## 4 Numerical and Simulation Results  
In this section, we present the simulation results for the outage probability and ergodic capacity for the considered systems. The outage probability of the horizontal cooperative system is studied for a depth of \(10\,\mathrm{m}\) and \(50\,\mathrm{m}\) from the sea surface. Additionally, the outage probability of a vertical cooperative system is studied for a link distance of \(60\,\mathrm{m}\). For all cases, the buoy is placed at the middle of the link. We assume the parameters of the Pacific Ocean at mid-latitudes [13] ([Figure 7.1]), and generate horizontal and vertical channel coefficients from a log-normal distribution under the assumption of weak optical turbulence. The sea temperatures from the surface to \(30\,\mathrm{m}\) depth (upper layer), and from \(30\,\mathrm{m}\) to \(60\,\mathrm{m}\) depth (lower layer), are considered to be \(16.4\,^{\circ}\mathrm{C}\) and \(15.6\,^{\circ}\mathrm{C}\), respectively. Unless otherwise specified, the values of the system configuration and environmental parameters are taken from Table 1. Depending on the considered parameters, the scintillation indexes of upper and lower layers are calculated as \(0.64\) and \(0.57\), respectively. Additionally, for the vertical cascaded link, the e2e scintillation index is calculated as \(0.92\)[31]. Our assumption of weak turbulence is supported by the fact that the scintillation index is less than \(1\)[19; 36; 41]. Figure 2 compares the outage probability (\(P_{o}\)) of the horizontal and vertical systems for the \(\mathcal{M}=1\) case, with the respective benchmark no-relay case. It can be seen that the simulation results overlap with the analytical results for all the cases, thereby validating the correctness of the derived analytical expression of the outage probability in (20). We present the outage probability curves for the vertical link (V) and two horizontal links at different depths, \(10\,\mathrm{m}\) (H-SD) and \(50\,\mathrm{m}\) (H-SD-D). It is observed that the relayed link clearly outperforms the direct SD link in all scenarios. Further, for both the relayed and no-relay cases, the horizontal system outperforms the vertical system. Furthermore, it can be seen that the performance of the horizontal system becomes better as we move deeper. The observed behavior is attributed to the increasing homogeneity of water at greater ocean depths, leading to diminished fluctuations in temperature, salinity, and refractive index. Reduced refractive index fluctuations lead to a smaller scintillation index value and less \begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & \multicolumn{1}{c}{**Values**} \\ \hline Wavelength & \(\lambda\) & \(530\,\mathrm{nm}\) \\ \hline Extinction coefficients & \(c\) & \(0.305\,\mathrm{m}^{-1}\) \\ \hline Correction coefficients & \(\rho_{\circ}\) & \(0.13\) \\ \hline Transmitter beam divergence angle & \(\mathcal{Q}_{F}\) & \(6^{\circ}\) \\ \hline Receiver aperture diameter & \(\mathcal{D}_{R}\) & \(5\,\mathrm{cm}\) \\ \hline Dissipation rate of mean square temperature & \(\chi_{T}\) & \(10^{-3}\,\mathrm{K}^{2}\mathrm{s}^{-3}\) \\ \hline Dissipation rate of turbulent kinetic energy & \(\varepsilon\) & \(10^{-2}\,\mathrm{m}^{2}\mathrm{s}^{-3}\) \\ \hline Relative strength of temperature and salinity fluctuation & \(\omega_{k}\) & \(-3\) \\ \hline Temperature & \(T\) & \(15.6\,^{\circ}\mathrm{C}\) to \(16.4\,^{\circ}\mathrm{C}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of spatial power spectrum. severe fading. As an example, for the targeted outage probability of \(10^{-4}\), using single relay results in significant SNR gains of 37.3 dB, 35.4 dB and 37.2 dB for the V-\(\mathcal{M}=1\), H-\(\mathcal{M}=1\) and H-D-\(\mathcal{M}=1\) cases, respectively, when compared to the respective benchmark cases.