## 3 Materials and Methods
### Ultrasonic Data Processing Approach to Estimate Delamination Depth  
An overview of the proposed ultrasonic data processing approach to estimate the concrete delamination depth is illustrated in Figure 5. The proposed approach is an optimization approach to obtain an optimal estimation of delamination depth by minimizing specific objective functions derived from Rayleigh and Rayleigh-Lamb equations. The data processing procedure is described below. Figure 4: Illustration of the contactless ultrasonic sensor array system to collect scanning data from concrete samples. CN board is a connector board that makes the connection between the output ends of the amplification board and the digital oscilloscope. As a preprocessing step, the Rayleigh wave velocity (\(c_{R}\)) of the tested concrete specimen was obtained using ultrasonic scanning data from a pristine region, \(u_{p}(t,x)\). To obtain \(c_{R}\), \(u_{p}(t,x)\) is converted to the frequency-wavenumber (\(f\)-\(k\)) domain using a 2-D Fourier transform, given by  
\[\overline{U}_{p}(f,k)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}u_{p}(t,x) e^{-l2\pi(f+kx)}dxdt, \tag{5}\]  
where \(\overline{U}_{p}(f,k)\) is the converted \(f\)-\(k\) domain ultrasonic data from a pristine concrete region. Then, \(c_{R}\) computed by  
\[c_{R}=\frac{f_{peak}}{k_{peak}}, \tag{6}\]  
where \(f_{peak}\) and \(k_{peak}\) are the peak frequency and wavenumber in the magnitude spectrum \(\left|\overline{U}_{p}(f,k)\right|\), respectively. In Step 1, the P- and S-wave velocities (\(c_{P}\) and \(c_{S}\), respectively) of the tested concrete specimen are updated by solving the following optimization problem:  
\[ \begin{array}{c}\underset{c_{P},\ c_{S}}{\min}L_{1}(c_{P},\ c_{S})\\ \text{subject to }c_{P}^{lower}\leq c_{P}\leq\ c_{P}^{upper}\\ c_{S}^{lower}\leq c_{S}\leq\ c_{S}^{upper},\end{array}  \tag{7}\]  
where \(c_{P(S)}^{lower}\) and \(c_{P(S)}^{upper}\) are the lower and upper bounds of \(c_{P(S)}\), respectively, and \(L_{1}\) is the first objective function in the proposed approach, given by  
\[L_{1}(c_{P},\ c_{S})=\left|\left(\frac{c_{R}}{c_{S}}\right)^{6}-\left(\frac{ c_{R}}{c_{S}}\right)^{4}+8\left(\frac{c_{R}}{c_{S}}\right)^{2}\left(3-2\left( \frac{c_{S}}{c_{P}}\right)^{2}\right)+16\left(\left(\frac{c_{S}}{c_{P}} \right)^{2}-1\right)\right|. \tag{8}\]  
The \(c_{P}^{lower}\), \(c_{P}^{upper}\), \(c_{S}^{lower}\), and \(c_{S}^{upper}\) values used in this study are 3200 m/s, 4000 m/s, 1700 m/s, and 2400 m/s, respectively. The ranges of \(c_{P}\) and \(c_{S}\) set by the lower and upper bounds are the typical wave velocity ranges of normal-strength concrete. The optimal solution of this optimization problem is obtained using the interior-point technique [21] implemented in a MATLAB function \"fmincon.\"  
In Step 2, the optimal delamination depth is obtained by solving the second optimization problem:  
\[ \begin{array}{c}\underset{d}{\min}L_{2}(d)\\ \text{subject to }d^{lower}\leq d\leq d^{upper},\end{array}  \tag{9}\]  
Figure 5: Overview of the proposed ultrasonic data processing approach to estimate the delamination depth of concrete. where \(d^{lower}\) and \(d^{upper}\) are the lower and upper bounds of the delamination depth, respectively. \(L_{2}\) is the second objective function given by  
\[L_{2}(d)=\frac{1}{N_{50}}\sum_{i=1}^{N_{50}}\Bigl{[}k_{data}^{50}(f_{i})-k^{50}( f_{i})\Bigr{]}^{2}+\frac{1}{N_{A0}}\sum_{i=1}^{N_{A0}}\Bigl{[}k_{data}^{A0}(f_{i})-k^{ A0}(f_{i})\Bigr{]}^{2}, \tag{10}\]  
where \(N_{50}\) and \(N_{A0}\) are the total numbers of fundamental symmetric and antisymmetric Lamb wave modes from the ultrasonic scanning data, respectively. \(k_{data}^{50}(f_{i})\) and \(k_{data}^{A0}(f_{i})\) are the S0 and A0 mode wavenumbers, respectively, identified in the dispersion curves obtained using the ultrasonic scanning data at the frequency \(f_{i}\), while \(k^{50}(f_{i})\) and \(k^{A0}(f_{i})\) are the analytically computed wavenumbers for the S0 and A0 modes at the frequency \(f_{i}\).