## Materials and Methods
### Mathematical Methods
#### 2.2.2 Virtual Refinement  
The greatest potential for reducing the required measuring time is a reduction in the number of measuring grid points. We briefly explain what happens to the results of the C-scan by the re-radiation method when the measuring grid is coarsened. Figure 6 depicts the calculated surfaces of the particleboard, with and without different flaw sizes. The measuring grid point distance was always \(\mathrm{d}x=\mathrm{d}y=2\) mm, which is less than \(\lambda/3\) (\(\lambda\approx 6.8\) mm) for the C-scan. The circular shape of the main lobe of the transmitter is depicted as the red area in each diagram. The flaws were placed with its center at the position \(x\approx 90\) mm and \(y\approx 70\) mm. At a size of 4 mm, the flaws are visible and can be identified at the yellow, green, and blue area at the flaw position. An amplitude decrease caused by the larger flaws measuring 10 and 20 mm is depicted within the blue area. The flaw in the size of 2 mm cannot be recognized because it is smaller than the detectability limit, which means that the maximum detectability of flaws of the Rayleigh-Sommerfeld diffraction integral is in a size of \(\lambda/2\) when \(\lambda\) is the wavelength of the propagation medium [25; 26]. Since the transmitter was operated with 50 kHz and the sound velocity of air is about 343 m/s, the wavelength is about 6.8 mm, and the 2 mm flaw is smaller than the maximum detectability. With the measuring grid point distance of 2 mm, 10,201 measuring grid points had to be measured for such a large measuring window of 200 \(\times\) 200 mm. Since each measurement needs 5 s, this is an enormous expenditure of time. Our proposed method to mitigate this large time expenditure is coarsening the measuring grid. Figure 7 depicts the results for no flaw and for the 20 mm flaw when the measuring grid get coarsened. Figure 7a,d are the \"default\" images. The top row is without a flaw, and below is with a 20 mm flaw. Figure 7b,e have a 10 mm measuring grid point distance, and Figure 7c,f a 14 mm grid point distance. The results in Figure 7b,e are influenced by much more noise. With some knowledge of the used transmitter, the circular form can be recognized. The 20 mm flaw is identifiable. However, a red area at \(y\approx 20\) mm can be seen due to a failure influenced back propagation due to less measuring grid points. The grid point distance is insufficient, and the images are not clear enough for a sufficient interpretation. When the measuring grid is coarsened down to the 14 mm grid, nothing can be detected. With this measuring grid point distance, only 196 measuring grid points need to be measured, reducing the experiment time by up to 98%. Figure 5: (**a**) Back propagated amplitude pressure field of several planes in a distance of 1 mm. At the rigth side is the measuring plane, and depicted at the left side is the air layer directly above the top surface of the particleboard with a 20 mm flaw. (**b**) Back propagated amplitude pressure field at the top of the particleboard with a 20 mm flaw. Figure 6: Back propagated amplitudes at the top of the particleboard. (**a**) Without a flaw, (**b**) with a 2 mm flaw, (**c**) with a 4 mm flaw, (**d**) with a 5 mm flaw, (**e**) with a 10 mm flaw, and (**f**) with a 20 mm flaw. The idea is to take these coarse grids and refine it in the measuring plane, as depicted in Figure 8. The black points refer to the measuring positions with the grid point distance of d\(x\) and d\(y\). The white points are the virtual grid points; with them, the grid has the new virtual grid point distance of d\(x_{Virt}\) and d\(y_{Virt}\). The idea is to fill these virtual measuring grid points with transient interpolated data and then apply the re-radiation method. This is due to the flaws having different geometrical forms in reality and finding a uniform function which covers all imaginable forms is not target-oriented. Shown here are the different interpolation methods employed.