# Gaussian Mixture Model for Marine Reverberations

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## Abstract

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## 1. Introduction

## 2. Theoretical and Statistical Distribution Characteristics of Reverberation

## 3. Statistical Modeling of Ocean Reverberation Data Based on the Gaussian Mixture Model (GMM) Method

#### 3.1. Gaussian Mixture Model (GMM) and Its Parameter Estimation Method (EM Algorithm)

#### 3.2. Improved EM Parameter Estimation Method

#### 3.2.1. Parameter Initialization Based on Reverberation Data

#### 3.2.2. GMM Parameter Estimation Based on EM Algorithm

#### 3.2.3. Model Evaluation

## 4. Simulation and Experiments Analysis

## 5. Verification Based on the Measured Data

#### 5.1. Method Validation

#### 5.2. Analysis of Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

EM | Expectation–maximization |

FOA | First Order Ambisonics |

GMM | Gaussian Mixture Model |

SαS | Symmetric Alpha–Stable |

GGMM | Generalized Gaussian Mixture Model |

BGMM | Bounded Gaussian Mixture Model |

BGGMM | Bounded Generalized Gaussian Mixture Model |

Probability density function | |

FH | Frequency histogram |

LFM | Linear Frequency Modulation |

CW | Continuous wave |

AIC | Akaike Information Criterion |

BIC | Bayesian Information Criterion |

MSE | Mean squared error |

## References

- Wang, L.; Wang, Q. The influence of marine biological noise on sonar detection. In Proceedings of the 2016 IEEE/OES China Ocean Acoustics (COA), Harbin, China, 9–11 January 2016. [Google Scholar]
- Tian, T. Shengna Jishu, 2nd ed.; Harbin Engineering University Press: Harbin, China, 2009. [Google Scholar]
- Faure, P. Theoretical Model of Reverberation Noise. J. Acoust. Soc. Am.
**1964**, 36, 259–266. [Google Scholar] [CrossRef] - Olishevski, B. Statistical Characteristics of Sea Reverberation, 2nd ed.; Science Press: Beijing, China, 1977. [Google Scholar]
- Middleton, D. New physical–statistical methods and models for clutter and reverberation: The ka–distribution and related probability structures. IEEE J. Ocean. Eng.
**1999**, 24, 261–284. [Google Scholar] [CrossRef] - Pernkopf, F.; Bouchaffra, D. Genetic–based EM algorithm for learning gaussian mixture models. IEEE Trans. Pattern Anal. Mach. Intell.
**2005**, 27, 1344–1348. [Google Scholar] [CrossRef] [PubMed] - Wei, H.; Wang, P. Gaussian mixture model for reverberation. Tech. Acoust.
**2007**, 26, 514–518. [Google Scholar] - Wang, P.; Wei, H.; Lou, L. Oceanic reverberation probability density modeling based on symmetric alpha–stable distribution. J. Harbin Eng. Univ.
**2021**, 42, 55–60. [Google Scholar] - Liu, W.; Wang, P.; Gu, X. Comparison of two EM algorithms for gaussian mixture parameter estimation. Tech. Acoust.
**2014**, 33, 539–543. [Google Scholar] - Liu, M.; Yu, Z. An improved expectation–maximum algorithm. J. Jilin Univ. (Sci. Ed.)
**2022**, 60, 1176–1182. [Google Scholar] - Najar, F.; Bourouis, S.; Bouguila, N. A comparison between different gaussian–based mixture models. In Proceedings of the 2017 IEEE/ACS 14th International Conference on Computer Systems and Applications (AICCSA), Hammamet, Tunisia, 30 October–3 November 2017. [Google Scholar]
- He, W.; Yu, R.; Zheng, Y.; Jiang, T. Image denoising using asymmetric gaussian mixture models. In Proceedings of the 2018 International Symposium in Sensing and Instrumentation in IoT Era (ISSI), Shanghai, China, 6–7 September 2018. [Google Scholar]
- Przyborowski, M.; Ślęzak, D. Approximation of the expectation–maximization algorithm for gaussian mixture models on big data. In Proceedings of the 2022 IEEE International Conference on Big Data (Big Data), Kyoto, Japan, 13–16 December 2022. [Google Scholar]
- Ma, B.; Gong, L.; Chen, X.; Liu, G. Study of characteristics of acoustic intensity in acoustic vector ocean reverberation based on CW pulse. J. Nav. Univ. Eng.
**2022**, 34, 102–106. [Google Scholar] - Ivakin, A.N.; Williams, K.L. Midfrequency acoustic propagation and reverberation in a deep ice–covered arctic ocean. J. Acoust. Soc. Am.
**2022**, 152, 1035–1044. [Google Scholar] [CrossRef] [PubMed] - Cao, F.; Zhang, X.; Han, J. Experimental analysis of statistical property of low frequency reverberation envelope in shallow water. In Proceedings of the 2021 OES China Ocean Acoustics, Heilongjiang, China, 14–17 July 2021. [Google Scholar]
- Wang, J.; Wang, C.; Cheng, T. Active sonar reverberation suppression based on beam space data normalization. In Proceedings of the 2017 IEEE International Conference on Signal Processing, Communications and Computing, Xiamen, China, 22–25 October 2017. [Google Scholar]
- Li, Q. Introduction to Sonar Signal Processing, 2nd ed.; Chinese Academy of Sciences: Beijing, China, 2000. [Google Scholar]
- Glodek, M.; Schels, M.; Schwenker, F. Ensemble gaussian mixture models for probability density estimation. Comput. Stat.
**2013**, 28, 127–138. [Google Scholar] [CrossRef] - Guo, H.; Chu, F.; Zhu, D. Research on gaussian mixture auto–regressive reverberation modeling and whitening algorithm. In Proceedings of the 2021 IEEE International Conference on Signal Processing, Communications and Computing, Xi’an, China, 17–20 August 2021. [Google Scholar]
- Jovanović, A.; Perić, Z.; Nikolić, J. The effect of uniform data quantization on GMM–based clustering by means of EM algorithm. In Proceedings of the 2021 20th International Symposium INFOTEH–JAHORINA, Sarajevo, Bosnia and Herzegovina, 17–19 March 2021. [Google Scholar]
- Kasim, F.A.B.; Pheng, H.S.; Nordin, S.Z.B. Gaussian mixture modelexpectation maximization algorithm for brain images. In Proceedings of the 2021 2nd International Conference on Artificial Intelligence and Data Sciences (AiDAS), Kuala Lumpur, Malaysia, 8–9 September 2021. [Google Scholar]
- Stepashko, V. Asymptotic properties of a class of criteria for best model selection. In Proceedings of the 2020 IEEE 15th International Conference on Computer Sciences and Information Technologies, Zbarazh, Ukraine, 23–26 September 2020. [Google Scholar]
- Wei, J.; Zhou, L. Model selection using modified AIC and BIC in joint modeling of paired functional data. Stat. Probab. Lett.
**2010**, 80, 1918–1924. [Google Scholar] [CrossRef] - Ding, J.; Tarokh, V.; Yang, Y. Bridging AIC and BIC: A new criterion for autoregression. IEEE Trans. Inf. Theory
**2017**, 64, 4024–4043. [Google Scholar] [CrossRef] - Mo, X.; Wen, H.; Yang, Y. A parameter estimation method of α stable distribution and its application in the statistical modeling of ice–generated noise. Acta Acust.
**2023**, 48, 319–326. [Google Scholar]

