## 4 Radiative Transfer in Coupled Atmosphere-Water (Including Snow/ice) Systems
### Radiative Transfer Equation--Polarized Radiation  
To generalize Equation (112) to apply to polarized radiation, we note that the multiple scattering term \(S^{\text{ms}}(\tau,u,\phi)=\frac{\omega(\tau)}{4\pi}\int_{0}^{2\pi}d\phi^{ \prime}\int_{-1}^{1}du^{\prime}p(\tau,u^{\prime},\phi^{\prime};u,\phi)I(\tau, u^{\prime},\phi^{\prime})\) in Equation (113) must be replaced by  
\[\mathbf{S}^{\text{ms}}(\tau,u,\phi)=\frac{\varpi(\tau)}{4\pi}\int_{0}^{2\pi} d\phi^{\prime}\int_{-1}^{1}du^{\prime}\mathbf{P}(\tau,u^{\prime},\phi^{ \prime};u,\phi)\mathbf{I}(\tau,u^{\prime},\phi^{\prime}) \tag{134}\]  
where \(\mathbf{I}(\tau,u^{\prime},\phi^{\prime})\) is the Stokes vector, and \(\mathbf{P}(\tau,u^{\prime},\phi^{\prime};u,\phi)\) is the scattering phase matrix (see Section 3.2). The first element of the vector \(\mathbf{S}^{\text{ms}}(\tau,u,\phi)\) represents the energy per unit solid angle, per unit frequency interval, and per unit time that is scattered by a unit volume in the direction \((u=\cos\theta,\phi)\). Hence, in a plane-parallel (slab) geometry, the integro-differential equation for polarized radiative transfer is expressed in terms of a Stokes vector \(\mathbf{I}(\tau,u,\phi)\) as  
\[u\frac{d\,\mathbf{I}(\tau,u,\phi)}{d\tau} = \mathbf{I}(\tau,u,\phi)-\mathbf{S}(\tau,u,\phi) \tag{135}\]  
where the source vector is  
\[\mathbf{S}(\tau,u,\phi) = \frac{\varpi(\tau)}{4\pi}\,\int_{0}^{2\pi}d\phi^{\prime}\,\int_{- 1}^{1}du^{\prime}\,\mathbf{P}(\tau,u^{\prime},\phi^{\prime};u,\phi)\mathbf{I}( \tau,u^{\prime  
The first term on the right hand side of Equation (137) describes the incident beam \(\mathbf{S}_{b}\) in direction \((-\mu_{0},\phi_{0})\), which is attenuated at depth \(\tau\) by a factor \(e^{-\tau/\mu_{0}}\) and undergoes single scattering into the direction \((u,\phi)\). For an unpolarized incident beam \(\mathbf{S}_{b}\) has the form  
\[\mathbf{S}_{b}=[I_{0}/2,I_{0}/2,0,0]^{T}\;\;\text{or}\;\;[I_{0},0,0,0]^{T} \tag{138}\]  
where the first or second expression corresponds to the choice of Stokes vector representation, \([I_{\parallel},I_{\perp},U,V]^{T}\) or \([I,Q,U,V]^{T}\). The second term on the right hand side of Equation (137) is due to thermal emission, which is unpolarized, and \(\mathbf{S}_{t}(\tau)\) is given by  
\[\mathbf{S}_{t}(\tau)=[B(T(\tau))/2,B(T(\tau))/2,0,0]^{T}\;\;\text{ or}\;\;\;[B(T(\tau)),0,0,0]^{T} \tag{139}\]  
where \(B\) is the Planck function, and where the first or second expression corresponds to the choice of Stokes vector representation. We have set \(\mu_{0}\equiv|u_{0}|\equiv|\cos\theta_{0}|\), where \(\theta_{0}\) is the polar angle of the incident light beam. The third term on the right hand side of Equation (137) describes radiation due to the incident beam \(\mathbf{S}_{b}\) that has been attenuated by the factor \(e^{-\tau_{a}/\mu_{0}}\) before reaching the air-water interface, undergoing Fresnel reflection given by the reflection matrix \(\mathbf{R}_{\text{F}}(-\mu_{0},m_{\text{rel}})\), attenuated by the factor \(e^{-(\tau_{a}-\tau)/\mu_{0}}\) to reach the level \(\tau\) in the atmosphere, and finally singly scattered from direction \((\mu_{0},\phi_{0})\) into direction \((u,\phi)\) described by the factor \(\frac{\phi(\tau)}{4\pi}\mathbf{P}(\tau,\mu_{0},\phi_{0};u,\phi)\). Thus, the incident beam propagates though the entire atmosphere and a portion of it is reflected upwards by the interface to reach depth \(\tau\) in the atmosphere, which explains the factor \(e^{-(2\tau_{a}-\tau)/\mu_{0}}\). In the lower slab (slab\({}_{2}\), water), the source term becomes  
\[\mathbf{Q}_{2}(\tau,u,\phi) = \frac{\varpi(\tau)}{4\pi}\,\mathbf{P}(\tau,-\mu_{0}^{w},\phi_{0} ;u,\phi)\mathbf{S}_{b}\,e^{-\tau_{a}/\mu_{0}} \tag{140}\] \[\times\,\mathbf{T}_{\text{F}}(-\mu_{0},m_{\text{rel}})\,\frac{ \mu_{0}}{\mu_{0}^{w}}e^{-(\tau-\tau_{a})/\mu_{0}^{w}}\] \[+\,\left[1-\varpi(\tau)\right]\mathbf{S}_{t}(\tau)\]  
where \(\mathbf{T}_{\text{F}}(-\mu_{0},m_{\text{rel}})\) is the Fresnel transmission matrix.