# A Denoising Method for Seismic Data Based on SVD and Deep Learning

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Singular Value Decomposition (SVD)

#### 2.2. Denoising Method Based on SVD and Deep Learning

#### 2.2.1. Selection of Deep Learning Networks

#### 2.2.2. Deeply Separable Convolution in MobileNetV2 Networks

#### 2.2.3. MobileNetV2 Network Training

#### 2.2.4. Algorithm Program Steps

Algorithm 1 Pseudocode of Seismic Denoising Algorithm |

Input:${D}_{m,n}$: seismic data have m traces, n samples, window parameters were set to a traces, b samples, (a is an odd number and b is an even number), rank of matrix K = 4; |

Output: ${W}_{m,n}:$ denoised seismic data; |

1: ${W}_{m,n}$ = ${D}_{m,n}$; |

2: for ($a/2$) + 1 < i $\le $ m − a/2 + 1, i + a do |

3: for (b-1)/2 + 1 < j $\le $ n − (b − 1)/2, j + 1 do |

4: Set window = a * b (a < m, b < n), obtain ${d}_{a,b}=D\left[i-\frac{a}{2}:i+\frac{a}{2}-1,j-\frac{b-1}{2}:j+\frac{b-1}{2}\right]$; |

5: Compute SVD function USV = svd(d), obtain ${U}_{a,a}$: left singular matrix,${S}_{a,b}$: singular value matrix,${V}_{b,b}$: right singular matrix; |

6: for 0 < k $\le $ K do |

7: Transform the kth RSV into 224*224 pictures; |

8: Using a trained MobileNetV2 network to predict the kth RSV; |

9: if V[:,k] = “effective” then |

10: S[k,k] = S[k,k]; |

11: A five-point smoothing is applied to the kth RSV; |

12: else V[:,k] = noise” |

13: S[k,k] = 0; |

14: end if |

15: end for |

16: ${w}_{a,b}$=$\sum _{k=1}^{K}{U}_{k}{S}_{k}{V}_{K}^{T}$; |

17: Extract the reconstruction matrix w[$:,\frac{b-1}{2}+1$] and insert it into the M[$i-\frac{a}{2}:i+\frac{a}{2}-1,j$] matrix data; |

