A Method for Estimating Source Depth Based on the Adjacent Mode Group Acoustic Pressure Field

Abstract

In order to effectively estimate the depth of the source in the acoustic pressure field, this study investigated the relationship between the distribution of acoustic pressure fields in different adjacent mode groups and the depth of the source in shallow waveguides and proposed a method to estimate the depth of the source on the basis of the adjacent mode group acoustic pressure field. We first derived and calculated the adjacent mode group acoustic pressure field of a typical shallow waveguide, then verified the accuracy of this derivation process through simulations. In addition, combined with singular value decomposition mode extraction, the adjacent mode group acoustic pressure field of the SACLANT experimental data was obtained and used as a comparative parameter for the method presented in this paper. By using the depth of the source as the estimation variable, a simulated annealing algorithm and related parameters were designed, and the feasibility of this method was verified through simulation and experiments. The proposed method achieved a higher localization accuracy without the need for accurate modeling of underwater acoustic channels. Under the conditions of the simulation environment, the average estimation error rate of the method was 0.24%, and with increases in the temperature coefficient and Markov chain length, the average estimation error rate of the method decreased. In the experimental environment, the average estimation error rate of the method was 0.45%. This study provides a method to obtain the depth of source in a shallow waveguide via the adjacent mode group acoustic pressure field.

1. Introduction

The research on source depth estimation in this study is based on passive detection technology, and passive depth localization of an acoustic source is an important research aspect. Specific applications include underwater source detection, marine search and rescue, navigation and tracking and so on. At present, the field of passive detection direction source depth estimation can be divided into two categories: preliminary estimation of source depths and accurate estimation of source depths. In terms of preliminary estimation, only the approximate depth of the source in the waveguide needs to be estimated. Emmetiere et al. [1] proposed a passive source depth classification method using a horizontal linear array. Conan et al. [2] studied depth classification of a source as a binary classification problem and proposed a decision index based on classical mode filters. Accurate estimation requires accurate acquisition of source depth [3,4,5]; thus, this study focused on the accurate estimation of source depth, and the singular points of the acoustic field were used as a bridge between the adjacent mode group (AMG) acoustic pressure field and the depth of the source. The AMG is the combination of modes of different orders, for example, (6,7)-AMG represents the combination of the sixth and seventh modes.

Waterhouse et al. [6] drew a streamline of underwater energy flow in 1985, tracked the energy flow of an acoustic field and determined that there are singular points near the energy streamline, which provided a new research direction for underwater acoustics. Later, Waterhouse and Skelton [7] carried out an experiment on the plane wave incident to an elastic spherical shell in water, which proved that the acoustic pressure at the vortex point was close to 0 Pa and the average particle velocity at the saddle point was close to 0 m/s. Chien [8] analyzed the streamline behavior near singular points in a two-dimensional acoustic field by using differential equations to describe the intensity of the streamline. On the basis of the assumption that the average particle velocity at the saddle point is close to 0 m/s during this period, it was determined that saddle points also occur at positions where the phase difference between the average particle velocity and acoustic pressure is an odd multiple of $\pi /2$. In 2001, Eliseevnin and Tuzhilkin [9] studied the singular points and acoustic power flow generated by the interference of two modes in an ideal waveguide in shallow water. The characteristics of acoustic power flow were more intuitively explored through acoustic power flow figures and grazing angle figures. In 2002, Neilsen [10] proposed a mode extraction method based on singular value decomposition, providing technical support for obtaining important mode distributions and conducting singular points analysis in the field of acoustics. In 2008, underwater singularities were truly detected [11]. Vladimir [12] demonstrated in 2019 that the distribution of singular points is related to the signal-to-noise ratio through experiments to detect them and that vortex points will shift over time, which could become a source of information for future changes in the marine environment. In 2023, Zexi Wang et al. [13] conducted simulation verification on singular points and inverted the depth of the source through the depth of the singular points.

In summary, AMG acoustic field singular points are generated by mode interference, with a periodic distribution in the distance direction, and these singular points contain rich environmental information, including waveguide information and source information. In addition, the distribution of singular points in the acoustic field corresponds to the distribution of acoustic pressure and particle velocity fields. Therefore, the AMG acoustic pressure field also contains rich source information.

