## 2 Materials and Methods  
### Optical Model of Rainy Conditions  
As shown in Figure 2, the thick clouds heavily reduce the intensity of natural light, which leads to a reduction in image brightness. In addition, along with the rain, there is an abundance of water spray in the air that generates scattering of light [16]. The reflected light of different bands has different transmissivity mapping. Hence, images taken through this kind of scattering media are characterized by a loss of contrast and changes in color. Even though there is serious image degradation, the specific atmospheric conditions on rainy days provide the possibility to image through the rain by employing circularly polarized information. Several studies on the propagation of light through water spray have demonstrated that for randomly oriented particles [15; 16; 17], the incident light (sunlight before into the water spray) [\(S_{0}(x,y)\), \(S_{1}(x,y)\), \(S_{2}(x,y)\), \(S_{3}(x,y)\)]T and the scattered light (reflected light from target surface) [\(S_{0}^{0}(x,y)\), \(S_{1}{}^{0}(x,y)\), \(S_{2}{}^{0}(x,y)\), \(S_{3}{}^{0}(x,y)\)]T are related by Equation (1):  
\[ \begin{split}&\left[S_{0}(x,y),S_{1}(x,y),S_{2}(x,y),S_{3}(x,y) \right]^{T}=L(\pi-i_{2})\cdot P(\psi)\cdot L(-i_{1})\cdot\left[S_{0}^{0}(x,y), S_{1}^{0}(x,y),S_{2}^{0}(x,y),S_{3}^{0}(x,y)\right]^{T}\\ &\quad=\left(\begin{array}{ccc}p_{11}(\psi)&p_{21}(\psi)\cos 2 i_{1}&-p_{21}(\psi)\sin 2i_{1}&0\\ p_{21}(\psi)\cos 2i_{2}&p_{22}(\psi)\cos 2i_{1}\cos 2i_{2}&-(p_{22}(\psi)\cos 2i_{2} \sin 2i\\ -p_{33}(\psi)\sin 2i_{2}\sin 2i_{1}&+p_{33}(\psi)\sin 2i_{2}\cos 2i_{1})&p_{ 43}(\psi)\sin 2i_{2}\\ p_{21}(\psi)\sin 2i_{2}&p_{22}(\psi)\sin 2i_{2}\cos 2i_{1}&p_{ 33}(\psi)\cos 2i_{2}\cos 2i_{1}\\ &+p_{33}(\psi)\times\sin 2i_{1}\cos 2i_{2}&-p_{22}(\psi)\sin 2i_{1}\sin 2i_{2} \\ 0&p_{43}(\psi)\sin 2i_{1}&p_{43}(\psi)\cos 2i_{1}&p_{44}(\psi)\end{array}\right)\left( \begin{array}{c}S_{0}^{0}(x,y)\\ S_{1}^{0}(x,y)\\ S_{2}^{0}(x,y)\\ S_{3}^{0}(x,y)\end{array}\right),\end{split}  \tag{1}\]  
Figure 1: Degree of polarization (DOP) as a function of optical depth in water with different diameter particles; (**a**) 2 \(\upmu\)m particle; (**b**) the size parameter (k\({}_{\rm a}\)) varies (Ref. [18]). where \(p\) is the phase matrix; \(\Psi\) means the scattering angle; \(i_{1}\) and \(i_{2}\) are the angles of rotation of the light scattered by water molecules; L is the rotation matrix; ((\((S_{1}(x,\,y))^{2}\)\(+\)\((S_{2}(x,\,y))^{2}\)\(+\)\((S_{2}(x,\,y))^{2}\)\()^{1/2}/S_{0}(x,\,y)\)) is the Degree of LP (DoLP); (\(S_{3}(x,\,y)/S_{0}(x,\,y)\)) is the Degree of CP (DoCP) [15]. The LP light (\([S_{0}\)\({}^{1}(x,\,y),\,S_{1}\)\({}^{1}(x,\,y),\,0,\,0]\)\({}^{T}\)) exists after a single scattering with sunlight as the incident light. It is also possible for CP light to be generated when the incident light experiences multiple scattering according to Equation (1). [PERSON] [16] also stated the fact that a rainy day or watery environment is a very specific condition that can generate more circularly polarized light through multi-scattering in a natural environment compared to a sunny day. Under natural conditions, multiple scattering is much more common than single scattering in media [15; 18]. Taking two-particle scattering as an example (Figure 3), the CP component \(S_{3}\)\({}^{2}(\psi,\,\rho)\) can be written as Equation (2):  
\[S_{3}^{2}(\psi,\rho)=\frac{2P_{21}(\psi)P_{43}(\psi)\sin\psi\cdot\sin\rho \cdot\cos\rho}{(1-\cos^{2}\psi\cdot\cos^{2}\rho)}. \tag{2}\]  
where \(\rho\) is the angle of the incident light plane and reflected light plane. According to typical single scattering by a droplet and Brewster's Law, the emergent light can reach an impressively high DoLP (more than 90%) when light is incident at an ideal position (generally, the incident angle stays between 38\({}^{\circ}\) and 43\({}^{\circ}\), and the emergent angle ranges from 48\({}^{\circ}\) to 60\({}^{\circ}\)) [16]. The light with high DoLP is more likely to generate CP light after scattering [15].