# A Comparison of Three Sediment Acoustic Models Using Bayesian Inversion and Model Selection Techniques

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## Abstract

**:**

## 1. Introduction

## 2. Geoacoustic Models and Acoustic Scattering Models

#### 2.1. Fluid Model

#### 2.1.1. EDFM

#### 2.1.2. Fluid Interface Roughness Scattering Model

#### 2.1.3. Fluid Volume Scattering Model

#### 2.2. Grain-Shearing Elastic Model

#### 2.2.1. VGS(λ)

#### 2.2.2. Elastic Scattering Model

#### 2.3. Poroelastic Model

#### 2.3.1. CREB

#### 2.3.2. Poroelastic Scattering Model

#### 2.4. Sensitivity Analysis

## 3. Bayesian Inference

#### 3.1. Parameter Inversion

#### 3.2. Convergence Criterion

#### 3.3. Model Selection

## 4. Experimental Measurements, Results, and Discussion

#### 4.1. Experiment Description

#### 4.2. Parameter Inversion

#### 4.2.1. Fluid Model

#### 4.2.2. Grain-Shearing Elastic Model

#### 4.2.3. Poroelastic Model

#### 4.3. Model Comparison and Selection

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Borgonovo indices for: (

**a**) fluid model; (

**b**) grain-shearing elastic model; (

**c**) poroelastic model.

**Figure 4.**Estimates of one-dimensional marginal posterior probability distributions (PPDs) for fluid model parameters.

**Figure 5.**Estimates of two-dimensional marginal PPDs for fluid model parameters (

**top right**) and correlation matrix map (

**bottom left**).

**Figure 7.**Estimates of two-dimensional marginal PPDs for grain-shearing elastic model parameters (

**top right**) and correlation matrix map (

**bottom left**).

**Figure 9.**Estimates of two-dimensional marginal PPDs for poroelastic model parameters (

**top right**) and correlation matrix map (

**bottom left**).

**Figure 10.**Comparison of uncertainties of ratios of compressional and shear sound speed in sediment to compressional sound speed in water and corresponding attenuation.

**Figure 11.**Comparison of uncertainties of backscattering strength predicted by samples based on different models and acoustic frequencies. Red asterisks represent corresponding experimental measurements, as shown in Figure 3.

**Figure 12.**Comparison of Bayes factors calculated for each pair of models through Bayesian inference applied to backscattering strength data. Number superimposed over each matrix element is the base 10 logarithm of the Bayes factor for a given pair of models.

Parameter | Symbol | Lower Bound | Upper Bound | Unit | |
---|---|---|---|---|---|

Common parameters | Roughness spectral exponent | ${\gamma}_{2}$ | 2 | 4 | dimensionless |

Roughness spectral strength | ${\omega}_{2}$ | 0.00001 | 0.0005 | ${\mathrm{m}}^{4}$ | |

Density fluctuation spectral exponent | ${\gamma}_{3}$ | 1 | 8 | dimensionless | |

Density fluctuation spectral strength | ${\omega}_{3}$ | 0.001 | 0.01 | ${\mathrm{m}}^{3}$ | |

Ratio of compressibility to density fluctuation | $\mu $ | −3 | 2 | dimensionless | |

Porosity | $\beta $ | 0.2 | 0.8 | dimensionless | |

Ratio of mass density of grains to water | ${\rho}_{r}$ | 2 | 3 | dimensionless | |

Ratio of bulk modulus of grains to water | ${K}_{r}$ | 5 | 30 | dimensionless | |

Fluid model parameters | Mean grain diameter | $d$ | $62.5\times {10}^{-6}$ | $1\times {10}^{-3}$ | m |

Tortuosity | $\alpha $ | 1 | 3 | dimensionless | |

Permeability | $\kappa $ | $6.5$ | $100$ | $\mathsf{\mu}{\mathrm{m}}^{2}$ | |

Grain-shearing elastic model parameters | Material exponent | $n$ | 0.02 | 0.2 | dimensionless |

Compressional rigidity coefficient | ${\gamma}_{1}$ | ${10}^{7}$ | ${10}^{9}$ | Pa | |

Shear rigidity coefficient | ${\gamma}_{t}$ | ${10}^{6}$ | ${10}^{8}$ | Pa | |

Compressional viscoelastic relaxation time | ${\tau}_{1}$ | ${10}^{-5}$ | ${10}^{-3}$ | s | |

Poroelastic model parameters | Mean grain diameter | $d$ | $62.5\times {10}^{-6}$ | $1\times {10}^{-3}$ | m |

Cementation exponent | $m$ | 1 | 4 | dimensionless | |

Pore shape factor | ${a}_{B}$ | 2 | 12 | dimensionless | |

Poisson’s ratio of grains | $\nu $ | 0.2 | 0.4 | dimensionless | |

Low-frequency asymptotic frame bulk modulus | ${K}_{bo}$ | 0 | $0.25\times {10}^{9}$ | Pa | |

High-frequency asymptotic increase | ${K}_{y}$ | 0 | $0.25\times {10}^{9}$ | Pa | |

Bulk relaxation frequency | ${f}_{k}$ | ${10}^{3}$ | $2\times {10}^{4}$ | Hz |

Parameter | Symbol | Value | Unit |
---|---|---|---|

Mass density | ${\rho}_{w}$ | 1000 | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

Bulk modulus | ${K}_{w}$ | $2.23\times {10}^{9}$ | $\mathrm{Pa}$ |

Dynamic viscosity | $\eta $ | 0.001 | $\mathrm{kg}/\mathrm{m}\text{}\mathrm{s}\text{}$ |

Compressional wave speed | ${c}_{w}$ | 1493 | $\mathrm{m}/\mathrm{s}$ |

${\mathbf{log}}_{10}{\mathit{B}}_{\left(\mathit{M}\mathit{o}{\mathit{d}}_{1},\mathit{M}\mathit{o}{\mathit{d}}_{2}\right)}$ | Evidence in favor of $\mathit{M}\mathit{o}{\mathit{d}}_{1}$ |
---|---|

<0 | $Mo{d}_{2}$ is favored |

0 to 0.5 | Not worth more than a bare mention |

0.5 to 1 | Substantial |

1 to 2 | Strong |

>2 | Decisive |

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**MDPI and ACS Style**

Zou, B.; Zhai, J.; Qi, Z.; Li, Z. A Comparison of Three Sediment Acoustic Models Using Bayesian Inversion and Model Selection Techniques. *Remote Sens.* **2019**, *11*, 562.
https://doi.org/10.3390/rs11050562

**AMA Style**

Zou B, Zhai J, Qi Z, Li Z. A Comparison of Three Sediment Acoustic Models Using Bayesian Inversion and Model Selection Techniques. *Remote Sensing*. 2019; 11(5):562.
https://doi.org/10.3390/rs11050562

**Chicago/Turabian Style**

Zou, Bo, Jingsheng Zhai, Zhanfeng Qi, and Zhaoxing Li. 2019. "A Comparison of Three Sediment Acoustic Models Using Bayesian Inversion and Model Selection Techniques" *Remote Sensing* 11, no. 5: 562.
https://doi.org/10.3390/rs11050562