## 2 System and Channel Models
### Air Bubble-Induced Fading Model  
Air bubble-induced fading is modeled using the generalized Gamma distribution and a mixture of the sum of two generalized Gamma distributions when considering small air bubbles and large air bubbles, respectively [22]. The generalized Gamma distribution is a versatile statistical model that includes the Gamma distribution, the Chi and Chi-squared distribution, the Exponential distribution, and the Weibull distribution, which are usually employed in modeling power fluctuations in radio-frequency (RF) and optical wireless systems [7; 23; 24]. In the context of underwater environments, the Exponential-Generalized Gamma mixture has been used to analyze the multihop UOWC systems performance under oceanic turbulence [25; 27]. Furthermore, in [22], the generalized Gamma distribution and the mixture of the sum of two generalized Gamma distributions provide an excellent agreement for a wide range of water turbidity scenarios, as well as different air bubble populations with experimental measurements under controlled laboratory conditions [22]. It should be noted that the small and large terms are defined in relation to the beam width of the transmitted beam. When the diameter of an air bubble is comparable to or bigger than such a beam width, it can result in a partial or complete light blockage. In such instances, this constitutes a scenario of large air bubbles. Concerning the PDF, large air bubbles exhibit a bimodal behavior due to the combined effects of light attenuation caused by large bubbles and power fluctuations induced by particles and small bubbles formed through bubble collisions. Conversely, scenarios involving smaller bubbles are assumed to represent instances of small air bubbles, which induce power fluctuations around a given received mean power. All details regarding fitting the PDF to the experimental measurements can be found in [22].  
The corresponding PDF for small air bubble-induced fading is given by ([22], Equation (7)).  
\[f_{h_{\rm small}}(h;a,d,p)=\frac{p}{a^{d}\Gamma(d/p)}\,\,h^{d-1}\,e^{-(h/a)^{p}}, \tag{2}\]  
Figure 1: Schematic of a UOWC link application between two AUVs under air bubble scenario due to oxygen systems of scuba divers and AUV propellers.  
where \(h\) is the random variable, \(\Gamma(\cdot)\) is the Gamma function, \(a\) is a positive scale parameter, and \(d\) and \(p\) are positive shape parameters [28]. In the same way, the considered PDF of large air bubble-induced fading is given by ([22], Equation (8))  
\[f_{h_{\text{large}}}(h;a_{1},d_{1},p_{1},a_{2},d_{2},p_{2},w)=w\cdot f_{h_{\text {small}}}(h;a_{1},d_{1},p_{1})+(1-w)\cdot f_{h_{\text{small}}}(h;a_{2},d_{2},p_ {2}), \tag{3}\]  
where \(w\) is the proportion between the blockage effect, the first term, and the power fluctuation effect, the second term, such that \(w\in[0,1]\).  
By fitting \(f_{h_{\text{small}}}(h)\) and \(f_{h_{\text{large}}}(h)\) channel models to experimental UOWC measurements, the specific values of the distribution parameters, as well as some parameters related to the UOWC channel such as the experimental propagation losses, \(L_{T}\), the extinction coefficient of the UOWC link, \(c\), the scintillation index, \(\sigma^{2}\), and the coefficient of determination, \(R^{2}\), are obtained with a 520 nm laser diode at a link distance of 3 m. For detailed information on the experimental equipment and the setup used in the measurements, including all relevant parameters and configurations, please refer to the comprehensive description provided in [22]. A summary of the parameters of each distribution and the remaining parameters derived from the measurement of the optical power fluctuations for different levels of commercial antacid is presented in Tables 1 and 2 when considering small and large air bubble scenarios, respectively. As reported, both proposed models accurately fit the random behavior of the received optical power for both small and large air bubbles, with a coefficient of determination \(R^{2}\) exceeding 0.95 for all evaluated scattering conditions.