## 4 Radiative Transfer in Coupled Atmosphere-Water (Including Snow/ice) Systems
### Radiative Transfer Equation--Polarized Radiation
The first term in Equation (140) is due to the incident beam \(\mathbf{S}_{b}\) that has been attenuated through the atmosphere by the factor \(e^{-\tau_{a}/\mu_{0}}\), transmitted into the water by the factor \(\mathbf{T}_{\text{F}}(-\mu_{0},m_{\text{rel}})\,\frac{\mu_{0}}{\mu_{0}^{w}}\), further attenuated by the factor \(e^{-(\tau-\tau_{a})/\mu_{0}^{w}}\) to reach depth \(\tau\) in the water, and singly scattered from the direction \((-\mu_{0}^{w},\phi_{0})\) into the direction \((u,\phi)\) which explains the factor \(\frac{\varpi(\tau)}{4\pi}\,\mathbf{P}(\tau,-\mu_{0}^{w},\phi_{0};u,\phi)\). The second term in Equation (140) is due to thermal emission in the water. Isolation of Azimuth Dependence  
We start by expanding the scattering phase matrix in a Fourier series:  
\[\mathbf{P}(u^{\prime},u;\phi^{\prime}-\phi)=\sum_{m=0}^{M}\big{\{}\mathbf{P} _{\text{c}}^{m}(u^{\prime},u)\cos m(\phi^{\prime}-\phi)+\,\mathbf{P}_{\text{s }}^{m}(u^{\prime},u)\sin m(\phi^{\prime}-\phi)\big{\}}. \tag{141}\]  
To isolate the azimuth dependence of the radiation field we expand the Stokes vector \(\mathbf{I}(\tau,u,\phi)\) in Equation (135) and the source term \(\mathbf{Q}_{1}(\tau,u,\phi)\) in Equation (137) or \(\mathbf{Q}_{2}(\tau,u,\phi)\) in Equation (140) in a Fourier series in a manner similar to the expansion of the scattering phase matrix in Equation (141):  
\[\mathbf{I}(\tau,u,\phi)=\sum_{m=0}^{M}\bigg{\{}\mathbf{I}_{\text{c}}^{m}(\tau,u)\cos m(\phi_{0}-\phi)+\,\mathbf{I}_{\text{s}}^{m}(\tau,u)\sin m(\phi_{0}- \phi)\bigg{\}} \tag{142}\]\[\mathbf{Q}_{p}(\tau,u,\phi)=\sum_{m=0}^{M}\left\{\mathbf{Q}_{\mathrm{cp}}^{m}(\tau,u )\cos m(\phi_{0}-\phi)\right.+\left.\mathbf{Q}_{\mathrm{sp}}^{m}(\tau,u)\sin m (\phi_{0}-\phi)\right\} \tag{143}\]  
where the subscript s or c denotes sine or cosine mode and the subscript p indicates the slab, p = 1 for slab\({}_{1}\), and p = 2 for slab\({}_{2}\). Using these expansions it can be shown that we obtain the following equations for the Fourier components (see [3] for details)  
\[u\frac{d\,\mathbf{I}(\tau,u,\phi)}{d\tau} = \mathbf{I}(\tau,u,\phi)-\mathbf{S}(\tau,u,\phi) \tag{135}\]0 \[u\frac{d\,\mathbf{I}(\tau,u,\phi)}{d\tau} = \mathbf{I}(\tau,u,\phi)-\mathbf{S}(\tau,u,\phi) \tag{135}\]1 \[u\frac{d\,\mathbf{I}(\tau,u,\phi)}{d\tau} = \mathbf{I}(\tau,u,\phi)-\mathbf{S}(\tau,u,\phi) \tag{135}\]2 (145) \[u\frac{d\,\mathbf{I}(\tau,u,\phi)}{d\tau} = \mathbf{I}(\tau,u,\phi)-\mathbf{S}(\tau,u,\phi) \tag{135}\]3  
The discrete ordinate method consists of replacing the integration over \(u^{\prime}\) by a discrete sum using Gaussian quadrature points \(u_{j}\) (the discrete ordinates) and corresponding weights \(w_{j}\). One obtains for each Fourier component:  
\[u\frac{d\,\mathbf{I}(\tau,u,\phi)}{d\tau} = \mathbf{I}(\tau,u,\phi)-\mathbf{S}(\tau,u,\phi) \tag{135}\]4 \[\qquad-\,\frac{\varpi(\tau)}{4}\,\sum_{ \begin{subarray}{c}j=-N\\ j\
eq 0\end{subarray} }^{N}\,w_{j}\bigg{\{}(1+\delta_{0m})\,\mathbf{P}_{ \mathrm{c}}^{m}(\tau,u_{j},u_{i})\,\mathbf{I}_{\mathrm{c}}^{m}(\tau,u_{j})\] \[\qquad-\,\mathbf{P}_{\mathrm{s}}^{m}(\tau,u_{j},u_{i})\,\mathbf{ I}_{\mathrm{s}}^{m}(\tau,u_{j})\bigg{\}}\,-\,\mathbf{Q}_{\mathrm{c}}^{m}( \tau,u_{i}),\quad i=\pm 1,\ldots,\pm N\] \[u_{i}\,\frac{d\mathbf{I}_{\mathrm{s}}^{m}(\tau,u_{i})}{d\tau}= \mathbf{I}_{\mathrm{s}}^{m}(\tau,u_{i})\] \[\qquad-\,\frac{\varpi(\tau)}{4}\,\sum_{ \begin{subarray}{c}j=-N\\ j\
eq 0\end{subarray} }^{N}\,w_{j}\bigg{\{}\mathbf{P}_{\mathrm{c}}^{m}(\tau,u_{j},u _{i})\,\mathbf{I}_{\mathrm{s}}^{m}(\tau,u_{j})\] (147) \[\qquad+\,\mathbf{P}_{\mathrm{s}}^{m}(\tau,u_{j},u_{i})\,\mathbf{ I}_{\mathrm{c}}^{m}(\tau,u_{j})\bigg{\}}\,-\,\mathbf{Q}_{\mathrm{s}}^{m}(\tau,u_{i}), \quad i=\pm 1,\ldots,\pm N.\]  
The convention for the indices of the quadrature points is such that \(u_{j}<0\) for \(j<0\), and \(u_{j}>0\) for \(j>0\). These points are distributed symmetrically about zero, i.e., \(u_{-j}=-u_{j}\). The corresponding weights are equal, i.e., \(w_{-j}=w_{j}\).