## 4 Results and Discussion
### Single Internal Defect
To improve the overall accuracy of the inversion, information of different frequencies should be taken into consideration. A lower frequency has a better tolerance for a relatively large-scale structure while a higher frequency should be considered for a small substructure [15]. Therefore, successive inversions with a moving frequency from low to high were applied to mitigate the nonlinearity of the inversion problem, resulting in a more robust result than a monochromatic inversion. In this study, a frequency group of seven discrete frequencies from 400 kHz to 1600 kHz were adopted because the excited and received signals showed an effective bandwidth of 480 kHz\(-\)1600 kHz, as shown in Figure 6b. Figure 9 shows the reconstructed tomographs when the inversion finished at frequencies of 0.4 MHz, 0.6 MHz, 0.8 MHz, 1 MHz, 1.2 MHz, 1.4 MHz and 1.6 MHz, respectively. Figure 8: (**a**) True model with a 3 mm circular hole and tomographs with different iterations: (**b**) \(k=5\); (**c**) \(k=10\); (**d**) \(k=20\); (**e**) \(k=40\). Figure 7: The evolution of the objective function with the number of iterations. Compared with the monochromatic inversion of 1 MHz, the overall noise was reduced based on the successive inversion of low frequencies. It could be seen that with the increase of the frequency of successive inversions, the accuracy of the 3 mm diameter circular hole was improved. The cross sections through the defect along the horizontal and vertical direction are presented in Figure 10. The FWI results showed a good agreement with the profile of the true defect in both directions. To quantitatively evaluate the reconstruction accuracy, the relative error of the full width at half maximum (FWHM) along the two directions along with their root mean square (RMS) error were considered. Therefore, the three indicators were defined as \(E_{V}=\frac{FWHM_{V}}{D_{def}}\), \(E_{H}=\frac{FWHM_{H}}{D_{def}}\) and \(E_{RMS}=\sqrt{\frac{E_{V}^{2}+E_{H}^{2}}{2}}\), where \(D_{def}\) denoted the diameter of the circular defect. Figure 11 shows the variations of the relative errors of the FWHM and their RMS error as a function of the successive-frequency inversions for circular holes with different diameters. Generally, the errors decreased as the successive-frequency increased and almost converged after the frequency of 1 MHz. It meant that successive-frequency inversion could indeed improve the accuracy of FWI tomography. Especially for the 3 mm diameter defect, the relative errors sharply decreased from 28.3% to 8.5%. When the defects were larger than 3 mm, the relative errors decreased to smaller than 2%. The relatively large error for the 3 mm diameter defect was due to the fact that 3 mm is close to half of the interrogating wavelength at 1 MHz, which is the theoretical resolution of the FWI [22]. However, the final error was smaller than 8%, indicating that a successive-frequency FWI could obtain an accurate tomograph for internal defects and the resolution approached a half wavelength. Figure 5.0 Tomographs of the 3 mm circular hole based on successive-frequency inversions at: (**a**) 0.4 MHz; (**b**) 0.6 MHz; (**c**) 0.8 MHz; (**d**) 1 MHz; (**e**) 1.2 MHz; (**f**) 1.4 MHz; (**g**) 1.6 MHz. Figure 5.1 The variation of relative errors of the FWHM as a function of successive-frequency inversions for circular holes: (a) 3 mm; (b) 4 mm; (c) 5 mm; (d) 6 mm. Figure 5.2 Cross sections of the reconstruction of the 3 mm diameter circular hole along the (a) horizontal and (b) vertical direction.