# A Method for Estimating Source Depth Based on the Adjacent Mode Group Acoustic Pressure Field
## 2 Theory
### Design of Source Depth Estimation Method  
The distributions of acoustic pressure field and singular points were obtained by using the environmental parameters in Table 1 and randomly selecting a mode combination. In Figure 8, it is easy to observe that the singular points on the right correspond to the dark region in the acoustic pressure field on the left, which proves that the distribution of the  
Figure 6: Mode extraction singular value results of the experimental data. Figure 7: Comparison of mode extraction results from the experimental data. AMG acoustic pressure field and the distribution of singular points in the acoustic field correspond to each other, because different sources will lead to different distribution of singular points. The Euclidean distances [21] between the AMG acoustic pressure field distributions at different source depths were different. The above experiments prove that the method in Figure 1 is feasible. Therefore, it is possible to use simulated annealing to estimate the depth of the source by using the AMG acoustic pressure field as an intermediate parameter. The cost function value is reduced through the simulated annealing algorithm [22], and the source depth estimation is the output. Under the same simulation environmental conditions listed in Table 1, the first two waveguide modes are known to be 6 and 7. For four different source depths, the acoustic pressure field of (6,7)-AMG in the seven periods of the interference structure of the field will have the distributions shown in Figure 9.  
Figure 8: Distributions of AMG acoustic pressure field and singular points (black \(\bullet\) represents the first set of vortex points, black \(\times\) represents the first set of saddle points and red \(\times\) represents the second set of saddle points, the darker the color, the lower the acoustic pressure value). Figure 9: Distributions of the (6,7)-AMG acoustic pressure field at different depths of sources. (**a**) The source depth is 20 m; (**b**) the source depth is 40 m; (**c**) the source depth is 60 m; (**d**) the source depth is 80 m (the darker the color, the lower the acoustic pressure value). The pairwise Euclidean distances between the AMG acoustic pressure field distributions at different source depths were calculated as 0.001 (between 20 m and 40 m), 0.00517 (between 40 m and 60 m), and 0.00375 (between 60 m and 80 m). There were differences among the different AMG acoustic pressure fields, which could be used as intermediate parameters. On the basis of the above, the AMG acoustic pressure field distribution was taken as an intermediate parameter, which corresponded to the AMG acoustic pressure field at the depth of the source (\(N\times M\) matrix). We defined the cost function on the basis of intermediate parameters:  
\[\text{Cost}(\Omega)=\frac{1}{N\times M}\sqrt{\sum_{i=1,j=1}^{i<N,j<M}\left(Z_{ (i,j)}^{e}-Z_{(i,j)}^{c}\right)^{2}} \tag{9}\]  
where \(\Omega\) represents the depth of the source to be estimated; \(N\) and \(M\) correspond to the horizontal and vertical pixels of the AMG acoustic pressure field image, respectively; \(Z_{(i,j)}^{e}\) represents the AMG acoustic pressure value corresponding to the intermediate parameter; and \(Z_{(i,j)}^{c}\) represents the AMG acoustic pressure value corresponding to the comparative parameter. When the contemporary valence function reached its minimum value, the \(\Omega\) value obtained was the estimated result of the source depth. A simulated annealing algorithm was used to iteratively obtain the optimal source depth results. The simulated annealing algorithm is essentially a double cycle through which the optimal solution is found [23]. The outer cycle is controlled by a temperature coefficient to change the temperature from high to low. In the inner cycle, the temperature is fixed; the new solution is obtained by adding random disturbance to the old solution, and the new solution is accepted according to a certain probability. The number of iterations of the inner cycle is called the Markov chain length.