## 3 The Zero-Crossing Reconstruction Method  
A spatial distribution of elastic properties of the modelled MICA paper sample can be found from the reconstructed spatial distribution of the phase velocity of the A\({}_{0}\) mode. For that the B-scan collected along the line \(y=0\) (Figure 6) can be exploited. Thus, at first the variations of the phase velocity of ultrasonic waves along the propagation path should be determined. The phase velocity usually is found from the distance \(\Delta x_{\text{ph}}\) which was covered by a particular phase point during the time interval \(\Delta t_{\text{ph}}\):  
\[c_{\text{ph}}=\Delta x_{\text{ph}}/\Delta t_{\text{ph}}. \tag{1}\]  
In general, the B-scan is the set of the signals measured at different distances (Figure 6), then the phase velocity can be estimated by taking two signals measured at neighbouring positions and estimating the propagation time of a particular phase point between those positions. As in the B-scan  
Figure 5: Ultrasonic pulses at two different distances from the 1 \(\times\) 5 mm\({}^{2}\) excitation zone: (a) 10 mm; (b) 40 mm. Figure 6: Simulated B-scan of the mcia MICA paper with hypothetic two rectangular zones with different elastic properties and thickness. The horizontal dashed red lines indicate borders of the rectangular darker zone. Figure 8: Mean value of the A\({}_{0}\) mode velocity versus distance \(x\). Figure 7: Spatial distributions of the A\({}_{0}\) mode velocity along B-scan across the rectangular darker zone of the simulated mica sample. The different colours correspond to the results obtained from different zero-cross lines in Figure 9.  
The diameter of this non-uniformity is 2 mm, e.g., comparable with the wavelength \(\lambda=(2-2.5)\) mm of the A\({}_{0}\) mode in the mica paper sample. The elastic parameters and thickness are the same as in the case of the previous rectangular non-uniformity. The simulated spatial distribution of the of-plane displacements of the mica paper at the instant \(t=225\) us is shown in Figure 10b. In the presented spatial distribution it is possible to see the ultrasonic signal reflected by the circular non-uniformity and propagating in a backward direction. The B-scan obtained along the line \(y=0\) mm crossing the centre of the circular non-uniformity is presented in Figure 11a. The reconstructed spatial distribution of the phase velocity obtained from the B-scan by the described zero-cross method is shown in Figure 11b. From the presented distribution follows that  
Figure 10: The sample of the mica paper with circular non-uniformity (**a**) and the snapshot of the spatial distribution of the ultrasonic pulsed field at the time instant \(225\) μs (**b**). Figure 9: The zero-crossing instants of the normal displacements waveforms along the B-scan line. velocities of the A\({}_{0}\) mode are different in the circular non-uniformity and in the rest of the mcia sample. Inside the non-uniformity the velocity is 110-115 m/s and outside 95-97 m/s. The spatial position of the ultrasound velocity peak exactly corresponds to the position of the circular non-uniformity. The presented example clearly shows ability of the described zero-crossing method to reconstruct correctly spatial distributions of A\({}_{0}\) mode phase velocity even in the case of small size, e.g., comparable with the wavelength, non-uniformities. We then analyse the case with a large number of circular non-uniformities of different diameters (Figure 12). This case is closest to the spatial distribution of zones with other properties of a real experimental mica specimen shown in Figure 13. The diameters of circular non-uniformities in simulation were selected randomly in the range 2-5 mm.