# 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  
The polarimetric scattering characteristics of surfaces can be completely described by the polarized bidirectional reflectance distribution function (pBRDF), which was defined by [PERSON] in 1995 [PERSON] (1995). While the actual surface of the object is rough, the actual normal \(z\) and the micro-surface normal do not coincide, and the size of the angle between the two lines can generally be thought of as meeting the Gaussian random distribution. This assumption forms the foundation of the traditional polarized bidirectional reflectance distribution function (pBRDF). The probability density function for micro-surface elements \(D(\omega)\) can be written as:  
\[D(\omega)=\frac{1}{2\pi\sigma^{2}\cos^{4}\alpha}\exp(\frac{-\tan^{2}\alpha}{2 \sigma^{2}}) \tag{1}\]  
where \(\sigma\) is the coefficient of surface roughness of the object's material. The higher the value of \(\sigma\), the closer the object's surface reflection is to diffuse reflection. As seen in Figure 1, \(k_{i}\) is the light wave vector that is incident, \(k_{r}\) is the light wave vector that is reflected, \(\Delta\phi=\phi_{i}-\phi_{r}\) is the relative azimuthal angle of the two light wave vectors, \(\theta_{i}\) and \(\theta_{r}\) are the angles of the incident light vector \(k_{i}\) and the reflected light vector \(k_{r}\) to the normal z, respectivelyz and n are the microplane and macroscopic normals of the rough material surface of the object with an \(\alpha\) angle, and \(\beta\) is the angle between the light wave vectors that are incident and the microplane norm, where the following geometric relationship between \(\alpha\) and \(\beta\) is satisfied. \[\cos(2\beta)=\cos\theta_{i}\cos\theta_{r}+\sin\theta_{i}\sin\theta_{r}\cos \Delta\phi \tag{2}\]  
\[\cos\alpha=\frac{\cos\theta_{i}+\cos\theta_{r}}{2\cos\beta} \tag{3}\]  
In order to continue to derive the geometric relationship between the angles within the microfacet element, the following two angles are introduced: The normal of the material's surface actually forms a plane with the incident light, as does the normal of the microplane element. Figure 1: Angular relationship of the micro-surface model. The incident light \(\mathbf{k}_{i}\) forms a plane with the normal of the actual surface of the material, and also forms a plane with the normal of the microfacet. The included angle between the two surfaces is \(\eta_{i}\), reflected light \(\mathbf{k}_{r}\) forms \(\eta_{r}\) in the same way. After calculation, we know that \(\eta_{i}\), \(\eta_{r}\), \(\theta_{i}\), \(\theta_{r}\), \(\beta\) satisfy the following equations:  
\[\cos(\eta_{i})=\left[\left(\cos\theta_{i}+\cos\theta_{r}\right)/\left(2\cos \beta\right)-\cos\theta_{i}\cos\beta\right]/\left(\sin\theta_{i}\sin\beta\right) \tag{4}\]  
\[cos(\eta_{r})=\left[\left(\cos\theta_{i}+\cos\theta_{r}\right)/\left(2\cos \beta\right)-\cos\theta_{r}\cos\beta\right]/\left(\sin\theta_{r}\sin\beta\right) \tag{5}\]  
In the process of reflection on the surface of rough objects, there is also a partial scattering of light being blocked by the cutoff-masking phenomenon, which will affect the scattering characteristics of the reflected light. Accordingly, the expression of the specular scattering model of the micro-surface element bidirectional reflection distribution function taking the masking function into account is as follows:  
\[f_{BRDF}(\theta_{i},\phi_{i},\theta_{r},\phi_{r},\lambda)=\frac{1}{2\pi}\frac {1}{4\sigma^{2}}\frac{1}{\cos^{4}\alpha}\frac{\exp\left[-\tan^{2}\alpha/2 \sigma^{2}\right]}{\cos\theta_{r}\cos\theta_{i}}G(\theta_{i},\phi_{i},\theta _{r},\phi_{r}) \tag{6}\]  
where \(\theta_{i}\) and \(\theta_{r}\) are the angles of the incident and reflected light norm to the surface of the macroscopic object, respectively, and \(\phi_{i}\) and \(\phi_{r}\) are the horizontal azimuth angles of the incident and reflected light, respectively.