# A Method for Estimating Source Depth Based on the Adjacent Mode Group Acoustic Pressure Field
## 4 Experiment  
In the experiment in Section 2.2, we performed SVD on SACLANT data [24] to extract mode information. After obtaining the mode distribution of the experimental data through singular value decomposition mode extraction, the first two modes with the highest proportions in the acoustic field--the second and fourth modes, as shown in Figure 7--were substituted into Equation (2) to obtain the experimental data (2,4)-AMG acoustic pressure field distribution results, as shown in Figure 13.  
Figure 12: Comparison of source depth estimation under different Markov chain lengths: **(a)** the Markov chain length is 500; **(b)** the Markov chain length is 1000; **(c)** the Markov chain length is 1500; **(d)** the Markov chain length is 2000.  
It can be seen from Figure 13 that the experimental data showed periodicity in the distance direction of the AMG acoustic pressure field. Since the acoustic pressure field distribution was obtained from 48 hydrophones in a vertical array, the AMG acoustic pressure field distribution and the singular point distribution did not completely correspond, but could still be used as a comparison parameter for the simulated annealing algorithm to estimate the source depth.  
On the basis of the method proposed in Section 2.1, we used the second- and fourth-order mode extraction results and substituted them into Equation (2) to iteratively obtain the Euclidean distance between the corresponding AMG acoustic pressure field distribution and the actual (2,4)-AMG acoustic pressure field distribution (Figure 13). This was accomplished by modifying the source depth, then performing the simulated annealing process in Algorithm 1 through Equation (4). Under the conditions of a temperature coefficient of 94% and a Markov chain length of 500, 15 groups of source depth estimation results were obtained through the experiments. As seen in the estimation comparison in Figure 14, the average source depth estimation value was 79.69 m, which was consistent with the actual source depth of 80 m, and the average estimation error of the experiment was only 0.45%.  
However, due to the diversity and uniqueness of other source depth estimation methods, the environment and experimental parameters [25] of each method are different, so it is difficult to conduct a controlled experiment with two different methods. The following briefly introduces the performance of the traditional method MFP and makes a comparison with the method proposed in this paper.  
In the first section, we mentioned that MFP requires accurate modeling of underwater acoustic channels [26]. As a result of the high sensitivity of the wave field to variations in environmental parameters, the use of this approach requires accurate knowledge of the ocean acoustic environment. MFP has a high sensitivity to environmental mismatch [27],  
Figure 14: Comparison of source depth estimation results of experimental data with actual values.  
Figure 13: (2,4)-AMG acoustic pressure distribution of the experimental data (the darker the color, the lower the acoustic pressure value).  
which makes it difficult to meet these requirements, resulting in large errors and limited practicability of the results. Compared with MPP, the method in this paper has lower requirements on the accuracy of environmental parameters when running a simulation, and environmental parameters do not need to be known if we collect mode information through vertical arrays in real marine environments. Vertical array data collection is closer to the specific experiment. From the experimental results in this paper, it is obvious that the results obtained by this method are more accurate and the error is smaller; the final average estimation error of the experiment was only 0.45%.