# A Cyclic Multi-Stage Implementation of the Full-Waveform Inversion for the Identification of Anomalies in Dams

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## Abstract

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## 1. Introduction

- Materials and methods: the formulation/algorithm of the forward model and its boundary conditions, the inverse model, and the inversion process.
- Numerical simulation: the geometry of the dam, the acquisition geometry, the wave parameters, and the cyclic multi-stage implementation under different conditions.
- Results: FWI results for different acquisition geometries, cyclic multi-stage inversion, and for different levels of noise.
- Conclusion: a summary of the applicability of our approach to identifying damages in dams.

## 2. Materials and Methods

#### 2.1. Forward Model

#### 2.2. Inverse Analysis

## 3. Numerical Simulation Example

## 4. Results

#### 4.1. Optimal Acquisition Geometry Selection

#### 4.2. Influence of Noise (Disturbance in Obtained Data) on the Reconstruction Quality/Error

#### 4.3. Cyclic Multi-Frequency Stage Inversion

#### 4.3.1. Influence of Noise (Disturbance in Obtained Data) on the Reconstruction Quality for Cyclic Multi-Frequency Stage Inversion

#### 4.3.2. Influence of Noise (Disturbance in Obtained Data) and Uncertainty in the Starting Model on the Reconstruction Quality for Cyclic-Multi-Frequency Stage Inversion

## 5. Conclusions

- The proposed FWI formulation is capable of effectively identifying and quantifying regions of weaknesses (i.e., heterogeneity) in both the dam structure and its foundation.
- The dam’s as-built material properties are used as a starting model for the inversion. This information is, in most cases, readily available, or can be easily estimated. If the material properties are completely unknown, it is recommended to underestimate and not to overestimate them.
- The efficiency of this method is influenced by the data acquisition geometry. Thus, we propose an acquisition setup which encloses regions in which critical damage is expected.
- The damaged regions are generally of lower velocities and smaller scale; thus, the ${V}_{\mathrm{s}}$ model, with its shorter wavelength, resolves the anomalies better than the ${V}_{\mathrm{p}}$ and $\rho $ models.
- A superimposition of the ${V}_{\mathrm{p}}$ and ${V}_{\mathrm{s}}$ models to identify damaged regions saturated with water increases the robustness of the method by leveraging the advantages of both models.
- The identified damages in the dam body had a better quality than the ones in the dam foundation for data corrupted with high noise levels. Thus, we propose, where possible, an acquisition setup which favors the recording of transmitted waves.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Scheme of a cyclic multi-stage full-waveform inversion with three full cycles (★ 1st cycle, • 2nd cycle, ■ 3rd cycle).

**Figure 3.**Distribution of seismic velocities and density in the air, water, foundation and dam structure: (

**A**) ${V}_{\mathrm{p}}$ distribution, (

**B**) ${V}_{\mathrm{s}}$ distribution (note shear waves only travel through solids) and (

**C**) $\rho $ distribution.

**Figure 4.**Acquisition setup showing distribution of sources (★) and sensors/receivers (•) in the domain. Sensors are placed at the bottom of the reservoir to pick up reflections resulting from anomalies in the foundation. (

**a**) Setup 1 (

**b**) Setup 2 (

**c**) Setup 3.

**Figure 5.**Source wavelet signal: wavelet in time domain (

**left**) and direct Fourier transform of the wavelet showing the 1000 Hz upper corner frequency (

**right**).

**Figure 6.**Sensor response due to propagated waves in medium. (

**Left**): Seismogram of all 165 sensor response (i.e., trace) in x-direction as a result of wave propagation from source 14. Trace 100–165 corresponds to sensors on the reservoir bottom; thus, it picks up the waves first. (

**Right**): Response of trace 53 located on the dam slope.

**Figure 7.**FWI model for acquisition setups with (

**A**) ${V}_{\mathrm{p}}$ distribution (

**B**) ${V}_{\mathrm{s}}$ distribution (note that shear waves only travel through solids) and (

**C**) $\rho $ distribution.

**Figure 8.**FWI residuals for acquisition setups as comparison with the true model for (

**A**) ${V}_{\mathrm{p}}$, (

**B**) ${V}_{\mathrm{s}}$ and (

**C**) $\rho $, respectively.

**Figure 9.**Cost function value evolution for each iteration: comparison of cost function (normalized) over 40 iterations for each acquisition setup.

**Figure 10.**Profile cut through the dam regions of reconstructed anomalies for acquisition setups 1 to 3 showing a comparison between the true model (—), FWI models (dashed lines: - - setup 1, - - setup 2, - - setup 3) and the initial model (⋯).

**Figure 11.**Cost function comparison for different noise levels: (

**a**) Cost function trend showing shorter trend for higher noise levels. (

**b**) Influence of noise on final cost function at the end of the simulation for each noise level, and also at end of the shortest simulation (13 iterations).

