## Materials and Methods
### Mathematical Methods
#### 2.2.2 Virtual Refinement
The computation took place in MATLAB (MathWorks, Natick, MA, US) and the interpolation with the command _interp3_. In this work, the five interpolation methods (\"linear\", \"nearest\", \"cubic\", \"spline\", and \"makima\") that this command already offers in MATLAB are used and compared. Figure 9 (top row) depicts snapshots of the time data in the measuring plane. Figure 9a is the very time-consuming measurement with a measurement grid point distance of 2 mm without flaw. Figure 9b the measurement with a grid point distance of 14 mm. This measurement needs only a fraction of time as the one with the 2 mm grid point distance but with decreased detectability. The virtual measuring grid points are depicted in Figure 9c. Figure 9c is a refined image of Figure 9b using the interpolation method \"nearest\" to a 2 mm grid point distance. It can be seen that these are the same data, and the grid is refined by copying the data to the virtual measuring grid points. Figure 9 (bottom row) depicts the back propagated amplitude distribution on the top of the particleboard from the upper row data. Figure 9c is the \"default\" and Figure 9e with the coarse measuring grid point distance of 14 mm. Figure 9f is the result of the refined measuring grid of Figure 9c. The circular shape of the main lobe of the transmitter can be clearly identified. Figure 7: Back propagated amplitude with different measuring grid point distances. (Top row) With no flaw on the particleboard; (lower row) with a 20 mm flaw. (**a**,**d**) with a 2 mm measuring grid, (**b**,**e**) with a 10 mm measuring grid, and (**c**,**f**) with a 14 mm measuring grid. Figure 10 shows a sequence of steps from the measurement of the pressure values to the final display of the pressure amplitude images. First, the measurement must be performed and the time data from each measurement point from the C-scan must be stored. This data can be stored as a 3D  
Figure 8: The idea of virtual measuring grid points. Figure 9: The upper row depicts snapshots of the time data in the measuring plane with different measuring grid point distances. (**a**) “Default” measuring grid with \(\mathrm{d}x=\mathrm{d}y=2\) mm. (**b**) The coarse measuring grid with a grid point distance of \(\mathrm{d}x=\mathrm{d}y=14\) mm; (**c**) the data from (**b**) refined with the interpolation method “nearest” to a measuring grid \(\mathrm{d}x_{Virt}=\mathrm{d}y_{Virt}=2\) mm. The lower row depicts the back propagated amplitude distribution at the top of the particleboard. (**d**) “default” measuring grid with \(\mathrm{d}x=\mathrm{d}y=2\) mm. (**e**) The coarse measuring grid with a grid point distance of \(\mathrm{d}x=\mathrm{d}y=14\) mm and (**f**) refined with the interpolation method “nearest” to a measuring grid \(\mathrm{d}x_{Virt}=\mathrm{d}y_{Virt}=2\) mm. matrix, where two dimensions represent the space and one, the time. After that, various material constants must be specified for the procedure, such as the speed of sound of the air, which can be calculated according to [PERSON] [27] after measuring humidity, air pressure, and temperature. In the experiments we carried out, the speed of sound of the air was 343 m/s. After entering the important parameters, the decision follows whether the detectability should be increased or not. If the detectability is to be increased, because a rough measuring grid was used, the existing 3D matrix must be enlarged in the two dimensions representing the space and filled with the interpolation results. This is followed by a discrete Fourier transformation and the application of re-radiation method. As a result, the pressure amplitudes at the previously defined distance to the measuring plane are displayed. The aforementioned method is used in the following section for the identification of flaws.