#
Bayesian Inversion for Geoacoustic Parameters in Shallow Sea^{ †}

^{1}

^{2}

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^{5}

^{6}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Prediction Modeling of a Shallow Sea Acoustic Field

_{1}, φ

_{p}

_{n}, and ψ

_{s}

_{n}are the displacement potential function in the water column, P-wave displacement potential function, and S-wave displacement potential function in the layered seabed, respectively. ρ

_{1}and c

_{1}are the density and sound speed in the water column. ρ

_{b}

_{n}, c

_{p}

_{n}, and c

_{s}

_{n}are the density, compression wave (P-wave) speed, and shear wave (S-wave) speed, respectively, of the n layer of seabed, α

_{p}

_{n}and α

_{s}

_{n}represent the attenuation of P-wave and S-wave, respectively; these five parameters in each layer of seabed are the expected inversion objects in this study. Moreover, z

_{s}is the depth of the point acoustic source, z = 0 and z = H are set as the sea surface and seabed in the model.

_{m}

_{n}=ω/c

_{m}(m=1,p,s and n=1…N) is the wave number of each layer and ω=2πf

_{0}is the angular frequency of the point acoustic source at f

_{0}.

_{0}and J

_{1}are the first order and second-order Bessel functions, respectively, and A, B, C, P

_{n}and S

_{n}are the indeterminate coefficients of the potential functions in each layer, the high mark ‘up’ and ‘down’ are used to representative the upgoing waves and downgoing waves in these potential functions. In the water column, the relationship between sound pressure p and potential function φ

_{1}is $p={\rho}_{1}{\omega}^{2}{\phi}_{1}$.

_{1}(continuous normal displacement, continuous normal stress, and zero tangential stress) and H

_{n}

_{+1}(continuous normal displacement, continuous normal stress, continuous tangential displacement, and continuous tangential stress), the relationship between these coefficients can be written as a (4N+1)th order matrix equations, which is shown in Equation (7).

_{n}and S

_{n}, can be solved by (b

_{ij})

_{(4N+1) ×1}= [(a

_{ij})

_{(4N+1) ×}

_{(4N+1)}]

^{−1}. (c

_{ij})

_{(4N+1) ×1}. Substituting b

_{ij}into Equations (4)–(6) further result in the potential functions in each layer. When replacing the coefficients in Equations (4)–(6), the expressions of displacement potential function in Figure 2 can be obtained. The expressions of sound pressure p in the water column, as the study object in inversion, can also be obtained with Equation (8).

_{l}= ξ

_{min}+ lΔξ, l = 0, 1, 2, …, (N

_{S}−1), r

_{j}= r

_{min}+ jΔr, j = 0, 1, 2, …, (N

_{S}−1), Δr · Δξ = 2π/N

_{S}.

## 3. Inversion Method

#### 3.1. Bayesian Inversion Theory

**d**with elements d

_{i}, and let

**m**represent the model vector composed of the awaiting inversion geoacoustic parameters m

_{i}. Both d

_{i}and m

_{i}are considered random variables that are related via Bayes’ rule [23]

**m**|

**d**) is the posterior probability density (PPD). P(

**d**|

**m**) is the conditional probability density function (PDF) of

**m**under given

**d**, P(

**m**) is the prior PDF of

**m**and representing the available parameter information independent of the data

**d**, P(

**d**) is the PDF of

**d**. As the P(

**d**) is independent of

**m**, and the P(

**d**|

**m**) can be regarded as the likelihood function L(

**d**|

**m**) for the measured data [23], the Equation (10) can be written as:

**m**) is determined by the form of the data and the statistical distribution of the data errors, including both measurement and model errors. Considering that it is difficult to obtain an independent estimate of the error statistics in practice, the assumption of unbiased Gaussian errors is used in processing, the form of the likelihood function is

**m**) is the cost function. After normalizing

**m**can be regarded as the inversion results. To interpret the M-dimensional PPD requires us to estimate the properties of the parameter value, uncertainties, and inter-relationships, such as the MAP model $\widehat{m}$, mean model $\overline{m}$ and marginal probability distributions P(m

_{i}|

**d**). These properties are respectively defined as:

#### 3.2. Cost Function

**m**)under the assumption of the Gaussian data errors [24].

^{f}and θ

^{f}is the magnitude and phase of the unknown complex source at each frequency. To remove the dependence on A

^{f}and θ

^{f}by setting the $\partial L(\mathbf{m})/\partial {A}^{f}=\partial L(\mathbf{m})/\partial {\theta}^{f}=0$ [25], the maximum likelihood solution of the source is

^{f}is the unknown variance at the fth frequency and

**I**is the identity matrix, the L(

**m**)can be simplified written as:

**m**) becomes

**m**), using a global optimization scheme.

_{0}= 100, cooling rate ξ = 0.99, the number of samples n = 10000, and the type of perturbation is non-uniform. For GA settings, the population of each generation is pop = 1000, the type of coding is binary coding, the maximum generation is 100, the type of selection is roulette wheel selection, the cross-over rate is 0.95, and the mutation rate is 0.05. GS settings corresponded to the SA settings. A block diagram of the research is shown in Figure 3.

