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Most of shallow water geoacoustic inversions based on modal dispersion cannot reliably estimate the deep geoacoustic parameters. Because these studies focused on the dispersions of water waves but ignored the dispersions of ground waves. Therefore, in this paper a Bayesian geoacoustic inversion is studied based on wideband modal dispersions of water waves and ground waves. Firstly, the modal dispersion curves with Airy phase components are discussed. Secondly, the Bayesian inversion theory and a novel sample-efficient inference algorithm, namely Variational Bayesian Monte Carlo, are introduced briefly. In the Bayesian inversion, the posterior probability densities of unknown parameters are inferred, which can provide the prediction closest to the observation data and the uncertainty of the prediction. Considering that the forward acoustic model is computationally intensive, the posterior analysis is carried out by using the Variational Bayesian Monte Carlo method. It is performed by finding the variational distribution closest to the target distribution and requires less computation time than the Markov chain Monto Carlo method. In the simulation study, a range-independent two-layer seabed, including the sediment layer and basement layer, is modeled, on the assumption that the water column is homogeneous. The function of shear wave in waveguide is ignored. The compressional sound speed of the sediment layer varies linearly from c1U to c1L between 0 and h1, while other geoacoustic parameters are constant. By comparing the inversion results with and without the information of ground waves for different signal-to-noise ratios, it can be concluded that the deep geoacoustic parameters are more sensitive to the dispersions of ground waves. And then, a shallow-water experimental study is carried out in the Bohai Sea of China. The average water depth is about 20 m. The wideband pulse signals are recorded by a hydrophone with a sensitivity of –170 dB re 1 V/μPa. The received signals include well-defined Airy phase components, and the modal dispersion curves of water waves and ground waves are extracted accurately. The experimental results indicate that the Bayesian inversion combining water and ground wave dispersions can not only estimate the deep geoacoustic parameters reliably, but also reduce the inversion uncertainties of other model parameters, such as the shallow geoacoustic parameters, water depth, etc. The estimated source-receiver range and water sound speed are close to their measured values. The modal dispersion curves predicted by the posterior mean samples are in good consistence with those extracted from the experimental data in different ranges. In addition, the good forecast of transmission loss also demonstrates the reliability of the joint inversion.
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Keywords:
- modal dispersion /
- geoacoustic inversion /
- water wave /
- ground wave
[1] 尚尔昌 2019 应用声学 38 468
Google Scholar
Shang E C 2019 Appl. Acoust. 38 468
Google Scholar
[2] Chapman N, Shang E C 2021 J. Theor. Comp. Acout. 29 2130004
Google Scholar
[3] Dosso S E, Dettmer J 2011 Inverse Probl. 27 055009
Google Scholar
[4] Shen Y, Pan X, Zheng Z, Gerstoft P 2020 J. Acoust. Soc. Am. 148 3279
Google Scholar
[5] 杨坤德, 马远良 2009 58 1798
Google Scholar
Yang K D, Ma Y L 2009 Acta Phys. Sin. 58 1798
Google Scholar
[6] Chapman N 2016 J. Mar. Sci. Eng. 4 61
Google Scholar
[7] Bonnel J, Chapman N 2011 J. Acoust. Soc. Am. 130 EL101
Google Scholar
[8] 郭晓乐, 杨坤德, 马远良 2015 64 174302
