## 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
Thus, we obtain a 1D Gaussian:  
\[p(\mu_{n},\sigma)=p(z_{x},z_{y})=\frac{1}{\pi\sigma^{2}}\exp{(-\frac{{z_{x}^{2 }+z_{y}^{2}}}{\sigma^{2}})}=\frac{1}{\pi\sigma^{2}}\exp{(-\frac{\tan^{2} \theta_{n}}{\sigma^{2}})}=\frac{1}{\pi\sigma^{2}}\exp{(-\frac{1-\mu_{n}^{2}}{ \sigma^{2}\mu_{n}^{2}})}. \tag{126}\]  
where  
\[\mu_{n}=\cos\theta_{n} = \frac{\mu+\mu^{\prime}}{\sqrt{2(1-\cos\Theta)}} \tag{127}\] \[\cos\Theta = -\mu\mu^{\prime}+\sqrt{1-\mu^{2}}\sqrt{1-\mu^{\prime 2}}\cos( \Delta\phi). \tag{128}\]  
_A Pseudo Two-Dimensional (Wind-Direction Dependent) Treatment of the BRDF_  
At the bottom of the atmosphere (\(\tau=\tau_{\text{atm}}\)), the upward reflected radiance \(I_{\text{refl}}^{+}(\tau_{\text{atm}},\mu^{\prime},\phi^{\prime})\) is connected to the downward incident diffuse radiance \(I_{\text{inc}}^{-}(\tau_{\text{atm}},\mu,\phi)\) and the attenuated direct radiance \(F_{0}e^{-\tau_{\text{atm}}/\mu_{0}}\) through the sea surface reflection that is described by the BRDF \(\rho(\mu,\mu^{\prime},\Delta\phi)\):  
\[I_{\text{refl}}^{+}(\tau_{\text{atm}},\mu^{\prime},\phi^{\prime})=\mu_{0}\ \rho(\mu_{0},\mu^{\prime},\phi^{\prime})\ F_{0}\ e^{-\tau_{\text{atm}}/\mu_{0}}+ \int_{0}^{2\pi}\int_{0}^{1}\mu\ \rho(\mu,\mu^{\prime},\Delta\phi)I_{\text{inc}}^{-}(\tau_{\text{atm}},\mu, \phi)d\mu d\phi \tag{129}\]  
where \(\Delta\phi=\phi^{\prime}-\phi\), and the solar azimuth angle was set to \(\phi_{0}=0^{\circ}\) so that \(\Delta\phi=\phi^{\prime}-\phi_{0}=\phi^{\prime}\) for the direct beam reflection \(\rho(\mu_{0},\mu^{\prime},\phi^{\prime})\). The 1D BRDF \(\rho(\mu,\mu^{\prime},\Delta\phi^{\prime})\) can be written as:  
\[\rho(\mu,\mu^{\prime},\Delta\phi) = \frac{1}{4\mu^{\prime}\mu(\mu_{n})^{4}}\cdot p(\mu_{n},\sigma) \cdot\rho_{\text{F}}\cdot s(\mu,\mu^{\prime},\sigma) \tag{130}\] \[z^{\prime}_{x}=\cos(\chi)z_{x}+\sin(\chi)z_{y}\qquad z^{\prime}_{y}=-\sin(\chi )z_{x}+\cos(\chi)z_{y}\]0 (131) \[z^{\prime}_{x}=\cos(\chi)z_{x}+\sin(\chi)z_{y}\qquad z^{\prime}_{y}=-\sin(\chi )z_{x}+\cos(\chi)z_{y}\]1  
Here \(\mu=\cos\theta\), \(\theta\) being the view zenith angle for the incident light, \(\mu^{\prime}=\cos\theta^{\prime}\), \(\theta^{\prime}\) being the view zenith angle for the reflected light, \(\Delta\phi\) is the relative azimuth angle, and \(\Theta\) is the scattering angle. In Equation (130), \(\rho_{\text{F}}\) is the Fresnel reflectance (see Equation (121)), \(s(\mu,\mu^{\prime},\sigma)\) describes the effect of shadowing, and \(p(\mu_{n},\sigma)\) is the surface slope distribution (Equation (126)). In a plane-parallel (1D) geometry, the radiance and the BRDF depend only on the difference \(\Delta\phi=\phi^{\prime}-\phi\) in azimuth between the direction of incidence \((\theta^{\prime},\phi^{\prime})\) and observation \((\theta,\phi)\). Hence, in a strict plane-parallel geometry, it is impossible to model a wind-direction dependent (azimuthally-asymmetric) BRDF. As a consequence, most treatments of water surface roughness effects are limited to a 1D treatment [76; 77]. In the 1D discrete ordinate method, the radiance \(I(\tau,\mu,\phi)\) and the BRDF \(\rho(\mu,\mu^{\prime},\Delta\phi)\) are expanded into a Fourier cosine series to isolate the azimuth dependence [14]. The pseudo two-dimensional treatment of the BRDF employs a 2D BRDF to compute the direct beam reflectance, but a 1D (Fourier expanded) BRDF to compute the reflectance due to diffuse, multiply scattered light. A post-processing step, which corrects the direct beam reflectance (1D \(\rightarrow\) 2D), is used for implementation. This post-processing method is similar to the Nakajima-Tanaka (NT) single-scattering correction [78], which retains the multiply scattered radiance, but corrects the singly scattered radiance without considering boundary reflection. In DISORT3, [PERSON] et al. [79] improved the NT procedure by adding a BRDF correction and the same strategy was used to add a 2D BRDF correction [80].