## 5 Conclusions  
In this paper, an ice-coupled ultrasonic tomography based on successive-frequency full waveform inversion technology was proposed to quantitively inspect the defects in complex-shaped parts. An ice-coupled ultrasound was able to treat the complex-shaped part as a regular structure and sharply reduced the mismatch of acoustic impedance compared with the immersion method. A series of numerical experiments were carried out to demonstrate the feasibility of the proposed method to quantitatively inspect the defects in a turbine blade. Firstly, the k-space pseudospectral method was applied to simulate ice-coupled ultrasonic testing for the turbine blade. A full matrix capture was then performed to record the ultrasonic signals with an evenly deployed transducer array. After data calibration, the recorded full wavefield was applied for a frequency-domain full waveform inversion based on the L-BFGS method. A single circular defect was applied to validate the  
Figure 16: Inversion results with an inaccurate initial model. (**a**) Tomograph and (**b**) horizontal profile for an initial model with a \(-2\%\) error. (**c**) Tomograph and (**d**) horizontal profile for an initial model with a \(+2\%\) error.  
proposed method and determine the iterative number and frequency. Compared with the monochromatic inversion, successive-frequency FWI could obtain more accurate results and the resolution approached a half wavelength. When the defects were larger than a half wavelength, the relative error was smaller than 2%. Extended studies on open notch cracks with different orientations and multiple adjacent defects proved that the proposed method was able to quantitatively detect various types of defects. Finally, inaccurate initial velocity models with a relative error of \(\pm\)2% were also investigated. Despite the relative error increasing slightly, the proposed method was able to reconstruct the defects even with a certain inaccuracy. The results demonstrated that the ice-coupled ultrasonic FWI method has a high potential to quantitatively inspect complex-shaped structures with an excellent resolution.  
It should be noted that a mode conversion in a practical ultrasonic test was not considered in this paper. In the future, an elastic wave equation should be considered in the forward model during the inversion process to obtain more accurate results and a practical experiment will be conducted.  
Conceptualization, M.Y.; Data curation, W.X. ([PERSON]); Formal analysis, W.X. ([PERSON]) and W.X. ([PERSON]); Funding acquisition, M.Y., X.J. and Y.C.; Investigation, W.X. ([PERSON]) and W.X. ([PERSON]); Methodology, M.Y.; Project administration, X.J. and Y.C.; Software, W.X. ([PERSON]); Supervision, M.Y. and X.J.; Validation, W.X. ([PERSON]); Writing--original draft, W.X. ([PERSON]) and W.X. ([PERSON]); Writing--review & editing, M.Y., X.J. and Y.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Key R&D Program of China (No. 2018 YFF01010500), National Natural Science Foundation of China (Nos. 51805097, 51975131, 11804059) and Guangdong Innovative and Entrepreneurial Research Team Program (No. 2016 ZT06G375).  
Not applicable.  
The authors declare no conflict of interest.