基于Curvelet变换的地震数据插值和去噪
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摘要
地震数据的插值重构是地震数据处理中一个非常关键的步骤,不成功的地震数据插值会导致多次波的错误预测,进而影响到抑制多次波的性能和地震成像的保真度。影响地震数据插值效果的主要因素有两个：(1)地震数据的采样方式(空间域的均匀采样或随机采样),(2)地震数据在变换域(如傅里叶、小波、Curvelet)的可压缩性。对于采集到的不完整的地震观测数据,可以通过抑制噪声和对丢失的地震道进行插值来提高地震数据的信噪比。本文利用地震数据在Curvelet域的可压缩性,提出了两种基于Curvelet变换的地震数据插值和去噪的方法。
论文的第二章简单回顾了Curvelet的发展、Curvelet的性质以及其离散算法的快速实现。
第三章针对含有随机噪声的不完整地震数据进行插值去噪。首先介绍了阈值去噪法,近来的研究表明,Curvelet阈值法能够得到比小波阈值法更好的随机噪声抑制效果,得到更高的信噪比。同时,阈值去噪法还克服了传统滤波法对有效地震信号损失较大这一缺点。据此本章提出了一种基于Curvelet阈值迭代的插值方法来实现对有损地震数据的插值和去噪。实验结果表明,在均匀采样和随机采样的情况下该方法都得到了较好的插值和去噪效果。
第四章把欠采样的地震数据插值问题转换为一个去噪问题,当规则抽取地震数据时其频谱发生混叠,产生相干噪声；在随机抽取地震数据的情况下,其频谱产生类白噪声谱。利用地震数据在Curvelet域系数的稀疏性、Curvelet本身的多方向性以及其时频域的局部性,有效地去除了由均匀抽样产生的相干噪声和随机抽样产生的类高斯噪声,最终实现了对地震数据的插值重构。实验结果表明,本章的插值方法对于有、无随机噪声的情况下都能得到较满意的插值结果,并在实际地震资料处理中得到了进一步的验证。
Seismic data interpolation is a crucial step in the seismic processing flow. For instance, unsuccessful interpolation leads to erroneous multiple predictions that adversely affect the performance of multiple elimination, and to imaging artifacts. Successful recovery of a signal from incomplete measurements depends mainly on two factors:(i) how sampling is done (e.g., uniform or non-uniform in the spatial domain), (ii) how compressible the signal is with respect to some prescribed transform (e.g., Fourier, wavelet, curvelet, etc.). In order to enhance seismic images with the available data by removing noise and filling in missing traces, we present two transform-based reconstruction methods that exploits the compressibility of seismic data in the curvelet domain.
In chapter 2, we first give an overview of the curvelet transform and its discrete implementation, the fast discrete curvelet transform (FDCT) and the property of curvelet transform.
Chapter 3 deals with the reconstruction of spatially-undersampled seismic data with random noise. We start by a brief review of denoising by simply coefficient shrinkage. Recent research has shown that the method of curvelet thresholding can suppress the random noise more effectively and achieve higher signal to noise ratio than the traditional methods and it overcomes the drawback that the conventional filtering approach may affect the effective wave when suppressing noise. In this chapter, we propose a new iterative interpolation idea based on curvelet thresholding to reconstruct seismic data with missing traces. Experiments demonstrate that this method can obtain fine result for both regular and irregular sampled seismic data.
In chapter 4, we turn the interpolation problem of coarsely sampled data into a denoising problem and propose a reconstructive method for incomplete seismic data based on this idea. We conclude by showing some reconstruction examples on synthetic and real data sets. Application of this method to interpolation problems on incomplete seismic data demonstrates that one can obtain excellent reconstructed result and effectively attenuate the random noise just as the case when there are no seismic data missing.
