# Deep Machine Learning for Path Length Characterization Using Acoustic Diffraction
## 1 Introduction
### Theory
\begin{table}
\begin{tabular}{c c c c c} \hline
**Frequency** & **Anomaly** & **Anomaly** & **Wavelength** & **Diffraction** \\  & **Width (mm)** & **Height (mm)** & **(mm)** & **Limit (mm)** \\ \hline
1 MHz & 6.47 & 0.68 & 5.9 & 7.09 \\ \hline
1 MHz & 2.17 & 0.68 & 5.9 & 0.79 \\ \hline
1 MHz & 6.47 & 1.32 & 5.9 & 7.09 \\ \hline
1 MHz & 2.17 & 1.32 & 5.9 & 0.79 \\ \hline
1 MHz & 6.47 & 6.46 & 5.9 & 7.09 \\ \hline
1 MHz & 2.17 & 6.46 & 5.9 & 7.07 \\ \hline
5 MHz & 6.47 & 0.68 & 1.2 & 34.88 \\ \hline
5 MHz & 2.17 & 0.68 & 1.2 & 3.92 \\ \hline
5 MHz & 6.47 & 1.32 & 1.2 & 34.88 \\ \hline
5 MHz & 2.17 & 1.32 & 1.2 & 3.92 \\ \hline
5 MHz & 6.47 & 6.46 & 1.2 & 34.88 \\ \hline
5 MHz & 2.17 & 6.46 & 1.2 & 34.77 \\ \hline \end{tabular}
\end{table} 
Table 3: Values for derivation of and Fraunhofer diffraction limit. Due to the complexities of multiple forms of diffraction and multiple diffraction ranges, it is difficult to produce an analytical function for the diffraction profile of the acoustic signal. Although an analytical expression is not easily expressed, the signal at a given observation point may still be characterized. According to the Huygens-Fresnel theory [22], every point on a wavefront is a source of spherical wavelets which mutually interfere. As such, the acoustic signal received at the top of the experiment can be expressed as the superposition of all secondary waves that initiated at the entrance plane of the acoustic anomaly. This concept is illustrated by the black lines of Figure 3.  
The Huygens-Fresnel theory allows us to presume that the sensor signal observed at any lateral location can be represented as the amplitude and phase of the acoustic wave at that location. The observed amplitude and phase for each point are those of the superposed waves at those points [23]. Since these values are directly impacted by the shape, size, and acoustic impedance of the anomaly, the collected signal contains information about the nature of the anomaly. This tells us that the spatial profile of the observed wave is uniquely determined by the properties of the anomaly which caused the diffraction. As discussed in [9], additional complexity is introduced as the signal crosses the barrier between the emitter's surface and the steel specimen. This is due to a portion of the signal being reflected back toward the emitter. The reflection must be removed to improve the quality of the received signal. This is not easily accomplished in the time domain; however, in the frequency domain, we can eliminate this effect by using the wave's transfer function. To calculate this transfer function, the FFT of the received wave is divided by the FFT of the original wave, this too is presented in [9]. The transfer function amplitude represents the reduction in the acoustic energy as it passes through the system while the transfer function phase indicates the change in acoustic path length. Together the amplitude and phase describe the full structural change of the wave caused by the medium. By analyzing the transfer function, we can extract features of an anomaly while ignoring the reflection barriers that are constant in the system. Therefore, the proposed machine learning methodologies utilize the transfer function's phase and amplitude data for analyzing the anomalies. Theoretically, it seems we can characterize the anomaly using the x-dependence of the sensor signal. In actuality, however, these signals are not easily differentiated. Figure 6 exhibits spatial profiles of the transfer function phase and amplitude data in which the anomaly size and acoustic path length varies. The human eye struggles to differentiate the different cases based on the spatial profile of this sensor data, particularly for the phase data which appear nearly random. It is easiest to see the similarities for matching acoustic path lengths among the amplitude data while very few similarities are found for the phase data. Finding trends in the data becomes more complex as the number of configurations increases.