## 1 Introduction
### Parameters in the Reflection Symmetric Case  
For reflection symmetric scatterers, an arbitrary backscattering coherency matrix can be expressed as follows [14]:  
\[\boldsymbol{T}=\overset{\rightharpoonup}{\boldsymbol{k}}_{p}\overset{ \rightharpoonup}{\boldsymbol{k}}_{p}=\left[\begin{array}{ccc}T_{11}&T_{12}&T_{ 13}\\ T_{12}^{*}&T_{22}&T_{23}\\ T_{13}^{*}&T_{23}^{*}&T_{33}\end{array}\right]=\boldsymbol{Q}(2\theta)^{ \text{T}}\left[\begin{array}{ccc}t_{11}&t_{12}&0\\ t_{12}^{*}&t_{22}&0\\ 0&0&t_{33}\end{array}\right]\boldsymbol{Q}(2\theta), \tag{1}\]  
where:  
\[\left\{\begin{array}{c}t_{11}=\langle\left|h+v\right|^{2}\rangle\\ t_{22}+t_{33}=\langle\left|h-v\right|^{2}\rangle\\ t_{12}=\langle\left(h+v\right)\left(h-v\right)^{*}\rangle\end{array}\right. \text{; }\boldsymbol{Q}(2\theta)=\left[\begin{array}{ccc}1&0&0\\ 0&\cos\ 2\theta&\sin\ 2\theta\\ 0&-\sin\ 2\theta&\cos\ 2\theta\end{array}\right].\]  
\(h\) and \(v\) are the con-diagonalization parameters, \(\overset{\rightharpoonup}{k}_{p}=\frac{1}{\sqrt{2}}\Big{[}\begin{array}{cc} S_{\text{HH}}+S_{\text{VV}}&S_{\text{HH}}-S_{\text{VV}}&2S_{\text{HV}}\end{array} \Big{]}^{\text{T}}\) is the Pauli-basis vector; and \(\langle\cdots\rangle\) denotes ensemble averaging. From (1), a parameter \(\alpha_{\text{B}}\) can be derived, which is a rotation invariant [13], as follows.  
\[\alpha_{\text{B}}=\tan^{-1}\bigg{(}\frac{T_{22}+T_{33}}{T_{11}}\bigg{)}=\tan^{ -1}\bigg{(}\frac{|\rho_{r}-1|^{2}+2|\rho_{r}|\cos\ \phi_{r}(1-|r_{\text{c}}|)}{|\rho_{r}+1|^{2}-2|\rho_{r}|\cos\ \phi_{r}(1-|r_{\text{c}}|)}\bigg{)}, \tag{2}\]where:  
\[\rho_{r}=|\rho_{r}|e^{j\phi_{r}}=\sqrt{\frac{\langle\left|S_{\rm VV}\right|^{2} \rangle}{\langle\left|S_{\rm HH}\right|^{2}\rangle}}e^{j(\langle\phi_{\rm VV} -\phi_{\rm HH}\rangle)};\]  
\[r_{c}=\frac{\langle S_{\rm HH}S_{\rm VV}^{*}\rangle}{\sqrt{\langle\left|S_{\rm HH }\right|^{2}\rangle\langle\left|S_{\rm VV}\right|^{2}\rangle}}.\]  
It is observed that \(\alpha_{\rm B}\) is determined by two statistical measures, i.e., \(\rho_{r}\) and \(r_{c}\), which are the ratio of the averaged co-polarizations and the co-polarization coherence, respectively. In the second-order coherency matrix, both parameters can be obtained directly. However, it is difficult to use the parameter \(\alpha_{\rm B}\) to describe the scattering coherence. Thus, we introduce a parameter \(\Delta\alpha_{\rm B}\) to relate the effect of \(r_{c}\), as follows:  
\[\Delta\alpha_{\rm B}=\alpha_{\rm B}-\alpha_{\rm av}, \tag{3}\]  
where:  
\[\alpha_{\rm av}=\tan^{-1}\Biggl{(}\frac{|\rho_{r}-1|^{2}}{|\rho_{r}+1|^{2}} \Biggr{)}. \tag{4}\]  
\(\Delta\alpha_{\rm B}\) can be used to measure the scattering randomness. Its sign is determined by the co-polarization phase difference (CPD). If all elemental scatterers in a resolution cell are consistent with one dominant scattering mechanism in both orientation and dielectric properties, where \(r_{c}\) is large, then \(\Delta\alpha_{\rm B}\) is close to \(0^{\circ}\); if scatterers vary considerably, where \(r_{c}\) is small, then \(\Delta\alpha_{\rm B}\) is far away from \(0^{\circ}\). For targets dominated by double-bounce scattering, since the physical model of double-bounce scattering is characterized by CPDs approaching to \(\pm\pi\)[14], then according to (3), \(\Delta\alpha_{\rm B}\) should be smaller than \(0^{\circ}\). In the next section, we analyze \(\Delta\alpha_{\rm B}\) and \(\alpha_{\rm B}\) for the reflection asymmetry case.