## 4 Angle Selection and Ray Categorization
### Ray Classification and Selection
Once the number of LEPs are determined, nodes utilize the quadrants in which the beam was transmitted and received to classify the ray connecting each pair. The balance of this section details such a process. Once the location of the LEP point is determined, nodes consider the quadrants of transmission and reception as well as the depth of the LEP point to determine the ray type. Specifically, since we are working in an obstacle free deep-water scenario, a transmitted signal will never face a directional reversal along the \(x\)-axis and thus a ray that is transmitted within the range of the first quadrant (\(Q1\)) will always be received either in the 2 nd or 3 rd quadrant (\(Q2\), \(Q3\)) as shown in Figure 1b. For rays transmitted within \(Q1\) and being received within \(Q3\) while the transmitter is located at a deeper point than the receiver, it is safe to assume that an even number of LEPs will be experienced. Fundamentally, if the number of LEP is zero, a node concludes that a direct ray has been received where the signal was intercepted by the receiver before a LEP point is experienced. Hence, nodes rule out experiencing reflection or LEPs since the receiver is located at a point where neither is possible. Moreover, since an  
Figure 9: Conceptual figure where the maximum horizontal distance for a given transmission angle is presented. odd number of LEPs cannot satisfy the condition on the receiving quadrant, experiencing any type of single LEP is ruled out. On the other hand, if rays are transmitted from \(Q1\) and received in \(Q2\) while knowing that the transmitter is at higher depth, a receiver concludes that an odd number of LEPs is experienced thus ruling out direct ray paths. Table 1 summarizes the number of LEPs observed when transmitting at different angles and depth configurations. The table emphasizes the fact that direct rays are only observed when transmission and reception angles adhere to \(Q1\) and \(Q3\) while the transmitter is located at a greater depth or when rays fall in \(Q4\), \(Q3\) and the transmitter is at a shallower location. In both cases reflections is also ruled out. Moreover, a direct ray cannot be observed for all other cases. Reflection, on the other hand, is observed only when the number of LEPs is greater than zero. Similarly, refracted rays with multiple LEPs may be observed depending on SSP conditions. Specifically, experiencing a differentiable LEP requires the SS along the propagation path to be on the average an increasing function, i.e., the ray must be traversing from a lower SS value to a higher one. Therefore, the gradient of the SSP controls the horizontal distance before an acoustic signal experiences a LEP, where generally lower SS gradients result in LEP point closer to the transmitter. Thus, to distinguish between the non-differentiable LEP from the differentiable ones, i.e., reflected from refracted rays experiencing a LEP, a node considers the number of experienced LEPs along the propagation path and their depths. If the depth of a LEP point corresponds to the sea surface or bottom, nodes conclude that the ray faced a reflection. On the other hand, if the LEP depth is lower than the surface the ray is gazing at, refraction causing a differentiable LEP is deduced. In summary, using the technique presented in this section, nodes calculate the number of LEPs faced in the propagation path and determine the type of ray that establishes a communication link. Moreover, by using such a technique, nodes do not have to go through the entire Bellhop algorithm [28] but rather uses simple and less complex equations to determine the expected number of LEPs along the propagation path. Once the rays are classified into their appropriate type, a node determines the usable range of each ray type, as discussed in the next section.