## 3 Applications
### Interference Fringe Extraction  
In order to invert \(c_{\text{s}}\) and \(d\) by Equation (16), we needed to extract fringes from the sound intensity interferogram in the range-frequency domain. This paper proposes a stripe extraction method, which is divided into four steps:  
Step 1: since there are dark and bright fringes in the sound intensity interferogram, we performed image threshold segmentation by the maximum between-class variance method (OTSU) in the first step [24]. \(I_{\text{OTSU}}\) of Figure 9 is the threshold segmentation result of Figure 4d, and OTSU could be used to approximately split the stripes into two categories, bright stripes and dark stripes. Nevertheless, the two interference fringes formed by \(\Delta t_{1}\) and \(\Delta t_{2}\) in Equation (1) made some points in the bright and dark fringes misclassified, which could cause certain interference to the classification results.  
Step 2: calculate the vertical gradient or horizontal gradient of \(I_{\text{OTSU}}\), and select the center of the interval of gradient change as the spatial coordinate of the interference fringe. The interference period of the fringe formed by \(\Delta t_{1}\) and \(\Delta t_{2}\) was comparatively small, and thus the gradient change interval in the two directions was relatively small. A certain threshold \(K\) can be set to filter the fine fringes. The gradient change interval of the dark stripes in \(I_{\text{OTSU}}\) in the frequency axis is:  
\[\left\{\begin{array}{c}\frac{\partial I_{\text{OTSU}}(r,f)}{\partial f}(f=f_ {1})=-1\\ \frac{\partial I_{\text{OTSU}}(r,f)}{\partial f}(f=f_{h})=1\end{array}\right. \quad\text{while}\quad(f_{H}-f_{L})\geq K,\ \ f_{\text{mid}}=\frac{f_{H}+f_{L}}{2} \tag{20}\]  
Figure 9: The dark stripe coordinate extraction process and results; (**a**) the threshold segmentation result, \(I_{\text{OTSU}}\), after OTSU; (**b**) the center coordinates of the dark stripes obtained after processing \(I_{\text{OTSU}}\) in step 2; (**c**) the center coordinates of different orders of stripes obtained after the area division; and (**d**) the comparison between the extraction results (colored solid lines) and theoretical results (black dotted lines).  
Here, we set \(K\) to 20 Hz. The frequency points of the dark stripes at each range point of \(I_{\text{OTSU}}\) were extracted by Equation (20), and the spatial position of the dark stripes could be obtained. The extracted result is shown in Figure 9b.  
Step 3: the polygonal area was used to roughly split the stripes of different orders, and the stripes were recognized based on the spatial coordinates extracted in step 2 which is shown in Figure 9b. The diving result of the polygonal area with eight points is shown in Figure 9c.  
Step 4: piecewise cubic Hermite interpolation polynomial (Pchip) was used to make the extracted coordinates continuous [25]. In order to solve the problem of floating up and down in coordinate extraction, we could obtain the average trend of the stripe coordinates by empirical mode decomposition (EMD) [26] combined with the interpolation results. Figure 9d shows that the extraction result was consistent with the theoretical value as a whole, and the black dashed line is the theoretical value calculated by Equations (3) and (5).  
Figure 10 shows the root mean square error (RMSE) of the extraction result in Figure 9d. Except for the 15 th order fringe, the RMSE of the other fringes was 3.5 to 6 Hz. We can see from the result of RMSE that the error of the extraction method was relatively small. Nonetheless, errors existed in the extraction method, which may inevitably add some errors to the inversion results of the acoustic parameters. As for whether the inversion error caused by the extraction method was acceptable, it is discussed from the following inversion results.