## 4. Simulation Results

_{p}, S-wave speed c

_{s}, the attenuation coefficient for the two speeds α

_{p}and α

_{s}, and seabed density ρ

_{b}. The true values of waveguide parameters in the simulation are presented in Table 1.

#### 4.1. The Analysis of the Acoustic Propagation Characteristics

_{j}, z, ω) was calculated by Equation (22).

_{p,}c

_{s,}ρ

_{b,}α

_{p}, and α

_{s}on acoustic propagation separately. The changing values of each parameter were set to the simulation true value deviation ±10%; when one of the above parameters changed, the other parameters remained fixed. In Figure 4a–e, the solid blue line represents the calculation results under the simulated true values, where the dotted black line and the dashed red line indicate the calculation results under the changing values. Figure 4f revealed the comparison of TLs’ anomalies when the value of each parameter was changed, and the change bounds were set between −10% to +10% of the true value [27].

_{p}and c

_{s}are varied. When ρ

_{b}changes, the variation in the Transmission Loss (TL) is relatively prominent. When the α

_{p}or α

_{s}is changed, the variation in the Transmission Loss (TL) is the least obvious. The anomaly values of the five parameters in Figure 4f reveal that, in the discus bounds, these parameters are in the descending order of degree of influence on TL are c

_{p}, c

_{s}, ρ

_{b}, α

_{p}, and α

_{s}. In this situation, the influence order of degree of five parameters on acoustic propagation characteristics can be preliminarily summarized as: c

_{p}>c

_{s}>ρ

_{b}>α

_{p}>α

_{s}.

#### 4.2. The Analysis of the Inversion Parameters’ Sensitivity

**m**) with the change of a single parameter. In each search bound of the parameter, E(

**m**) touches the minimum value only at the true simulated value of the parameter, which can avoid the impact of the local optimum solution on the optimization of the cost function in the subsequent algorithm. Nonetheless, with the change of those five parameters, the range of E(

**m**) variation is different. It can be seen from Figure 5f, that within the search bounds of five parameters, the parameters are in the descending order of degree of influence on E(

**m**) are c

_{p}, c

_{s}, ρ

_{b}, α

_{p}and α

_{s}. Therefore, the influence order of the degree of five geoacoustic parameters on p(r, z) can be defined as: c

_{p}> c

_{s}> ρ

_{b}> α

_{p}> α

_{s}. This further verifies our results discussed in Section 4.1.

#### 4.3. The Inversion Results

_{p}, c

_{s}> ρ

_{b}> α

_{p}and α

_{s}(Figure 6 and Figure 7). Further, the results are lining with the findings presented in Section 4.1 and Section 4.2. Comparing Figure 6 and Figure 7, we can find that for the same parameter the PPD in Figure 7 is wider.

_{p}, c

_{s}and ρ

_{b}, the inversion results are similar in the two different marine environments, such as Figure 6a–c, Figure 7a–c, Figure 8a–c,e,f and Figure 9a–c,e,f. It is summarized that noise has less effect on the inversion results of sensitive parameters.

## 5. Results of Measured Data

#### 5.1. Introduction to the Scaling Experiment

_{s}= z/N, z′ = z/N, r′ = r/N, and f′ = Nf. In this case, the relationship between the original pressure p and the scaled pressure p′ can be obtained: p′(r′, z′, ω′) = Np(r, z, ω), and both the fluctuation and distribution of the original acoustic field remain unchanged in the scaled acoustic field [29].

_{s}and receive depth z

_{r}are 20/0.1 m, 20/1 m, 20/10 m, 20/100 m and 10/0.1 m, 10/1 m, 10/10 m, 10/100 m respectively. The values of various acoustic parameters are set to be the same as the true value in Section 4.

