## 3 Results  
### Simulation  
To verify the performance of the proposed method, a simulation for ultrasonic FMC imaging in an oblique incidence situation was conducted. The simulation was designed for immersion detection of a steel specimen with side-drilled holes (SDHs). The simulation was carried out on the CIVA platform (version 2020, the French Atomic Energy Commission, Paris, France). As shown in Figure 4, the specimen was made of steel, with a sound velocity of 5889 m/s, and the reflectors were a set of SDHs with diameters of 1 mm. The SDHs were arranged in a horizontal line with a step of 3 mm, and the vertical distance of the SDHs from the surface of the specimen was 10 mm. The coordinate origin was set on the upper surface of the specimen. The \(x\)-axis was parallel to the specimen's surface while the \(z\)-axis was along the depth direction. The ultrasonic transducer was set to be a linear array with the detailed parameters listed in Table 1. The array and specimen were immersed in water, with a sound velocity of 1483 m/s, and the angle between the array and the horizontal plane was 10 degrees. The distance between the center of the array and the upper surface of the specimen was 20 mm. The source signal was a Hanning windowed signal with one and a half cycles. The FMC data were sampled at 50 MHz frequency with a time range of  
Figure 3: The flowchart of the proposed method. 20 us, and then imported into MATLAB (version 2018b, MathWorks, Natick, MA, USA) for post-processing. The raw dataset size was 4000 \(\times\) 64 \(\times\) 64 pixels. As well as the proposed method, the time-domain TFM was also used to process the FMC data, for comparison. The time-domain TFM utilizes the Fermat principle to calculate the refraction points at the interface, then accurately estimates the wave propagation time. The processing computer had an AMD Ryzen 5 3600 CPU and an NVIDIA GeForce GTX 970 GPU. All the algorithms used in the simulation and the experiment were accelerated by the GPU in MATLAB. Figure 5a shows the result reconstructed by the time-domain TFM. It can be seen that the tilted coordinate was well compensated. However, the shape of the SDHs in the reconstructed image was distorted, and this increased as the angle of the incident beam increased. Figure 5c shows an enlarged image of SDH A in the image reconstructed by TFM. The shape of SDH A is no longer a circle. This distortion is introduced by the heuristic approach used in the TFM, where the ray-based method used for the beam path calculation results in errors in the oblique incidence compensation [16]. The result reconstructed by the proposed method is shown in Figures 5b,d is the local enlarged image of SDH A. Apparently, the proposed method yields a more accurate result than time-domain TFM. It can compensate well for the wave-field rotation introduced by the oblique incidence and avoid the distortion of the reflectors. The proposed method provides a more rigorous solution to the inverse problem, and the array performance exhibits less dependence on the lateral position x at any given depth z than the TFM. Furthermore, Figure 5b shows a lower background noise than the result for TFM. Specifically, the signal-to-noise ratio (SNR) of Figure 5b is 23.15, while the SNR of Figure 5a is 9.86. This means that the proposed method can achieve a better SNR than the TFM. \begin{table}
\begin{tabular}{c c} \hline \hline
**Array Parameter** & **Value** \\ \hline Number of elements & 64 \\ Element pitch & 0.6 mm \\ Element width & 0.6 mm \\ Element length & 10 mm \\ Center frequency & 5 MHz \\ \(-\)6 dB bandwidth & 78\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the phased array used in the simulation and experiment. Figure 4: The geometry of oblique incidence situation for simulation.