## 4 Angle Selection and Ray Categorization
### Ray Types  
Transmitted acoustic signals in an underwater environment generally propagate in an inhomogeneous medium that is bounded by the sea surface and bottom. Such bounded inhomogeneous medium often advocates different propagation paths for rays transmitted at different angles and depths. To predict rays suitable for tracking underwater nodes and/or to maintain a communication link among mobile or drifted nodes, it is inadequate to localize the node and know its location only but rather it is important to know the type of ray establishing such a connection, i.e., direct, reflected or refracted with LEPs. Specifically, when separation among a node pair varies due to mobility or drift, different ray types are required to maintain the communication link since each ray can only cover the underwater medium partially. In addition, medium inhomogeneity not only controls refraction of transmitted signals, but it also dictates the number of angles at which the receiver node can be reached. Thus, a node ought to have some means of classifying rays and know the range covered by each ray type to maintain communication with drifted or mobile nodes.  
Since the underwater environment is bounded by the sea surface and bottom, it is inevitable for rays to get reflected from these surfaces or refract to the point such that the propagation direction along the vertical axis is inverted. Essentially, reflected rays bounce from the sea surface and/or bottom and may also face refraction along their journey. The point of reflection introduces a LEP point where the signal propagation path is non-differentiable, i.e., a discontinuity in the first derivative is observed. On the other hand, non-reflected rays can be categorized as either rays experiencing a LEP due to medium inhomogeneity or rays that reach the receiver before either a surface reflection or LEP is experienced. Specifically, the former type includes rays that do not bounce from any surface but faced refraction that causes a directional inversion along the \(z\)-axis and thus generating LEPs along their propagation path. For a refracted ray, the first derivative of the propagation path always generates a continuous signal along the entire path including the LEP point. On the other hand, the latter type is referred to as the direct ray and is defined as the ray with the shortest propagation path between  
Figure 8: Conceptual representation of connecting rays where two distinct vertical subspaces are shown with their respective connecting-rays. To map the 3D environment (**a**) into 2D (**b**), we rotate each vertical plane along the \(z\)-axis until all the vertical planes merger into a single plane referred to as the process of superimposing subspaces.  
a communicating pair that face neither a LEP nor reflection from any surface, i.e., faced zero LEPs. A direct ray may still be refracted leading to a curvy propagation path but on the average is either an increasing or decreasing function in \(\Omega\). Based on the above definitions, the node starts to classify rays using the measured SS, angle of transmission and the quadrant (\(Q\)) at which the ray is transmitted and received as discussed in the next section.