#### 5.2. Procedure and Data Processing of the Scaling Experiments

_{s}= 87 mm, the depth of acoustic sensor z

_{r}= 10 mm, the depth of water layer H = 182 mm. The sound speed c

_{1}= 1450.212 m·s

^{−1}(obtained by the empirical formula with the water temperature in tank, the temperature is measured by the temperature recorder of Star-Oddi company [30]).

_{s}= 20 MHz. The starting point was 60 m away from the source position. The sampling rate of each measurement point was repeated ten times to reduce the influences of random error. After completing the one-point measurement, the walkway drives the acoustic sensor to move away from the transmitter by 2 mm, and repeated the same for 719 points. Figure 13a shows the TL measured in the experiment. Figure 13b shows the time-domain diagram of the 50th to 150th receiving signal received during the experiment. The red lines in Figure 13b represent the arrival time of the direct signal, the surface reflection signal, and the bottom reflection signal at the reception point from the 50th to the 150th. The measurement parameters are shown in Table 3. It can be inferred from Figure 13b that the arrival time of each path signal achieved by the simulation is consistent with the actual arrival time, which verifies the reliability of the measurement experiment.

#### 5.3. Analysis of Inversion Results

_{p}, c

_{s}and ρ

_{b}are relatively narrow in their prior search bounds, which proves c

_{p}, c

_{s}, ρ

_{b}are more sensitive to the cost function, and have fewer uncertainties. The PPDs of α

_{p}, α

_{s}are flat in their prior search bounds, which means that α

_{p}and α

_{s}are not sensitive to the cost function. The sharpness of the probability density curve shows the sensitivity of the cost function to every parameter: c

_{p}, c

_{s}> ρ

_{b}> α

_{p}, and α

_{s}. The inversion results obtained via the measurement experiment data are consistent with the simulation results discussed in Section 4.

^{−3}. It is very near to the inversion value, which proves the viability of the inversion results. Thus, it is concluded that: cp, cs, and ρb are more sensitive and have fewer uncertainties to acoustic pressure than αp and αs.

#### 5.4. The Verification

## 6. Conclusions

_{p}), S-wave speed (c

_{s}), seabed density (ρ

_{b}), p-wave attenuation (α

_{p}), and s-wave attenuation (α

_{s}) are inverted. In this paper, we used simulated data and the experimental data to verify the feasibility of the inversion method. The Posterior Probability Densities (PPDs) of the five parameters are obtained using the cost function based on the Bayesian inversion theory. The PPDs and 2D marginal PPDs provide parameter estimates and uncertainties. It is summarized that c

_{p}, c

_{s}, and ρ

_{b}in the seabed have fewer uncertainties and are more sensitive to acoustic pressure than α

_{p}and α

_{s}.

_{p}> c

_{s}> ρ

_{b}> α

_{p}> α

_{s}.

^{3}, which is very close to the inversion value (1.2451 ± 0.0758g/m

^{3}). All these prove the credibility of the inversion method.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The influence of the five geoacoustic parameters on acoustic propagation, (

**a**)-(

**e**) corresponds to the c

_{p,}c

_{s,}ρ

_{b,}α

_{p}and α

_{s}respectively, (

**f**) reveals the comparison of TLs’ anomalies when the value of each parameter was changed.

**Figure 5.**The sensitivity of the cost function (E(

**m**)) for the five geoacoustic parameters. (

**a**) to (

**e**)corresponds to the c

_{p,}c

_{s,}ρ

_{b,}α

_{p}, and α

_{s}, respectively. (

**f**) corresponds to the comparison of five parameter’s influence on E(

**m**).

**Figure 6.**Five geo-acoustic parameters’ Posterior Probability Density (PPD) in the noiseless sea environment. The red lines mean the true value of each parameter in simulation. The green segments represent the mean value of the inversion results and their variance.

**Figure 7.**Five geo-acoustic parameters’ PPD in the noisy sea environment. The red lines mean the true value of each parameter in simulation. The green segments represent the mean value of the inversion results and their variance.

**Figure 8.**2D marginal PPDs between different parameters in the noiseless sea environment. White dashed lines mark the true values of each parameter in the simulation.

**Figure 9.**2D marginal PPDs between different parameters in the noisy sea environment. White dashed lines mark the true values of each parameter in the simulation.

**Figure 10.**The verification of simulation results. The solid blue line means the Transmission Loss (TL) measured in the simulation and the dashed red line represents the TL simulated by inversion result. (

**a**) the comparison of TL in the noiseless sea environment; (

**b**) the comparison of TL in the noisy sea environment.

**Figure 11.**Transmission losses under the four scaling conditions. The solid blue line represents the frequency 150 × 0.1 Hz, the dashed red line represents the frequency 150 × 1 Hz, the dotted black line represents the frequency 150 × 10 Hz, and the dashed dotted green line represents the frequency 150 × 100 Hz.