Google Scholar
Guo X L, Yang K D, Ma Y L 2015 Acta Phys. Sin. 64 174302
Google Scholar
[9] Duan R, Chapman N, Yang K D, Ma Y L 2016 J. Acoust. Soc. Am. 139 70
Google Scholar
[10] 李佳蔚, 鹿力成, 郭圣明, 马力 2017 66 204301
Google Scholar
Li J W, Lu L C, Guo S M, Ma L 2017 Acta Phys. Sin. 66 204301
Google Scholar
[11] Lin Y T, Lynch J F, Chotiros N, Chen C F, Newhall A, Turgut A, Schock S G, Chiu C S, Bartek L, Liu C S 2004 IEEE J. Oceanic Engineer. 29 1231
Google Scholar
[12] Wan L, Badiey M, Knobles D P, Wilson P S 2018 J. Acoust. Soc. Am. 143 EL199
Google Scholar
[13] Blei D M, Kucukelbir A, McAuliffe J D 2017 J. Am. Stat. Assoc. 112 859
Google Scholar
[14] Acerbi L 2018 arXiv: 1810.05558 v2 [stat. ML]
[15] Che Y F, Wu X, Pastore G, Li W, Shirvan K 2021 Ann. Nucl. Energy 153 108046
Google Scholar
[16] Jensen F B, Kuperman W A, Porter M B, Schmidt H 2011 Computational Ocean Acoustics (Vol. 2) (New York: Springer) pp337–452
[17] Dosso S E, Nielsen P L, Wilmut M J 2006 J. Acoust. Soc. Am. 119 208
Google Scholar
[18] Porter M B 1991 The KRAKEN Normal Mode Program (La Spezia: SACLANT Undersea Research Center) Technical Report SM-245
[19] Bonnel J, Thode A, Wright D, Chapman N 2020 J. Acoust. Soc. Am. 147 1897
Google Scholar
[20] Daubechies I, Lu J F, Wu H T 2011 Appl. Comput. Harmon. A. 30 243
Google Scholar
[21] Dosso S E, Wilmut M J, Lapinski A L S 2001 IEEE J. Oceanic Engineer. 26 324
Google Scholar
[22] 王鹏, 贾凯, 吴建政, 胡日军 2015 海洋地质与第四纪地质 35 23
Wang P, Jia K, Wu J Z, Hu R J 2015 Marine Geology Quaternary Geology 35 23
[23] 张剑, 李日辉, 王中波, 张训华, 黄龙, 孙荣涛 2016 海洋地质与第四纪地质 36 1
Zhang J, Li R H, Wang Z B, Zhang X H, Huang L, Sun R T 2016 Marine Geology & Quaternary Geology 36 1
[24] Li Z L, Zhang R H 2004 Chin. Phys. Lett. 21 1100
Google Scholar
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图 1 地声参数反演流程图
Figure 1. Flow chart of geoacoustic inversion.
图 2 3层水平分层介质波导模型
Figure 2. Three-layer waveguide model.
图 3 1阶模态群速度曲线 (a)不同的
$ {h_1} $ ; (b)不同的${c_{{\text{1 L}}}}$ ; (c)不同的${c_2}$ ; (d)不同的${\rho _2}$ , 圆圈给出了艾里相频率的位置Figure 3. Group velocity curves of Mode 1: (a) Different values of
$ {h_1} $ ; (b) different values of${c_{{\text{1 L}}}}$ ; (c) different values of${c_2}$ ; (d) different values of${\rho _2}$ . The circles indicate the Airy phase frequencies.
图 4 仿真信号 (a)归一化时域波形; (b)时频图, 其中1阶模态的底波、水波和Airy相结构清晰可辩
Figure 4. Simulation signal: (a) Normalized time domain waveform; (b) time-frequency diagram. The ground wave, water wave, and Airy phase component of Mode 1 are well-defined.
图 5 不同信噪比条件下底波频散曲线的单次提取结果(a)和提取误差标准差(b)
Figure 5. Single extraction results (a) and standard deviations of extraction errors (b) of ground wave dispersion curve under different SNRs.
图 6 两种数据条件下部分参数的一维边缘后验概率密度函数 (a)
$ {h_1} $ ; (b)${c_{{\text{1 L}}}}$ ; (c)${c_2}$ ; (d)${\rho _2}$ Figure 6. 1 D marginal posterior probability densities of some parameters for Data 1 and Data 2: (a)
$ {h_1} $ ; (b)${c_{{\text{1 L}}}}$ ; (c)${c_2}$ ; (d)${\rho _2}$ .
图 7 实验描述 (a)实验地点; (b)水体声速剖面
Figure 7. Description of the experiment: (a) Experimental site; (b) sound speed profile in water.
图 8 实验信号 (a)归一化时域波形; (b)时频图, 其中1阶模态的底波、水波和Airy相结构清晰可辩
Figure 8. Experimental signal: (a) Normalized time domain waveform; (b) time-frequency diagram. The ground wave, water wave, and Airy phase component of Mode 1 are well-defined.
图 9 频散曲线的实测结果和后验预测结果 (a)数据条件1, 水波和底波频散数据, 插图为低频部分的局部放大; (b)数据条件2, 水波频散数据
Figure 9. Measurements and posterior predictions of dispersion curves: (a) Data 1, water waves and ground waves, the enlarged view of the low frequency part is inset; (b) Data 2, water waves.