For the study of source depth estimation, the traditional method is matched field processing (MFP), but MFP has many drawbacks [14]. In MFP, the underwater acoustic channel needs to be modeled; whether the model can accurately describe the sound field will directly determine the accuracy of the location result. Recently, scholars have put forward more new methods and theories. In 2020, Wenbo W et al. [15] proposed a method based on a convolution neural network (CNN) and conventional beamforming (CBF), which overcame the limitation of real sound-speed profiles and source bandwidths on the accuracy of multispectral transformation for depth estimation (MSTDE). Although a neural network can estimate the source depth, it requires a large number of data sets and training of these data sets. The randomness of the distribution of ambient noise in the frequency domain also greatly affects the output results. In 2022, Zhai Duo et al. [16] used different cost functions and proposed three methods to estimate the source depth by matching the normalized cross-spectral density, temporal envelope and time delay of the smooth-averaged cross-correlation function for the case of two horizontally separated hydrophones or arrays. However, the average absolute error in the source depth of the proposed method was still large and the proposed method was more suitable for the case of a low-frequency band and an environment with large negative sound speed gradient. In 2023, Yanqun W et al. [17] proposed an efficient two-step depth estimation scheme, using narrowband and broadband constructive and deconstructive striation patterns due to interference between the direct and sea-surface-reflected arrivals at a horizontal linear array at the bottom of deep water. Due to the limited source range spans, the distortion of the beam intensity striations and range estimation errors, the performance of the algorithm will be degraded.

On the basis of previous research, this study simulated and analyzed how to determine the AMG acoustic pressure field and related it to the depth of the source. We studied the method of estimating the depth of the source based on the AMG acoustic pressure field and ultimately applied it to the experimental data. Without the need for accurate modeling of underwater acoustic channels, the proposed method achieved higher localization accuracy. In addition, the proposed method has low sensitivity to environmental mismatch, and vertical array data collection is closer to the specific experiment, which makes it applicable in more seas and increases flexibility.

The main content of this paper is as follows: Section 2 derives AMG acoustic pressure field, obtains mode information through singular value decomposition and explains the design idea of the algorithm. Section 3 analyzes the influence of different parameters on the simulated annealing algorithm. Section 4 conducts experiments using SACLANT data and obtains experimental results. In addition, the advantages and disadvantages of the proposed algorithm are analyzed, and the proposed algorithm is compared with MFP. Finally, Section 5 concludes this paper.

2. Theory

Before introducing the theory, the basic steps of the method presented in this paper are briefly introduced. Figure 1 shows the basic steps of the proposed method. Firstly, we need to collect waveguide acoustic pressure data through a shallow water vertical array. Then, singular value decomposition (SVD) is used for mode extraction. After acquiring the accurate mode distribution $Z$, the horizontal vector $R$ can be obtained through Equation (1) to obtain the AMG acoustic pressure field. Finally, we used the simulated annealing algorithm to estimate the depth of the source using the AMG acoustic pressure field as an intermediate parameter.

2.1. AMG Acoustic Pressure Field Acquisition

The distribution of an acoustic pressure field is a combination of numerous AMG acoustic pressure fields, which leads to a more complex distribution of singular points in the entire waveguide. The AMG acoustic pressure field can be obtained on the basis of mode extraction and the decomposition of the acoustic pressure field in normal mode theory, and can be decomposed into vector product forms in the horizontal and vertical directions:

$$P\left(r,z\right)=Z\left(z\right)R\left(r\right)$$

(1)

It is assumed that in the far field of the ideal shallow waveguide, the upper layer is a uniform water layer, and the lower layer is the hard uniform seafloor. ${\rho }_1$ is the density of waveguide, ${ \rho }_2$ is the density of the bottom, the speed of sound in the waveguide is $c_1$ and the speed of sound at the bottom is $c_2$. Here, the reference system consists of horizontal and vertical axes $r$ and $z$, and the origin is at the sea level. If the acoustic pressure field $P$ is known and the accurate mode distribution $Z$ is obtained through mode extraction, the horizontal vector $R$ can be obtained through Equation (1) to obtain the simulated AMG acoustic pressure field. The known simulated acoustic pressure is [18]:

$$P\left(r,z\right)=2\pi \omega \rho \sum _{n=l,q}\sin \left({\xi }_{l}z\right)F(z_0,b_n){\mathrm{H}}_0^{\left(1\right)}\left(b_{n}r\right)$$

