## 3 Proposed Method for Zenith Angle Estimation from Mixed Reflected Light
### Zenith Angle Estimation Model Based on Two Types of Reflected Light  
The relationship between the amplitude and the phase of incident, as well as reflected and refracted light, can be derived based on the electromagnetic theory of light [21]. This relationship is further described by [PERSON]'s laws, where a plane wave with a vector direction can be decomposed into s and p components, as illustrated in Figure 4.  
The reflection coefficients \(r_{s}\) and \(r_{p}\), and the transmission coefficients \(t_{s}\) and \(t_{p}\) for the \(s\) and \(p\) components, can be defined as follows:  
\[\begin{array}{l}r_{s}=\frac{E_{0\cup}}{E_{0\cup}}=\frac{n_{1}\cos\theta_{i} -n_{2}\cos\theta_{i}}{n_{1}\cos\theta_{i}+n_{2}\cos\theta_{i}}\\ r_{p}=\frac{E_{0\cup}}{E_{0\cup}/}=\frac{n_{2}\cos\theta_{i}-n_{1}\cos\theta_ {i}}{n_{2}\cos\theta_{i}+n_{1}\cos\theta_{i}}\\ t_{s}=\frac{E_{0\cup}}{E_{0\cup}}=\frac{2n_{1}\cos\theta_{i}}{n_{1}\cos\theta_ {i}+n_{2}\cos\theta_{i}}\\ t_{p}=\frac{E_{0\cup}/}{E_{0\cup}/}=\frac{2n_{1}\cos\theta_{i}}{n_{2}\cos \theta_{i}+n_{1}\cos\theta_{i}}\end{array} \tag{7}\]  
where \(n_{1}\) and \(n_{2}\) represent the refractive indices of different media, and \(n_{1}=1\) for the light incident from the air. According to [PERSON]'s law,  
\[n_{1}\sin\theta_{i}=n_{2}\sin\theta_{i} \tag{8}\]  
Figure 3: The degree of polarization of diffuse reflection in relation to the refractive index and the angle of incidence [17].  
The reflectance \(R\) and transmittance \(T\) are defined as follows:  
\[R_{s}=r_{s}^{2},R_{p}=r_{p}^{2} \tag{9}\] \[T_{s}=1-R_{s},T_{p}=1-R_{s}\]  
According to the definition of the polarization degree, the degree of polarization of the reflected and refracted light can be expressed as follows [19]:  
\[P_{r}=\left|\frac{R_{p}-R_{s}}{R_{p}+R_{s}}\right|,P_{t}=\left|\frac{T_{p}-T_{s }}{T_{p}+T_{s}}\right| \tag{10}\]  
The light received by the detector is a mixture of specular and diffuse reflected light. \(S_{m}\), \(S_{r}\), and \(S_{t}\) represent the intensity of the mixed light received by the detector, the intensity of the specular reflected light, and that of the diffuse reflected light, respectively; \(P_{m}\), \(P_{r}\), and \(P_{t}\) denote the degree of polarization of the mixed light, the specular reflected light, and the diffused reflected light, respectively. Therefore, we can obtain the following:  
\[S_{m}P_{m}=S_{r}P_{r}+S_{t}P_{t} \tag{11}\]  
\[S_{m}=S_{r}+S_{t} \tag{12}\]  
\[\frac{S_{r}}{S_{t}}=\frac{R_{p}+R_{s}}{T_{p}+T_{s}} \tag{13}\]  
Furthermore, we can obtain the following:  
\[\frac{S_{r}}{S_{t}}=\frac{P_{m}-P_{t}}{P_{r}-P_{m}}=\frac{R_{p}+R_{s}}{T_{p}+T _{s}} \tag{14}\]  
Considering Equations (9)-(14), we can obtain the following:  
\[\frac{S_{r}}{S_{t}}=\frac{r_{p}^{2}+r_{s}^{2}}{2-r_{p}^{2}-r_{s}^{2}} \tag{15}\]  
Figure 4: A diagram of the \(s\) and \(p\) components [20].  
Substituting [PERSON]'s law into the equation, the relationship between the degree of polarization of the mixed reflected light from the object surface, the refractive index of the object surface, and the zenith angle can be derived as follows:  
\[\frac{\left(\cos\theta-n\cos[arc\sin(\frac{\sin\theta}{n})]\right)^{2}}{\left( \cos\theta+n\cos[arc\sin(\frac{\sin\theta}{n})]\right)^{2}}+\frac{\left(n\cos \theta-\cos[arc\sin(\frac{\sin\theta}{n})]\right)^{2}}{\left(n\cos\theta+ \cos[arc\sin(\frac{\sin\theta}{n})]\right)^{2}} \tag{16}\]  
The relationship is depicted in the Figure 5 below.