图 10 待反演参数的一维(数据条件1和条件2)和二维(数据条件1)边缘后验概率密度函数
Figure 10. 1D (Data 1 and Data 2) and 2D (Data 1) marginal posterior probability densities of unknown parameters.
图 11 针对数据条件1的归一化后验参数协方差矩阵
Figure 11. Normalized posterior covariance matrix of unknown parameters for Data 1.
图 12 针对数据条件1的VBMC收敛情况 (a)证据下界ELBO; (b) KL散度
Figure 12. Convergences of the VBMC method for Data 1: (a) ELBO; (b) KL divergence.
图 13 针对数据条件1的压缩波声速和密度剖面后验估计结果 (a) 2层海底模型; (b) 3层海底模型
Figure 13. Posteriori estimates of compressed-wave sound speed profiles and density profiles for Data 1: (a) Two-layer seabed model; (b) three-layer seabed model.
图 14 另一组宽带脉冲信号时频图和频散曲线预测结果(实线) (a)接收距离3.42 km; (b)接收距离5.35 km; (c)接收距离6.51 km
Figure 14. Time-frequency diagrams and dispersion curve predictions (solid lines) of other broadband pulse signals: (a) The range is 3.42 km; (b) the range is 5.35 km; (c) the range is 6.51 km.
图 15 传播损失的理论计算结果和实测结果比较, 声源深度6 m, 接收深度11.5 m (a)频率200 Hz; (b)频率315 Hz
Figure 15. Comparisons of the theoretical and experimental transmission loss at source depth 6 m and receiver depth 11.5 m: (a) The frequency is 200 Hz; (b) the frequency is 315 Hz.
表 1 VBMC算法步骤
Table 1. Steps of the VBMC algorithm.
算法 : 变分贝叶斯蒙特卡罗(Acerbi, 2018) 1: $t \leftarrow 0$, 初始化GMM模型参数${\phi _0}$和${K_0}$ 2: When 收敛条件未满足, do 3: $t \leftarrow t + 1$ //迭代 4: for $ 1 \ldots {n_{\text{active}} }$ do 5: 采样新的样本点$\boldsymbol{\theta} _{\rm new} \leftarrow {\rm argmax}_{\boldsymbol{\theta}} [a({\boldsymbol\theta})]$ //$a({\boldsymbol{\theta }})$
为采样函数6: 计算样本点的$ \log p(\boldsymbol{d}, {\boldsymbol{\theta} _{\text{new}} }) $值, 并将其加入训练集 7: $\log p({\boldsymbol{d}}, {\boldsymbol{\theta}} ) \leftarrow$训练GP模型, 得到最优的模型超参数 8: ${K_t} \leftarrow $更新GMM模型参数 9: ${\phi _t} \leftarrow $利用随机梯度下降法对ELBO寻优, 更新GMM
模型参数10: end 
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表 2 反演参数空间和先验区间
Table 2. Parameter spaces and prior bounds for inversion.
待反演参数 单位 仿真参数值 先验区间 沉积层厚度$ {h_1} $ m 47 [10, 80] 沉积层密度${\rho _1}$ ${\text{g/c}}{{\text{m}}^{\text{3}}}$ 1.5 [1, 2] 基底层密度${\rho _2}$ ${\text{g/c}}{{\text{m}}^{\text{3}}}$ 2.1 [1.5, 2.5] 沉积层上部声速$ {c_{1{\text{U}}}} $ ${\text{m/s}}$ 1675 [1600, 1700] 沉积层下部声速$ {c_{1{\text{L}}}} $ ${\text{m/s}}$ 1675 [1650, 1750] 基底层声速$ {c_2} $ ${\text{m/s}}$ 1765 [1700, 1800] 水深dw m 21 [15, 25] 收发距离$r$ km 6.24 [4, 8] 时间因子$ {\delta _{\text{t} }} $ s — [–0.5, 0.5] 水体声速cw ${\text{m/s}}$ 1511 [1500, 1520]
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表 3 针对数据条件1的后验统计结果
Table 3. Posterior statistical results for Data 1.