(2)

$$F\left(z_0,b_n\right)=\frac{{\xi }_n\sin \left({\xi }_n{z}_0\right)}{{\xi }_{n}H-\sin \left({\xi }_{n}H\right)\cos \left({\xi }_{n}H\right)-{\left({\rho }_1/{\rho }_2\right)}^2\tan \left({\xi }_{n}H\right){\mathrm{s}\mathrm{i}\mathrm{n}}^2\left({\xi }_{n}H\right)}$$

(3)

$${\xi }_n=\sqrt{k_1^2-b_n^2}$$

(4)

$$k_i=\omega /c_i,i=\mathrm{1,2}$$

(5)

Here, $H$ is the depth of the waveguide ($0\le z\le H$), $\omega$ is the cyclic frequency of the signal, $z_0$ is the depth of the source, $n$ is the serial number of the normal wave, $b_n$ is the $n$th eigenvalue and ${\mathrm{H}}_0^{\left(1\right)}\left(x\right)$ is the zero-order Hankel function of the first kind.

By replacing $\sin \left({\xi }_{l}z\right)$ in Equation (2) with the mode distribution $Z$ obtained through mode extraction, the acoustic pressure can be obtained through mode extraction. Through the above steps, we can obtain the acoustic pressure field, as shown in Figure 2b. Figure 2 shows the simulated acoustic pressure field and calculated acoustic pressure field obtained by (6,7)-AMG when the source is at the bottom of the sea; the darker the color, the lower the acoustic pressure value; the simulated shallow water environmental parameters [13] are shown in

Table 1. Simulation of shallow ocean environment parameters.

Parameters	Value
Source Frequency	50 (Hz)
Number of Media	1
Top Option	Vacuum
Waveguide Depth	150 (m)
Bottom Option	Acoustic: Elastic
Lower Limit of Phase Velocity	0 (m/s)
Upper Limit of Phase Velocity	2000 (m/s)
Receiver Depths	0–150 (m)
Sound Speed: Waveguide	1500 (m/s)
Sound Speed: Bottom	2000 (m/s)
Rho: Water	1 (g/cm3)
Rho: Bottom	2 (g/cm3)

. Here, the Euclidean distance is used to calculate the similarity of two images to achieve image matching. The Euclidean distances in Figure 2a,b are only 0.0072 and the difference is small, which verifies that obtaining the acoustic pressure field distribution of AMGs through mode extraction and calculation is feasible.

2.2. Mode Extraction

Singular value decomposition (SVD) can decompose the measurement matrix of any row full rank into the following:

$$A_{N\times M}=US{V}^{†}$$

(6)

where the complex matrix $A$ of N × M can be decomposed into three matrices, $S$ is a semi-positive diagonal matrix, $U,V$ are orthogonal matrices and $†$ represents the conjugate transpose. The acoustic pressure data $P$ are collected through a vertical array, and the matrix $P$ is decomposed by SVD:

$$P=e^{i\pi ∕4}\overline{\mathsf{\Phi }}\mathsf{\Sigma }R$$

(7)

where the column vector (${\overline{\mathsf{\varphi }}}_n$) of $\overline{\mathsf{\Phi }}$ corresponds to different order mode functions and the diagonal elements of $\mathsf{\Sigma }$ consist of the singular values of the matrix $P$. Usually, in practical applications, the obtained sound pressure matrix $P$ is not a square matrix, so we use the cross spectral density matrix $C$ for singular value decomposition:

$$C=P{P}^{†}\approx \mathsf{\Phi }{\mathsf{\Lambda }}^2{\mathsf{\Phi }}^{†}$$

(8)

Similarly, $\mathsf{\Lambda }$ is a diagonal matrix, where the singular values gradually decrease on the diagonal, and the column vector of $\mathsf{\Phi }$ is the mode corresponding to each singular value.

