Source localization by Matching the Multipath Arrival Angles based on Sparse Bayesian Learning

In the deep-sea direct arrival region, a significant multipath characteristic is presented by the propagation of acoustic signals. This characteristic is closely linked to the distance and depth of the sound source and can be utilized for source localization. In this study, the changes in incidence angle characteristics of multipath signals at different distances and depths are analysed initially. Subsequently, the high-resolution azimuthal spectra are obtained using the Sparse Bayesian Learning (SBL) method. The azimuthal spectra are then matched with multiple incidence angles using the Gaussian kernel function, facilitating the localization of sound sources. Throughout this paper, the impacts of different distance, depth, and SNR conditions on the model are assessed through simulations. Furthermore, the model’s validity is confirmed by utilizing experimental data from an explosion sound source.


Introduction
Vertical hydroacoustic arrays with strong array gain, positioned near the ocean floor, can help locate sound sources near the surface over medium and long distances.This takes advantage of the fact that sound travels well in the deep-sea region just above the seabed.Conventional methods for passively detecting the sound source are matched field processing (MFP).MFP creates an ambiguity surface that shows the range and depth of the source by matching the measured acoustic array data with a dictionary of replica sound fields.Replica sound fields are computed based on the underwater acoustic propagation model using specific environmental param [1][2] .Common examples of MFP include the Bartlett processor [3] and the high-resolution minimum-variance distortionless-response (MVDR) processor [4] .The Bartlett processor is robust in mismatched environmental param condition, but it produces more noticeable sidelobes in the ambiguity surface.MVDR effectively reduces these sidelobes, although localization errors may increase if the acoustic fields environmental param mismatched [5] .
The acoustic propagation of near-surface source in deep sea environment presents multipath attributes.This is particularly evident when the receiver is situated close to the seafloor and the sound velocity near the receiver is higher than that of the sea surface [6] .This phenomenon is termed the Reliable Acoustic Path (RAP).In RAP channels, the direct arrivals from near-surface sources can be received at considerable distances.Multiple paths, including direct (D), surface-reflected (SR), bottomreflected (BR) paths and others, contribute to the overall multipath arrival characteristics.By extracting the multipath characteristics from the measured signals, such as arrival angles, time delays, or interference structures, the matching the multipath arrival characteristics (MMAC) localization method exploits these mapping relationships to determine the range and depth of the source.In literature [7] a deep-sea vertical array is used to localizing the sound source.It is achieved by extracting the D path and SR path arrival angles and matching to the candidate source angles calculated by Ray model.However, distinguish the multipath angles from the azimuthal spectrum is feasible only under simple circumstances.In the presence of multi-target scenarios and near-surface conditions, the process necessitates manual intervention due to increased complexity.
Compressed sensing has brought about a transformation in the conventional 'Nyquist' sampling principle and is now widely employed in signal processing [8][9] .By utilizing a sparse spatial model, compressed sensing is applied to estimate orientation.This approach offers improved higher resolution compared to traditional methods [10] .The Sparse Bayesian Learning (SBL) algorithm, a compressed sensing method based on probabilistic statistics, provides superior estimation accuracy and robustness [11]   .In this study, SBL is used to estimate azimuthal spectra for vertical arrays.The Gaussian kernel function is used to match multipath angles for source localization.In comparison to the literature [7] the proposed approach eliminates the need to distinguish the multipath angles from the azimuthal spectrum and demonstrates higher adaptability in complex scenarios.

Mutipath Angels versus Source Range and Depth
According to ray theory, within the deep-sea direct arrival zone, the acoustic signal propagation from the source to the sensor displays multipath characteristics.As demonstrated in Figure 1, the incidence angles for the D-path, SR-path, and BR-path consistently shift monotonically with respect to both distance and depth.Figure 1 utilizes the Munk sound speed profile, depicting a sea depth of 5000 m, with hydrophone stationed at a depth of 4000 m.It can be seen that the angles of incidence for the D, SR and BR paths increase gradually with distance.Similarly, the angles for D and BR increase with depth, whereas the SR angle decreases proportionally with depth.Consequently, the incidence angles of the D, SR, and BR paths hold utility for the purpose of source localization.

Azimuthal Spectrum Estimation based on SBL
Employing sparse Bayesian learning (SBL) facilitates the estimation of the azimuthal spectrum for the sound source's incidence angle.The measured signal of hydroacoustic arrays Y can be expressed as: Assuming that each component in S is independent and follows a Gaussian distribution with mean 0 and variance i  , the hyperparam ( ) , the posterior probability of S can be expressed as [10] : assuming the noise n follows a Gaussian distribution with mean 0 and variance 2  .Thus, the power spectrum Y follows the a Gaussian distribution with mean of AS and variance of 2  I .And the marginal probability density function of the observed power spectrum Y can be obtained from the full probability formula: Where,

Localization by Matching Multipath Arrival Angles.
The correlation between the incidence angles of multipath signals and the source range and depth forms the foundation for sources localization.However, the estimation of arrival angles from the azimuth spectrum is intricate, particularly within complex marine environment.Therefore, the Gaussian kernel function is employed to estimate the correlation ratio between the azimuth spectrum and the Multipath angles.
The sparse azimuth spectrum ( ) A  can be estimated using SBL and can be represented as follows: ( ) ( ) Where,  represents the discrete sequence of angles, Where,  the variance in Gaussian kernel function.If the angle of incidence value  is not estimated, the correlation ratio can be directly determined using the azimuthal spectrum and simplified to: Here,