**Figure 12.**(

**a**) Experimental measurement system, (

**b**) The movable micro-worktable for measurement equipment.

**Figure 13.**(

**a**) The TL measured in the experiment; (

**b**) The arrival time of the signal. The red line represents the arrival time of the direct signal, the dashed red line means the arrival time of surface reflection signal, and the dotted red line means the arrival time of the bottom reflection signal.

**Figure 14.**Five geo-acoustic parameters’ PPD. Red lines mean the inversion result of each parameter by GA. Green segments represent the mean value of the inversion results and their variance.

**Figure 15.**2D marginal PPDs between different parameters. White dashed lines mark the Genetic Algorithm (GA) results of each parameter.

**Figure 16.**The verification approaches, (

**a**) solid blue line means the TL measured in experiment and the dashed red line represents the TL simulated by inversion result; (

**b**) solid blue line means the normalized signal amplitude in the time domain at the 470th reception point (one meter from the starting point), and the dashed red line represents the normalized signal amplitude in the time domain simulated by inversion result. The after off effects of the transmitter are highlighted by arrows.

Parameters | Simulated Value |
---|---|

Depth H/m | 100 |

Sound speed c_{1}/m·s^{−1} | 1500 |

Sea-water density ρ_{1}/ g·cm^{−3} | 1.025 |

P-wave speed c_{p}/m·s^{−1} | 2000 |

S-wave speed c_{s}/ m·s^{−1} | 1000 |

Seabed density ρ_{b}/g·cm^{−3} | 1.5 |

P-wave attenuation α_{p}/dB·λ^{−1} | 0.5 |

S-wave attenuation α_{s}/ dB·λ^{−1} | 0.5 |

Parameters | True Values | Search Bounds | Inversion Values (Noiseless) | Inversion Values (Noisy) |
---|---|---|---|---|

c_{p}/m·s^{−1} | 2000 | 1800, 2200 | 2000.9367 ± 14.5002 | 2001.7484 ± 25.9427 |

c_{s}/m·s^{−1} | 1000 | 900, 1100 | 996.5609 ± 9.1982 | 1000.9983 ± 11.7670 |

ρ_{b}/g·cm^{−3} | 1.5 | 1.35, 1.65 | 1.5040 ± 0.0121 | 1.5063 ± 0.0152 |

α_{p}/dB·λ^{−1} | 0.5 | 0.45, 0.55 | 0.4974 ± 0.0046 | 0.5023 ± 0.0053 |

α_{s}/dB·λ^{−1} | 0.5 | 0.45, 0.55 | 0.4989 ± 0.0050 | 0.5015 ± 0.0.0067 |

z_{s} /mm | z_{r} /mm | H /mm | c_{1}/m·s^{−1} |
---|---|---|---|

87 | 84 | 182 | 1450.212 |

Parameters | Search Bounds | Inversion Values |
---|---|---|

c_{p}/m·s^{−1} | 2200–2500 | 2397.3563 ± 31.9997 |

c_{s}/m·s^{−1} | 1100–1300 | 1187.9400 ± 6.8722 |

ρ_{b}/g·cm^{−3} | 1.0–1.8 | 1.2451 ± 0.0758 |

α_{p}/dB·λ^{−1} | 0.1–1.1 | 0.6616 ± 0.2489 |

α_{s}/dB·λ^{−1} | 0.1-1.1 | 0.8705 ± 0.1468 |

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## Share and Cite

**MDPI and ACS Style**

Zheng, G.; Zhu, H.; Wang, X.; Khan, S.; Li, N.; Xue, Y.
Bayesian Inversion for Geoacoustic Parameters in Shallow Sea. *Sensors* **2020**, *20*, 2150.
https://doi.org/10.3390/s20072150

**AMA Style**

Zheng G, Zhu H, Wang X, Khan S, Li N, Xue Y.
Bayesian Inversion for Geoacoustic Parameters in Shallow Sea. *Sensors*. 2020; 20(7):2150.
https://doi.org/10.3390/s20072150

**Chicago/Turabian Style**

Zheng, Guangxue, Hanhao Zhu, Xiaohan Wang, Sartaj Khan, Nansong Li, and Yangyang Xue.
2020. "Bayesian Inversion for Geoacoustic Parameters in Shallow Sea" *Sensors* 20, no. 7: 2150.
https://doi.org/10.3390/s20072150