**Figure 13.**Profile cut for ${V}_{\mathrm{s}}$ through the dam in regions of reconstructed anomalies for different noise levels showing a comparison between the true model (—), FWI model (- -) and the initial model (⋯). (

**a**) 0% noise (

**b**) 1% noise (

**c**) 2% noise (

**d**) 5% noise (

**e**) 10% noise.

**Figure 14.**Reconstruction error considering 0%, 1%, 2%, 5% and 10% noise in the data. (

**a**) ${V}_{\mathrm{p}}$ reconstruction error (

**b**) ${V}_{\mathrm{s}}$ reconstruction error.

**Figure 15.**FWI results at iteration stages 6 and 28 considering a noise level of 2%. (

**b**) shows better anomaly identification despite having a higher reconstruction error. (

**a**) Reconstruction at 6th iteration (

**b**) Reconstruction at 28th iteration (final).

**Figure 16.**FWI results for cyclic multi-frequency inversion anomaly identification. (

**a**) 1% noise; (

**b**) 5% noise (

**c**); 10% noise.

**Figure 17.**Reconstruction error for a cyclic multi-frequency stage inversion considering different noise levels in the data. (

**a**) ${V}_{\mathrm{p}}$ reconstruction error (

**b**) ${V}_{\mathrm{s}}$ reconstruction error.

**Figure 18.**Cyclic multi-frequency inversion anomaly identification considering certain difference in dam as-built material properties in the starting model having additionally 2% noise in the data. (

**a**) −10% difference in starting model (

**b**) +2% difference in starting model (

**c**) +10% difference in starting model.

Material | ${\mathit{V}}_{\mathbf{p}}$ [m/s] | ${\mathit{V}}_{\mathbf{s}}$ [m/s] | $\mathit{\rho}$ [kg/m${}^{3}$] |
---|---|---|---|

Dam body | 3500 | 2200 | 2000 |

Dam tunnel | 0 | 0 | 1.25 |

Dam foundation | 4500 | 2700 | 2550 |

Water | 1500 | 0 | 1000 |

Air/Vacuum | 0 | 0 | 1.25 |

**Table 2.**FWI computational time after 13 iterations and number of iterations per frequency stage number both for consideration of noise in data.

Noise Level [%] | Computation Time [h] | Number of Iterations of | |||
---|---|---|---|---|---|

after 13 Iterations | Stage 1 | Stage 2 | Stage 3 | Stage 4 | |

0 | 0.68 | 14 | 6 | 6 | 6 |

1 | 0.85 | 7 | 15 | 6 | 5 |

2 | 0.83 | 7 | 15 | 6 | 6 |

5 | 0.75 | 5 | 7 | 5 | 5 |

10 | 0.78 | 4 | 3 | 3 | 3 |

Cycle 1 | Cycle 2 | Cycle 3 | |||
---|---|---|---|---|---|

Frequency [Hz] | ${\mathit{t}}_{\mathbf{lim}}$ [s] | Frequency [Hz] | ${\mathit{t}}_{\mathbf{lim}}$ [s] | Frequency [Hz] | ${\mathit{t}}_{\mathbf{lim}}$ [s] |

0.4 | 0.03 | 0.4 | 0.06 | 0.6 | 0.1 |

0.6 | 0.03 | 0.6 | 0.06 | 0.8 | 0.1 |

0.4 | 0.03 | 0.8 | 0.06 | 1.0 | 0.1 |

0.6 | 0.06 | 0.8 | 0.1 | ||

0.4 | 0.06 | 0.6 | 0.1 | ||

0.8 | 0.1 | ||||

1.0 | 0.1 |

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**MDPI and ACS Style**

Alalade, M.; Reichert, I.; Köhn, D.; Wuttke, F.; Lahmer, T.
A Cyclic Multi-Stage Implementation of the Full-Waveform Inversion for the Identification of Anomalies in Dams. *Infrastructures* **2022**, *7*, 161.
https://doi.org/10.3390/infrastructures7120161

**AMA Style**

Alalade M, Reichert I, Köhn D, Wuttke F, Lahmer T.
A Cyclic Multi-Stage Implementation of the Full-Waveform Inversion for the Identification of Anomalies in Dams. *Infrastructures*. 2022; 7(12):161.
https://doi.org/10.3390/infrastructures7120161

**Chicago/Turabian Style**

Alalade, Muyiwa, Ina Reichert, Daniel Köhn, Frank Wuttke, and Tom Lahmer.
2022. "A Cyclic Multi-Stage Implementation of the Full-Waveform Inversion for the Identification of Anomalies in Dams" *Infrastructures* 7, no. 12: 161.
https://doi.org/10.3390/infrastructures7120161