## 4 2D Spatial-Temporal Spectrum Method  
For identification of one or a few non-uniform areas in the sample under investigation we propose a method based on the analysis of a 2D spatial-temporal spectrum of the B-scan of normal displacements [26]. This spectrum is obtained by the 2D Fourier transform \(FFT_{2D}\) of the B-Scan data \(\xi(t,x)\):  
\[U\Big{(}f,\ \lambda^{-1}\Big{)}=FFT_{2D}[\xi(t,\ x)], \tag{3}\]  
where \(f\) is the frequency and \(\lambda\) is the wavelength. The 2D spatial-temporal spectrum calculated using the B-scan measured along the line crossing the single circular non-uniformity (Figure 11) is presented in Figure 17. The colour scale of the amplitudes is shown on the right side of the image. In this figure two zones indicated by two rectangles show two waves \(A^{\prime}_{0}\) and \(A^{\prime\prime}_{0}\) propagating with different velocities. Their amplitudes are shown by the colour scale on the right of Figure 17. The largest amplitudes are at a frequency of 45 kHz and spread along spatial frequencies \(\lambda^{-1}\). The mode \(A^{\prime}_{0}\) is concentrated around the spatial frequency \(\lambda^{-1}\) = 0.47 mm\({}^{-1}\) that corresponds approximately to the phase velocity 95 m/s. The \(A^{\prime\prime}_{0}\) mode is spread around the lower spatial frequency \(\lambda^{-1}\) = 0.37 mm\({}^{-1}\) and the corresponding phase velocity approximately is 120 m/s. From the presented results it is possible to see that the amplitude of the \(A^{\prime}_{0}\) mode is higher than of the \(A^{\prime\prime}_{0}\) mode. The presented results show that the discussed method enables revealing the presence of a few modes instead of one mode which usually propagates in uniform thin samples. The phase velocity \(c_{\rm ph}\) of the A\({}_{0}\) mode can be found from the well-known relation \(c_{\rm ph}\ (f)=\lambda(f)f\), where the wavelength \(\lambda\) is found from the 2D spatial temporal spectrum at the selected frequency \(f\). Thus, this method is based on the exploitation of different propagation velocities of those waves. The waves with different phase velocities can be separated by a 2D spatial-temporal band-pass filter:  
\[U_{f}\Big{(}f,\ \lambda^{-1}\Big{)}=U\Big{(}f,\ \lambda^{-1}\Big{)}\cdot H_{f }\Big{(}f,\ \lambda^{-1}\Big{)}, \tag{4}\]  
where \(H_{f}\ (f,\ \lambda^{-1})\) is the spatial-temporal response of the filter. The cut-off spatial and temporal frequencies of the filter are chosen according to the mode which should be separated. The cut-off spatial  
Figure 17: 2D spatial-temporal spectrum of the B-scan modes in the case of a single circular non-uniformity. frequencies \(\lambda^{-1}\) are 0.35 mm\({}^{-1}\) and 0.53 mm\({}^{-1}\) shown in Figure 17. Correspondingly, the temporal cut-of frequencies are 30 kHz and 60 kHz. The filtered B-Scan \(\xi_{f}(t,x)\) is obtained by the inverse 2D Fourier transform:  
\[\xi_{f}(t,\ x)=FFT_{2D}^{-1}\Big{[}U_{f}\big{(}f,\ \lambda^{-1}\big{)}\Big{]}. \tag{5}\]  
From the analysis presented follows that the modes with different phase velocities can be separated applying 2D filtering of the 2D spectrum. However, if the spatial or temporal narrow band filtering is used then according to the uncertainty principle the spatial resolution \(\Delta x\) is lost:  
\[\lambda_{x}^{-1}.x\approx 1, \tag{6}\]  
where \(\Delta\lambda_{x}^{-1}\) is the bandwidth of the 2D filter in the spatial domain. It means that in this case it is impossible to get spatial dependence of the phase velocity versus distance \(x\) and only the average ultrasound velocity \(\overline{c}\) along the selected distance \(\Delta x=x_{2}-x_{1}\) can be obtained:  
\[\overline{c}=\frac{1}{\Delta x}\ \int_{x_{1}}^{x_{2}}c(x)dx. \tag{7}\]  
Let us analyse the case with a big number of circular non- uniformities of different diameters (Figure 12).