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October 2018, 12(5): 1157-1172. doi: 10.3934/ipi.2018048

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

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

Received  March 2017 Revised  April 2018 Published  July 2018

Fund Project: 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

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.

Citation: Fengmin Xu, Yanfei Wang. Recovery of seismic wavefields by an lq-norm constrained regularization method. Inverse Problems & Imaging, 2018, 12 (5) : 1157-1172. doi: 10.3934/ipi.2018048
References:
[1]

E. J. CandesJ. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

[2]

E. J. Candes and M. B. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, 25 (2008), 21-30.

[3]

J. J. CaoY. F. WangJ. T. Zhao and C. C. Yang, A review on restoration of seismic wavefields based on regularization and compressive sensing, Inverse Problems in Science and Engineering, 19 (2011), 679-704. doi: 10.1080/17415977.2011.576342.

[4]

R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008), 1-14. doi: 10.1088/0266-5611/24/3/035020.

[5]

X. J. ChenF. M. Xu and Y. Y. Ye, Lower bound theory of nonzero entries in solutions of $l_2$-$l_p$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852. doi: 10.1137/090761471.

[6]

S. ChenD. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM Journal on Scientific Computing, 20 (1998), 33-61. doi: 10.1137/S1064827596304010.

[7]

Y. H. Dai and R. Fletcher, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming, Numerische Mathematik, 100 (2005), 21-47. doi: 10.1007/s00211-004-0569-y.

[8]

D. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[9]

A. J. W. Duijndam and M. A. Schonewille, Non-uniform fast Fourier transform, Geophysics, 64 (1999), 539-551.

[10]

V. B. Ewout and P. F. Michael, Probing the pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008), 890-912. doi: 10.1137/080714488.

[11]

M. A. T. FigueiredoR. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 586-597. doi: 10.1109/JSTSP.2007.910281.

[12]

T. Goldstein and S. Osher, The split Bregman method for $l_1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.

[13]

G. Hennenfent and F. J. Herrmann, Simply denoise: Wavefield reconstruction via jittered undersampling, Geophysics, 73 (2008), V19-V28. doi: 10.1190/1.2841038.

[14]

F. J. Herrmann and G. Hennenfent, Non-parametric seismic data recovery with curvelet frames, Geophysical Journal International, 173 (2008), 233-248. doi: 10.1111/j.1365-246X.2007.03698.x.

[15]

F. J. HerrmannD. L. WangG. Hennenfent and P. P. Moghaddam, Curvelet-based seismic data processing: a multiscale and nonlinear approach, Geophysics, 73 (2008), A1-A6. doi: 10.1190/1.2799517.

[16]

S. J. KimK. KohM. LustigS. Boyd and D. Gorinevsky, An interior-point method for large-scale $l_1$-regularized least squares, IEEE Journal on Selected Topics in Signal Processing, 8 (2007), 1519-1555.

[17]

M. J. LaiY. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_q$ minimization, SIAM Journal on Numerical Analysis, 51 (2013), 927-957. doi: 10.1137/110840364.

[18]

B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records, Geophysics, 69 (2004), 1560-1568. doi: 10.1190/1.1836829.

[19]

S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.

[20]

N. Meinshausen and B. Yu, Lasso-type recovery of sparse representations for high-dimensional data, Annals of Statistics, 37 (2009), 246-270. doi: 10.1214/07-AOS582.

[21]

M. Naghizadeh and M. D. Sacchi, Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data, Geophysics, 75 (2010), WB189-202. doi: 10.1190/1.3509468.

[22]

B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM Journal on Computing, 24 (1995), 227-234. doi: 10.1137/S0097539792240406.

[23]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ., 1970.

[24]

M. D. SacchiT. J. Ulrych and C. J. Walker, Interpolation and extrapolation using a high-resolution discrete Fourier transform, IEEE Transactions on Signal Processing, 46 (1998), 31-38. doi: 10.1109/78.651165.

[25]

M. D. SacchiD. J. Verschuur and P. M. Zwartjes, Data reconstruction by generalized deconvolution, Expanded Abstracts 74th Annual Meeting SEG, Denver, USA (Denver, Oct. 2004), (2004), 1989-1992. doi: 10.1190/1.1843303.

[26]

R. Tibshirani, Regression shrinkage and selection via the lasso, Journal Royal Statistical Society B, 58 (1996), 267-288.

[27]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems, John Wiley and Sons, New York, 1977.

[28]

J. A. Tropp and A. C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Transactions on Information Theory, 53 (2007), 4655-4666. doi: 10.1109/TIT.2007.909108.

[29]

E. van den Berg and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2009), 890-912. doi: 10.1137/080714488.

[30]

Y. F. Wang, Computational Methods for Inverse Problems and Their Applications, Higher Education Press, Beijing, 2007.

[31]

Y. F. Wang and S. Q. Ma, Projected Barzilai-Borwein methods for large scale nonnegative image restorations, Inverse Problems in Science and Engineering, 15 (2007), 559-583. doi: 10.1080/17415970600881897.

[32]

Y. F. WangS. F. Fan and X. Feng, Retrieval of the aerosol particle size distribution function by incorporating a priori information, Journal of Aerosol Science, 38 (2007), 885-901.

