## 3 Modified SBR-FBSSA Method for Composite Scattering
### Modified SBR Algorithm
\tag{12}\]  
where \(\eta_{1}\) and \(\eta_{2}\) are the impedances of the air and rough surface medium, respectively; \(k_{1}\) and \(k_{2}\) are wave numbers corresponding; \(\theta_{i}\) and \(\theta_{t}\) are incident and transmission angles, simultaneously, \(\theta_{i}\) and \(\theta_{t}\) obey Snell's Law. Similarly, for h-polarization, the expressions are  
\[\left\{ \begin{array}{l}R_{h00}=\frac{\eta_{2}\cos\theta_{i}-\eta_{1}\cos \theta_{i}}{\eta_{2}\cos\theta_{i}+\eta_{1}\cos\theta_{i}}\\ R_{h10}=\frac{\left(\eta_{2}\sin\theta_{i}(1-R_{h00})-\eta_{1}\frac{k_{1}\cos \theta_{i}}{2\cos\theta_{i}}\sin\theta_{i}(1+R_{h00})\right)}{\left(\eta_{2} \cos\theta_{i}+\eta_{1}\cos\theta_{i}\right)}\\ R_{h102}=\frac{-\eta_{1}\big{(}1-k_{1}^{2}/k_{2}^{2}\big{)}(1+R_{h00})}{2(\eta _{2}\cos\theta_{i}+\eta_{1}\cos\theta_{i})\cos\theta_{i}}\\ R_{h20}=\frac{R_{h00}}{\cos^{2}\theta_{i}}-R_{h10}\frac{\eta_{2}\sin\theta_{i }+\eta_{1}\frac{k_{1}\cos\theta_{i}}{2\cos\theta_{i}}\sin\theta_{i}}{\eta_{2} \cos\theta_{i}+\eta_{1}\cos\theta_{i}}\end{array} \right. \tag{13}\]  
According to the GO method, the \(i\)-th incident electric field at each reflection point can be used to compute the \(i\)-th reflected electric field. Meanwhile, the \((i+1)\)-th incident electric field is obtained using the \(i\)-th reflected electric field. The field and path tracing can be realized utilizing such sequential calculation. After the ray tracing, the scattering field is calculated by the PO method  
\[E^{\rm s}\approx\frac{ik}{4\pi}\cdot\frac{e^{-ikr}}{r}[\hat{\bf s}\times(\mathbf{M _{\rm s}}+\eta_{0}\hat{\bf s}\times\mathbf{J_{\rm s}})]\cdot\Delta\mathbf{A}\cdot I \tag{14}\]  
where \(\mathbf{J_{\rm s}}\) and \(\mathbf{M_{\rm s}}\) denote electric current and magnetic current on the facet, \(k\) is the wave number, \(\hat{\bf s}\) represents the unit vector of observation direction, and \(r\) indicates the distance from the original point to a point located in the facet. \(\Delta\mathbf{A}\) denotes the area of the facet; \(I\) can be calculated by the Gordon method in [37]. applicable conditions, [PERSON] [39] pointed out that the appropriate requirements of these two methods could not cover all rough scale cases. Small slope approximation (SSA) is suitable for the electromagnetic scattering problem from rough surfaces satisfying the KA and SPM and for the transition type of rough surfaces. What's more, in the facet-based model which contains the SSA, the facets are treated as rough surfaces instead of smooth plates [39]. For the SSA, the scattering amplitude of the facet \(S_{n}\) in the local coordinate system is given as  
\[S_{l}(k_{sl},k_{ii})=\frac{2(q_{sl}q_{il})^{1/2}}{q_{sl}+q_{il}}B(k_{sl},k_{il}) \times\int\frac{d\tau_{l}}{\left(2\pi\right)^{2}}\exp[-\mathrm{i}(p_{sl}-p_{il })\cdot\mathbf{\tau}_{l}+\mathrm{j}(q_{sl}+q_{il})h(\mathbf{\tau}_{l})]d\mathbf{\tau}_{l} \tag{18}\]  
where \(B\) is local polarization factors, the relation between \(B\) and global polarization factors \(F\) is as follows [40]  
\[ \begin{bmatrix}F_{VV}&F_{VH}\\ F_{HV}&F_{HH}\end{bmatrix} = \begin{bmatrix}\hat{\mathbf{v}}_{s}\cdot\hat{\mathbf{v}}_{s}^ {{}^{\prime}}&\hat{\mathbf{h}}_{s}\cdot\hat{\mathbf{h}}_{s}^{{}^{\prime}}\\ \hat{\mathbf{v}}_{s}\cdot\hat{\mathbf{h}}_{s}^{{}^{\prime}}&\hat{\mathbf{h}}_{s}\cdot\hat {\mathbf{v}}_{s}^{{}^{\prime}}\end{bmatrix}\begin{bmatrix}B_{vv}&B_{vib}\\ B_{\mathrm{m}}&B_{hh}\end{bmatrix}\begin{bmatrix}\hat{\mathbf{v}}_{i}\cdot\hat{ \mathbf{v}}_{i}^{{}^{\prime}}&\hat{\mathbf{h}}_{i}\cdot\hat{\mathbf{h}}_{i}^{{}^{\prime}} \\ \hat{\mathbf{v}}_{i}\cdot\hat{\mathbf{h}}_{i}&\hat{\mathbf{h}}_{i}\cdot\hat{\mathbf{v}}_{i}^{{}^{ \prime}}\end{bmatrix}  \tag{19}\]  
where \( \begin{bmatrix}\hat{\mathbf{h}}_{i}^{{}^{\prime}}\hat{\mathbf{v}}_{i}^{{}^{\prime}} \hat{\mathbf{h}}_{s}^{{}^{\prime}}\hat{\mathbf{v}}_{s}^{{}^{\prime}}\end{bmatrix} \) and \( \begin{bmatrix}\hat{\mathbf{h}}_{i}\hat{\mathbf{v}}_{i}\hat{\mathbf{h}}_{s}\hat{\mathbf{v}}_{ s}\end{bmatrix} \) are unit polarization vectors in local- and global-coordinate systems.