## 3 The Radiative Transfer Equation
### RTE Numerical Solutions
#### 3.4.3 Stochastic Model  
Various stochastic models [24; 25; 109] based on the probabilistic character of photon trajectory have been developed [29]. The research team of [24] suggested a stochastic model of UOWC systems to describe the spatial-temporal probability distribution of light passing into turbid coastal water, both with and without scattering. This model has been validated using MCS just for non-scattering and single-scattering components of UOWC systems. With a long transmission distance or turbid water, multiple scattering is the dominant behaviour of scattering behaviour. The same research group of [25] looked at all components involving non-scattering, single scattering, and multiple scattering and then proposed a more comprehensive stochastic channel model. A stochastic-analytic method was utilised to conclude the model, which entails calculating the PDFs of two crucial variables, scattering angle \(\theta_{s}\) and step distance \(\tau\), with respect to each photon. They reported that the proposed stochastic channel model requires less computation than the time-consuming MCS method. Thus, it helps evaluate the efficiency of UOWC systems.  
Furthermore, the authors of [25] studied a highly aligned geometry of UOWC links, each having a 532 nm parallel light source with a \(0^{\circ}\) divergence angle and a receiver with apertures' size of 0.25, 0.5, and 0.75 m. Their work is based on the following newly derived stochastic modelling equation:  
\[P_{n}(t,x,y)=e^{-c\tau t}\prod_{i=1}^{n-1}\biggl{[}\int_{0}^{\infty}d\mu_{i-1 }\int_{0}^{\pi}d\theta_{i}\int_{0}^{2\pi}d\varphi_{i}\biggr{]}\cdot\ \biggl{(}\frac{b}{2\pi}\biggr{)}^{n}\int_{0}^{\infty}d\mu_{n-1}\prod_{i=1}^{n- 1}\bigl{[}f_{0}(\theta_{i})\sin\theta_{i}\ I_{i}^{\prime}\bigr{]}F \tag{55}\]  
In reviewing the reported results in [25], we can observe that the spatial and temporal profiles of the proposed model match well with MCS in turbid seawater such as coastal and harbor water. The derivation in [25] assumed that all photons have the same launching direction confined by the transmitter's divergence angle and elevation angle given in Equation (16)--[25]. Such an assumption implies that their formulation assumed a point optical source and does not account for the optical source profile. Such an assumption is plausible for a collimated beam and not otherwise.  
It should be noted that the stated receiver aperture sizes are beyond the practical optical receiver system limits. The possible reason for large receiver aperture sizes is to secure a lower path loss (PL) for the underlying scenario. Hence, the reported temporal response for the large aperture is greater than that of the smaller aperture for the same FOV scope. While a large receiver aperture leads to a higher optical SNR, it causes a tail elongation of the received optical pulse. This optical pulse distortion causes the build-up of an ISI that eventually degrades the quality of the electronic received signal at the input of the decision unit (see Figure 3A). Such signal degradation reduces the system BER unless we optimise the electronic equaliser-filter unit before the decision unit [50]. In conclusion, from an overall receiver performance perspective, a wide-scope inclusion of multiple scattering is not a good approach to compensate for the propagation PL.