# Cooperative Terrestrial-Underwater FSO System: Design and Performance Analysis
## 3 Channel Model
### Terrestrial Link  
The three main atmospheric phenomena that affect optical wave propagation and give rise to attenuation factor \(h_{1}\) described in the previous section are (i) path loss \(h_{l}\), (ii) geometric spread and pointing errors \(h_{p}\), and (iii) atmospheric turbulence \(h_{a}\). Therefore, the total channel attenuation is modeled as the product of these channel factors as  
\[h_{1}=h_{l}h_{p}h_{a}, \tag{7}\]  
where \(h_{l}\) is deterministic and \(h_{p}\) and \(h_{a}\) are random variables (RVs) whose modeling is described in the following sections. Path loss \(h_{l}\) occurs due to atmospheric absorption and scattering caused by the molecules that form the air and by solid and liquid particles suspended in the atmosphere. This path loss can be computed by the exponential law of Beers-Lambert [31] as  
\[h_{l}=\exp{(-aL)}, \tag{8}\]  
where \(L\) is the propagation distance and \(a\) is the attenuation coefficient which depends on the size and distribution of the scattering particles and the wavelength utilized. Since atmospheric conditions change very slowly over time (especially compared to the transmitted symbol period), they can be considered constant over long periods. Thus, in our analysis, it is assumed that atmospheric path loss \(h_{l}\) has a deterministic nature. In line-of-sight links, pointing errors are due to the dynamic misalignment between the transmitter and the receiver as a result of meteorological phenomena such as strong gusts of wind or other phenomena of different origin that cause equipment vibration. For example, in our link, as the repeater node is located on the surface of the sea, it can be affected by wave motion causing pointing errors. However, as indicated in the previous section, this particular effect is not taken into account in our analysis. Pointing errors are particularly critical in narrow beam links as they cause deep fading of the received signal. Even without pointing errors, there is always a geometric loss due to the broadening of the beam as it propagates through the atmosphere. As depicted in Figure 1, at distance \(L\), the beamwidth defined by its waist \(\omega_{l}\) can be obtained as \(\omega_{l}\simeq\theta_{T}L\), where \(\theta_{T}\) is the divergence angle of the laser and \(L\) is the path length of the propagation path. Since the beamwidth is generally larger than the size of the receiver aperture, this leads to overflow loss. Therefore, geometric loss depends mainly on the ratio at the receiver between the beam waist, \(\omega_{l}\), and the receiver aperture radius, \(r\), \(\omega_{l}/r\). To consider both phenomena, here, we follow the general model proposed in [32]. Thus, assuming a Gaussian beam profile with a beam waist, \(\omega_{l}\), and a circular aperture receiver of radius, \(r\), the attenuation due to the geometric spread and pointing errors can be approximated as the Gaussian form  
\[h_{p}\approx A_{0}\exp{\left(-\frac{2\rho^{2}}{\omega_{leq}^{2}}\right)}, \tag{9}\]  
where, as shown in Figure 1, \(\rho\) is the radial pointing error, \(A_{0}\) is the fraction of collected power without pointing error, i.e., only due to geometric spread, and \(\omega_{leq}^{2}\) is the equivalent beam width. Here, \(A_{0}\) and \(\omega_{leq}^{2}\) are given by \(A_{0}=\text{erf}(v)^{2}\) and \(\omega_{leq}^{2}=\omega_{l}^{2}\sqrt{\tau}\text{erf}(v)/\left[2v\exp(-v^{2})\right]\), respectively; \(\text{erf}(\cdot)\) is the error function; and \(v=\sqrt{\pi}r/(\sqrt{2}\omega_{l})\). Moreover, considering independent identical Gaussian distribution for horizontal \(x\) and vertical \(y\) displacements in the receiver plane, radial error \(\rho=\sqrt{x^{2}+y^{2}}\) is modeled as a Rayleigh distribution with a jitter variance at the receiver, \(\sigma_{\rho}^{2}\). Under these assumptions, the channel coefficient, \(h_{p}\), can be seen as a function of the radial displacement, \(\rho\), which is an RV.