**Figure 4.**Waveform plot, probability density function (PDF) curve, and mean square error of non–Gaussian random sequences. (

**a**) Waveform plot; (

**b**) comparative PDF curves based on different models in graphical format; and (

**c**) mean square error plots of the fitting results from different models.

**Figure 5.**Simulation of the reverberation data of an LFM signal, the PDF curve, and its mean square error plot. (

**a**) Waveform plot; (

**b**) comparative PDF curves based on different models in graphical format; and (

**c**) mean square error plots of the fitting results from different models.

**Figure 6.**Simulation of the reverberation data of a CW signal, the PDF curve, and its mean square error plot. (

**a**) Waveform plot; (

**b**) comparative PDF curves based on different models in graphical format; and (

**c**) mean square error plots of the fitting results from different models.

**Figure 7.**The reverberation data obtained in Experiment 1, along with probability density function (PDF) curves and their mean square errors. (

**a**) Waveform plots; (

**b**) comparative PDF curve comparisons based on different models; and (

**c**) mean square error plots for the fitting results from different models.

**Figure 8.**The first section of reverberation data, PDF curves, and their mean square errors in Experiment 2. (

**a**) Waveform plots; (

**b**) comparative PDF curve comparisons based on different models; and (

**c**) mean square error plots for the fitting results from different models.

**Figure 9.**The second section of reverberation data, PDF curves, and their mean square errors in Experiment 2. (

**a**) Waveform plots; (

**b**) comparative PDF curve comparisons based on different models; and (

**c**) mean square error plots for the fitting results from different models.

**Figure 10.**The third section of reverberation data, PDF curves, and their mean square errors in Experiment 2. (

**a**) Waveform plots; (

**b**) comparative PDF curve comparisons based on different models; and (

**c**) mean square error plots for the fitting results from different models.

Distribution | G–D | SαS–D | GM–D | ||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | $\left[\mathit{\mu},\mathit{\sigma}\right]$ | $\left[\mathit{\alpha},\mathit{\beta},\mathit{\gamma},\mathit{\mu}\right]$ | $\left[{\mathit{\lambda}}_{\mathit{k}},{\mathit{\mu}}_{\mathit{k}},{\mathit{\sigma}}_{\mathit{k}}\right]$ | ||||||

Estimation | 0.096 | 1.710 | 1.579 | 0.149 | 1.011 | 0.141 | 0.648 | 0.185 | 2.015 |

