# Accurate Sparse Recovery of Rayleigh Wave Characteristics Using Fast Analysis of Wave Speed (FAWS) Algorithm for Soft Soil Layers

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{1}-norm minimization algorithm of CS theory. The traditional frequency-wavenumber transform technique and in-site downhole observation are employed as the comparison of the proposed technique. The experimental results indicate the proposed FAWS algorithm has a good agreement with both the results of conventional even-spaced geophone array and the in-site measurements, which provides an effective and efficient way for accurate non-destructive evaluation of the surface wave dispersion curve of the soil.

## 1. Introduction

## 2. Materials and Methods

_{1}-norm minimization algorithm, which is then transformed to the dispersion curve of surface wave velocity.

#### 2.1. Compressive Sensing Framework

#### 2.2. Wave Field Imaging

^{−1}, that is, $\omega =2\pi f$, ${\omega}_{p}$ is the central angular frequency, and ${Q}_{p}$ is the the quantity Q-factor of the filter; higher ${Q}_{p}$ produces narrower bandwidth.

## 3. Results and Discussion

#### 3.1. Conventional Even-Spaced Array Results

#### 3.2. Results of the Proposed Fast Analysis of Wave Speed (FAWS) Method

_{1}-norm minimization algorithm of CS theory (Equation (10)). The reconstruction results are shown in Figure 7. The two modes can be obviously separated in the wavenumber-frequency spectrum. The reconstructed results are demonstrated in an image whose amplitudes is mapped by a white-red colortable: the higher magnitude, the darker red color. It can be seen the wavenumber-frequency spectrum have two peaks: one peak is observed at 50 Hz with the wavenumber of 1 ${\mathrm{m}}^{-1}$, the other is in the range of 50 Hz to 60 Hz with the wavenumber around 2 ${\mathrm{m}}^{-1}$, which are in good agreement with the conventional 2D-FFT algorithm.

#### 3.3. In-Site Downhole Testing Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Feng, L.; Yi, X.; Zhu, D.; Xie, X.; Wang, Y. Damage detection of metro tunnel structure through transmissibility function and cross correlation analysis using local excitation and measurement. Mech. Syst. Signal Process.
**2015**, 60, 59–74. [Google Scholar] [CrossRef] - Shen, S.-L.; Wu, H.-N.; Cui, Y.-J.; Yin, Z.-Y. Long-term settlement behaviour of metro tunnels in the soft deposits of Shanghai. Tunn. Undergr. Space Technol.
**2014**, 40, 309–323. [Google Scholar] [CrossRef] - Goel, R.K.; Singh, B.; Zhao, J. Underground Infrastructures: Planning, Design, and Construction; Butterworth-Heinemann: Oxford, UK, 2012. [Google Scholar]
- Fang, Q.; Zhang, D.; Wong, L.N.Y. Shallow tunnelling method (STM) for subway station construction in soft ground. Tunn. Undergr. Space Technol.
**2012**, 29, 10–30. [Google Scholar] [CrossRef] - Ding, L.; Zhou, C. Development of web-based system for safety risk early warning in urban metro construction. Autom. Constr.
**2013**, 34, 45–55. [Google Scholar] [CrossRef] - Wang, M. An overview of development of railways, tunnels and underground works in China. Tunn. Constr.
**2010**, 4, 351–364. [Google Scholar] - Foti, S.; Parolai, S.; Albarello, D.; Picozzi, M. Application of surface-wave methods for seismic site characterization. Surv. Geophys.
**2011**, 32, 777–825. [Google Scholar] [CrossRef] - Foti, S.; Lai, C.G.; Rix, G.J.; Strobbia, C. Surface Wave Methods for Near-Surface Site Characterization; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Jones, R. Surface wave technique for measuring the elastic properties and thickness of roads: Theoretical development. Br. J. Appl. Phys.
**1962**, 13, 21. [Google Scholar] [CrossRef] - Jongmans, D.; Demanet, D. The importance of surface waves in vibration study and the use of Rayleigh waves for estimating the dynamic characteristics of soils. Eng. Geol.
**1993**, 34, 105–113. [Google Scholar] [CrossRef] - Nazarian, S. In situ shear wave velocity from spectral analysis of surface waves. In Proceedings of the 8th World Conference on Earthquake Engineering, San Francisco, CA, USA, 21–28 July 1984; pp. 31–38. [Google Scholar]
- Maraschini, M.; Ernst, F.; Foti, S.; Socco, L.V. A new misfit function for multimodal inversion of surface waves. Geophysics
**2010**, 75, G31–G43. [Google Scholar] [CrossRef] - Socco, L.V.; Foti, S.; Boiero, D. Surface-wave analysis for building near-surface velocity Models—Established approaches and new perspectives. Geophysics
**2010**, 75, 83–102. [Google Scholar] [CrossRef] - Harley, J.B.; Moura, J.M. Sparse recovery of the multimodal and dispersive characteristics of Lamb waves. J. Acoust. Soc. Am.
**2013**, 133, 2732–2745. [Google Scholar] [CrossRef] [PubMed] - Di Ianni, T.; De Marchi, L.; Perelli, A.; Marzani, A. Compressive sensing of full wave field data for structural health monitoring applications. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2015**, 62, 1373–1383. [Google Scholar] [CrossRef] [PubMed] - Harley, J.B. Predictive guided wave models through sparse modal representations. Proc. IEEE
**2016**, 104, 1604–1619. [Google Scholar] [CrossRef] - Xu, K.; Minonzio, J.-G.; Ta, D.; Hu, B.; Wang, W.; Laugier, P. Sparse SVD method for high-resolution extraction of the dispersion curves of ultrasonic guided waves. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2016**, 63, 1514–1524. [Google Scholar] [CrossRef] [PubMed] - Drémeau, A.; Courtois, F.; Bonnel, J. Reconstruction of dispersion curves in the frequency-wavenumber domain using compressed sensing on a random array. IEEE J. Ocean. Eng.
**2017**, 42, 914–922. [Google Scholar] [CrossRef] - Esfandabadi, Y.K.; De Marchi, L.; Testoni, N.; Marzani, A.; Masetti, G. Full wavefield analysis and damage imaging through compressive sensing in Lamb wave inspections. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2018**, 65, 269–280. [Google Scholar] [CrossRef] [PubMed] - Wang, W.; Bao, Y.; Zhou, W.; Li, H. Sparse representation for Lamb-wave-based damage detection using a dictionary algorithm. Ultrasonics
**2018**, 87, 48–58. [Google Scholar] [CrossRef] [PubMed] - Jiang, B.; Jia, P.; Zhao, W.; Wang, W. The application of compressive sampling in rapid ultrasonic computerized tomography (UCT) technique of steel tube slab (STS). PLoS ONE
**2018**, 13, e0190281. [Google Scholar] [CrossRef] [PubMed] - Jiang, B.; Zhao, W.; Wang, W. Improved ultrasonic computerized tomography method for STS (Steel Tube Slab) structure based on compressive sampling algorithm. Appl. Sci.
**2017**, 7, 432. [Google Scholar] [CrossRef] - Donoho, D.L. Compressed Sensing. IEEE Trans. Inf. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Candès, E.J.; Wakin, M.B. An introduction to compressive sampling. IEEE Signal Process. Mag.
**2008**, 25, 21–30. [Google Scholar] [CrossRef] - Candès, E.J.; Romberg, J.; Tao, T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory
**2006**, 52, 489–509. [Google Scholar] [CrossRef] - Candes, E.J.; Romberg, J.K.; Tao, T. Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math.
**2006**, 59, 1207–1223. [Google Scholar] [CrossRef] [Green Version] - Baraniuk, R.G. Compressive sensing. IEEE Signal Process. Mag.
**2007**, 24, 118–121. [Google Scholar] [CrossRef] - Malioutov, D.; Cetin, M.; Willsky, A.S. A sparse signal reconstruction perspective for source localization with sensor arrays. IEEE Trans. Signal Process.
**2005**, 53, 3010–3022. [Google Scholar] [CrossRef] [Green Version] - Mesnil, O.; Ruzzene, M. Sparse wavefield reconstruction and source detection using compressed sensing. Ultrasonics
**2016**, 67, 94–104. [Google Scholar] [CrossRef] [PubMed] - ASTM. Standard Test Methods for Downhole Seismic Testing; ASTM International: West Conshohocken, PA, USA, 2017. [Google Scholar]

