## 2 Theory and Modeling
### The Transmission Matrix of Wave Facets  
The transmission matrix of wave facets \(\mathbf{t}\) is used in the situation where an incident light is transmitted into direction \((\mu,\varphi)\) from direction \((\mu^{\prime},\ \varphi^{\prime})\). \(\mathbf{t}\) is written as follows:  
\[\mathbf{t}(\mu^{\prime},\ \varphi^{\prime},\mu,\varphi)=\ S(\mu,\mu^{\prime})\frac{ \pi p(\mu_{n},\varphi_{n})}{4\big{|}\mu\big{|}\mu^{\prime}\big{|}\mu_{n}}\mathbf{R} (\pi-\chi_{2})\mathbf{T}\mathbf{F}(\theta_{i})\mathbf{R}(-\chi_{1})\times\frac{n_{t}^{2}cos \theta_{t}cos\theta_{i}}{\left(n_{t}cos\theta_{t}-n_{i}cos\theta_{i}\ \right)^{2}}, \tag{15}\]  
where \(S(\mu,\mu^{\prime})\) is the shadowing function expressed in the same way as Equation (23). \(n_{i}\) and \(n_{t}\) are refractive indices of the incident and transmitted media, respectively. \(\mathbf{T}\mathbf{F}(\theta_{i})\) is the Fresnel transmission matrix. For the air-incident case (Figure 2b), the light is refracted into the sea from the atmosphere medium (\"air-sea\"). The transmission matrix \(\mathbf{t}\mathbf{aw}(\mu^{\prime},\ \varphi^{\prime},\mu,\varphi)\) is expressed as follows:  
\[\mathbf{t}\mathbf{aw}(\mu^{\prime},\ \varphi^{\prime},\mu,\varphi)=\ S(\mu,\mu^{\prime}) \frac{\pi p(\mu_{n},\varphi_{n})}{4\big{|}\mu\big{|}\mu^{\prime}\big{|}\mu_{n} }\mathbf{R}(\pi-\chi_{2})\mathbf{T}\mathbf{F}_{AW}\big{(}\theta_{i}^{a}\big{)}\mathbf{R}(-\chi _{1})\times\frac{m^{2}\text{cos}\theta_{t}^{a}\text{cos}\theta_{i}^{a}}{\left( m\text{cos}\theta_{t}^{a}-\text{cos}\theta_{i}^{a}\right)^{2}}. \tag{16}\]  
The air-to-sea Fresnel transmission matrix \(\mathbf{T}\mathbf{F}_{AW}\big{(}\theta_{i}^{a}\big{)}\) is expressed as follows:  
\[\mathbf{T}\mathbf{F}_{AW}\Big{(}\theta_{i}^{a}\Big{)}=\frac{mcos\theta_{t}^{a}}{\cos \theta_{t}^{a}}\left( \begin{array}{cccc}t_{11}^{a}&t_{12}^{a}&0&0\\ t_{12}^{a}&t_{11}^{a}&0&0\\ 0&0&t_{33}^{a}&t_{34}^{a}\\ 0&0&t_{34}^{a}&t_{33}^{a}\end{array} \right) \tag{17}\]  
where  
\[\left\{ \begin{array}{l}t_{11}^{a}(\theta_{i}^{a})=0.5\times[(t_{1}^{a}( \theta_{i}^{a}))^{2}+(t_{r}^{a}(\theta_{i}^{a}))^{2}]\\ t_{11}^{a}(\theta_{i}^{a})=0.5\times[(t_{1}^{a}(\theta_{i}^{a}))^{2}-(t_{r}^{a }(\theta_{i}^{a}))^{2}]\\ t_{33}^{a}(\theta_{i}^{a})=Re(t_{i}^{a}(\theta_{i}^{a})\times t_{r}^{a}( \theta_{i}^{a}))\\ t_{34}^{a}(\theta_{i}^{a})=Im(t_{i}^{a}(\theta_{i}^{a})\times t_{r}^{a}( \theta_{i}^{a}))\end{array} \right. \tag{18}\]  
In Equation (18), \(t_{I}^{a}\big{(}\theta_{i}^{a}\big{)}\) and \(t_{r}^{a}\big{(}\theta_{i}^{a}\big{)}\) are the transmission coefficients of the parallel and vertical components of transmitted light, respectively. \[\left\{ \begin{array}{l}t_{I}^{a}\big{(}\theta_{i}^{a}\big{)}=\frac{2 mcos \theta_{i}^{a}}{m^{2}cos\theta_{i}^{a}+\sqrt{m^{2}-sin^{2}\theta_{i}^{a}}}\\ t_{r}^{a}\big{(}\theta_{i}^{a}\big{)}=\frac{2 cos\theta_{i}^{a}}{\alpha cos\theta_{ i}^{a}+\sqrt{m^{2}-sin^{2}\theta_{i}^{a}}}\end{array} \right.. \tag{19}\]  
For the water-incident case (Figure 3b), the light is refracted into the atmosphere from the sea medium (\"sea-air\"). The transmission matrix \(\mathbf{t}\mathbf{wa}(\mu^{\prime},\ \varphi^{\prime},\mu,\varphi)\) is expressed as follows:  
\[\mathbf{t}\mathbf{wa}(\mu^{\prime},\ \varphi^{\prime},\mu,\varphi)=\ S(\mu,\mu^{\prime}) \frac{\pi p(\mu_{n},\varphi_{n})}{4\big{|}\mu\big{|}\mu^{\prime}\big{|}\mu_{n}} \mathbf{R}(\pi-\chi_{2})\mathbf{T}\mathbf{F}_{WA}\big{(}\theta_{i}^{w}\big{)}\mathbf{R}(-\chi _{1})\times\frac{cos\theta_{t}^{w}cos\theta_{i}^{w}}{\left(mcos\theta_{t}^{w}-cos \theta_{i}^{w}\right)^{2}}. \tag{20}\]The sea-to-air Fresnel transmission matrix \(\mathbf{TF_{WA}}\Big{(}\partial_{i}^{w}\Big{)}\) is expressed as follows:  
\[\mathbf{TF_{WA}}\Big{(}\partial_{iw}^{i}\Big{)}=\frac{\cos\theta_{w}^{i}}{m\cos \theta_{w}^{i}}\times\left( \begin{array}{cccc}t_{11}^{a}&t_{12}^{a}&0&0\\ t_{12}^{a}&t_{11}^{a}&0&0\\ 0&0&t_{33}^{a}&t_{34}^{a}\\ 0&0&t_{34}^{a}&t_{33}^{a}\end{array}