The experimental data used in the study presented in this paper were from a sea trial conducted at the SACLANT center in the Mediterranean Sea near the west coast of Italy in October 1993 [19]. This experiment collected mainly waveguide acoustic pressure data through a shallow water vertical array and was conducted in a flat waveguide area between 120 m and 140 m of the isobath. In the test, the acoustic pressure data were obtained through a vertical array of 48 hydrophones spanning a 94 m aperture; three groups of test data were obtained, as shown in

Table 2. SACLANT simulation data.

Data Name	Meaning	Data Quantity
A2601-1	170 Hz static source on 26 October	59.6 MB
P2701	170 Hz movable source on 27 October	59.6 MB
A2601-2	350 Hz static source on 26 October	29.8 MB

.

The three sets of data correspond to static source test data with a source frequency of 170 Hz (A2601-1), moving source test data with a source frequency of 170 Hz (P2701) and static source test data with a source frequency of 350 Hz (A2601-2). This study used mainly experimental data from a moving source with a source frequency of 170 Hz, as this set of data was consistent with the data requirements of singular value decomposition mode extraction. The data composition was $48\times N$, where 48 corresponds to the number of vertical array hydrophones, and $N$ is related to the time of data acquisition. The longer the elapsed time, the greater the value of $N$.

In order to verify the accuracy of the mode extraction results, a one-to-one simulation was conducted in the Acoustic Toolbox v2.2 (AT, an open-source software for sound field modeling calculations developed by Michael Porter) based on the waveguide environment corresponding to the SACLANT data, the sound speed profile [20] of which is shown in Figure 3. The source was placed at a depth of 80 m in the waveguide, with the corresponding simulation mode distribution shown in Figure 4.

Due to the correlation between the depth of the source and the order of the waveguide mode, the greater the value of the corresponding mode at the depth of the source, the greater the effect of that mode. In order to obtain better mode extraction results, it was necessary to limit the data used. According to the description of extraction conditions in singular value decomposition mode extraction [10], in order to fully sample the modes, all 48 hydrophones were taken as the acoustic pressure monitoring range. To ensure that the signal source passed through a sufficiently large range interval, all data during the time period that passed by the test vessel were classified as experimental data (48 × 602,112). The mode extraction and singular value results obtained through singular value decomposition are shown in Figure 5 and Figure 6.

The number of blue circles in the test data mode extraction results (Figure 5) was 48, which corresponded to the number of vertical array hydrophones, and the vertical coordinates corresponded to the depth of the 48 hydrophones. From the experimental data mode extraction of singular value results (Figure 6), the percentage differences between singular values were 10.71%, 11.35%, 12.5%, 9.50%, 18.55%, and 8.61%. Strictly speaking, 18.55% is not satisfactory. However, due to the error in the experiment, the rationality of some data could not be guaranteed. However, the rest of the data met the requirements for singular value results (5~15%). Generally speaking, the experiment was acceptable.

Figure 7 illustrates the comparison of simulated waveguide modes and mode extraction results. The mode order obtained by comparing the mode extraction results with the simulation was 2, 4, 7, 5, 3, 5, 6, and the mode amplitude ratios were 0.104, 0.102, 0.067, 0.005, 0.046, 0.005, 0.048, respectively. The order of mode extraction was affected mainly by environmental biases, among which the vertical array in the experimental environment may have shifted, resulting in inconsistent data times. Additionally, the simulation environment was too ideal. So, there are differences between the theoretical result and the mode extraction results at the seventh mode when $z$ is lower than 20.7 m in the third subplot in Figure 7. However, by comparing the mode distributions, the results of the mode extraction corresponded to the simulation results.

2.3. 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 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.

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:

$$\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}\left(\mathsf{\Omega }\right)=\frac{1}{N\times M}\sqrt{\sum _{i=1,j=1}^{i<N,j<M}{(Z_{\left(i,j\right)}^e-Z_{\left(i,j\right)}^c)}^2}$$

(9)

where $\mathsf{\Omega }$ 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; $Z_{\left(i,j\right)}^e$ represents the AMG acoustic pressure value corresponding to the intermediate parameter; and $Z_{\left(i,j\right)}^c$ represents the AMG acoustic pressure value corresponding to the comparative parameter. When the contemporary valence function reached its minimum value, the $\mathsf{\Omega }$ 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.

