## 4 Angle Selection and Ray Categorization
### Angle Discretization and Selection  
We consider each transmission angle as a separate communication channel as described in Section 3.1, where the aim is to find channels with the least overlaps and avoid generating shadow zones. Since beam overlapping at large distances is inevitable due to reflections and LEPs, we select beams that avoid both overlapping and shadow regions up to a distance \((d_{s})\) where LEP and/or reflections do not occur. Recall from Section 3.1, that nodes can generate identical beams in shape and size and transmit them at distinct directions with a beam width of \(\psi\) that is dictated by the transducer capability. Two distinct transmitted beams could (i) overlap, (ii) create a shadow zone or (iii) stage contiguously. We formally define two beams to be overlapping if their center angles deviation is less than \(\psi\), i.e., \(k_{i}-k_{j}<\psi\) for \(k_{i}\
eq k_{j}\). On the other hand, two beams form a shadow zone if \(k_{i}-k_{j}>\psi\), i.e., the angular deviation between two neighboring beams is higher than \(\psi\). A contiguous beam corresponds to \(k_{i}-k_{j}=\psi\). Leveraging these formal definitions, a node starts the beam selection process where the goal is to select angles that are contiguous to avoid both overlapping beams and shadow zone. Nodes strive to find the smallest subset \(\kappa\) of beams, i.e., \(\kappa\subset K\), such that (i) any two adjacent beams in \(\kappa\) are contiguous, i.e., has angular deviation of \(\psi\), and (ii) the union of all beams in \(\kappa\) avoids shadow zones within a distance \(d_{s}\). To find the beams in \(\kappa\), nodes define a hypothetical sphere referred in our work as the Transmission Sphere (TS). The radius of such TS is the distance at which any generated acoustic beam \((k_{i})\) transitions from the near field region to the far field region of the transducer. Since the assumed transducer enables transmission at distinct directions in all 3D space, nodes can eliminate shadow zones by selecting beams where the union of all beam's footprints onto the TS are capable of completely covering its surface with minimal overlaps. To find the beams constituting the set \(\kappa\), nodes maps the TS surface into a grid, where each grid element represents the footprint of a transmitted beam. Fundamentally, nodes divide the TS surface into cells that closely resembles the departing beam's footprint onto the TS. Since a beam is visualized as having a conical shape, the intersection of the TS with the departing beam ideally forms a circular shape that is inscribed onto the TS. However, stacking non-overlapping and tangent circular grid elements onto the TS surface will generate shadow regions since the union of all non-overlapping circles cannot cover the entire sphere surface. Moreover, having overlapping circles inherits the problems of overlapping beams. On the other hand, having irregularly shaped cells significantly complicates the analysis. Since the choice of the grid element significantly impacts the complexity of beam tracing in an inhomogeneous habitat, we pursue a geodesic grid that strikes a balance between avoiding beam overlapping, eliminating shadow region and faithfully representing the departing beam in shape and size [26]. The simplest and basic geodesic grid is shown in Figure 4a that is obtained by inscribing an icosahedron into the TS surface where each face is represented as a triangle. However, a triangle is quite far from the ideal circular beam-footprints onto the TS and the fact that an icosahedron in its basic form poorly maps the sphere makes it an inappropriate choice. However, subdividing the icosahedron by iteratively bisecting the triangles and projecting them onto the sphere generates a smoother surface approximating the TS as shown in Figure 4b. In the figure, each point on the grid has six edges connecting it to its neighboring points forming a hexagon shape footprint except for 12 points that have five edges that form a pentagon.