# Space Object and Background Polarization Models and Detectability Analysis
## 2 Basic Theory of the Polarization Bidirectional Reflection Distribution Function Model
### Bidirectional Reflection Distribution Function Model Based on Microlanes
\[f=f_{dd}+k_{s}f_{sp} \tag{10}\]  
The body scattering component \(f_{dd}\) and the associated expressions, and \(\rho_{d}\) was proposed in the [PERSON] work to elucidate the impact of body scattering in reflected light on the polarization state [14], as shown in following equations:  
\[f_{dd_{i,j}}=\frac{\cos\theta_{r}}{\pi}\rho_{d}\cdot M_{i,j}^{d}(\theta_{i}, \theta_{r},\Delta\phi,\eta) \tag{11}\]  
\[\rho_{d}=(1-R_{i})\frac{(1-k_{2})R_{\infty}}{1-k_{2}R_{\infty}} \tag{12}\]  
\[R_{i}=\frac{r_{s}+r_{p}}{2} \tag{13}\]  
\(M_{i,j}^{d}(\theta_{i},\theta_{r},\Delta\phi,\eta)\) is the Muller matrix of the body scattering effect, the Fresnel reflectance of the light that comes from the first medium and hits the target surface is known as \(R_{i}\), which can be calculated from the Fresnel reflection coefficients \(r_{s}\) and \(r_{p}\). \(k_{2}\) represents the Fresnel reflectance of the scattered light that was incident on the target surface from the initial medium. \(R_{\infty}\) is the target surface's diffuse reflectance at roughly infinite thickness [15]. The expressions of \(r_{s}\) and \(r_{p}\) are shown as follows:  
\[r_{s}=\frac{(A-\cos\beta)^{2}+B^{2}}{(A+\cos\beta)^{2}+B^{2}} \tag{14}\]  
\[r_{p}=r_{s}\frac{(A-\sin\beta\cos\beta)^{2}+B^{2}}{(A+\sin\beta\cos\beta)^{2}+ B^{2}} \tag{15}\]  
\[A=\sqrt{\frac{\sqrt{C}+D}{2}} \tag{16}\]  
\[B=\sqrt{\frac{\sqrt{C}-D}{2}} \tag{17}\]  
\[C=4n^{2}k^{2}+D^{2} \tag{18}\]  
\[f=f_{dd}+k_{s}f_{sp} \tag{10}\]0  
where \(n\) is the refractive index, and \(k\) is the absorption coefficient. \(N=n+ik\) is the complex refractive index of the material. The refractive index \(n\), defined as the ratio of the speed of light in a vacuum to the speed of light in a medium, reflects the degree of deflection of light as it travels through the medium. The extinction coefficient k is related to the exponential attenuation of light waves as they pass through the medium and can be thought of as representing the absorption of light by matter. Based on the microsurface metatheory in Section 2.1 and considering the masking effect existing on the actual rough material surface, the representation of the surface scattering component \(f_{sp}\) is deduced as \[f=f_{dd}+k_{s}f_{sp} \tag{10}\]1  
where \(i,j\) denote the \(i+1\) th row and \(j+1\) th column in the Mueller matrix, and the value interval is an integer from 0 to 3; \(\Delta\phi\) is the relative azimuth angle \(\Delta\phi=\phi_{i}-\phi_{r}\); \(\sigma\) is the material surface roughness. The Kubelka-Munk model was chosen to model the body scattering in the target scattered light [16; 17], and the multiparametric polarization bi-directional reflection distribution function can be modeled by combining the surface scattering component \(f_{sp}\) expression and the body scattering component \(f_{dd}\) expression as  
\[f=f_{dd}+k_{s}f_{sp} \tag{10}\]2  
Since the circular polarization component in natural light is very small and negligible, its Stokes vector when natural light is incident can be expressed as \(E_{i}=\left[100\right]^{T}\), and consequently the Stokes vector of the reflected light is as in Equation (22):  
\[L_{r}= \begin{bmatrix}f_{sp_{0}}f_{sp_{0}}f_{sp_{0}}\\ f_{sp_{10}}f_{sp_{11}}f_{sp_{12}}\\ f_{sp_{20}}f_{sp_{21}}f_{sp_{22}}\end{bmatrix}\begin{bmatrix}1\\ 0\\ 0\end{bmatrix} = \begin{bmatrix}f_{sp_{0}}\\ f_{sp_{10}}\\ f_{sp_{20}}\end{bmatrix}  \tag{22}\]  
where \(L_{r}\) is the reflected irradiance Stokes vector parameter and \(f_{i,j}\) is the pBRDF Mueller matrix element. Assuming that the radiation incident on the material surface is unpolarized natural light, the polarization of the scattered light is known as DOP.