## 4 Radiative Transfer in Coupled Atmosphere-Water (Including Snow/ice) Systems
### Radiative Transfer Equation--Unpolarized Radiation
#### 4.1.3 A Wind-Blown (Rough) Air-Water Interface--Pseudo-Two-Dimensional BRDF Treatment  
A calm (flat) atmosphere-water interface occurs only for very low wind speeds. A wind-roughened water surface occurs more frequently and is therefore more realistic. Consider a Cartesian coordinate system \((x,y,z)\) in which \(z\) is the vertical coordinate. To calculate the slope distribution, \(p(z_{x},z_{y})\), we consider a plane wave incident on a rough surface characterized by a Gaussian random height distribution \(z=f(x,y)=f({\bf r}_{\perp})\) where \(f({\bf r}_{\perp})=f(x,y)\) with mean height \(\langle z\rangle=\langle f(x,y)\rangle\equiv\langle f({\bf r}_{\perp})\rangle=0\). We now focus on a particular tilted surface facet that makes a polar angles \(\theta_{n}\) with respect to the vertical direction and a relative azimuth angle \(\alpha\). Let the incident solar radiance \(I_{i}\) be at a zenith angle \(\theta_{0}\), the reflected radiance \(I_{r}\) be at zenith angle \(\theta\), and the relative azimuth between \(I_{i}\) and \(I_{r}\) be at angle \(\Delta\phi\). Then the slope of the tilted surface facet has components \(z_{x}\) and \(z_{y}\) defined by:  
\[z_{x}=\frac{\partial z}{\partial x}=\frac{\partial f(x,y)}{\partial x}=\sin \alpha\tan\theta_{n}\qquad z_{y}=\frac{\partial z}{\partial y}=\frac{\partial f (x,y)}{\partial y}=\cos\alpha\tan\theta_{n}.\]  
For an anisotropic distribution of slope components (dependent on the wind direction), we define new slope components as follows:  
\[z^{\prime}_{x}=\cos(\chi)z_{x}+\sin(\chi)z_{y}\qquad z^{\prime}_{y}=-\sin(\chi )z_{x}+\cos(\chi)z_{y}\]  
where \(\chi=\phi_{\rm s}-\phi_{W}=\) rotation from the sun-observation system \((x,y,z)\) and \(\phi_{W}=\) the wind direction. The slope distribution can be written as a Gram-Charlier series [75]:  
\[p(z^{\prime}_{x},z^{\prime}_{y})=\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp{[- \frac{1}{2}(\frac{{z^{\prime}_{x}}^{2}}{\sigma_{x}^{2}}+\frac{{z^{\prime}_{y}} ^{2}}{\sigma_{y}^{2}})]}\Big{[}1-\Delta(\xi,\eta)\Big{]} \tag{124}\]  
where \(\xi=\frac{{z^{\prime}_{x}}}{\sigma_{x}}\), \(\eta=\frac{{z^{\prime}_{y}}}{\sigma_{y}}\), \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) are variances of \(z^{\prime}_{x}\) and \(z^{\prime}_{y}\), and the function \(\Delta(\xi,\eta)\) represents the departure of the slope distribution from a strict two-dimensional (2D) Gaussian due to skewness and peakedness. Please note that in the absence of skewness and peakedness (\(\Delta(\xi,\eta)=0\)) the Gram-Charlier series reduces to a 2D Gaussian distribution:  
\[p(z^{\prime}_{x},z^{\prime}_{y})=\frac{1}{2\pi\sigma_{x}\sigma_{y}}\exp{[- \frac{1}{2}(\frac{{z^{\prime}_{x}}^{2}}{\sigma_{x}^{2}}+\frac{{z^{\prime}_{y}} ^{2}}{\sigma_{y}^{2}})]}. \tag{125}\]Furthermore, for an isotropic slope distribution \(\chi=0\) so that \(z_{x}^{\prime}=z_{x}\) and \(z_{y}^{\prime}=z_{y}\), \(\sigma_{x}=\sigma_{y}\). Therefore \(\sigma^{2}=\sigma_{x}^{2}+\sigma_{y}^{2}=2\sigma_{x}^{2}=2\sigma_{x}\sigma_{y}\), and hence  
\[\frac{1}{2}(\frac{{z_{x}^{\prime}}^{2}}{\sigma_{x}^{2}}+\frac{{z_{y}^{\prime}} ^{2}}{\sigma_{y}^{2}})|=\frac{{z_{x}^{2}+z_{y}^{2}}}{2\sigma_{x}\sigma_{y}}= \frac{{z_{x}^{2}+z_{y}^{2}}}{\sigma^{2}};\ \ \text{and}\ \ \ z_{x}^{2}+z_{y}^{2}=\tan^{2}\theta_{n}(\sin^{2} \alpha+\cos^{2}\alpha)=\tan^{2}\theta_{n}\]  
since \(z_{x}=\sin\alpha\tan\theta_{n}\) and \(\ z_{y}=\cos\alpha\tan\theta_{n}\).