Recovery of seismic wavefields by an lq-norm constrained regularization method

Fengmin Xu; Yanfei Wang

doi:10.3934/ipi.2018048

Article Contents

2018, Volume 12, Issue 5: 1157-1172. Doi: 10.3934/ipi.2018048

This issue Previous Article Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations Next Article On finding a buried obstacle in a layered medium via the time domain enclosure method

Recovery of seismic wavefields by an l_q-norm constrained regularization method

Fengmin Xu^1, and
Yanfei Wang^2,3,4, ,

1.
School of Economics and Finance, Xi'an Jiaotong University, Xi'an 710049, China
2.
Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
3.
University of Chinese Academy of Sciences, Beijing 100049, China
4.
Institutions of Earth Science, Chinese Academy of Sciences, Beijing 100029, China

* Corresponding author: Yanfei Wang
* Corresponding author: Yanfei Wang

Received: February 28, 2017

Revised: March 31, 2018

Published: September 2018

This work is supported by National Natural Science Foundation of China under grant numbers 91630202, Strategic Priority Research Program of the Chinese Academy of Science (Grant No.XDB10020100) and 11571271.

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Abstract

Reconstruction of the seismic wavefield from sub-sampled data is an important problem in seismic image processing, this is partly due to limitations of the observations which usually yield incomplete data. In essence, this is an ill-posed inverse problem. To solve the ill-posed problem, different kinds of regularization technique can be applied. In this paper, we consider a novel regularization model, called the $l_2$-$l_{q}$ minimization model, to recover the original geophysical data from the sub-sampled data. Based on the lower bound of the local minimizers of the $l_2$-$l_{q}$ minimization model, a fast convergent iterative algorithm is developed to solve the minimization problem. Numerical results on random signals, synthetic and field seismic data demonstrate that the proposed approach is very robust in solving the ill-posed restoration problem and can greatly improve the quality of wavefield recovery.

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Mathematics Subject Classification: Primary: 90C90, 86A22, 86-08; Secondary: 65J20.

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Figure 1. $\phi(\alpha, q)$ for $\alpha, \, \, q\in(0, 1)$

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Figure 2. (a) The input signal and the restoration; (b) difference between the true and restored signals; (c) the input signal and the restoration; (d) difference between the true and restored signals

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Figure 3. (a) The input signal and the restoration for a small regularization parameter $\alpha = 1.0\times 10^{-4}$; (b) difference between the true and restored signals for a small regularization parameter $\alpha = 1.0\times 10^{-4}$; (c) the input signal and the restoration for a large regularization parameter $\alpha = 0.5$; (d) difference between the true and restored signals for a large regularization parameter $\alpha = 0.5$

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Figure 4. Seismogram

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Figure 5. (a) Incomplete data; (b) recovery results; (c) frequency of the sub-sampled data; (d) frequency of the restored data

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Figure 6. (a) Difference between the restored data and the original data; (b) recovery results using the Fourier transform based method; (c) frequency of the restored data using the Fourier transform based method; (d) difference between the restored data using the Fourier transform based method and the original data

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Figure 7. (a) The field data; (b) seismic data with missing traces; (c) restored seismic data; (d) frequency of the restored seismic data

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Figure 8. (a) Restored seismic data using the Fourier transform based method; (b) frequency of the restored seismic data using the Fourier transform based method

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Figure 9. (a) The field data; (b) seismic data with missing traces; (c) restored seismic data; (d) frequency of the restored seismic data

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Figure 10. (a) Restored seismic data using the Fourier transform based method; (b) frequency of the restored seismic data using the Fourier transform based method

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