# Study of Sea Fog Environment Polarization Transmission Characteristics
## 3 Polarization Transmission of Marine Fog Simulation Based on RT3  
### Radiative Transfer Vector Equation  
The radiative transfer equation (RTE) is a fundamental equation that defines how electromagnetic waves travel through a medium and are redistributed through absorption and scattering. It is separated into vector and scalar radiative transfer equations. Polarization is the phenomenon in which the vibration vector of a shear wave is skewed in certain directions. We are aware that a scalar has no direction, but polarization information is a vector with direction. Using the scalar radiation transmission formula to calculate the polarization transmission equation will result in the loss of polarization information. Therefore, the vector radiative transfer equation with the Stokes vector can be employed to solve the polarization transfer equation in order to increase calculation precision. The light scattering and polarization transmission process of polarized light in the sea fog layer can be represented by the following vector radiative transfer equation [13]:  
\[\begin{array}{l}\pi\frac{dI(z,\varphi,\theta)}{d\tau}-i(z,\kappa,\varphi)+ \frac{\varphi}{d\tau}\int\limits_{z_{0}-1}^{2\pi}\left(z,\kappa,\varphi, \varphi^{\prime}\right)I(z,\varphi^{\prime},\varphi^{\prime})\mu^{\prime} \varphi^{\prime}+\frac{\omega}{2\pi}i\tau_{0}=\left(\frac{\pi}{\kappa_{0}} \right)n(z,\kappa,\varphi^{\prime},-\kappa_{0},\varphi_{0})\left](z,\kappa, \varphi^{\prime},\varphi_{0},\varphi_{0})\left](z,\kappa,\varphi^{\prime}, \varphi_{0},\varphi_{0})\left](z,\kappa,\varphi^{\prime},\varphi_{0},\varphi_{0 })\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{ \prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z, \kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0}) \right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{ \prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z, \kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z, \kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z, \kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi _{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{ \prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z, \kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi _{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappa,\varphi^{ \prime},\varphi_{0})\right](z,\kappa,\varphi^{\prime},\varphi_{0})\right](z,\kappaWhen the scattering particle is spherical, the scattering phase matrix \(P\) has just four variables, and its value can be determined using joint Legendre. \(P\) can be expressed as:  
\[P= \begin{bmatrix}a_{1}&b_{1}&0&0\\ b_{1}&a_{2}&0&0\\ 0&0&a_{3}&b_{2}\\ 0&0&-b_{2}&a_{4}\end{bmatrix}  \tag{15}\]  
In Equation (16), \(L\) equals:  
\[L= \begin{bmatrix}1&0&0&0\\ 0&\cos 2i&\sin 2i&0\\ 0&-\sin 2i&\cos 2i&0\\ 0&0&0&1\end{bmatrix}  \tag{16}\]  
In general, we employ the Fourier transform to decompose the Stokes vector \(I\) and Mueller matrix to improve the dependability of radiation transmission's intermediate variables. This is the particular expansion form:  
\[I=\sum\limits_{m=0}^{m} \begin{bmatrix}I_{m}\cos m(\phi-\phi_{0})\\ Q_{m}\cos m(\phi-\phi_{0})\\ U_{m}\sin(\phi-\phi_{0})\\ \sin(\phi-\phi_{0})\end{bmatrix}  \tag{17}\]  
\[M=\frac{1}{2}C_{0}+\sum\limits_{m=1}^{m}\left[C_{m}\cos m(\phi-\phi\prime)+S_{ m}\sin m(\phi-\phi\prime)\right] \tag{18}\]  
In Formula (18), \(C_{m}\) and \(S_{m}\) represent m-order Fourier mode coefficients.