# Cooperative Terrestrial-Underwater FSO System: Design and Performance Analysis
## 3 Channel Model
### Terrestrial Link
Hence, the PDF of \(h_{p}\) can be seen as a random variable transformation problem, which leads to the following expression [32]:  
\[f_{h_{p}}(h_{p})=\frac{\xi_{p}^{2}}{A_{0}^{\xi_{0}}}h_{p}^{\xi_{p}^{2}-1}, \tag{10}\]where \(\xi_{\rho}=\omega_{\mathit{leg}}/(2\sigma_{\rho})\) denotes the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation. In order to model intensity fluctuations caused by atmospheric turbulence, the statistical Gamma-Gamma distribution is assumed here because of its mathematical tractability and accuracy in characterizing a wide range of scenarios from weak to strong turbulence. Thus, and following [17], the probability density function (PDF) of \(h_{a}\) is written as  
\[f_{h_{a}}(h_{a})=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma (\beta)}h_{a}^{(\alpha+\beta)/2-1}K_{\alpha-\beta}\left(2(\alpha\beta h_{a})^{ 1/2}\right), \tag{11}\]  
where \(K_{p}(x)\) is the modified Bessel function of the second kind, and \(\Gamma(x)\) is the Gamma function, with \(\alpha\) representing the effective number of large-scale cells of the scattering process, and with \(\beta\) denoting the effective number of small-scale cells. These parameters provide information on the level of intensity of the atmospheric turbulence and, since the atmospheric link is modeled by Gamma-Gamma distribution, the irradiance variance, \(\sigma_{a}^{2}\), depends on these parameters as follows: \(\sigma_{a}^{2}=\alpha^{-1}+\beta^{-1}+(\alpha\beta)^{-1}\). We note that large \(\alpha\) and \(\beta\) values are associated with weak turbulence regimes, while small values are related to a strong turbulence level. This is equivalent to saying that a regime of weak fluctuations is associated with \(\sigma_{a}^{2}<1\), moderate fluctuations with \(\sigma_{a}^{2}\approx 1\), whereas strong fluctuations are characterized by \(\sigma_{a}^{2}>1\). Such parameters, \(\alpha\) and \(\beta\), can be obtained as follows [16]:  
\[\alpha=\Big{[}\exp\left(0.49\sigma_{R}^{2}(1+1.11\sigma_{R}^{12/5})^{-7/6} \right)-1\Big{]}^{-1} \tag{12}\]  
\[\beta=\Big{[}\exp\left(0.51\sigma_{R}^{2}(1+0.69\sigma_{R}^{12/5})^{-5/6} \right)-1\Big{]}^{-1}, \tag{13}\]  
where \(\sigma_{R}^{2}\) is the Rytov variance which, in turn, can be obtained for downlinks as follows [16]:  
\[\sigma_{R}^{2}=2.25k^{\frac{1}{6}}\sec^{\frac{11}{6}}(\theta)\int_{0}^{Z}C_{n }^{2}(z)z^{\frac{5}{6}}\,dz, \tag{14}\]  
assuming that the receiver is at sea level. In this expression, \(C_{n}^{2}(z)\) is the index of refraction structure parameter at altitude \(z\), \(k=2\pi/\lambda\) is the optical wave number, \(Z\) is the transmitter height and \(\theta\) is the zenith angle, respectively. We note that \(C_{n}^{2}\) is a characteristic parameter of atmospheric turbulence that gives a measure of the strength of the turbulence. Due to its dependence on height, this parameter is usually considered constant for scenarios where the signal propagates horizontally. However, for vertical or inclined links, the value of \(C_{n}^{2}(z)\) must be obtained from models [16]. Once the three factors included in Equation (7) are individually discussed, the statistical characterization of composite channel \(h_{1}\) can be achieved. Thus, from [32], the PDF of \(h_{1}\) is given by  
\[f_{h_{1}}(h_{1})=\frac{2\xi_{p}^{2}(\alpha\beta)^{\frac{\alpha+\beta}{2}}}{(A_ {o}h_{l})^{\xi_{p}}\Gamma(\alpha)\Gamma(\beta)}h_{1}^{\xi_{2}^{2}-1}\int_{ \frac{h_{1}}{A_{0}\eta_{l}}}^{\infty}h_{a}^{\frac{\alpha+\beta}{2}-1-\xi_{p}^ {2}}K_{\alpha-\beta}(2\sqrt{\alpha\beta h_{a}})\,dh_{a}, \tag{15}\]  
where \(\alpha\) and \(\beta\) parameters include the information on the strength of the turbulence, \(\xi\) contains the severity of the pointing error, \(A_{0}\) denotes the geometric spread attenuation and \(h_{l}\) is the path loss. We note that such a joint PDF is used later in Section 3.3 to derive the probability of error, taking into account both atmospheric and underwater links, by calculating the cumulative distribution function (CDF).