# Direct Method for Identification of Two Coefficients of Acoustic Equation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Two-Dimensional Acoustic Inverse Problem

## 3. Obtaining the Density and the Speed of Sound

- 1.
- 2.
- 3.
- 4.
- Recover the density $\rho (x,y)=\frac{\sigma (x,y)}{c\left(x\right)}$.

## 4. One-Dimensional Acoustic Inverse Problem

## 5. Numerical Results

## 6. Discussion

- 1.
- 2.
- Recover ${c}_{1}(z,y)$ by solving the linearized problem

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The 1D case—the result of the reconstruction in travel-time coordinates: (

**a**) the reconstruction of the acoustic impedance; (

**b**) the reconstruction of the speed of sound; (

**c**) the reconstruction of the density of the medium.

**Figure 2.**The 1D case—the result of the reconstruction of the speed of sound in the original coordinates.

**Figure 5.**The 2D smooth case—density reconstruction: (

**a**) 3 sources/receivers; (

**b**) 7 sources/receivers; (

**c**) 11 sources/receivers.

**Figure 6.**The 2D case—density reconstruction: (

**a**) exact solution; (

**b**) computed solution (five sources/receivers); (

**c**) computed solution (nine sources/receivers).

**Figure 7.**The 2D case—density reconstruction (noised data): (

**a**) exact solution; (

**b**) computed solution (15 sources/receivers, noiseless data); (

**c**) computed solution (15 sources/receivers, 5% noise in the data); (

**d**) computed solution (7 sources/receivers, 5% noise in the data).

**Figure 9.**The 2D case—the result of the density reconstruction: (

**a**) exact solution; (

**b**,

**c**) solution obtained by G–L–K method; ((

**b**) 5 sources/receivers; (

**c**) 11 sources/receivers); (

**d**) solution obtained by optimization approach (8 sources/receivers).

Depth, km | 0.17 | 0.47 | 0.87 | 1.070 | 1.320 | 1.600 | 2.100 | 2.200 | 2.300 |
---|---|---|---|---|---|---|---|---|---|

${v}_{s}\left(z\right)$, km/s | 0.9 | 1.7 | 3.1 | 3.5 | 2.7 | 3.2 | 2.85 | 3.4 | 2.8 |

$\rho \left(z\right)$, ${10}^{3}$ kg/${\mathrm{m}}^{3}$ | 2.1 | 2.4 | 2.65 | 2.75 | 2.5 | 2.7 | 2.6 | 2.75 | 2.6 |

Time (s) | G–L–K Method | Optimization | |||
---|---|---|---|---|---|

2 sources | 5 sources | 8 sources | 11 sources | 8 sources | |

${N}_{x}=100$ | 1.37 | 5.22 | 11.87 | 23.016 | 300 |

${N}_{x}=200$ | 5.17 | 20.81 | 52.17 | 89.25 | 1700 |

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**MDPI and ACS Style**

Novikov, N.; Shishlenin, M.
Direct Method for Identification of Two Coefficients of Acoustic Equation. *Mathematics* **2023**, *11*, 3029.
https://doi.org/10.3390/math11133029

**AMA Style**

Novikov N, Shishlenin M.
Direct Method for Identification of Two Coefficients of Acoustic Equation. *Mathematics*. 2023; 11(13):3029.
https://doi.org/10.3390/math11133029

**Chicago/Turabian Style**

Novikov, Nikita, and Maxim Shishlenin.
2023. "Direct Method for Identification of Two Coefficients of Acoustic Equation" *Mathematics* 11, no. 13: 3029.
https://doi.org/10.3390/math11133029