# Designing an Adaptive Underwater Visible Light Communication System +
## 3 UVLC Channel Modeling
The radiative transfer equation (RTE), which considers the linear motion and angular scattering of photons, is given by  
\[\left[\frac{1}{v}\frac{\partial}{\partial t}+\overrightarrow{r}\cdot \
abla\right]I(t,\overrightarrow{r},\overrightarrow{n})=-cl(t, \overrightarrow{r},\overrightarrow{n}+\int_{4\pi}\beta(\overrightarrow{r}, \overrightarrow{n}\overrightarrow{n}^{\prime})I(t,\overrightarrow{r}, \overrightarrow{n})d\overrightarrow{n}^{\prime}E(t,\overrightarrow{r}, \overrightarrow{n}) \tag{3}\]  
where \(v\) is the speed of light, \(t\) is time, \(\overrightarrow{n}\) is the direction vector, \(\overrightarrow{r}\) is the position vector, \(\
abla\) is the divergence operator concerning the position vector \(\overrightarrow{r}\), \(I\) is the radiance, \(\beta\) is the volume scattering function (VSF), and \(E\) is the source radiance [21]. The RTE accounts for both the attenuation of light (due to absorption and scattering) and the redistribution of light through scattering processes. It can be solved through the  
Figure 3: System model and link geometry of the UVLC link [20]. Monte Carlo approach, which works by sending a large number of photons individually through the water body [22], but, despite being accurate and easy to program, it is time-exhausting. The stochastic model is not only easy to program but also time-efficient. It follows an analytical approach to find the scattering function in order to compute the path loss and the CIR. The position of a photon obtained from this model can be calculated by the probability density function (PDF) but the scattering angle is challenging to deduce from the cumulative distribution function (CDF). Hence, the Henyey-Greenstein (HG) function is used in this model to deduce the scattering phase function, which is critical in finding the path loss and signal quality at the receiving end [23; 24; 25]. The HG equation is given by [26; 27]  
\[P_{HG}(\theta)=\frac{1-g^{2}}{4\pi(1+g^{2}-2 gcos(\theta))^{3/2}} \tag{4}\]  
where \(P_{HG}\) is the probability density function of the scattering phase function and \(g\) is the particle asymmetry factor, which depends on the medium characteristics and is equal to the average of \(cos(\theta)\). The scattering angle \(\theta\) can be found by [27]  
\[\theta=cos^{-1}(\frac{1}{2g}[1+g^{2}-(\frac{1-g^{2}}{1+g-2g\xi})^{2}]) \tag{5}\]  
where \(\xi\) is a random variable between 0 and 1. From the scattering angle, the power at the receiver can be deduced as [28]  
\[P_{r}=\frac{P_{t}e^{-cz}D^{2}cos(\phi)}{4z^{2}tan^{2}(\theta)} \tag{6}\]  
where \(P_{t}\) is the transmitted power, \(\theta\) is the scattering angle, \(D\) is the diameter of the receiver, and \(\phi\) is the inclination angle (angle between the receiver's normal with respect to the light beam) [28]. The quality of a signal is evaluated based on its SNR, symbolized as \(S\). \(S\) at the receiver is given by  
\[S=\frac{P_{r}^{2}}{P_{noise}^{2}} \tag{7}\]  
where \(P_{noise}\) denotes the noise power. The SNR \(S\) can be used to find the BER. Different modulation schemes have different BER formulas. The BER for the pulse position modulation (PPM) is given by [29]  
\[\eta_{BER}=\frac{1}{2}erfc\Big{[}\frac{1}{2\sqrt{2}}\sqrt{\frac{SLlog_{2}L}{2} }\Big{]} \tag{8}\]  
where \(L\) is the PPM level (number of slots needed to encode the message bits) and \(erfc\) is the complimentary error function, which is given by [30; 31]  
Figure 2:0  
We adopt the stochastic model due to its lower computational complexity. In this paper, we study the effect of \(\phi\) in Equation (6) on the system SNR and BER performance. Based on Equations (6) and (7), we can conclude that the alignment between the transmitter (TX) and receiver (RX) plays a crucial role. This alignment is the central focus of our study.