## 1 Introduction  
Recent advances in acoustic underwater networks (AUN) have enabled the development of applications such as environmental state monitoring, search and rescue, seabed profiling, and coastal surveillance [1]. In an AUN, a mix of mobile, tethered and freely floating nodes must establish and maintain communication links in an ad-hoc manner while ensuring certain quality of service [2]. Furthermore, forming and maintaining a connected network topology for these applications proves beneficial where nodes must collect and exchange data to accomplish their mission and to set new objectives. Surveying the area of a crashed airplane is an example where nodes in an AUN must collaborate to enable localization and retrieval of the Blackbox [3]. According to [PERSON] and [PERSON], acoustic signals are deemed the best communication means among nodes that are more than 100 m apart due to their low attenuation and absorption in underwater environments [4]. Fundamentally, as the frequency of the transmitted signal increases, the underwater attenuation and absorption losses rapidly grow. On the other hand, utilizing frequencies less than five kilohertz to overcome these losses degrades the ability of achieving high data rates among communicating pairs. Thus, frequencies in the range of tens of kilohertz are favored as they strike a balance between achievable bitrate and the encountered underwater losses. Moreover, directional transmissions, rather than omni, are preferred since they overcome underwater attenuation and consume less energy. However, surface and bottom reflections, high propagation delays, drifting of underwater nodes and refraction of acoustic signals still pose major challenges to the design and operation of AUNs. Underwater Propagation of Acoustic Signals: The underwater environment is characterized as an inhomogeneous medium, such inhomogeneity is due to point-to-point changes in underwater characteristics such as temperature, salinity, and pressure (depth) [5]. Essentially, underwater inhomogeneity is mainly manifested by means of observing the sound speed profile (SSP) of the water column. Such variation of sound speeds (SS) along the propagation path of acoustic signals causes refraction and often yields a continuous change in the gazing angle of a propagating signal. Thus, to find the propagation path of transmitted acoustics signals, ray tracing mechanisms are leveraged to infer the trajectory of each ray. Although, current ray tracing algorithms yield a good approximation of the actual acoustic signals, they require estimating a propagation path within each layer and then concatenating them to come up with the full path. Moreover, maintaining communication with mobile or drifted nodes often requires estimation of the best communication angle that is usually not necessarily contiguous to the previous one. To illustrate, let us consider the example in Figure 1a, where initially node \(n_{R1}\) lies on the path of the ray transmitted by node \(n_{T}\) at an angle \(\phi_{T}=12^{\circ}\). As node \(n_{R1}\) drifts horizontally to the right, \(n_{T}\) shifts to the ray transmitted at angle \(\phi_{T}=10^{\circ}\) to maintain communication. When \(n_{R1}\) drifts further to the right, transmitting at smaller angles will not establish communication with \(n_{R1}\) since all these rays will refract until a local extremum point (LEP) is disclosed along the propagation path prohibiting the ray from reaching \(n_{R1}\), as illustrated in Figure 1a. Thus, node \(n_{T}\) must use a different type of ray to reach \(n_{R1}\) where a ray with a single LEP is to be used to reach \(n_{R1}\). Therefore, it is not enough for nodes to know the location of neighboring nodes, but they also need to classify rays into different types and determine the ranges of each one to enable switching among rays to maintain a communication link. Figure 1a indicates that when SSP conditions are favorable to produce multiple differentiable LEPs, the angle of transmission plays an important role in determining the location of the LEP.