论文的第二章简单回顾了Curvelet的发展、Curvelet的性质以及其离散算法的快速实现。
第三章针对含有随机噪声的不完整地震数据进行插值去噪。首先介绍了阈值去噪法,近来的研究表明,Curvelet阈值法能够得到比小波阈值法更好的随机噪声抑制效果,得到更高的信噪比。同时,阈值去噪法还克服了传统滤波法对有效地震信号损失较大这一缺点。据此本章提出了一种基于Curvelet阈值迭代的插值方法来实现对有损地震数据的插值和去噪。实验结果表明,在均匀采样和随机采样的情况下该方法都得到了较好的插值和去噪效果。
第四章把欠采样的地震数据插值问题转换为一个去噪问题,当规则抽取地震数据时其频谱发生混叠,产生相干噪声；在随机抽取地震数据的情况下,其频谱产生类白噪声谱。利用地震数据在Curvelet域系数的稀疏性、Curvelet本身的多方向性以及其时频域的局部性,有效地去除了由均匀抽样产生的相干噪声和随机抽样产生的类高斯噪声,最终实现了对地震数据的插值重构。实验结果表明,本章的插值方法对于有、无随机噪声的情况下都能得到较满意的插值结果,并在实际地震资料处理中得到了进一步的验证。
Seismic data interpolation is a crucial step in the seismic processing flow. For instance, unsuccessful interpolation leads to erroneous multiple predictions that adversely affect the performance of multiple elimination, and to imaging artifacts. Successful recovery of a signal from incomplete measurements depends mainly on two factors:(i) how sampling is done (e.g., uniform or non-uniform in the spatial domain), (ii) how compressible the signal is with respect to some prescribed transform (e.g., Fourier, wavelet, curvelet, etc.). In order to enhance seismic images with the available data by removing noise and filling in missing traces, we present two transform-based reconstruction methods that exploits the compressibility of seismic data in the curvelet domain.
In chapter 2, we first give an overview of the curvelet transform and its discrete implementation, the fast discrete curvelet transform (FDCT) and the property of curvelet transform.
Chapter 3 deals with the reconstruction of spatially-undersampled seismic data with random noise. We start by a brief review of denoising by simply coefficient shrinkage. Recent research has shown that the method of curvelet thresholding can suppress the random noise more effectively and achieve higher signal to noise ratio than the traditional methods and it overcomes the drawback that the conventional filtering approach may affect the effective wave when suppressing noise. In this chapter, we propose a new iterative interpolation idea based on curvelet thresholding to reconstruct seismic data with missing traces. Experiments demonstrate that this method can obtain fine result for both regular and irregular sampled seismic data.
In chapter 4, we turn the interpolation problem of coarsely sampled data into a denoising problem and propose a reconstructive method for incomplete seismic data based on this idea. We conclude by showing some reconstruction examples on synthetic and real data sets. Application of this method to interpolation problems on incomplete seismic data demonstrates that one can obtain excellent reconstructed result and effectively attenuate the random noise just as the case when there are no seismic data missing.
引文
[1]刘冰.基于Curvelet变换的地震数据去噪方法研究[D].吉林大学硕士学位论文.2008.
[2]G Hennenfent. Sampling and reconstruction of seismic wavefields in the curvelet domain, Ph. D. Thesis, BRITISH COLUMBIA,2008.
[3]张军华,吕宁,田连玉,等.地震资料去噪方法技术综合评价[J].地球物学进展.2006,21(2)：546-553.
[4]S. Mallat. A wavelet tour of signal processing, Academic Press, San Diego, CA,1998.
[5]D.L. Donoho and I.M. Johnstone, "Minimax Estimation via Wavelet Shrinkage," Annals of Statistics,1998,26(3):879-921.
[6]Candes, E. J., Ridgelets:theory and applications, Ph.D. Thesis, Statistics, Stanford 1998.
[7]Candes E J, Donoho D L. Curvelets:A surprisingly effective nonadaptive representation of objects with edges[M]. TN:Vanderbilt University Press,1999.
[8]仝中飞.Curvelet阈值迭代法在地震数据去噪和插值中的应用研究[D].长春：吉林大学,2009.
[9]付燕.人工地震信号去噪方法研究[D].西安：西北工业大学博士学位论文.2002.
[10]Candes E J, Demanet L, Donoho D L, et al. Fast discrete curvelet transforms[R]. Applied and computational mathematics, California Institute of Technology,2005:1-43.
[11]Herrmann F J, Verschuur E. Curvelet domain multiple elimination with sparseness constraints[C]. Society of Exploration Geophysicists, Expanded Abstracts, p1333-1336,2004.
[12]G. Hennenfent and F. J.Herrmann. Seismic denoising with nonuniformly sampled curvelets, IEEE Computing in Science and Engineering, vol.8, pp.16-25,2006.
