# Deep Machine Learning for Path Length Characterization Using Acoustic Diffraction
## 4 Discussion
As the above vector Figure 5 indicates, the 5 MHz case makes the boundary 5 times greater than the 1 MHz case, meaning that the 5 MHz case diffraction is more Fresnel-like. At the same frequency, the 6.47 mm hole makes the boundary 9 times greater than the 2.17 mm hole. It is clearly seen in Figure 16 that the samples suffering the most are those with the acoustic path lengths of 64.736, 125.66, and 614.99. These are the samples that are collected at a frequency of 5 MHz. Because half of the data collected at this frequency are expected to fall in the Fresnel range, and the other in the Fraunhofer range, the data corresponding to these acoustic path lengths are likely to vary wildly in terms of the behavior of the wave. Because of the wildly different profiles, particularly of the phase data, which are more difficult to predict, one area of improvement may be to create an ensemble model for handling the phase data. This would mean training on CNN on Fresnel data and another on Fraunhofer data, then using the models together to make predictions on incoming data. This would allow more accurate predictions about the phase data while still relieving the operator from having to know whether the data were collected in a Fresnel or Fraunhofer regime. As to (b), the z-dependence of the radius of curvature seems to affect the performance. The acoustic signal from the emitter is blocked by the hole due to the high reflectivity at the entrance of the hole. At the exit of the hole, a portion of the acoustic wave is missing over the diameter of the hole. According to [PERSON] principle, we can view that the sensor signal is the superposition of the element waves on the exit plane of the disc. This means that the received acoustic signal is the superposed signal without a hole minus the actual superposed signal leaving the exit plane of the disc. Depending on the distance between the plane of the acoustic emitter and the end of the hole (i.e., the hole length), the radius of curvature of the missing wave at the disc's exit plane varies. It is likely that the greater the dependence of the curvature of the missing wave at the disc's exit plane on the hole thickness, the more sensitive the sensor signal to the hole thickness, i.e., it is easier to infer the hole thickness from the sensor signal. This can be examined by inspecting the Rayleigh length of the signals in relation to the thickness of the hole. The Rayleigh length is of importance as this gives the distance at which the wavefront reaches maximum curvature or minimum radius of curvature. Figure 17 shows the above factors. Here, the left graph plots the physical hole thickness vs. hole thickness relative to the Fresnel-Fraunhofer boundary, and the middle and right graphs plot the physical hole thickness vs. hole thickness relative to the Rayleigh length for the hole diameter 2.17 mm and 6.47 mm cases, respectively. The left graph clearly indicates that the specimens with the larger anomaly are least Fraunhofer, meaning farthest from the Fraunhofer regime, while the small anomaly leads to data which are more Fraunhofer, meaning these specimens lead to diffraction which falls deep within the Fraunhofer regime. Particularly, the 1 MHz data with the small, 2.17 mm, anomaly are highly Fraunhofer while the 5 MHz data with the large, 6.47 mm, anomaly are far from Fraunhofer. The middle and right graphs indicate in all cases the hole thickness is orders of magnitude smaller than the Rayleigh length. Since the radius of curvature rapidly decreases with the z-distance toward the Rayleigh length, where the radius of curvature takes the minimum value, these two graphs depict the range where the radius of curvature (the phase) is sensitive to the z-distance (the hole thickness). The steeper the line, the more sensitive the phase is to the hole thickness. The LSTM model performed between the phase and amplitude CNN models with a mean-square error of 0.00067 and an R-square statistic of 0.995.