[33]

Y. F. WangJ. J. CaoY. X. YuanC. C. Yang and N. H. Xiu, Regularizing active set method for nonnegatively constrained ill-posed multichannel image restoration problem, Applied Optics, 48 (2009), 1389-1401. doi: 10.1364/AO.48.001389.

[34]

Y. F. Wang, Sparse optimization methods for seismic wavefields recovery, Proceedings of the Institute of Mathematics and Mechanics, 18 (2012), 42-55.

[35]

Y. F. WangJ. J. Cao and C. C. Yang, Recovery of seismic wavefields based on compressive sensing by an $l_1$-norm constrained trust region method and the piecewise random sub-sampling, Geophysical Journal International, 187 (2011), 199-213.

[36]

Y. F. Wang, C. C. Yang and J. J. Cao, On Tikhonov regularization and compressive sensing for seismic signal processing Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150008, 24pp. doi: 10.1142/S0218202511500084.

[37]

Y. F. Wang, A. G. Yagola and C. C. Yang (editors), Computational Methods for Applied Inverse Problems, (published in "Series: Inverse and Ill-Posed Problems Series 56"), Walter de Gruyter, 2012. doi: 10.1515/9783110259056.

[38]

Y. X. Yuan, Numerical Methods for Nonliear Programming, Shanghai Science and Technology Publication, Shanghai, 1993.

[39]

H. Zou, The adaptive lasso and its oracle properties, Journal of the American Statistical Association, 101 (2006), 1418-1429. doi: 10.1198/016214506000000735.

[40]

P. M. Zwartjes and M. D. Sacchi, Fourier reconstruction of nonuniformly sampled, aliased seismic data, Geophysics, 72 (2007a), V21-V32. doi: 10.1190/1.2399442.

[41]

P. M. Zwartjes and A. Gisolf, Fourier reconstruction with sparse inversion, Geophysical Prospecting, 55 (2007), 199-221. doi: 10.1111/j.1365-2478.2006.00580.x.

show all references

References:
[1]

E. J. CandesJ. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509. doi: 10.1109/TIT.2005.862083.

[2]

E. J. Candes and M. B. Wakin, An introduction to compressive sampling, IEEE Signal Processing Magazine, 25 (2008), 21-30.

[3]

J. J. CaoY. F. WangJ. T. Zhao and C. C. Yang, A review on restoration of seismic wavefields based on regularization and compressive sensing, Inverse Problems in Science and Engineering, 19 (2011), 679-704. doi: 10.1080/17415977.2011.576342.

[4]

R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008), 1-14. doi: 10.1088/0266-5611/24/3/035020.

[5]

X. J. ChenF. M. Xu and Y. Y. Ye, Lower bound theory of nonzero entries in solutions of $l_2$-$l_p$ minimization, SIAM Journal on Scientific Computing, 32 (2010), 2832-2852. doi: 10.1137/090761471.

[6]

S. ChenD. Donoho and M. Saunders, Atomic decomposition by basis pursuit, SIAM Journal on Scientific Computing, 20 (1998), 33-61. doi: 10.1137/S1064827596304010.

[7]

Y. H. Dai and R. Fletcher, Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming, Numerische Mathematik, 100 (2005), 21-47. doi: 10.1007/s00211-004-0569-y.

[8]

D. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[9]

A. J. W. Duijndam and M. A. Schonewille, Non-uniform fast Fourier transform, Geophysics, 64 (1999), 539-551.

[10]

V. B. Ewout and P. F. Michael, Probing the pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2008), 890-912. doi: 10.1137/080714488.

[11]

M. A. T. FigueiredoR. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems, IEEE Journal of Selected Topics in Signal Processing, 1 (2007), 586-597. doi: 10.1109/JSTSP.2007.910281.

[12]

T. Goldstein and S. Osher, The split Bregman method for $l_1$ regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), 323-343. doi: 10.1137/080725891.

[13]

G. Hennenfent and F. J. Herrmann, Simply denoise: Wavefield reconstruction via jittered undersampling, Geophysics, 73 (2008), V19-V28. doi: 10.1190/1.2841038.

[14]

F. J. Herrmann and G. Hennenfent, Non-parametric seismic data recovery with curvelet frames, Geophysical Journal International, 173 (2008), 233-248. doi: 10.1111/j.1365-246X.2007.03698.x.

[15]

F. J. HerrmannD. L. WangG. Hennenfent and P. P. Moghaddam, Curvelet-based seismic data processing: a multiscale and nonlinear approach, Geophysics, 73 (2008), A1-A6. doi: 10.1190/1.2799517.

[16]

S. J. KimK. KohM. LustigS. Boyd and D. Gorinevsky, An interior-point method for large-scale $l_1$-regularized least squares, IEEE Journal on Selected Topics in Signal Processing, 8 (2007), 1519-1555.

[17]

M. J. LaiY. Xu and W. Yin, Improved iteratively reweighted least squares for unconstrained smoothed $l_q$ minimization, SIAM Journal on Numerical Analysis, 51 (2013), 927-957. doi: 10.1137/110840364.

