## 3 Simulation and Analyses
### Verification of the Micro-Crack Signal on the Rough Surface  
Several simulations were conducted to demonstrate the significant degradation of the signal-to-noise ratio under the influence of surface roughness. As an example, the front surface is the rough surface while the micro-crack is on the smooth back surface. The depth of the crack is 400 \(\upmu\)m, and width 250 \(\upmu\)m. The front-surface roughness \(\mathbf{R_{a}}\) is 25 \(\upmu\)m and \(\mathbf{R_{sm}}\) 400 \(\upmu\)m. Figure 5 shows the propagation of the acoustic wave for ultrasonic detection of micro-crack. The coherent and incoherent waves caused by the interaction of the incident wave with the rough interface are shown in Figure 5a-c. It can be seen clearly that the reflected wave beam from the rough coupled interface inside water is no longer focused. The incoherent wave inside the solid is scattered in all directions. Figure 5d shows that the energy intensity of the reflected wave from the crack reduces significantly. The pulse-echo of the crack transmits from the rough solid-water interface. The energy intensity further weakens because of multiple scattering, as shown in Figure 5e,f. The phenomenon agrees with the previous analysis. Figure 6 shows several raw A-scan signals acquired from the simulation models. The size of micro-cracks on the smooth back surface in the models is the same as that in Figure 5. The roughnesses \(\mathbf{R_{a}}\) on the front surface are set to 3.2 \(\upmu\)m, 12.5 \(\upmu\)m, and 25 \(\upmu\)m, respectively. The signals at the center correspond to the micro-crack, as shown in Figure 6a. The noise signals occurring elsewhere are attributed to scattering from the rough surface. Figure 6b,c show the echo received from rougher surfaces; the amplitude of the crack signal decreases with increase in \(\mathbf{R_{a}}\). When the front surface roughness \(\mathbf{R_{a}}\) is 25 \(\upmu\)m, the signal of the crack becomes nearly indistinguishable from the noise (Figure 6a). The reflected wave signals obtained from the front surface get wider with increase in \(\mathbf{R_{a}}\) due to scattering in different directions. Figure 6d presents the reference signal when no micro-crack is on the rough back surface, and there is no apparent micro-crack signal (Figure 6c). The results correspond to previous analyses that the energy intensity of transmitted wave reduces because of multiple scattering. Furthermore, the amplitude of the noise signal increases slightly. Therefore, the signal-to-noise of the micro-crack signal degrades significantly with rougher surface. The surface roughness-induced attenuation of the coherent reflected longitudinal wave and transmitted shear wave are characterized by the time-domain signal distortion of the micro-crack. It is necessary to distinguish the characteristics between the micro-crack signal and noise signal. For most ultrasonic nondestructive testing equipment, accurate identification of the crack mainly depends on the signal-to-noise ratio of the pulse-echo signal. A scalar parameter SARF is defined as the signal amplitude ratio factor to clearly demonstrate the signal-to-noise ratio. SARF can be expressed as:  
\[SARF=\frac{\overline{As_{max}}}{\overline{An_{max}}} \tag{7}\]  
where \(AA_{max}\) is the maximum amplitude of the echo signal of the crack and \(An_{max}\) is the maximum amplitude of the noise signal, as shown in Figure 6b. \(\overline{As_{max}}\) and \(\overline{An_{max}}\) are mean amplitudes of the crack echo signal and noise signal, respectively, obtained from multiple simulations or measurements. As can be seen, when SARF approaches 1, the crack signal is completely immersed in the noise signal. For most industrial ultrasonic inspections, SARF is set to be greater than 3.16 to characterize the detection capacity of micro-cracks, which means the signal-to-noise ratio is larger than 10 dB. Figure 5: Scattering diagram of acoustic wave during ultrasonic detection process of micro-crack when the front surface is rough and back surface is smooth.