# Deep Machine Learning for Path Length Characterization Using Acoustic Diffraction
## 4 Discussion  
The very low mean-square error of 0.00016 and very high R-square statistic of 0.998 exhibited by the Amplitude CNN were expected due to the high correlation between the amplitude values, low number of outliers, and similarities in the amplitude profile. The accuracy of the model can be explained by looking at the high levels of correlation both between the amplitude values at each lateral point and with the acoustic path lengths themselves in Figure 14. This high correlation coupled with the larger distribution of normal values seen in Figure 13 points toward data with few outliers and values which will be meaningful to the determination of acoustic path length. Additionally, examination of the transfer function amplitude profile of Figure 6 shows that the amplitude trends for samples exhibiting the same acoustic path length often exhibit similarities in the width of the peaks in the amplitude profile and strength of the signal despite having differently sized acoustic anomalies. These similarities are often difficult to discern immediately and require the review of many samples as the peaks may not always align due to very minor  
Figure 16: Actual acoustic path lengths (x) vs. predicted acoustic path lengths (y) with a regression line to demonstrate how close the model is to perfect accuracy. In each figure we see the predictions from left to right of: the path length 12.95 (1 MHz data), the path length 25.13 (1 MHz data), the path length 64.74 (5 MHz data), the path length 123 (1 MHz data), the path length 125.66 (5 MHz data), and then the path length 614.99 (5 MHz data). The derivation of these path length values are displayed in Table 4. It is immediately evident in the LSTM and Phase CNN graphs that the 1 MHz data are exhibiting less error in prediction than the 5 MHz data. Figure 15: Comparison of training loss (mse) per epoch for LSTM, Phase CNN, and Amplitude CNN. changes in the placement of the specimen. All of these factors point to a model being highly likely to learn trends in the data despite the challenges of interpreting these data that are experienced by humans. The increased mean-square error of 0.003 and reduced R-square statistic of 0.971 of the phase CNN model may also be explained by these figures. It is noted that there are fewer features that correspond to the acoustic path length and that the transfer function phase values are less correlated with one another. It is also clear from the box-and-whisker charts in Figure 13 that the phase data experience far less normal values and many outliers with a fairly large range. This seems to indicate that the phase data are behaving in a less predictable manner. This makes sense because the phase is a more complex function of x than the amplitude. Even in the Fraunhofer regime, the phase has ripples to either side of the central lobe. The phase profile in Figure 6 also supports this theory as the samples with corresponding acoustic path lengths rarely align in the phase profile. The decreased performance of the phase model appears to be explained by the physics behind the model. The performance of the model can be discussed based on (a) diffraction and (b) the z-dependence of the radius of curvature. In point (b), the z-axis is along the propagation of the acoustic signal (perpendicular to the \(x\)-axis). As to (a), the impact of diffraction on the lateral variation of the phase plays a role. We note that the x-dependence of the phase is much more complex in the Fresnel range than in the Fraunhofer range. Because of the complexity, a tiny error in the x-position upon the scanning procedure can lead to a huge difference in the phase value. The boundary between Fresnel and Fraunhofer ranges can be approximated by the aperture size, D, and the wavelength, lambda (see Figure 5). The data utilized by the models have two aperture sizes corresponding to the hole diameter of 2.17 mm and 6.47 mm.