## 3 The Radiative Transfer Equation  
The energy conservation of a beam travelling in a uniform medium may be described using RTE. The solution of the RTE can be obtained analytically as well as numerically, as mentioned before. Finding a precise analytical solution is challenging because the RTE is an integral-differential equation with multiple independent variables. Analytical solutions to the RTE, among them being BLL, Single-Scattering Approximation (SSA), Two-Stream Approximation (TSA), Small-Scattering Angle Approximation (SSAA), and diffusion approximation, provide valuable insights into light behaviour in UOWC channels. Nevertheless, these solutions are not generic, and they have specialised applications and limits. For example, the SSAA, which implies the assumption of LOS channel link configuration, is appropriate for underwater environments with small scattering angles, making it useful in clear water with few scattering particles. The TSA is effective for homogeneous underwater environments with clearly established optical properties but may not correctly reflect diverse environments or complex scattering events. The diffusion approximation is valid where many scattering events predominate, but it may lose accuracy in areas with strong absorption as a result [29]. The study in [87] provided an analytical solution to the RTE using the adjusted Stokes vector. The system accounts for multiplied scattering as well as influences on light polarisation. The numerical findings of this model reveal that the ISI and BER are data rate and link length functions. Such a finding can be utilised to anticipate numerous optical performance measures about UOWC models, like the farthest distance for communication at a given data rate and BER at the photodiode input. Studies [12; 13] presented a beam-spread function (BSF) that describes the overall scattering profile of the optical power as a function of distance and water turbidity for laser-based UOWC by analytically resolving the RTE. To simplify the derivation, the LOS link configuration (i.e., small scattering angle) assumption was used by assuming that the phase function sharply peaked at narrow angles and that there was sufficient absorption to eliminate the contribution of scattering events at broad angles. Additionally, it is assumed that the irradiance distribution is symmetric along the beam axis, and it accounts for all multiple-scattering events occurring at small forward angles. This analytical system demonstrates the connection between output optical power and link range for various transceiver pointing accuracies. Experiments in a water tank also verified it. Using the Stokes vector, an analytic solution was established in [87] by considering light polarisation and multiple scattering. Because of the difficulties of acquiring a precise analytical solution to RTE [88], which is usually accomplished via creating hypotheses along with approximations [89], RTE solutions in the numerical form of RTE have received increased interest. According to [PERSON] et al. [10], the following numerical approaches are common for solving RTEs: the MC method, Discrete Ordinate Model (DOM), invariant-imbedding method, and the stochastic model. The primary distinction between these strategies is how the RTE is solved mathematically, how boundary conditions are handled close to the water's surface, and how long the resulting computer run time takes. The four numerical solutions are discussed in Sections 3.4 and 4, and Table 5[90] summarises the upsides and downsides of these approaches. The IOP set listed in Table 3 provides a full description of the optical properties of a water volume as needed for understanding and predicting underwater optical beams within the radiative transfer theory (RTT) framework. The total values of the various IOP variables are relevant to radiative transfer modelling and, consequently, to RTE formulation. However, the cumulative and convolutive impacts of each of these variables are by the various constituents of a water medium.