## 2 System and Channel Model  
We consider a horizontal and vertical UWVLC cooperative communication system as shown in Figure 1. In the horizontal communication, we consider two cases to study the impact of temperature variation with depth on the performance. In the first case, the communication takes place at a depth of less than 10 m, and in the second case, the communication takes place at a depth of 50 m. The source \((\mathcal{S})\) and destination \((\mathcal{D})\) are horizontally aligned and relay positions are assumed to remain static. In the case of vertical communication, the \(\mathcal{S}\) is placed at the sea surface and the \(\mathcal{D}\) is at a depth of 60 m. The communication takes place with the help of relays placed on the buoys, which are located at the center of source and destination in all cases. One relay \((\mathcal{R}_{m})\) out of \(\mathcal{M}-\) relays is selected based on the maximum signal-to-noise ratio (SNR). The \(\mathcal{S}\) and \(\mathcal{D}\) are equipped with one laser and one photo detector, whereas all relays have one pair of lasers and photo detectors. We consider the on-off keying (OOK) modulation scheme and it is assumed that the channel information is perfectly known at all the nodes. Further, it is assumed that the transmitting and receiving nodes are precisely aligned. In the vertical communication in Figure 1b, we consider a cascaded layered structure having different temperatures [24]. The sea water temperature is non-uniform with the depth [13]. As the temperature stays the same for longer distances at a given depth in horizontal communication [12; 13], we consider two cases of horizontal communication at different depths. The communication takes place in a dual-hop mode in two phases. In the first phase, the \(\mathcal{S}\) transmits the information signal \(x\) through the \(\mathcal{SR}_{m}\) link, and the \(\mathcal{M}\) relays detect the signal. In the second phase, the detected signal at the buoy corresponding to the maximum end-to-end (e2e) SNR is decoded, re-modulated and sent to the \(\mathcal{D}\) through the \(\mathcal{RD}\) link. The relay-assisted dual-hop system is assumed to work in half-duplex mode considering perfect channel state information available at \((\mathcal{M})\) relays and \((\mathcal{D})\). The received signal \(y_{R}\) at the \(m\)th relay is expressed as  
\[y_{R_{m}}=\eta_{\circ}P_{s}h_{SR_{m}}I_{SR_{m}}x+w_{R_{m}}\, \tag{1}\]  
where \(1\leq m\leq\mathcal{M}\). \(\eta_{\circ}\) is the electrical-to-optical conversion efficiency coefficient. The \(P_{s}\) is the source power. The \(I_{SR}\) and \(h_{SR}\) are the path loss and optical turbulence-induced fading coefficient of the \(\mathcal{SR}_{m}\) link, respectively. The \(w_{R_{m}}\) is the independent and identically distributed (iid) zero mean additive white Gaussian noise (AWGN) with variance \(\sigma^{2}_{w_{R_{m}}}\). The received signal at the node \(\mathcal{D}\) is represented as  
\[y_{D}=\eta_{\circ}P_{B}h_{RD}I_{RD}\hat{x}+w_{D}, \tag{2}\]  
where \(\hat{x}\) is the transmitted symbol by the buoy with power \(P_{R}\). We assume that the total assigned power \(P_{t}\) is equally divided between the source and relay \((P_{s}=P_{r}=P_{t}/2)\). The fading coefficients of \(\mathcal{SR}_{m}\) and \(\mathcal{RD}_{m}\) links are modeled by the log-normal distribution considering the weak oceanic turbulence [19; 21]. The \(w_{D}\) is the zero mean AWGN with variance \(\sigma^{2}_{w_{D}}\). The PDF of log-normal distribution is represented as [21]  
\[f_{h_{j}}(h)=\frac{1}{\sqrt{2\pi}\ \sigma_{j}h}\exp\bigg{(}\frac{-(\ln(h)- \mu_{j})^{2}}{2\sigma_{j}^{2}}\bigg{)}. \tag{3}\]  
where \(j\in\{\mathcal{SR}_{m},\mathcal{R}_{m}\mathcal{D}\}\). The noise variance of \(w_{R_{m}}\) and \(w_{D}\) in (1) and (2), respectively, is assumed to be equal, \(\sigma^{2}_{R_{m}}=\sigma^{2}_{w_{D}}=\sigma^{2}_{w}\). In the vertical system, we consider a multilayered concatenated structure. The channel coefficient of each layer also follows the log-normal distribution.