In general, the termination temperature is 0 °C, but this increases the time complexity and requires more iterations. As long as the termination temperature approaches 0 °C, the end state is basically the same, so only an acceptably low temperature is required. The initial temperature was 2000 °C, the end temperature was 10⁻¹⁴ °C, the temperature coefficient was 90%, the length of the Markov chain was 100 (number of internal cycles) and the tolerance was a dynamic value. The equation is as follows:

$$\mathsf{\Theta }\left(\mathsf{\Omega }\right)=\left\{\begin{array}{ll}{e}^{\frac{\mathrm{Cost}\left({\mathsf{\Omega }}^{\prime }\right)-\mathrm{Cost}\left(\mathsf{\Omega }\right)}{\mathsf{\Omega }}}& (\mathrm{Cost}\left({\mathsf{\Omega }}^{\prime }\right)<\mathrm{Cost}\left(\mathsf{\Omega }\right))\\ 1& (\mathrm{Cost}\left({\mathsf{\Omega }}^{\prime }\right) \ge \mathrm{Cost}\left(\mathsf{\Omega }\right))\end{array}\right.$$

(10)

where $\mathsf{\Omega }$ represents the current source depth. If the cost of internal circulation at the current temperature is lower than the cost of the previous internal circulation, the current procedural source depth is accepted; if the cost of the internal circulation at the current temperature is higher than that of the last internal circulation, the exponential function is used to calculate the current tolerance to choose the source depth. The direction change of depth after the current internal circulation is determined by a Gaussian random number, and the range of change is determined by the combination of the current temperature and the random number.

Algorithm 1 summarizes the basic process of estimating the depth of the source through the simulated annealing algorithm, starting from the distribution of the AMG acoustic pressure field. In the simulation environment of Table 1, the estimated source depth obtained by setting the source depth to 100 m was ${\mathsf{\Omega }}_0=97.26$ m.

Algorithm 1: A method for estimating source depth based on the AMG acoustic pressure field.

Input: Comparison parameters (AMG acoustic pressure field). Initialization: Set the initial temperature, Markov chain length, and temperature coefficient and randomly set the initial source depth. Process: Obtain the latest iteration number of the inner loop and check whether it reaches the Markov chain length; If the number of internal circulations reaches the length of the Markov chain after cooling down, calculate the cost of this internal circulation depth according to Equation (4) and skip to Step 4; If the number of internal circulations does not reach the length of Markov chain, calculate the cost under the depth of this internal circulation according to Equation (4); Calculate the tolerance according to Equation (5); If the tolerance is 1, accept the current depth and skip to Step 9; If the tolerance is not 1, use probability values to select the depth; Accept Step 6 and skip to Step 9; Abandon Step 6; Determine whether the threshold temperature has been reached and terminate if the conditions are met; otherwise, return to Step 1. Over Output: $\mathsf{\Omega }$

3. Simulation

In order to further verify the feasibility of the method, we conducted method estimation at 15 source depths (set as 10 m, 20 m, …, 150 m, respectively), and the final comparison of the source depths estimation is shown in Figure 10.

In Figure 10, the horizontal coordinate corresponds to the true depth of the source, the vertical coordinate corresponds to the predicted depth of the source, the blue line corresponds to the true depth of the source and the orange line corresponds to the estimated depth of the source.

We calculated the experimental error of the estimated results in Figure 10 using the following average error rate formula:

$$∆=\frac{1}{N}\sum _{i=10,s=10,i=i+s}^{N\times s}\frac{\left|Z_i^e-Z_i^t\right|}{Z_i^t}$$

(11)

where $N$ is the number of test source depths (here, $N$ = 15), $s$ is the interval depth of test source depths (here, $s$ = 10 m), $Z_i^e$ is the estimated source depth obtained through the simulated annealing algorithm and $Z_i^t$ is the true source depth. We calculated the average error rate of the current experiment as $∆=4.83\%$ using Equation (11).