18: end for |

19: end for |

20: Return ${W}_{m,n}$. |

## 3. Results

#### 3.1. Synthetic Data

#### 3.2. Field Data

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Lin, H.; Wang, S.; Li, Y. A Branch Construction-Based CNN Denoiser for Desert Seismic Data. IEEE Geosci. Remote Sens. Lett.
**2021**, 18, 736–740. [Google Scholar] [CrossRef] - Liu, D.; Wang, W.; Chen, W.; Wang, X.; Zhou, Y.; Shi, Z. Random-Noise Suppression in Seismic Data: What Can Deep Learning Do? In SEG Technical Program Expanded Abstracts 2018; Society of Exploration Geophysicists: Houston, TX, USA, 2018; pp. 2016–2020. [Google Scholar]
- Liu, W.; Duan, Z. Seismic Signal Denoising Using f-x Variational Mode Decomposition. IEEE Geosci. Remote Sens. Lett.
**2020**, 17, 1313–1317. [Google Scholar] [CrossRef] - Liu, G.; Chen, X.; Du, J.; Wu, K. Random noise attenuation using f-x regularized nonstationary autoregression. Geophysics
**2012**, 77, V61–V69. [Google Scholar] [CrossRef] - Gao, Z.; Zhang, S.; Cai, J.; Hong, L.; Zheng, J. Research on Deep Convolutional Neural Network Time-Frequency Domain Seismic Signal Denoising Combined with Residual Dense Blocks. Front. Earth Sci.
**2021**, 9, 571. [Google Scholar] [CrossRef] - Liu, Y.; Li, B. Streaming orthogonal prediction filter in the t-x domain for random noise attenuation. Geophysics
**2018**, 83, F41–F48. [Google Scholar] [CrossRef] - Xuehua, C.; Zhenhua, H.E. Improved S-Transform and Its Application in Seismic Signal Processing. J. Data Acquis. Process.
**2005**, 20, 449–453. [Google Scholar] - Xu, M.J.; Gao, J.Y.; Hu, H.; Zhou, M.J. Wavelet transform based on GCV criterion and CUDA technology for ground roll attenuation. Prog. Geophys.
**2018**, 33, 760–768. [Google Scholar] - Huo, S.; Luo, Y.; Kelamis, P.G. Simultaneous sources separation via multidirectional vector-median filtering. Geophysics
**2012**, 77, V123–V131. [Google Scholar] [CrossRef] - Yang, Y.; Lu, J.; Wang, Y. Vertical Seismic Profile Wavefield Separation Using Median Filtering Constrained by the Linear Radon Transform. Appl. Sci.
**2018**, 8, 1494. [Google Scholar] [CrossRef] [Green Version] - Li, B.; Huang, H.; Wang, T.; Wang, M.; Wang, P. Research on Seismic Signal Classification and Recognition Based on EEMD and CNN. In Proceedings of the 2020 IEEE 3rd International Conference on Electronics and Communication Engineering (ICECE), Xi’An, China, 14–16 December 2020; pp. 83–88. [Google Scholar]
- Wang, C.; Wang, Y. Robust singular value decomposition filtering for low signal-to-noise ratio seismic data. Geophysics
**2021**, 86, V233–V244. [Google Scholar] [CrossRef] - Bekara, M.; van der Baan, M. Local singular value decomposition for signal enhancement of seismic data. Geophysics
**2007**, 72, V59–V65. [Google Scholar] [CrossRef] - Chen, K.; Sacchi, M.D. Robust reduced-rank filtering for erratic seismic noise attenuation. Geophysics
**2015**, 80, V1–V11. [Google Scholar] [CrossRef] - Brankovic, M.; Gildin, E.; Gibson, R.L.; Everett, M.E. A Machine Learning-Based Seismic Data Compression and Interpretation Using a Novel Shifted-Matrix Decomposition Algorithm. Appl. Sci.
**2021**, 11, 4874. [Google Scholar] [CrossRef] - Feng, Q.; Li, Y. Denoising Deep Learning Network Based on Singular Spectrum Analysis—DAS Seismic Data Denoising with Multichannel SVDDCNN. IEEE Trans. Geosci. Remote Sens.
**2022**, 60, 1–11. [Google Scholar] [CrossRef] - Xi, C.; Mi, B.; Dai, T.; Liu, Y.; Ning, L. Spurious signals attenuation using SVD-based Wiener filter for near-surface ambient noise surface wave imaging. J. Appl. Geophys.
**2020**, 183, 104220. [Google Scholar] [CrossRef] - Zhang, Y.; Lin, H.; Li, Y.; Ma, H. A Patch Based Denoising Method Using Deep Convolutional Neural Network for Seismic Image. IEEE Access
**2019**, 7, 156883–156894. [Google Scholar] [CrossRef] - Wu, X.; Liang, L.; Shi, Y.; Fomel, S. FaultSeg3D: Using synthetic data sets to train an end-to-end convolutional neural network for 3D seismic fault segmentation. Geophysics
**2019**, 84, IM35–IM45. [Google Scholar] [CrossRef] - Jiang, J.; Ren, H.; Zhang, M. A Convolutional Autoencoder Method for Simultaneous Seismic Data Reconstruction and Denoising. IEEE Geosci. Remote Sens. Lett.
**2022**, 19, 1–5. [Google Scholar] [CrossRef] - Shi, Y.; Wu, X.; Fomel, S. SaltSeg: Automatic 3D salt segmentation using a deep convolutional neural network. Interpretation
**2019**, 7, SE113–SE122. [Google Scholar] [CrossRef] - Yin, X.; Liu, F.; Cai, R.; Yang, X.; Zhang, X.; Ning, M.; Shen, S. Research on Seismic Signal Analysis Based on Machine Learning. Appl. Sci.
**2022**, 12, 8389. [Google Scholar] [CrossRef] - Sandler, M.; Howard, A.; Zhu, M.; Zhmoginov, A.; Chen, L.C. MobileNetV2: Inverted residuals and linear bottlenecks. In Proceedings of the Computer Vision and Pattern Recognition, Salt Lake City, UT, USA, 18–23 June 2018. [Google Scholar]
- Krizhevsky, A.; Sutskever, I.; Hinton, G.E. ImageNet classification with deep convolutional neural networks. Commun. ACM
**2017**, 60, 84–90. [Google Scholar] [CrossRef] [Green Version] - Simonyan, K.; Zisserman, A. Very Deep Convolutional Networks for Large-Scale Image Recognition. arXiv
**2014**, arXiv:1409.1556. [Google Scholar] - He, K.; Zhang, X.; Ren, S.; Sun, J. Deep residual learning for image recognition. In Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Las Vegas, NV, USA, 27–30 June 2016; pp. 770–778. [Google Scholar]
- Huang, G.; Liu, Z.; van der Maaten, L.; Weinberger, K.Q. Densely connected convolutional networks. In Proceedings of the 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Honolulu, HI, USA, 21–26 July 2017; pp. 2261–2269. [Google Scholar]
- Howard, A.G.; Zhu, M.; Chen, B.; Kalenichenko, D.; Wang, W.; Weyand, T.; Andreetto, M.; Adam, H. MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications. arXiv
**2017**, arXiv:1704.04861. [Google Scholar]

