## 3 The Radiative Transfer Equation
### Radiative Transfer Equation--Solution Option Overview  
RTE is used to model light propagation through water, considering interactions with water molecules, dissolved substances, and suspended particles in the context of a water medium. By applying the RTE in the UOWC environment, researchers and engineers can better understand the behaviour of photons' propagation in seawater environments and design more effective communication systems for applications such as underwater sensing, monitoring, and data transmission. As we stated, this must be preceded by understanding the necessary set of modelling elements in general and IOP associated with seawater type, absorption coefficients, scattering coefficients, and turbulence effects [5; 6; 7; 8; 9; 10; 11].  
The RTE, also called the n-squared law for radiance, is a crucial law of geometrical radiometry. Its derivation is based on two mediums separated by a transparent surface. UOWC systems are primarily concerned with the time-independent RTE of horizontally homogeneous aquatic environments coupled with a fixed refraction index. The derivation of RTE starts from [PERSON]'s law, but it is complex and lengthy and can be found in [66; 91; 92]. As demonstrated in [66], the usual form of the RTE is  
\[\cos\theta\frac{dL(z,\theta,\varnothing)}{c\;dz}=S(\theta,\varnothing)-L(z, \theta,\varnothing)+w_{0}\int_{4\pi}\widetilde{\beta}\big{(}\left\{\;\theta^{ \prime},\varnothing^{\prime}\right\}\rightarrow\left\{\theta,\varnothing\right\} \big{)}\;L(z,\;\theta^{\prime},\varnothing^{\prime})d\Omega\big{(}\theta^{ \prime},\varnothing^{\prime}) \tag{43}\]  
where \(L(z,\theta,\varnothing)\) denotes the light irradiance through the path \((\theta,\varnothing)\), \(w_{0}\) represents the single-scattering albedo, \(c\) stands for the attenuation coefficient defined in Equation (1), \(\Omega\) is a solid angle, \(\widetilde{\beta}(\theta,\varnothing)\) is the SPF from an incident direction \((\;\theta^{\prime},\varnothing^{\prime})\) to a final direction \((\theta,\varnothing)\), and \(S(\theta,\varnothing)\) is an effective source function.  
The LHS of Equation (43) represents a change in energy over a differential length. This is equal to the energy of any internal sources minus the extinction losses caused by attenuating the first and second terms on the RHS, respectively [66]. However, the third term on the RHS represents the contributions from photons that had previously been scattered away from the main beam of light \(S(\theta,\varnothing)\), only to be scattered back towards the point of reception. This term is crucial in determining the temporal dispersion of a UOWC link, which implies that this term is the core of MCS, which defines the numerical solution of the RTE in Equation (43).  
The 1D, time-independent RTE is a linear two-point boundary value problem. This implies that the boundary conditions describe the radiance at the top and bottom of the ocean (the two spatial points \(\left\{\;\theta^{\prime},\varnothing^{\prime}\right\}\) and \(\left\{\theta,\varnothing\right\}\)), and the propagation of radiance within the medium between the boundaries is described by the linear integral-differential RTE [5; 6; 7].