3.1. The Influence of the Temperature Coefficient on the Simulated Annealing Algorithm

In order to reduce the error rate of source depth estimation, all parameters related to the simulated annealing algorithm, except for the temperature coefficient, were controlled, and the error rate of the method was reduced by increasing the temperature coefficient in advance. This was necessary due to the influence of the temperature coefficient on the cooling rate and the number of simulated annealing algorithm iterations. Figure 10 shows the estimated results with a temperature coefficient of 90%. The temperature coefficients were modified to 92%, 94%, 96% and 98%, respectively, and simulated annealing was performed to obtain the results. The comparison of source depth estimation under different temperature coefficients is shown in Figure 11.

As shown in Figure 11, as the temperature coefficient increased, the estimated depth of the source gradually matched the true depth of the source. We calculated the average error rate of the experiment under four different temperature coefficients: ${∆}_{92\%}=4.05\%$, ${∆}_{94\%}=2.77\%$, ${∆}_{96\%}=1.78\%$ and ${∆}_{98\%}=0.24\%$; the higher the temperature coefficient, the better the depth estimation result. However, when starting from the temperature coefficient, the overall algorithm time complexity was O(n), so as the temperature coefficient increased, the execution efficiency of the method decreased.

3.2. The Influence of the Markov Chain Length on the Simulated Annealing Algorithm

In the simulated annealing algorithm, in addition to the temperature coefficient, the length of the Markov chain affects the source depth estimation. The length of the Markov chain is the internal cycle process at each temperature (the state sequence passed in the search process). The longer the Markov chain, the more sufficient the algorithm searches and the greater the possibility of obtaining the global optimal solution. Therefore, parameters other than the length of the Markov chain were controlled. The lengths were set to 500, 1000, 1500, and 2000, respectively, with a temperature coefficient of 90%, and a simulated annealing analysis was conducted. The estimation results of the source depth are shown in Figure 12.

It can be seen from Figure 12 that with an increase in the Markov chain length, the source depth estimation results obtained by simulated annealing gradually became consistent with the real source depth. The average error rates corresponding to different Markov chain lengths were ${∆}_{500}=3.95\%$, ${∆}_{1000}=2.19\%$, ${∆}_{1500}=1.54\%$ and ${∆}_{2000}=1.41\%$; the longer the Markov chain, the better the depth estimation results. However, the time complexity of the algorithm was O(n²) when the Markov chain length was taken as the starting point, so as the Markov chain length increased, the method’s execution efficiency was reduced.

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.

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], which makes it difficult to meet these requirements, resulting in large errors and limited practicability of the results. Compared with MFP, 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%.

5. Conclusions

This study simulated and analyzed the effectiveness of obtaining the AMG acoustic pressure field and combined it with singular value decomposition mode extraction technology to obtain the AMG acoustic pressure field based on SACLANT test data. These data were used as a comparative parameter for the source depth estimation method based on the AMG acoustic pressure field proposed in this paper. In addition, in this study, we conducted a simulation and experimental verification using the proposed source depth estimation method. Under the conditions of the simulation environment, the average estimation error rate of the method was as low as 0.24%, and with increases in the temperature coefficient and Markov chain length, the average estimation error rate of the method decreased. In the experimental environment, setting the algorithm-related parameters resulted in an average estimation error of 0.45% for the method.

Without the need for accurate modeling of underwater acoustic channels, the proposed method achieved a higher localization accuracy. However, the method proposed in this paper has also some limitations. For example, we need to make sure that the mode extraction process is accurate; if the mode information contains errors, this will lead to AMG acoustic pressure field deviation, resulting in increased experimental error. In addition, the proposed method uses SVD to extract the mode information and uses a simulated annealing algorithm to find the optimal solution iteratively, which increases the computation amount.

Although our method was validated through simulation and experiments, further research on reducing the size of the vertical array, using other iterative algorithms to reduce the computation or improving the precision of mode extraction is required. The smaller the vertical array, the less difficult it is to operate. Inevitably, SVD is used to extract accurate mode information, but in order to reduce the computation amount, other iterative algorithms can be selected instead of the simulated annealing algorithm to estimate the source depth. Thus, comparative experiments will be conducted in the future between this method and other excellent source depth estimation methods.