## 4 Numerical and Simulation Results
Moreover, the H-\(\mathcal{M}=1\) and H-D-\(\mathcal{M}=1\) systems perform better than the V-\(\mathcal{M}=1\) system by 5.2 and 7 dB in the case of a relayed link, and by 4.1 dB and 7.1 dB in the case of no relay links, respectively. The horizontal system at a depth of 50 m has an SNR gain of 1.8 dB and 2.7 dB in the relayed and non-relayed systems, respectively, over the horizontal system at a depth of 10 m due to a fall in temperature, which reduces the turbulence strength. In Figure 3, the analytical and simulated outage probability curves are plotted with respect to \(\psi_{\circ}\) for \(1\leq\mathcal{M}\leq 5\). As expected, the simulation and analytical results are overlapping. We present the curves for the vertical cascaded link and compare it with a horizontal link at a depth of 10 m. It is observed that for the target outage probability of \(10^{-4}\), employing 2 relays results in an SNR gain of 8.4 dB and 6 dB for the vertical link and horizontal link, respectively, when compared to the single relay case. It is further observed that, for \(\mathcal{M}\geq 3\), one extra relay is required in the vertical link compared to the horizontal link to achieve the same outage probability for a given SNR and link distance. In Figure 4, the effect of the mean square temperature (\(X_{T}\)) on the outage probability at different depths is studied. It is observed that for a target outage probability of \(10^{-4}\), for \(X_{T}=(10^{-3},\,0.75\times 10^{-3},\,0.5\times 10^{-3})\)\(K^{2}S^{-3}\), an SNR of \((66.1,58.4,50.1)\) dB is required at the depth of 10 m, and an SNR of \((62.7,\,56.1,\,48.3)\) dB is required at the depth of 50 m. The presented results reveal that the outage performance improves with the reduction in the mean square temperature. Further, for the considered values of \(X_{T}\), a marginal reduction in the required SNR is observed at 50 m depth when compared to 10 m depth. Figure 2: Outage probability versus average SNR (\(\psi_{\circ}\)) dB for SD and relayed link. Symbols V, H and SD represent the vertical, horizontal and direct non-relayed link from source to destination, respectively. Figure 5 shows the ergodic capacity versus \(\psi_{\circ}\) plots for the vertical relayed and non-relay cases. The overlapping analytical and simulated plots validate the correctness of the derived ergodic capacity expression given in (30). We observe a significant increase in the capacity for the relayed case when compared to the no-relay case for both the horizontal and vertical links. Results demonstrate that at an SNR of 30 dB, a capacity of \(7.5\)\(\mathrm{bps}/\mathrm{Hz}\) and \(14\)\(\mathrm{bps}/\mathrm{Hz}\) is achieved, respectively, for the no-relay and relayed link cases with \(\mathcal{M}=1\). Further, increasing the number of relays beyond one results in the additional capacity gains of \(1,0.5,0.3\)\(\mathrm{bps}/\mathrm{Hz}\) for \(\mathcal{M}=2,3,\) and \(4\), respectively. Figure 4: Outage probability \((P_{o})\) versus average SNR (\(\psi_{\circ}\)) dB for different dissipation rates of mean square temperature (\(X_{T}\)). Figure 3: Outage probability \((P_{o})\) versus average SNR (\(\psi_{\circ}\)) dB for different number of relays \(\mathcal{M}\).