# O2O: An Underwater VLC Approach in Baltic and North Sea
## 3 The Channel Condition Model  
The attenuation coefficient \(I_{l}\) depends on the line of sight distance \(dt_{NB}\) between the transceivers and the extinction coefficient \(c(\lambda)\). Moreover, \(c(\lambda)\) depends on the water medium with corresponding ocean water \(N\) and \(B\). The notations \(NandB\) are denoted as the North and Baltic ocean parameters. In this study, we consider both of the water mediums as clear ocean waters for the simulation. Therefore, \(I_{l}\) for both oceans follows [PERSON]'s law as,  
\[I_{l_{NB}}=\exp\{-c_{NB}(\lambda)dt_{NB}\}. \tag{3}\]  
In this work, a GG probability distribution function is used to model turbulence channel fading for the communication link. It is noteworthy that the GG fading model is used to moderate-to-strong turbulence channel conditions. The expression for probability density function (PDF) of turbulence channel condition is given as [21],  
\[ \begin{split}& f_{I_{l}(I_{l_{NB}})}=2\frac{(\alpha_{NB}\beta_{NB}) \frac{(\alpha_{NB}+\beta_{NB})}{2}}{\Gamma(\alpha_{NB})\Gamma(\beta_{NB})}(I_{ l_{NB}})^{\frac{(\alpha_{NB}+\beta_{NB})}{2}-1}\\ & K_{\alpha_{NB}-\beta_{NB}}\left(2\sqrt{\alpha_{NB}\beta_{NB}I_ {l_{NB}}}\right),\end{split}  \tag{4}\]  
where \(\Gamma(\cdot)\) is the GG function and \(K_{(\alpha_{NB}-\beta_{NB})}(\cdot)\) is denoted as the modified Bessel function of the second kind, \(\alpha_{NB}\) and \(\beta_{NB}\) are the large and small scale eddies variances, respectively. The parameters \(\alpha_{NB}\) and \(\beta_{NB}\) particles are modeled as (5) and (6) on the given practical values at different temperature salinity, and depth as in [22]. In UVLC link, we consider turbulence fading under spherical wave model, therefore the large and small scale factors formulated as follows ([23] Equation (47)),  
\[\alpha_{NB}=\left[\exp\left(\frac{0.17\chi_{I_{NB}}^{2}}{\left(1+0.167\chi_{I_ {NB}}^{\frac{2}{5}}\right)^{\frac{7}{6}}}\right)-1\right]^{-1}, \tag{5}\]  
Figure 2: The proposed UVLC system model, where a source in Baltic sea propagates the signal through optical beam towards to the receiver based in North ocean. The investigate communication scenario is considered in strong channel conditions. \[\beta_{NB}=\left[exp\left(\frac{0.225\chi_{I_{NB}}^{2}}{\left(1+0.259d^{2}\chi_{I_{ NB}}^{\frac{5}{6}}\right)}\right)-1\right]^{-1}. \tag{6}\]  
In (5) and (6), \(\chi_{I_{NB}}^{2}\) is represented the scintillation index and known as the Rytov variance. However, the Rytov variance is calculated (Equation (22) in [2]) as follows,  
\[\chi_{I_{NB}}^{2}=8\pi^{2}k_{0}^{2}d_{0}\int_{0}^{1}\int_{0}^{\infty}\kappa \Phi_{n}(\kappa)\left\{1-cos\left[\frac{d_{0}\kappa^{2}}{k_{0}}(\xi-\xi^{2}) \right]\right\} \tag{7}\] \[\times[1+C_{1}(\kappa\eta)^{\frac{2}{3}}]\left(\omega^{2}\exp(-A_ {T}\delta)+d_{r}\exp(-A_{S}\delta)\right.\] \[\left.-\omega(d_{r}+1)\exp(-A_{TS}\delta)\right)d\kappa d\xi,\]  
where, in above expression (7) the wave number \(k_{0}=2\pi/\lambda\), link distance \(d_{0}\), magnitude of the spatial frequency \(\kappa\), and spectrum spectrum model \(\Phi_{n}(\kappa)\) are defined, respectively. Additionally, \(\xi\), \(\eta\), \(C_{0}\), \(C_{1}\) are represented as normalized distance variable, Kolmogorov micro-scale length, and constants with relative strength of temperature and salinity \(\omega\), and eddy diffusivity ratio \(d_{r}\). Furthermore, the parameters \(A_{T}=C_{0}C_{1}^{-2}P_{T}^{-1}\), \(A_{S}=C_{0}C_{1}^{-2}P_{S}^{-1}\) are dependent on temperature and salinity Prandtl numbers, while \(\delta=1.5C_{1}^{2}(\kappa\eta)^{45}+C_{1}^{2}(\kappa\eta)^{2}\) is used in (7), respectively. Throughout the expression in (7), the large and small scale parameters are calculated for the given temperature and salinity as in [1], and further used in the investigated BER, and outage probability of closed form expressions as (18) and (21).