{1}\sin\theta_{i}(1-R_{v00})-\eta_{2}\frac{k_{1}\cos \theta_{i}}{2\cos\theta_{i}}\sin\theta_{i}(1+R_{v00})\right)}{\left(\eta_{1} \cos\theta_{i}+\eta_{2}\cos\theta_{i}\right)}\\ R_{v02}=\frac{-\eta_{2}\big{(}1-k_{1}^{2}/k_{2}^{2}\big{)}(1+R_{v00})\cos \theta_{i}}{2(\eta_{1}\cos\theta_{i}+\eta_{2}\cos\theta_{i})}\\ R_{v20}=\frac{R_{v02}}{\cos^{2}\theta_{i}}-R_{v10}\frac{\eta_{1}\sin\theta_{i} +\eta_{2}\frac{k_{1}\cos\theta_{i}}{2\cos\theta_{i}}\sin\theta_{i}}{\eta_{1} \cos\theta_{i}+\eta_{2}\cos\theta_{i}}\end{array} \right.