0.352 | −0.068 | 0.881 | |||||||

MSE | 2.2 × 10^{−4} | 2.4 × 10^{−5} | 1.8 × 10^{−5} |

Distribution | G–D | SαS–D | GM–D | ||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | $\left[\mathit{\mu},\mathit{\sigma}\right]$ | $\left[\mathit{\alpha},\mathit{\beta},\mathit{\gamma},\mathit{\mu}\right]$ | $\left[{\mathit{\lambda}}_{\mathit{k}},{\mathit{\mu}}_{\mathit{k}},{\mathit{\sigma}}_{\mathit{k}}\right]$ | ||||||

Estimation | −0.144 | 0.271 | 1.430 | 0.112 | −0.148 | −0.123 | 0.385 | −0.123 | 2.091 |

0.615 | −0.158 | 0.267 | |||||||

MSE | 0.0420 | 0.0062 | 0.0009 |

Distribution | G–D | SαS–D | GM–D | ||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | $\left[\mathit{\mu},\mathit{\sigma}\right]$ | $\left[\mathit{\alpha},\mathit{\beta},\mathit{\gamma},\mathit{\mu}\right]$ | $\left[{\mathit{\lambda}}_{\mathit{k}},{\mathit{\mu}}_{\mathit{k}},{\mathit{\sigma}}_{\mathit{k}}\right]$ | ||||||

Estimation | −0.144 | 0.271 | 1.430 | 0.112 | −0.148 | −0.123 | 0.248 | 0.004 | 0.013 |

0.692 | 0.692 | 0.037 | |||||||

0.060 | 0.031 | 0.051 | |||||||

MSE | 0.5546 | 0.0550 | 0.0139 |

Data | Figure 6a | Figure 7a | Figure 8a | Figure 9a |
---|---|---|---|---|

${K}_{H}$ | 1 | 1 | 3 | 3 |

${K}_{H}-MSE$ | 0.0107 | 0.0170 | 0.0189 | 0.0230 |

${K}_{AIC}$ | 6 | 5 | 4 | 6 |

${K}_{AIC}-MSE$ | 0.0015 | 0.0015 | 0.0013 | 0.0006 |

${K}_{BIC}$ | 6 | 5 | 4 | 6 |

${K}_{BIC}-MSE$ | 0.0031 | 0.0015 | 0.0013 | 0.0006 |

${K}_{B}$ | 3 | 3 | 4 | 5 |

${K}_{H}-MSE$ | 0.0031 | 0.0027 | 0.0013 | 0.0013 |

Parameter | SαS–D [$\mathsf{\alpha},\mathsf{\beta},\mathsf{\gamma},\mathsf{\mu}$] | GM–D [${\mathsf{\lambda}}_{\mathit{k}}{,\mathsf{\mu}}_{\mathit{k}},{\mathsf{\sigma}}_{\mathit{k}}$] | |||||
---|---|---|---|---|---|---|---|

Data | |||||||

Figure 6a | 1.627 | 0.151 | 0.133 | 0.024 | 0.767 | 0 | 0.023 |

0.206 | 0.0138 | 0.015 | |||||

0.027 | −0.037 | 0.100 | |||||

Figure 7b | 1.077 | 0.030 | 0.112 | 0.093 | 0.741 | −0.087 | 0.149 |

0.254 | −0.096 | 0.057 | |||||

0.005 | −0.656 | 0.253 | |||||

Figure 8b | 1.606 | −0.028 | 0.09 | −0.09 | 0.514 | 0.062 | 0.098 |

0.061 | −0.062 | 0.281 | |||||

0.354 | 0.077 | 0.253 | |||||

0.061 | 0.782 | 0.096 | |||||

Figure 9b | 1.537 | 0.010 | 0.220 | 0.053 | 0.255 | 0.420 | 0.196 |

0.036 | 0.847 | 0.068 | |||||

0.032 | −0.752 | 0.059 | |||||

0.372 | 0.060 | 0.115 | |||||

0.305 | −0.272 | 0.221 |

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## Share and Cite

**MDPI and ACS Style**

Sun, T.; Wen, Y.; Zhang, X.; Jia, B.; Zhou, M.
Gaussian Mixture Model for Marine Reverberations. *Appl. Sci.* **2023**, *13*, 12063.
https://doi.org/10.3390/app132112063

**AMA Style**

Sun T, Wen Y, Zhang X, Jia B, Zhou M.
Gaussian Mixture Model for Marine Reverberations. *Applied Sciences*. 2023; 13(21):12063.
https://doi.org/10.3390/app132112063

**Chicago/Turabian Style**

Sun, Tongjing, Yabin Wen, Xuegang Zhang, Bing Jia, and Mengwei Zhou.
2023. "Gaussian Mixture Model for Marine Reverberations" *Applied Sciences* 13, no. 21: 12063.
https://doi.org/10.3390/app132112063