# Contactless Ultrasonic Wavefield Imaging to Visualize Near-Surface Damage in Concrete Elements
## 2 Materials and Methods
### Frequency-Wavenumber (f-k) Domain Wavefield Data Processing  
An overview of the _f-k_ domain wavefield data processing approach is shown in Figure 2. The data processing approach described in this section generally follows the data processing procedure \begin{table}
\begin{tabular}{c c} \hline
**Case** & **The Number of Impact Damage Points** \\ \hline
2 & 2 (D1 \& D2) \\
3 & 3 (D1 to D3) \\
4 & 4 (D1 to D4) \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
Figure 1: The contactless scanning ultrasound measurement hardware (**left**) and the concrete sample (**right**) that shows impact damage locations; ultrasonic scan area indicated by red box. Impact damage points were introduced sequentially from the position D1 to D4. that we proposed in [19]. However, the dimension of the forward and inverse Fourier transforms (FT) carried out in the process is increased by one to consider a 3-D data structure of the collected wavefield data: one dimension in time \(t\) and two more dimensions in space \(x,y\). The wavefield data processing procedure is described here. First, a 3-D FT is applied to the collected wavefield data to convert the data from time-space (\(t\)-\(x\)-\(y\)) to frequency-wavenumber (\(f\)-\(k_{x}\)-\(k_{y}\)) domains using  
\[\overline{S}\big{(}f,k_{x},k_{y}\big{)}=\int_{-\infty}^{\infty}\int_{-\infty} ^{\infty}\int_{-\infty}^{\infty}s(t,x,y)e^{-2\pi(ft+k_{x}x+k_{y}y)}dydxdt, \tag{1}\]  
where \(s(t,x,y)\) is the original time-space domain wavefield and \(\overline{S}\big{(}f,k_{x},k_{y}\big{)}\) the converted frequency-wavenumber domain wavefield. Then, \(\overline{S}\big{(}f,k_{x},k_{y}\big{)}\) is decomposed into two wavefield components: the forward propagating wavefield \(\overline{S}_{FP}\big{(}f,k_{x},k_{y}\big{)}\), and the non-propagating oscillatory field \(\overline{S}_{NPO}\big{(}f,k_{x},k_{y}\big{)}\). This is achieved by multiplying \(\overline{S}\big{(}f,k_{x},k_{y}\big{)}\) with filtering masks \(M_{FP}\big{(}f,k_{x},k_{y}\big{)}\) and \(M_{NPO}\big{(}f,k_{x},k_{y}\big{)}\), respectively, through  
\[\overline{S}_{FP}\big{(}f,k_{x},k_{y}\big{)}=\overline{S}\big{(}f,k_{x},k_{y} \big{)}M_{FP}\big{(}f,k_{x},k_{y}\big{)} \tag{2}\]  
and  
\[\overline{S}_{NPO}\big{(}f,k_{x},k_{y}\big{)}=\overline{S}\big{(}f,k_{x},k_{y }\big{)}M_{NPO}\big{(}f,k_{x},k_{y}\big{)}. \tag{3}\]  
Full detail about the masking functions are provided below in Section 3.2. The two decomposed wavefield components are then reconverted into time-space domain using an inverse 3-D FT through  
\[s_{FP}(t,x,y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\overline{S}_{FP}\big{(}f,k_{x},k_{y}\big{)}e^{2\pi(ft+k_{x}x+k_{y}y)} dk_{y}dk_{x}df \tag{4}\]  
Figure 2: An overview of the proposed \(f\)-\(k\) domain wavefield data processing approach. \[s_{NPO}(t,x,y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}\overline{S}_{NPO}\Big{(}f,k_{x},k_{y}\Big{)}e^{2\pi(f+k_{x}x+k_{y}y)}dk_ {y}dk_{x}df. \tag{5}\]  
In the next step, the time-cumulative wave energies, \(E_{FP}(x,y)\) and \(E_{NPO}(x,y)\), are computed up to time \(T\) for both wavefield components by  
\[E_{FP}(x,y)=\int_{0}^{T}[s_{FP}(t,x,y)]^{2}dt \tag{6}\]  
and  
\[E_{NPO}(x,y)=\int_{0}^{T}[s_{NPO}(t,x,y)]^{2}dt. \tag{7}\]  
Here, \(T\) is defined such that the wavefield data are long enough to include sufficient scattering content, but short enough not to include direct acoustic wave noise. The same value of \(T\), 450 \(\upmu\)s, is used to process all the wavefields collected from the pristine and damaged sample cases. To make use of spatial distribution of the non-propagating wave energy for damage location, the non-propagating wave energy is computed in time-space domain after reconverting those data from Fourier domain.