( )
ii A   − can be regard as the azimuth spectrum ,which only exhibits a peak at the arrival angle i  .Thus, the correlation ratio of azimuth spectrum ( ) To avoid redundant calculations of the same peak in the azimuthal spectrum, the corresponding peak should be removed from the azimuthal spectrum.This is denoted as: Where, ˆr  represent the correlation angle to ( )

Localisation results for sound sources at different distances and depths
To analyze the influence of changes in range and depth on the positioning accuracy of MMAC-SBL, the simulation employs the Munk sound velocity profile.The sea depth is set at 4200 m, and the vertical array with 16 elements ranges in depth from 3900 to 4012.5 m with a spacing of 7.5 m.The acoustic source target emits a broadband continuous signal characterized by a distinct line spectral feature within the signal processing frequency band of 20 to 500 Hz.The positioning errors corresponding to changes in the source depth at distances of 5 km, 10 km, and 15 km are illustrated in Figure 2(a) and (b).and the source distance are at 5 km, 10 km, and 15 km.From the figures, it becomes evident that MMAC-SBL is prone to substantial localization errors when the depth of the sound source is below 50 m.As the depth increases, however, accurate estimation of the distance can generally be achieved, and the localization error for the depth remains below 20 m.Figures 2(c) and (d) respectively illustrate the localization error against variations in the distance at the depth of 5m, 50m and 200m.These figures demonstrate that MMAC-SBL struggles to achieve accurate source localization when the source distance is close.This challenge primarily stems from the SBL algorithm's lower angular resolution in the endfire direction, leading to diminished localization accuracy.Nevertheless, as the distance from the source increases, MMAC-SBL is capable of achieving sound source localization.Overall, MMAC-SBL, despite not reaching the accuracy of MFP-Bartlett when disregarding sound field mismatch and signalto-noise ratio effects, can still be utilized for sound source localization.

Influence of SNR
To investigate the impact of SNR variations on the algorithm, the source sets at 10 km range with a depth of 200 m in simulation.SNR spans from -20 dB to 20 dB , with each SNR value being simulated 20 times.Other simulation conditions remain consistent with previous settings.Fig. 3 presents the average absolute errors in distance and depth localization results across varying SNR levels.Notably, the depth localization error for MMAC-SBL remains below 10 m when SNR exceeds -3 dB.Likewise, the distance localization error is contained within 0.5 km provided SNR is greater than -7 dB.In the absence of considerations for acoustic paramter mismatches , the resilience of MMAC-SBL to SNR is inferior to that of MFP-Bartlett.However, MMAC-SBL attains precise localization with an SNR surpassing -3 dB.

Experimental results
The 2020 South China Sea experimental dataset was employed for verification.The sound source is the explosion bombs at depths of 50m and 200m.The vertical array spanning a depth range from 4030m to 4135m with a spacing of 7.5 m.The sea depth measured 4193m.Results of 200m deep explosion source are presented in Figure 4(a) and 4(b).Notably, both the MMAC-SBL and MFP-Bartlett models achieve precise localization result.The Mean Absolute Error (MAE) for range localization of MMAC-SBL(0.378km) slightly better than that of MFP-Bartlett (0.453km).Fig. 4(b) demonstrates occasional notable errors in depth localization for both MMAC-SBL and MFP-Bartlett, reaching a maximum discrepancy close to 200m.MMAC-SBL showcases enhanced accuracy in depth localization, reporting an MAE of 32.21m compared to MFP-Bartlett's 84.43m.For the localization of a 50m deep explosion source, results in Figure 4(c) and 4(d) depict the localization results of range and deep respectively.The MMAC-SBL distance localization boasts an MAE of 0.237km, slightly surpassing MFP-Bartlett's 0.290km.In depth localization, MFP-Bartlett registers an MAE of 25.93m, while MMAC-SBL delivers higher accuracy with an MAE of 6.87m.

Discussion and Conclusion
In this paper, a source localization method using vertical arrays is presented.Firstly, alterations in the incidence angles of the D, SR, and BR paths are analysed across different distances and depths.Secondly, the high-resolution azimuthal spectrum of the multipath incidence angles is achieved through the utilization of the SBL model.Lastly, the azimuthal spectrum and multipath angles are aligned using the Gaussian kernel method.The effectiveness of the model is subsequently validated through simulations and experimental data.

1 .
(a)D and SR path angels (b)BR path angles Figure Variation of multipath incidence angles with distance and depth

S
represents the spectrum in the direction of m  with an angle of incidence.n represents the noise vector.
maximizing the log probability function of the observed variable, the parameter

iA
represents the azimuthal spectral magnitude of the direction i  , I represents the number of estimated orientations, and ( )  represents the unit sampling sequence.If the incident angle is i  in the azimuthal spectrum, and one of the multipath angels is r  with the incident angles of D path D  , SR path SR  and BR path BR  can be determined using Equation (7), remarked as


are approximately equal, the azimuth spectrum solved by SBL is a large peak positioned between two incident angles, which can lead to increased error when using MMAC.Therefore, when the difference between D  and SR  is less than the threshold angle g  , the correlation ratio should be SR C =1.Consequently, for the grid position of ( ) , ss rz , the fuzzy function should be:

Figure 2 .Figure 3 .
Figure 2. Effect of range and depth variations on positioning errors )