## 3 Analysis  
To quantify the visualized raster orientation, an in-house MALAB script was used to analyze the fully captured acoustic waveform. The raster was determined with respect to the index direction, labeled \(x_{1}\) in Figure 1. Data were taken at \(N_{1}\) discrete points in \(x_{1}\in\{0,L_{1}\}\) and \(N_{2}\) discrete points in \(x_{2}\in\{0,L_{2}\}\) that were equally spaced such that \(\Delta x_{1}=x_{1,j+1}-x_{1,j}\) for all \(j\in\{1,2,\ldots,N_{1}\}\) and \(\Delta x_{2}=x_{2,k+1}-x_{2,k}\) for all \(k\in\{1,2,\ldots,N_{2}\}\). The digitized signal corresponding to the acoustic waveform was defined as \(F\big{(}t_{i},x_{1,j},x_{2,k}\big{)}\) where \(t_{i}\), \(x_{1,j}\), and \(x_{2,k}\) are discrete values in, respectively, time \(t\), the index direction (\(x_{1}\)), and the scan direction (\(x_{2}\)). In the present study, a 100 MHz digitizer was utilized; thus  
Figure 1: Images of (**a**) a representative component with \(\times_{1}\) and \(\times_{2}\) directions and (**b**) a depicted scan path and dimensions of the component. \(t_{i+1}-t_{i}=\frac{1}{100\text{ MHz}}=10\text{ ns}\), and a raster step size of \(\Delta x_{1}=\Delta x_{2}=0.1\text{ mm}\) was used. The data were shifted in time such that  
\[\overset{\sim}{t}(t,x_{1},x_{2})=t-t_{0}(x_{1},x_{2}) \tag{1}\]  
where \(t_{0}(x_{1},x_{2})\) refers to the time at which the initial front wall echo was received by the transmitting transducer. This time shift has the effect of aligning each A-scan in time in reference to the front wall echo at each \((x_{1},x_{2})\) location, resulting in the array \(F\big{(}\overset{\sim}{t}(t,x_{1},x_{2}),x_{1,k},x_{2,l}\big{)}\). Next, a Gaussian smoothing technique was used such that  
\[\hat{F}\big{(}t,x_{1,j},x_{2,k}\big{)}=\frac{1}{2\pi\sigma_{x_{1}}\sigma_{x_{2 }}}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}e^{\frac{-( \overset{\sim}{x}_{1}-x_{1})^{2}}{2\sigma_{x_{1}}^{2}}}F\big{(}\overset{\sim} {t}(t,\overset{\sim}{x}_{1},\overset{\sim}{x}_{2}),\overset{\sim}{x}_{1}, \overset{\sim}{x}_{2}\big{)}d\overset{\sim}{x}_{1}d\overset{\sim}{x}_{2} \tag{2}\]  
where the distribution parameters \(\sigma_{x_{1}}\) and \(\sigma_{x_{2}}\) define the extent of the spatial smoothing of the captured acoustic signal. In the present study, \(\sigma_{x_{1}}\) and \(\sigma_{x_{2}}\) were set to three times the raster spacing, specifically 0.3 mm. It was found that 0.3 mm yielded acceptable results without loss of the individual raster features. Next, a spatially averaged signal \(\overline{F}(t)\) was created as  
\[\overline{F}(t)=\frac{1}{L_{1}L_{2}}\int\limits_{0}^{L_{2}}\int\limits_{0}^{L _{1}}\hat{F}(t,x_{1},x_{2})dx_{1}dx_{2} \tag{3}\]  
A typical result of the spatially averaged signal is shown in Figure 2. Notice in this figure that individual peaks can be observed, with the peak around \(t=0\) corresponding to the front wall reflection of the part and the peak near \(t=6\) us corresponding to the backwall reflection of the part. From the averaged acoustic signal shown in Figure 2a, the time at which the individual \(N\) local maxima of \(\overline{F}(t)\) occur can be identified. Except for the front wall and the back wall, each of the observed local maxima are caused by the acoustic reflection of the interface between the individual deposited layers, with a couple of notable exceptions discussed herein. The most relevant exceptions are the peaks near the front surface of the part, i.e., the second and third peaks near \(t=0.5\) us and \(t=0.7\) us seen in Figure 2a. This occurs due to the mode shape change of the transmission frequency of the transducer near 10 MHz and the natural frequency caused by the repeated layers that drive the frequency toward 6 MHz in the present study. This mode shape shift causes peaks in the reflected acoustic waveform that do not necessarily correspond to an interface between layers. As such, peaks closer than the known layer height were discarded in the present analysis but will be noted in the report.