## 5 Performance Validation
### Performance Results
From the figure, we can see that when the transmission angle is error free, less than 5% of the rays are wrongfully classified. As we introduce a Gaussian error \(\mathcal{N}(0,\sigma)\) we find that on the average, the error grows almost linearly with the increasing value of \(\sigma\). Thus, even when the node orientation step produces an error, the effect of such an error on the ray classification technique is bounded and does not grow to a level that the technique is not usable. Furthermore, since the expected error in detecting the orientation described in Section 4.2 often results in a maximum error equivalent to the smallest beam width achievable by the transducer and its capability to steer such a beam, the orientation error is envisioned to be small. Figure 12: Propagation path of selected acoustic signals shown in different deployment configurations. Part (**a**,**d**) show the SSP for the figures in the same row while parts (**b**–**e**) show the node distribution for each deployment configuration and the corresponding refracted rays only among pairs. The error resulting from an erroneous depth (labeled \"Depth Error\") estimation is also simulated and presented in Figure 13 where we have added a Gaussian error \(\mathcal{N}(0,\sigma)\) to the actual depth of the transmitter node. Figure 13 shows that the effect of gaussian error up to \(\sigma=20\) m is negligible. Such observation is supported by the fact that in a randomly deployed topology, the chances of having two nodes close to a bouncing surface, communicating at transmission angles less than five degrees and use a straight-line propagation model is quite slim. Moreover, for rays with differentiable LEPs we need to determine the error produced by a pressure sensor. Specifically, since the depth is obtained by a pressure sensor, the accuracy of the depth is dependent on the hardware used by a node. The depth sensor \"OceanDEPTH\" that is manufactured by \"OceanTools\" produces an error of 0.25% [32]. Thus, the value of the error for depth up to three kilometers for such a sensor will be \(e=7.5\) m. We utilize such an error value to plot the experienced error in \(x_{LEP}\) and show the result in Figure 14. Specifically, when the straight-line approximation is utilized the error is represented in Figure 14 while the maximum error bound for the differentiable LEP is presented in Figure 14.  
In summary, our algorithm is quite robust and correctly classifies rays even with errors introduced in the angle and depth. The algorithm is more sensitive to errors in the transmission angle while the effect of a depth error is generally insignificant. For the depth-error cases, the corresponding classification errors for the straight-line model has been found to be depe  
Figure 14: The error observed in the \(x_{LEP}\) where (**a**) is the error when Equation (5) used while (**b**) shows the maximum error bound when Equation (6) is used. Both figures assume that the error by the pressure sensor is \(e=7.5\) m.  
Figure 13: The classification error trend when the angle of transmission and depth are subjected to an additive gaussian error with zero mean and standard deviation depicted on the \(x\)-axis. angle; such effect diminishes for higher angles. The depth-caused differentiable LEP error, on the other hand, is bounded by a maximum cap. Finally, when utilizing a pressure sensor with different accuracy, the generated error pattern will only be a linear translation along the \(y\)-axis of the cases shown in Figure 14.