## 4 Angle Selection and Ray Categorization
### Underwater Regions Based on Ray Types  
The ray classification technique discussed in Section 4.5 determines the type of ray given an established communication link, yet, it cannot provide the range beyond which one type of ray cannot be used by a transmitter to reach a receiver. To determine the range of each ray type, nodes leverage the SSP and the possible transmission angles to determine the location of LEPs. Since the transmission angle is bounded in the open interval \(\phi_{T}\in(-90,90)\), nodes first estimate the location of LEPs for all transmitted signals. Specifically, using Equation (4) the range of angles (\(\
u_{LEP}\)) at which \(c_{LEP}\leq c_{m}\) is first calculated. For each angle within \(\
u_{LEP}\), the node leverages Equation (6) and notes the location of the LEPs. Utilizing the maximum angular value in \(\
u_{LEP}\), a node applies Equation (5) to determine the location of a non-differentiable LEP. As shown in Figure 10, connecting all LEPs noted by node \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Quadrant**} & \multirow{2}{*}{**Depths**} & \multicolumn{2}{c}{**Number of LEPs**} \\ \cline{1-1}
**Tx** & & **Rx** & \(n\in\mathbb{Z}^{*}\), i.e., Set of Non-Negative Integers** \\ \hline
2,3 & 1,4 & x \({}^{1}\) & Impossible \\ \hline \multirow{3}{*}{1} & \multirow{3}{*}{3} & \(z_{T}\geq z_{R}\) & \(2n\) \\ \cline{2-3}  & & \(z_{T}<z_{R}\) & \(2n+2\) \\ \hline
1 & 2 & x & \(2n+1\) \\ \hline
4 & 3 & x & \(2n+1\) \\ \hline \multirow{3}{*}{4} & \multirow{3}{*}{2} & \(z_{T}>z_{R}\) & \(2n+2\) \\ \cline{2-3}  & & \(z_{T}\leq z_{R}\) & \(2n\) \\ \hline \hline \end{tabular} \({}^{1}\) x represents a don’t-care value.  
\end{table} 
Table 1: Ray classification based on node depths and quadrants of the transmitted and reception. yields a contour \((\lambda_{LEP})\) that can be leveraged to identify four different zones. The first zone is labeled in Figure 10 as Region 1 where only direct and surface reflected rays can establish a communication link. In the second zone, Region 2, only reflected and rays facing a single LEP can reach the receiver, i.e., direct ray connections are ruled out. For the third zone, labeled as Region 3, the direct ray is forced to spread out to the extent that it drops below the receiver sensitivity especially for long range communication. In particular, a beam bounded by Ray 5 and Ray 6 does not spread much before reaching \(\lambda_{LEP}\). However, as soon as the beam approaches the contour, a rapid stretch is exerted onto the beam's wave-front by the SSP as shown by the size of the lines representing the wave-front in Figure 10. Considering the spreading losses only, the law of energy conservation states that the energy must be evenly distributed along the curve representing the wave-front of a beam. Furthermore, since a line approaching \(\lambda_{LEP}\) from the left and right significantly vary in size, the beam's energy must spread evenly across such lines. As a consequence, beams transmitted with a finite energy must spread rapidly and hence the acoustic energy at the receiver transducer located after \(\lambda_{LEP}\) will be significantly lowered and becomes undetectable. Thus, for Region 3, neither direct rays nor rays facing a differentiable LEP can reach and only reflected rays can cover such a region. The final zone is the white space shown in Figure 10; such a zone is not reachable by \(n_{T}\) using any angle, i.e., a permanent shadow zone. In summary, the SSP of the water column dictates the propagation path of an acoustic signal. Therefore, whenever the SSP changes nodes need to obtain a new set of reading to reflect the most recent SSP. Based on the observed refraction and reflection of rays, the underwater environment can be categorized into four regions based on the type of ray observed in each one. A region in which none of the rays can reach is referred to as the permanent shadow zone. To identify the location of the permanent shadow zone, a node must find the ray path of two distinct rays.