## 3 Methodology
The shallow water channel can be modeled using the Pekeris model [18], as in Figure 1, where the speed of sound propagation is constant in the whole layer. In Figure 1, \(\mathrm{d}_{1}\) is the depth of the source node, \(\mathrm{d}_{2}\) is the depth of the receiver, \(\mathrm{h}\) is the depth of the area considered and \(R\) the transmission range. The distance along the direct path is calculated as [18]:  
\[\mathrm{D}_{00}=\sqrt{R^{2}+(\mathrm{d}_{1}-\mathrm{d}_{2})^{2}} \tag{9}\]  
The distance along different rays is given by:  
\[\mathrm{D}_{\mathrm{sb}}=\sqrt{R^{2}+\left[2\mathrm{b}\mathrm{h}+\mathrm{d}_{1 }-\left(-1\right)^{\mathrm{s-b}}\mathrm{d}_{2}\right]^{2}} \tag{10}\]  
\[\mathrm{D}_{\mathrm{bs}}=\sqrt{R^{2}+\left[2\mathrm{b}\mathrm{h}-\mathrm{d}_{1 }+\left(-1\right)^{\mathrm{b-s}}\mathrm{d}_{2}\right]^{2}} \tag{11}\]  
where \(\mathrm{D}_{\mathrm{sb}}\) is the path calculated when the first reflection occurs at the surface (s as number of surface reflections), while \(\mathrm{D}_{\mathrm{bs}}\) represents the length of the path when the first reflection occurs at the sea bottom (b as bottom reflections). Equation (10) must be satisfied the condition: \(0\leq\mathrm{s-b}\leq 1\), while the Equation (11) must be satisfied the condition: \(0\leq\mathrm{b-s}\leq 1\). If we consider a rough surface and this roughness is small compared to the wavelength, the reflection loss would be expressed through the scattering process. The reflectivity caused by a rough boundary [14], is given by Equation (12):  
\[R^{\prime}(\theta)=R(\theta)\times e^{-0.5\mathrm{a}^{2}} \tag{12}\]  
Figure 1: Pekeris model of shallow water acoustic channel. \[\text{a}=2\times\text{k}\times\sigma\times\sin(\theta) \tag{13}\]  
\[\sigma=\sqrt{0.324\times 10^{-5}v^{5}} \tag{14}\]  
where k is acoustic wave length (\(\text{k}=2\pi/\lambda\)), \(R^{\prime}(\theta)\) is the new reflectivity coefficient reduced because of scattering at the rough interface, \(\theta\) is the incident angle, a expresses [PERSON]'s roughness parameter Equation (13), and \(\sigma\) represent the rms roughness of the surface Equation (14), while \(v\) is the wind speed (m/s). In a smooth ocean surface, the reflectivity coefficient would be [14]:  
\[R^{\prime}(\theta)=-e^{-0.5\text{x}a^{2}} \tag{15}\]  
The surface lost [19], is only used when the roughness coefficient is small, it is calculated as:  
\[\mathrm{C}(\mathrm{r},\mathrm{f},\mathrm{d},\mathrm{T},\mathrm{S},\mathrm{p} \mathrm{H},\mathrm{w},\mathrm{s},\mathrm{P}_{\mathrm{tx}})=\mathrm{B}\times \mathrm{log}_{2}(1+\mathrm{SNR}(\mathrm{r},\mathrm{f},\mathrm{d},\mathrm{T}, \mathrm{w},\mathrm{s},\mathrm{P}_{\mathrm{tx}})) \tag{8}\]0  
Rayleigh model is used for modeling the loss caused by the sea bottom reflection. This model considers the density and the speed of sound in the interface between water and sea bottom. Simplified Rayleigh model considers that the scattering phenomenon is not as important as in the sea surface which means the loss is frequency independent. To calculate the sea bottom loss we use [15]:  
\[\mathrm{C}(\mathrm{r},\mathrm{f},\mathrm{d},\mathrm{T},\mathrm{S},\mathrm{p} \mathrm{H},\mathrm{w},\mathrm{s},\mathrm{P}_{\mathrm{tx}})=\mathrm{B}\times \mathrm{log}_{2}(1+\mathrm{SNR}(\mathrm{r},\mathrm{f},\mathrm{d},\mathrm{T}, \mathrm{w},\mathrm{s},\mathrm{P}_{\mathrm{tx}})) \tag{8}\]1  
where m = p1/p and n = c/c1 being p and c density and sound speed in sea water, respectively; while p1 and c1 are the density and sound speed in the seabed, respectively. The angles of incidence \(\theta\) are not constant.