[18]

B. Liu and M. D. Sacchi, Minimum weighted norm interpolation of seismic records, Geophysics, 69 (2004), 1560-1568. doi: 10.1190/1.1836829.

[19]

S. G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, 1998.

[20]

N. Meinshausen and B. Yu, Lasso-type recovery of sparse representations for high-dimensional data, Annals of Statistics, 37 (2009), 246-270. doi: 10.1214/07-AOS582.

[21]

M. Naghizadeh and M. D. Sacchi, Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data, Geophysics, 75 (2010), WB189-202. doi: 10.1190/1.3509468.

[22]

B. K. Natarajan, Sparse approximate solutions to linear systems, SIAM Journal on Computing, 24 (1995), 227-234. doi: 10.1137/S0097539792240406.

[23]

R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ., 1970.

[24]

M. D. SacchiT. J. Ulrych and C. J. Walker, Interpolation and extrapolation using a high-resolution discrete Fourier transform, IEEE Transactions on Signal Processing, 46 (1998), 31-38. doi: 10.1109/78.651165.

[25]

M. D. SacchiD. J. Verschuur and P. M. Zwartjes, Data reconstruction by generalized deconvolution, Expanded Abstracts 74th Annual Meeting SEG, Denver, USA (Denver, Oct. 2004), (2004), 1989-1992. doi: 10.1190/1.1843303.

[26]

R. Tibshirani, Regression shrinkage and selection via the lasso, Journal Royal Statistical Society B, 58 (1996), 267-288.

[27]

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems, John Wiley and Sons, New York, 1977.

[28]

J. A. Tropp and A. C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Transactions on Information Theory, 53 (2007), 4655-4666. doi: 10.1109/TIT.2007.909108.

[29]

E. van den Berg and M. P. Friedlander, Probing the Pareto frontier for basis pursuit solutions, SIAM Journal on Scientific Computing, 31 (2009), 890-912. doi: 10.1137/080714488.

[30]

Y. F. Wang, Computational Methods for Inverse Problems and Their Applications, Higher Education Press, Beijing, 2007.

[31]

Y. F. Wang and S. Q. Ma, Projected Barzilai-Borwein methods for large scale nonnegative image restorations, Inverse Problems in Science and Engineering, 15 (2007), 559-583. doi: 10.1080/17415970600881897.

[32]

Y. F. WangS. F. Fan and X. Feng, Retrieval of the aerosol particle size distribution function by incorporating a priori information, Journal of Aerosol Science, 38 (2007), 885-901.

[33]

Y. F. WangJ. J. CaoY. X. YuanC. C. Yang and N. H. Xiu, Regularizing active set method for nonnegatively constrained ill-posed multichannel image restoration problem, Applied Optics, 48 (2009), 1389-1401. doi: 10.1364/AO.48.001389.

[34]

Y. F. Wang, Sparse optimization methods for seismic wavefields recovery, Proceedings of the Institute of Mathematics and Mechanics, 18 (2012), 42-55.

[35]

Y. F. WangJ. J. Cao and C. C. Yang, Recovery of seismic wavefields based on compressive sensing by an $l_1$-norm constrained trust region method and the piecewise random sub-sampling, Geophysical Journal International, 187 (2011), 199-213.

[36]

Y. F. Wang, C. C. Yang and J. J. Cao, On Tikhonov regularization and compressive sensing for seismic signal processing Mathematical Models and Methods in Applied Sciences, 22 (2012), 1150008, 24pp. doi: 10.1142/S0218202511500084.

[37]

Y. F. Wang, A. G. Yagola and C. C. Yang (editors), Computational Methods for Applied Inverse Problems, (published in "Series: Inverse and Ill-Posed Problems Series 56"), Walter de Gruyter, 2012. doi: 10.1515/9783110259056.

[38]

Y. X. Yuan, Numerical Methods for Nonliear Programming, Shanghai Science and Technology Publication, Shanghai, 1993.

[39]

H. Zou, The adaptive lasso and its oracle properties, Journal of the American Statistical Association, 101 (2006), 1418-1429. doi: 10.1198/016214506000000735.

[40]

P. M. Zwartjes and M. D. Sacchi, Fourier reconstruction of nonuniformly sampled, aliased seismic data, Geophysics, 72 (2007a), V21-V32. doi: 10.1190/1.2399442.

[41]

P. M. Zwartjes and A. Gisolf, Fourier reconstruction with sparse inversion, Geophysical Prospecting, 55 (2007), 199-221. doi: 10.1111/j.1365-2478.2006.00580.x.

Figure 1.  $\phi(\alpha, q)$ for $\alpha, \, \, q\in(0, 1)$
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
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 $\alpha = 0.5$
Figure 4.  Seismogram
Figure 5.  (a) Incomplete data; (b) recovery results; (c) frequency of the sub-sampled data; (d) frequency of the restored data
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
Figure 7.  (a) The field data; (b) seismic data with missing traces; (c) restored seismic data; (d) frequency of the restored seismic data
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
Figure 9.  (a) The field data; (b) seismic data with missing traces; (c) restored seismic data; (d) frequency of the restored seismic data
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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