# Cooperative Terrestrial-Underwater FSO System: Design and Performance Analysis
## 3 Channel Model
### Underwater Link  
The propagation of an optical beam underwater is very challenging due to the different characteristics and properties of the water bodies, ranging from shallow to deep water, which give rise to phenomena such as absorption and scattering. These effects collectively affect the underwater optical beam and are more or less relevant depending on the underwater environment through which it propagates. In addition, underwater wireless optical communications are affected not only by absorption and scattering but also by ocean turbulence, which is the main cause of fading in wireless optical communications. Thus, the total attenuation of the underwater link is modeled as the product of two factors as  
\[h_{2}=h_{lg}h_{u}, \tag{16}\]  
where the first factor, \(h_{lg}\), includes path loss and geometric loss and it is assumed to be deterministic, and the second factor, \(h_{u}\), is irradiance fluctuation due to ocean turbulence which is a random variable. This turbulence together with scattering and movements of the transmitter and receiver can cause severe pointing errors in the underwater links. However, in the case of scattering, because it causes the beam broadening to be increased, it provides a natural mechanism to mitigate potential angular pointing errors due to jitter at the expense of a higher attenuation due to geometric spread [33]. We did not consider the effect of pointing errors in \(h_{2}\). The \(h_{lg}\) factor can be obtained as follows:  
\[h_{lg}\simeq\exp{(-Fc_{T}z)}\times k_{1}\text{erf}\bigg{(}\frac{\sqrt{\pi}r}{ \sqrt{2}\omega_{z}}\bigg{)}^{2}, \tag{17}\]  
where the first term represents the path loss and the second term represents the geometric loss. The path loss is given by the exponential Beers-Lambert law, where \(z\) represents the depth of the underwater link, \(c_{T}\) is the extinction coefficient and \(F\) is a coefficient that takes into account the increase in received power due to scattering. In this work, we assume \(F=1\) while the values of \(c_{T}\) are taken from [33]. Furthermore, in the second term corresponding to geometric loss, \(r\) is the detector radius, \(\omega_{z}\simeq k_{2}\theta_{T}z\) is the beam waist at depth \(z\), \(\theta_{T}\) is the divergence angle of the transmitted beam, and \(k_{1}\) and \(k_{2}\) are two coefficients that model the increase in beam spreading due to scattering [33]. On the other hand, ocean turbulence or ocean scintillation occurs as a result of random variations in the refractive index of the propagation medium caused by currents, turbulent eddies, temperature and salinity changes, among other factors. In this paper, this phenomenon, given random factor \(h_{u}\), is characterized by the Weibull model [9; 23], recently proposed to characterize salinity- or temperature-induced fading in submarine optical channels. Taking into account the Weibull model for \(h_{u}\) and the losses, \(h_{lg}\), defined in Equation (17), the behavior of the joint underwater channel, \(h_{2}\), can be modeled by the following PDF:  
\[f_{h_{2}}(h_{2})=\frac{K}{\Lambda}\bigg{(}\frac{h_{2}}{h_{lg}\Lambda}\bigg{)} ^{K-1}\text{exp}\left[-\bigg{(}\frac{h_{2}}{h_{lg}\Lambda}\bigg{)}^{K}\right], \tag{18}\]  
where \(K>0\) corresponds to the shape parameter relative to the scintillation index of irradiance fluctuations, and \(\Lambda>0\) is the scaling parameter relative to the mean value of the irradiance [34].