# Contactless Ultrasonic Wavefield Imaging to Visualize Near-Surface Damage in Concrete Elements
## 3 Results
### Concrete Damage Visualization  
The complete wavefield data processing approach described in Section 2.3 was applied to the measured time-space wavefield data shown in Figure 4. Figure 6a,b show the filtering masks used to extract forward propagating surface waves (\(M_{FP}\)) and non-propagating oscillatory fields (\(M_{NPO}\)), respectively. Each mask is defined in _f-k\({}_{x}\)-k\({}_{y}\)_ (3-D), but in the figure it is shown in _kx-k\({}_{y}\)_ (2-D) at \(f\) = 43 kHz for visualization purposes. In the figures, white and black colors represent complete pass and reject of the masking function, respectively. Assuming that the majority of the wave components propagating in the pristine concrete sample takes the form of forward propagating surface waves, \(M_{FP}\) shown in Figure 6a is defined as  
\[M_{FP}\Big{(}f,k_{x},k_{y}\Big{)}=\left\{ \begin{array}{ll}1&\text{if}\left| \overline{S}_{p}\Big{(}f,k_{x},k_{y}\Big{)}\right|>0.05\,\text{max}\Big{(} \big{|}\overline{S}_{p}\big{|}\Big{)}\\ 0&\text{otherwise},\end{array} \right. \tag{9}\]  
where \(\big{|}\overline{S}_{p}\big{|}\) is the _f-k_ magnitude spectrum (3-D) of the wavefield from the pristine concrete sample. Here, the _f-k_ region having magnitude greater than 5% of \(\text{max}\Big{(}\big{|}\overline{S}_{p}\big{|}\Big{)}\) is assumed to capture most of the forward propagating surface wave energy. Then, \(M_{NPO}\) shown in Figure 6b is defined to suppress forward propagating and back-scattered surface waves using  
\[M_{NPO}\Big{(}f,k_{x},k_{y}\Big{)}=\left\{ \begin{array}{ll}1-M_{FP}\Big{(}f, k_{x},k_{y}\Big{)}&\text{if}\,k_{x}>0\\ 1-M_{BS}\Big{(}f,k_{x},k_{y}\Big{)}&\text{if}\,k_{x}\leq 0,\end{array} \right. \tag{10}\]  
where \(M_{BS}\) is the _f-k_ domain mask to extract back-scattered surface waves that is obtained by flipping \(M_{FP}\) with respect to zero-\(k_{x}\) line (\(k_{x}=0\)). Although back-scattered surface waves can indicate presence of damage, we suppress them to extract non-propagating wave components that can have higher damage localization capability. The extracted forward propagating wavefields using \(M_{FP}\) (Figure 6a) for all the concrete sample cases are shown in Figure 7, and the extracted non-propagating oscillatory fields using \(M_{NPO}\) (Figure 6b) in Figure 8. The extracted forward propagating surface waves for all the concrete sample cases show similarity in amplitudes and geometric patterns, demonstrating insensitivity of forward propagating components to concrete damage. A significant distinction between each sample case is observed from the extracted non-propagating oscillatory fields however, as shown in Figure 8. The pristine concrete sample (Case 0) case, shown in Figure 8a, indicates a very small amount of non-propagating wave energy throughout the scanned region. The damaged sample cases (Cases 1 to 4), shown in Figure 8b-e, exhibit non-propagating wave components with high oscillatory energies nea  
Figure 5: Normalized _f-k_ (_f-k\({}_{x}\)-k\({}_{y}\)_) domain magnitude spectra of the wavefield data at the center frequency of excitation (_f_ = 43 kHz): (**a**) Case 0 (pristine), (**b**) Case 1 (D1 only), (**c**) Case 2 (D1 and D2), (**d**) Case 3 (D1 to D3) and (**e**) Case 4 (D1 to D4). The color bar indicates normalized spectral amplitude. Normalized non-propagating wave energy maps, \(E_{\text{NPO}}^{N}(x,y)\), for all the tested concrete sample cases are shown in Figure 9. As observed from the extracted non-propagating oscillatory fields in Figure 8, the damaged sample cases (Figure 9b-e) show higher \(E_{\text{NPO}}^{N}\) values nearby the impact locations, while the pristine sample case (Figure 9) shows negligible \(E_{\text{NPO}}^{N}\) values throughout the scanned area. Figure 6: Filtering masks to extract (**a**) forward propagating surface waves (\(M_{FP}\)) and (**b**) non-propagating oscillatory fields (\(M_{\text{NPO}}\)). In the images, the white color represents complete pass and the black color complete reject of the masking function.