**Figure 2.**Magnitude of the proposed bandpass filter bank: (

**a**) second-order bandpass transfer function; and (

**b**) bandpass filter bank.

**Figure 4.**(

**a**) Olympic station in Shenyang metro line 9; (

**b**) schematic of the experimental excitation and geophone array; and (

**c**) in-site measurements setup.

**Figure 5.**Raw waveforms records: (

**a**) waveforms acquired by the even-spaced geophone array; and (

**b**) time-frequency spectrogram of the geophone located at 5 m.

**Figure 6.**Identification results of the soil properties computed by even-spaced geophone array: (

**a**) f-k spectrum obtained by the two-dimensional fast Fourier transform (2D-FFT) method; and (

**b**) dispersion image of the soil properties.

**Figure 7.**Identification results of the soil properties computed by the proposed FAWS approach and random-placed geophone array: (

**a**) distribution of the randomly activated geophones; (

**b**) f-k spectrum; and (

**c**) dispersion image of the soil properties.

Bore Hole Number | Soil Type | Depth (m) | Shear Wave Velocity (m/s) | Compression Wave Velocity (m/s) |
---|---|---|---|---|

ZDBC-01 | Topsoil | 2.9 | 170 | 445 |

Silt and clay | 5.3 | 200 | 529 | |

Sand | 7.8 | 269 | 709 | |

ZDBC-02 | Topsoil | 2.2 | 154 | 424 |

Silt and clay | 4.3 | 195 | 532 | |

Sand | 8.5 | 244 | 634 | |

ZDBC-03 | Topsoil | 3 | 161 | 431 |

Silt and clay | 5.2 | 199 | 523 | |

Sand | 6.8 | 261 | 681 | |

ZDBC-04 | Topsoil | 3.4 | 172 | 439 |

Silt and clay | 6 | 187 | 488 | |

Sand | 5.6 | 257 | 678 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Jiang, B.; Song, J.; Wang, W.
Accurate Sparse Recovery of Rayleigh Wave Characteristics Using Fast Analysis of Wave Speed (FAWS) Algorithm for Soft Soil Layers. *Appl. Sci.* **2018**, *8*, 1204.
https://doi.org/10.3390/app8071204

**AMA Style**

Chen Z, Jiang B, Song J, Wang W.
Accurate Sparse Recovery of Rayleigh Wave Characteristics Using Fast Analysis of Wave Speed (FAWS) Algorithm for Soft Soil Layers. *Applied Sciences*. 2018; 8(7):1204.
https://doi.org/10.3390/app8071204

**Chicago/Turabian Style**

Chen, Zhuoshi, Baofeng Jiang, Jingjing Song, and Wentao Wang.
2018. "Accurate Sparse Recovery of Rayleigh Wave Characteristics Using Fast Analysis of Wave Speed (FAWS) Algorithm for Soft Soil Layers" *Applied Sciences* 8, no. 7: 1204.
https://doi.org/10.3390/app8071204