## 2 System Model
Such a scattering is largely independent of wavelength and depends on impurities and turbidity in the open water, and it results in a significant inter-symbol interference (ISI) (if the bit rate is not lowered to accommodate for temporal scattering) that limits seriously the performance of any underwater optical wireless communication system. Therefore, its effect is more pronounced in coastal areas than open oceans. As detailed in [17], a single-input/single-output underwater optical wireless communication system is considered employing non-uniform OOK modulation with IM/DD due to its low cost and implementation simplicity. Let us suppose a green laser diode operating at 532 nm at the transmitter side and a circular receiver aperture at the receiver side. A direct consequence of this modulation scheme is that the input signal is non-negative as it is proportional to light intensity. Thus, the average optical power transmitted for any sequence of bits of information (each of them transmitted in a temporal pulse of a bit period of duration) is affected by the oceanic path loss and fading due to oceanic turbulence (\(h_{2}\) in our system model). Then, a convolution operation of that sequence of pulses and a finite-length sequence of real coefficients denoting the equivalent discrete-time impulse response of the system is performed, resulting an additional summation term in (2) representing ISI, an effect associated with multipath interference. This latter interference is produced when an optical signal reaches the detector after encountering multiple scattering objects or multiple reflections from other underwater bodies. This eventually results in waveform time dispersion (time spreading) and decreases the data rate due to ISI. Nevertheless, the amount of multipath interference depends upon system specifications and the propagation environment. Hence, for shallow water environments, optical waves reflected from the surface or bottom generate multiple signals at the detector. However, for deep oceans, these surface and bottom reflections can be ignored [16]. Since it is coming from multipath underwater propagation, that ISI term (denoted by \(i_{ISI}\)), is also affected by \(h_{2}\) in a multiplicative manner in the following manner. \[i_{S_{2}}(t)=GR_{2}h_{2}(t)i_{S_{1}}(t)+h_{2}(t)i_{ISI}(t)+i_{N_{2}}(t). \tag{3}\]  
Namely, such an equivalent discrete-time impulse response of the system is constructed by the convolution of the transmitted filter, the fading free-channel impulse response with unity area and the employed matched filter. For the sake of simplicity, we model this ISI via its variance, \(\sigma_{\mathrm{s}}^{2}\). In Section 4, we describe the method for incorporating this ISI in a mathematically tractable manner to derive analytical expressions associated with the error probability of the complete TU-FSO system. At this point, we want to remark that, in pure sea water or in clear oceans, initially, both absorption and turbulence will be limiting factors, and as the water approaches closer to the land, where organic matter and suspended particulates are present, scattering dominates. In this respect, a rate adaptive transmission scheme such as the ones analyzed in this manuscript can mitigate the adverse effect of the randomly changing underwater environment [57]. Additionally, we are aware that our chosen cooperative communication scheme (amplify and forward) requires the use of the same modulation format in air and undersea. Some free-space optical communication systems do not use OOK and either use DPSK, a bandwidth efficient coherent modulation, or photon efficient orthogonal modulations (such as PPM and FSK). However, most fielded undersea terminals (and the ones currently purchasable) implement OOK at 1 GHz or lower due to limitations in available COTS (commercially available off-the-shelf) components in blue-green. It is, therefore, likely that any relay that wants to communicate between aerial and undersea systems would need a modulation format change.