待反演参数 均值 方差 95%可信区间 实测值 $ {h_1} $ 46.65 3.77 [39.69, 54.21] — ${\rho _1}$ 1.52 0.09 [1.35, 1.69] — ${\rho _2}$ 2.12 0.19 [1.75, 2.44] — $ {c_{1{\text{U}}}} $ 1675.89 9.40 [1654.35, 1690.32] — $ {c_{1{\text{L}}}} $ 1673.52 8.04 [1660.41, 1691.35] — $ {c_2} $ 1765.03 15.78 [1731.16, 1789.77] — $ {d_{\text{w}}} $ 20.69 0.45 [19.70, 21.41] 19—22 $r$ 6.24 0.30 [5.63, 6.83] 6.07 $ {\delta _{\text{t}}} $ 0.280 0.002 [0.276, 0.284] — $ {c_{\text{w}}} $ 1510.98 4.13 [1503.08, 1517.91] 1511—1513
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[1] 尚尔昌 2019 应用声学 38 468
Google Scholar
Shang E C 2019 Appl. Acoust. 38 468
Google Scholar
[2] Chapman N, Shang E C 2021 J. Theor. Comp. Acout. 29 2130004
Google Scholar
[3] Dosso S E, Dettmer J 2011 Inverse Probl. 27 055009
Google Scholar
[4] Shen Y, Pan X, Zheng Z, Gerstoft P 2020 J. Acoust. Soc. Am. 148 3279
Google Scholar
[5] 杨坤德, 马远良 2009 58 1798
Google Scholar
Yang K D, Ma Y L 2009 Acta Phys. Sin. 58 1798
Google Scholar
[6] Chapman N 2016 J. Mar. Sci. Eng. 4 61
Google Scholar
[7] Bonnel J, Chapman N 2011 J. Acoust. Soc. Am. 130 EL101
Google Scholar
[8] 郭晓乐, 杨坤德, 马远良 2015 64 174302
Google Scholar
Guo X L, Yang K D, Ma Y L 2015 Acta Phys. Sin. 64 174302
Google Scholar
[9] Duan R, Chapman N, Yang K D, Ma Y L 2016 J. Acoust. Soc. Am. 139 70
Google Scholar
[10] 李佳蔚, 鹿力成, 郭圣明, 马力 2017 66 204301
Google Scholar
Li J W, Lu L C, Guo S M, Ma L 2017 Acta Phys. Sin. 66 204301
Google Scholar
[11] Lin Y T, Lynch J F, Chotiros N, Chen C F, Newhall A, Turgut A, Schock S G, Chiu C S, Bartek L, Liu C S 2004 IEEE J. Oceanic Engineer. 29 1231
Google Scholar
[12] Wan L, Badiey M, Knobles D P, Wilson P S 2018 J. Acoust. Soc. Am. 143 EL199
Google Scholar
[13] Blei D M, Kucukelbir A, McAuliffe J D 2017 J. Am. Stat. Assoc. 112 859
Google Scholar
[14] Acerbi L 2018 arXiv: 1810.05558 v2 [stat. ML]
[15] Che Y F, Wu X, Pastore G, Li W, Shirvan K 2021 Ann. Nucl. Energy 153 108046
Google Scholar
[16] Jensen F B, Kuperman W A, Porter M B, Schmidt H 2011 Computational Ocean Acoustics (Vol. 2) (New York: Springer) pp337–452
[17] Dosso S E, Nielsen P L, Wilmut M J 2006 J. Acoust. Soc. Am. 119 208
Google Scholar
[18] Porter M B 1991 The KRAKEN Normal Mode Program (La Spezia: SACLANT Undersea Research Center) Technical Report SM-245
[19] Bonnel J, Thode A, Wright D, Chapman N 2020 J. Acoust. Soc. Am. 147 1897
Google Scholar
[20] Daubechies I, Lu J F, Wu H T 2011 Appl. Comput. Harmon. A. 30 243
Google Scholar
[21] Dosso S E, Wilmut M J, Lapinski A L S 2001 IEEE J. Oceanic Engineer. 26 324
Google Scholar
[22] 王鹏, 贾凯, 吴建政, 胡日军 2015 海洋地质与第四纪地质 35 23
Wang P, Jia K, Wu J Z, Hu R J 2015 Marine Geology Quaternary Geology 35 23
[23] 张剑, 李日辉, 王中波, 张训华, 黄龙, 孙荣涛 2016 海洋地质与第四纪地质 36 1
Zhang J, Li R H, Wang Z B, Zhang X H, Huang L, Sun R T 2016 Marine Geology & Quaternary Geology 36 1
[24] Li Z L, Zhang R H 2004 Chin. Phys. Lett. 21 1100
Google Scholar
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