## 2 Theory and Modeling
### The Reflection Matrix of Wave Facets  
The reflection matrix of wave facets \(\mathbf{r}\) describes a situation where incident light is reflected into direction \((\mu,\varphi)\) from direction \((\mu\prime\), \(\varphi\prime)\), where \(\mu=\cos(\theta)\), \(\mu\prime=\cos(\theta\prime)\), \(\theta\) and \(\theta\prime\) are zenith angles measured from the positive z axis, and \(\varphi\) and \(\varphi\prime\) are azimuth angles measured from the positive \(x\)-axis in the anticlockwise direction when looking downward. \(\mathbf{r}\) is written as follows:  
\[\mathbf{r}(\mu^{\prime},\varphi^{\prime},\ \mu,\varphi)=\ S(\mu,\mu^{\prime}) \frac{\pi p(\mu_{n},\varphi_{n})}{4\big{|}\big{|}\mu\big{|}\mu^{\prime}\big{|} \mu_{n}}\mathbf{R}(\pi-\chi_{2})\mathbf{R}\mathbf{F}(\theta_{i})\mathbf{R}(-\chi_{1}), \tag{3}\]  
where \(p(\mu_{n},\varphi_{n})\) is given by Equation (1), and \(S(\mu,\mu^{\prime})\) is the shadowing function accounting for the fact that the lidar light may be blocked by wave facet slopes [27; 28]. \(S(\mu,\mu^{\prime})\) is expressed as follows:  
\[S(\mu,\mu^{\prime})=\frac{1}{1+\Lambda\!\left(\big{|}\mu\big{|}\right)+\Lambda \!\left(\big{|}\mu^{\prime}\big{|}\right)}, \tag{4}\]  
The air-to-air Fresnel reflection matrix \(\mathbf{RF_{AA}}\!\left(\theta_{i}^{a}\right)\) is a function of the incidence angle \(\theta_{i}^{a}\) in the reflection plane and is expressed as follows:  
\[\mathbf{RF_{AA}}\!\left(\theta_{i}^{a}\right)=\left( \begin{array}{cccc}r_{11}^{a }&r_{12}^{a}&0&0\\ r_{12}^{a}&r_{11}^{a}&0&0\\ 0&0&r_{33}^{a}&r_{34}^{a}\\ 0&0&r_{34}^{a}&r_{33}\\ \end{array} \right) \tag{9}\]  
where  
\[\left\{ \begin{array}{l}r_{11}^{a}\left(\theta_{i}^{a}\right)=0.5\,\times[ \left(r_{l}^{a}\left(\theta_{i}^{a}\right)^{2}+\left(r_{r}^{a}\left(\theta_{i} ^{a}\right)\right)^{2}\right]\\ r_{11}^{a}\left(\theta_{i}^{a}\right)=0.5\,\times[\left(r_{l}^{a}\left(\theta_ {i}^{a}\right)^{2}-\left(r_{r}^{a}\left(\theta_{i}^{a}\right)\right)^{2} \right]\\ r_{33}^{a}\left(\theta_{i}^{a}\right)=Re\!\left(r_{l}^{a}\left(\theta_{i}^{a} \right)\times r_{r}^{a}\left(\theta_{i}^{a}\right)\right)\\ r_{34}^{a}\left(\theta_{i}^{a}\right)=Im\!\left(r_{l}^{a}\left(\theta_{i}^{a} \right)\times r_{r}^{a}\left(\theta_{i}^{a}\right)\right)\end{array} \right. \tag{10}\]  
In Equation (10), \(Re\!\left(\right)\) and \(Im\!\left(\right)\) are the real and imaginary parts, respectively. The asterisk denotes the complex conjugate. \(r_{l}^{a}\left(\theta_{i}\right)\) and \(r_{r}^{a}\left(\theta_{i}\right)\) are the reflection coefficients of the parallel and vertical components of reflected light, respectively. \[\left\{ \begin{array}{l}r_{l}^{a}\!\left(\theta_{i}^{a}\right)=\frac{m^{2} \cos\theta_{i}^{a}-\sqrt{m^{2}-sin^{2}\theta_{i}^{a}}}{m^{2}\cos\theta_{i}^{a }+\sqrt{m^{2}-sin^{2}\theta_{i}^{a}}}\\ r_{r}^{a}\!\left(\theta_{i}^{a}\right)=\frac{\cos\theta_{i}^{a}-\sqrt{m^{2}-sin ^{2}\theta_{i}^{a}}}{\cos\theta_{i}^{a}+\sqrt{m^{2}-sin^{2}\theta_{i}^{a}}} \end{array} \right., \tag{11}\]  
where \(m\) is the water refractive index. \(\theta_{i}^{a}\) can be calculated using the formulas in the Appendix A.  
Figure 2: Geometry of the propagation of the lidar light across a rough sea surface for the air-incident case. (**a**) Reflection “air to air”. (**b**) Refraction “air to sea”. The notations are as follows: \(\theta_{i}^{a}\), \(\theta_{r}^{a}\), and \(\theta_{t}^{a}\) are respectively the incidence angle, reflection angle, and refraction angle relative to the wave facet normal vector \(\overset{\rightarrow}{n}\) in the scattering plane. \(\overset{\rightarrow}{s}\), \(\overset{\rightarrow}{v}\), and \(\overset{\rightarrow}{t}\) are the unit vectors of the incident, reflected, and transmitted light, respectively. \(\theta^{\prime}\) is the zenith angle of the incident light, \(\theta\) is the zenith angle of the exit light (reflected light for the reflection matrix and transmitted light for the transmission matrix), and \(\varphi\) is the azimuth angle.