**Figure 1.**(

**a**) Synthetic data; (

**b**) the left singular vector diagram corresponding to the effective signal after SVD; (

**c**) the right singular vector diagram corresponding to the effective signal.

**Figure 4.**The general construction of the bottleneck layer of the MobileNetV2 network, referenced from [23].

**Figure 6.**The trained MobilenetV2 network can predict the six right singular vectors as (

**a**–

**f**), respectively.

**Figure 8.**Filtered results were achieved using conventional SVD filtering with (

**a**) K = 1 and (

**b**) K = 2; (

**c**) improved SVD filtering.

**Figure 9.**(

**a**) Original field seismic data; filtered results achieved using conventional SVD filtering with (

**b**) K = 1 and (

**c**) K = 2; (

**d**) improved SVD filtering.

**Figure 10.**(

**a**) Original field seismic data; filtered results achieved using conventional SVD filtering with (

**b**) K = 1 and (

**c**) K = 2; (

**d**) improved SVD filtering.

Model | Batch Size | Run Time (h) | ImageNet Accuracy (%) |
---|---|---|---|

restnet50 | 8 | 8.77 | 99.91 |

MobileNetV2 | 16 | 3.25 | 99.94 |

Input | Network Layer | Expansion Multiplier | Output Channels | Number Repetitions | Stride |
---|---|---|---|---|---|

${224}^{2}\times 3$ | conv2d | - | 32 | 1 | 2 |

${112}^{2}\times 32$ | bottleneck | 6 | 16 | 1 | 1 |

${112}^{2}\times 16$ | bottleneck | 6 | 24 | 2 | 2 |

${56}^{2}\times 24$ | bottleneck | 6 | 32 | 3 | 2 |

${28}^{2}\times 32$ | bottleneck | 6 | 64 | 4 | 2 |

${14}^{2}\times 64$ | bottleneck | 6 | 96 | 3 | 1 |

${14}^{2}\times 96$ | bottleneck | 6 | 160 | 3 | 2 |

${7}^{2}\times 160$ | bottleneck | 6 | 320 | 1 | 1 |

${7}^{2}\times 320$ | conv2d 1 $\times $ 1 | - | 1280 | 1 | 1 |

${7}^{2}\times 1280$ | avgpool 7 $\times $ 7 | - | - | 1 | - |

$1\times 1\times 1280$ | conv2d 1 $\times $ 1 | - | k | - |

Method | Seismic Data with Noise | Conventional SVD,K = 1 | Conventional SVD,K = 2 | Improved SVD |
---|---|---|---|---|

SNR (dB) | −10.2592 | −6.8822 | −6.5176 | 8.1007 |

Method | Original Data | Conventional SVD, K = 1 | Conventional SVD, K = 2 | Improved SVD |
---|---|---|---|---|

SNR (dB) | 11.3336 | 18.5297 | 16.8173 | 65.7588 |

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**MDPI and ACS Style**

Ji, G.; Wang, C.
A Denoising Method for Seismic Data Based on SVD and Deep Learning. *Appl. Sci.* **2022**, *12*, 12840.
https://doi.org/10.3390/app122412840

**AMA Style**

Ji G, Wang C.
A Denoising Method for Seismic Data Based on SVD and Deep Learning. *Applied Sciences*. 2022; 12(24):12840.
https://doi.org/10.3390/app122412840

**Chicago/Turabian Style**

Ji, Guoli, and Chao Wang.
2022. "A Denoising Method for Seismic Data Based on SVD and Deep Learning" *Applied Sciences* 12, no. 24: 12840.
https://doi.org/10.3390/app122412840