# Deep Machine Learning for Path Length Characterization Using Acoustic Diffraction
## 4 Discussion
The advantage of this model is that it is able to use both phase and amplitude data and uses less preprocessing, as no images have to be formed. It is possible that this model is suffering the sample problems as the phase CNN suffers in that it may need to process Fresnel and Fraunhofer data separately. Similar to the future improvements for the phase CNN, this model may be split to create an ensemble in order to attempt to improve the performance to take advantage of the details held in both the phase and amplitude data. Another approach that may be taken on the LSTM model is to skip the preprocessing which calculates the phase and amplitude and allow the model to take in simply the transfer function of the wave. This would preserve some information that may be getting lost in the present model. We test the theorized model improvements by using only the data taken with the 2.17 mm anomaly, as we know that this data predominantly fall in the Fraunhofer range, the only exception being the data with a path length of 614.99. In this case, the largest diameter is the thickness of the 6.47 mm disc, so the 2.17 mm data collected for this specimen are still in the Fresnel range. With only these data, the phase CNN and the LSTM model were again trained and tested to validate that removal of the Fresnel data leads to improved  
Figure 17: Hole-thickness vs. relative to Fresnel-Fraunhofer boundary (**left**), Hole-thickness vs. Rayleigh length for 2.17 mm hole diameter (**center**), Hole-thickness vs. Rayleigh length for 6.47 mm hole diameter (**right**). ability to learn on the data. Using only these data leads to the phase CNN achieving a mean-square error of 0.0002 and an R-square statistic of 0.998; this is a large improvement from the previous R-square score of 0.971. Similarly, the LSTM model sees an improvement now achieving a mean-square error of 0.00027 and R-square 0.998. Figure 18 demonstrates the goodness of fit of the new predictions. The LSTM model does not improve to the level of the CNN models, even with the simplified dataset. The LSTM is likely suffering from the additional complexity of leveraging both phase and amplitude data. As seen in the CNN models, the phase data fit best to a ReLU function, meaning it presents a linear trend, while the amplitude data fit better to a hyperbolic tangent, making the trend nonlinear. Further work is likely needed to determine how best to improve the LSTM model; however, Figure 19 depicts the large-scale improvement seen in the phase CNN when the Fresnel and Fraunhofer data are split. Here, it is evident that (1) the model when trained on only 1 MHz data is more accurate than when trained on just 5 MHz data if both the 2.17 mm and 6.47 mm holes are present and (2) when the 6.47 (more Fresnel-like data) is removed, both the 1 MHz and 5 MHz predictions are improved. (1) is consistent with the above argument that the 1 MHz data are more Fraunhofer-like than the 5 MHz data, and the sensitivity of the phase to the thickness is higher with 1 MHz (Figure 17). From (2), we can say that in the cases where both anomaly sizes are present, the data is compromised by the 6.47 mm data. This yields promising results in support of creating an ensemble of CNN models to separate the predictions of the more simplistic Fraunhofer and the more complex Fresnel data. Figure 18: Actual acoustic path lengths (x) vs. predicted acoustic path lengths (y) on only the data with a 2.17 mm anomaly (Fraunhofer regime) with a regression line to demonstrate how close the model is to perfect accuracy plotted for both the phase CNN **(left)** and the LSTM model **(right)**.In each figure, we see the predictions from left to right of the following: 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).