# 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
\(G(\theta_{i},\phi_{i},\theta_{r},\phi_{r})\) is the influence of the ing effect [10] effect from the masking effect. Assuming \(\phi_{i}=0\), the shadowing function can be reduced to a function \(G(\theta_{i},\theta_{r},\phi_{r})\) with respect to \(\theta,i\theta_{r},\phi_{r}\), and the expression is derived using the approximation formula as follows [11]:  
\[G(\theta_{i},\theta_{r},\phi_{r})=\frac{1+\frac{\omega_{p}|\tan\theta_{ip}\tan \theta_{p}|}{1+\sigma_{r}\tan\beta_{p}}}{\left[\left(1+\omega_{p}\tan^{2} \theta_{ip}\right)\left(1+\omega_{p}\tan^{2}\theta_{rp}\right)\right]} \tag{7}\]  
where the angle \(\theta_{ip},\theta_{rp},\beta_{p}\) designates, in accordance with the following geometric relationships, the spherical projection of the angle \(\theta_{i},\theta_{r},\beta\). \[\left\{ \begin{array}{l}\omega_{p}=\sigma_{p}(1+\frac{u_{p}\sin\alpha}{\sin \alpha+v_{p}\cos\alpha})\\ \tan\theta_{ip}=\tan\theta_{i}\frac{\sin\theta_{i}+\sin\theta_{r}\cos\varphi_{ r}}{2\sin\alpha\cos\beta}\\ \tan\theta_{rp}=\tan\theta_{r}\frac{\sin\theta_{r}+\sin\theta_{r}\cos\varphi_{ r}}{2\sin\alpha\cos\beta}\\ \tan\beta_{p}=\frac{\left|\cos\theta_{i}-\cos\beta\right|}{2\sin\alpha\cos\beta} \end{array} \right. \tag{8}\]  
where \(\sigma_{p}\), \(\sigma_{r}\), \(u_{p}\) and \(v_{p}\) are empirical parameters connected to the surface-related material roughness coefficients and are often given the following empirical values [12]: \(\sigma_{p}=0.0136,\sigma_{r}=0.0136,u_{p}=9.0,v_{p}=1.0\). Multi-Parameter Polarization Bidirectional Reflection Distribution Function Model of Rough Material Surface  
The bidirectional reflection distribution function can describe the reflection process of light rays on the surface of anisotropic objects more comprehensively. Not only the original intensity of the light will change, but so will its polarization state, and the polarization bidirectional reflection distribution function model is:  
\[M(\theta_{i},\phi_{i};\theta_{r},\phi_{r};\lambda)=\frac{dL_{r}(\theta_{i},\phi _{i};\theta_{r},\phi_{r};\lambda)}{dE_{i}(\theta_{i},\phi_{i};\lambda)} \tag{9}\]  
where \(dE_{i}\) and \(dL_{r}\) are the Stokes vectors of incident and scattered light. Based on this, the polarization bidirectional reflection distribution function \(M(\theta_{i},\phi_{i};\theta_{r},\phi_{r};\lambda)\) is created. This function is a crucial addition to the bidirectional reflection distribution function because it can reflect how the polarization state of light changes during the reflection process on an object's surface. The target's polarization characteristics can be determined using the polarization bidirectional reflection distribution function, and the algorithm can inverse the substance's fundamental physical properties. Surface scattering and body scattering are the two main types of light scattered from material surfaces. This study assumes that body scattering and multiple scattering of light off rough material surfaces are both parts of the same non-polarized process and concurrently models and analyzes them. The Mueller matrix [13], which may describe the scattering polarization characteristics of the target material, can be used to determine the relationship between the incident light and the scattered light Stokes vector using the polarization bi-directional reflection distribution function. In order to distinguish the effects brought on by the two scattered lights during the surface scattering of the material, the specular coefficient \(k_{s}\) is introduced in this paper, as well as a pBRDF model that includes the surface scattering component \(f_{sp}\) of the material.