## 4 Results and Discussion  
### Single Internal Defect  
At first, a single internal defect was embedded in a turbine blade to perform data processing and a parametric study of ultrasonic FWI tomography. For simplicity, a circular hole with a diameter of 3 mm was located at (40,120). In the first simulation, an ultrasonic wave was excited with the first transducer located at (1,6). The corresponding wave propagations at different time slots are shown in Figure 5. From the simulated wavefield, the ultrasonic wave reflected at the ice-steel interface and diffracted with the defect. At \(t=6\) us, the incident wave began to reach the edge of the blade. The reflection from the ice-steel interface can be seen in Figure 5, indicated by a red arrow. The transmitted wave and several interface waves propagated inside the blade. When \(t=21\) us, the direct wave arrived at the receiving transducers as shown in Figure 5. At \(t=26\) us, the propagating waves interacted with the defect, generating diffracted waves, which can be seen in Figure 5. Finally, the ultrasonic wavefield was recorded by all of the 36 transducers. Without a loss of generality, the waveform recorded by the 19 th receiver placed at (84,6) and its normalized amplitude spectrum are shown in Figure 6. The first wave package was the direct wave from the 1 st transducer to the 19 th transducer. In addition, there were two more waves due to the reflection at the boundary and the diffraction from the internal defects. The path of the diffraction wave from the defect is depicted in Figure 5d with blue dashed lines. The corresponding time-of-flight was about 26 us. Thus, the second wave package was the diffracted wave from the defect, followed by multiple waves reflected from the lower boundary of the blade. All of the reflections from the ice-steel interface and the diffraction signal from the defect can be seen in the A-scan signal in Figure 6a. Due to the complex geometry of the turbine blade, reflection signals from the boundaries were complicated and the energy was very high, making the diffracted wave from the internal defect difficult to identify. Moreover, compared with the excitation, the amplitude spectrum was distorted to a certain degree due to the existence of wave scattering phenomena, as shown in Figure 6b. Nevertheless, the recorded signal still contained information at frequencies around 1 MHz with a large bandwidth. The number of iterations is one of the key parameters in the inversion process. With the increase of iterations, the accuracy of the inversion result will be improved but the computing time will increase linearly. Therefore, it is important to determine the number of iterations to balance the accuracy and the computation cost. The variation of the misfit function, as described by Equation (3), as a function of iteratio  
Figure 5: Ultrasonic wave propagation excited with the 1 st transducer at different time slots: (**a**) t = 6 μs; (**b**) = 8 μs; (**c**) = 21 μs; (**d**) = 26 μs. Figure 6: The recorded signal of the 19 th transducer excited with the 1 st transducer. (**a**) The A-scan waveform and (**b**) the corresponding amplitude spectrum. Obviously, the misfit monotonically decreases as the iterative number increases. After 20 iterations, the misfit was small enough (lower than 1%), indicating that the accuracy of the inversion result was high enough. The FWI tomographs for the different iterations are also shown in Figure 8. When the iteration was set to as few as \(k=5\), the quality of the FWI image was poor and the defect was difficult to observe. After 10 iterations, the defect became clearly visible but the detailed information of the reconstructed defect changed. When the number of iterations was larger than 20, the inversion results became stable. Hence, the number of iterations was set as 20 for the following studies. Although the inversion result converged after 20 iterations, there was still a remarkable noise in the final result.