## 2 System Model  
In this work, we analyze an amplify-and-forward (AF) based dual-hop TU-FSO system, as shown in Figure 1. In both hops, we consider point-to-point optical links using an intensity modulation with direct detection (IM/DD) scheme. The time-dependent photocurrent at the terrestrial FSO detector output is written as follows:  
\[i_{S_{1}}(t)=R_{1}h_{1}(t)P_{t}(t)+i_{N_{1}}(t), \tag{1}\]  
where \(R_{1}\) is the detector's responsivity, with \(h_{1}\) being the normalized (\(E[h_{1}]=1\)) scintillation affecting the irradiance, and following a Malaga (\(\mathcal{M}\)) statistical distribution, as detailed in [26; 45]. Furthermore, the average optical transmitted power is written as \(P_{t}\), whereas it is assumed that the detector current noise signal, \(i_{N}\), is mainly caused by a zero-mean additive white Gaussian noise (AWGN) with variance \(\sigma_{n}^{2}\). On the other hand, the terrestrial detector in the FSO link works as a relay node for cooperative communications. In this respect, it processes the received information data; then, such information is sent to the final destination. If this latter one is assumed to be the underwater FSO detector, then its time-dependent photocurrent can be then expressed as follows. \[i_{S_{2}}(t)=GR_{2}h_{2}(t)i_{S_{1}}(t)+i_{N_{2}}(t). \tag{2}\]  
In (2), \(G\) is the channel state information (CSI)-assisted relay gain at the terrestrial FSO detector, as explained in [46], whilst \(h_{2}\) denotes the normalized scintillation coefficient induced by the salinity and/or temperature gradient following a Weibull distribution. Moreover, parameters \(R_{2}\), and \(i_{N_{2}}\) were defined in the same sense as \(R_{1}\) and \(i_{N_{1}}\) were described for \(i_{S_{1}}\), but now corresponding to the underwater channel. For the sake of simplicity, we have considered that the ocean ambient noise, \(i_{N_{2}}\), is AWGN. Nevertheless, this assumption can be improved with the inclusion of an impulsive noise [47; 48; 49] described by the heavy tail rather than the exponential tail, since the AWGN model ignores the impulsive appearance of electromagnetic interference, oceanic noise or noises caused by humans using other machines, especially for scenarios with shallow water. Without loss of generality, note that weather-induced attenuation has not been considered in this paper. Although that effect also degrades the performance of FSO systems in the manner shown in [50], however, due to its deterministic nature, such atmospheric  
Figure 1: AF-based TU-FSO system model, based on the one in [44]. attenuation acts merely as a scaling factor as indicated in [51]. Moreover, for analogous reasons, we have not considered the effect of absorption in water. As in the FSO link, absorption can be included as a pure loss term, as detailed in [52], acting as a scaling factor. In addition, a comparison of underwater wireless optical communication system configurations and its associated performance can be found at [53]. For instance, for a laser diode wavelength of 450 nm employing OOK, a range of 20 m can be achieved for a data rate of 1.5 Gbps, as reported in [53]. More recently, a 100 m range through tap-water channel was demonstrated in [54], using OOK and a laser of 520 nm and 500 Mbps of data rate. Better performance can be obtained when employing a different modulation technique. Thus, as an illustrative example, in 2019, a commercial underwater optical communication product called BlueComm-200 was released, achieving a range of 150 m (>200 m for moderate to low turbidity dark water) with a data rate of 10 Mbps employing a 450 nm LED and a photomultiplier [55]. Finally, optical scattering [19; 29; 56] from suspended particles was incorporated in the model in a simple and straightforward manner.