[13]Herrmann F J, Deli Wang, G Hennenfent, et al. Seismic data processing with curvelets: A multiscale and nonlinear approach[C]. Society of Exploration Geophysicists, Expanded Abstracts, p2220-2224,2007.
[14]R. Neelamani, A. Baumstein, D. Gillard, M. Hadidi, W. Soroka, Coherent and random noise attenuation using the curvelet transform, the leading edge,27 (2),240-248 (2008).
[15]H. Chauris, J. Ma, Seismic imaging in the curvelet domain:achievements and perspectives,71st EAGE conference and Exhibition, Amsterdam, Netherlands,8-11 June 2009.
[16]H. Douma, M. de Hoop, Leading-order seismic imaging using curvelets, Geophysics,
72(6),S231-S248(2007).
[17]H. Chauris, T. Nguyen, Seismic demigration/migration in the curvelet domain, Geophysics,73 (2), S35-S46 (2008).
[18]F. Herrmann, D. Wang, D. Verschuur, Adaptive curvelet-domain primary-multiple separation, Geophysics,73 (3), A17-A21 (2008).
[19]E. Candes, J. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math.,59,1207-1233 (2005).
[20]D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (4),1289-1306(2006).
[21]Richard Baraniuk, A Lecture on Compressive Sensing. To appear in IEEE Signal Processing Magazine, July 2007.
[22]Candes E J, Wakin M B, An Introduction To Compressive Sampling. IEEE Signal Processing Magazine,March 2008
[23]F. J. Herrmann and G. Hennenfent. Non-parametric seismic data recovery with curvelet frames:Geophysical Journal International,2008.
[24]E. Candes, D. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise singularities, Comm. Pure Appl. Math.,57,219-266 (2004).
[25]E. Le Pennec and S.Mallat, Sparse Geometrical Image Approximation with Bandelets, IEEE Transaction on Image Processing,2004.
[26]孙延奎.小波分析及其应用[M].北京：机械工业出版社,2005：31-36，157-158
[27]王建英,尹忠科,张春梅.信号与图像的稀疏分解及其初步应用[M].成都：西南交通大学出版社,2006：49-139.
[28]Mallat S. and Zhang Z. Matching pursuit with time-frequency dictionaries. IEEE Trans. On Signal Processing,1993,41(12):3397-3415.
[29]Bergeau F., and Mallat S. Matching pursuit of images. Proceedings of IEEE-SP, 1994:330-333.
[30]冈萨雷斯等著.数字图像处理[M].阮秋琦等译.第2版.北京：电子工业出版社2005：375-379
[31]E. Candes, D. Donoho, Continuous curvelet transform:Ⅰ. Resolution of the wavefront set, Appl. Comput. Harmon. Anal.,19,162-197 (2003).
[32]E. Candes, D. Donoho, Continuous curvelet transform:Ⅱ. Discretization and frames, Appl. Comput. Harmon. Anal.,19,198-222 (2003).
[33]J. Ma, G Plonka, A review of curvelets and recent applications, IEEE Sig. Proc.Mag, submitted, (2008)
[34]G Hennenfent and F. J. Herrmann Irregular sampling:from aliasing to noise:Presented at the EAGE 69th Conference & Exhibition,2007b.
[35]G Hennenfent and F. J., Random sampling:new insights into the reconstruction of coarsely-sampled wavefields:Presented at the SEG International Exposition and 77th Annual Meeting.2007c.
[36]H. Shan, J. Ma, H. Yang, Comparisons of wavelets, contourlets, and curvelets for seismic denoising, J. Applied Geophysics,2009.
[37]Brian Russell, Dan Hampson, Joong Chun. Noise elimination and the radon transform, Part1[J].The Leading Edge,1990,9(11):18-23.
[38]Hennenfent G, Herrmann F J. Simply denoise:Wavefield reconstruction via jitred undersampling[J].Geophysics,2008,73(3):V19-V28.
[39]S. Tong, L. Wang, Marine Linear Noise Suppression in the Curvelet Domain, [J]IEEE computer society,2009.
[40]L.Shan, J. Fu, Curvelet Transform and its Application in Seismic Data Denoising, [J]IEEE computer society,2009.
[41]G. Y. Chen and T. D. Bui, Multiwavelet Denoising using Neighbouring Coefficients, IEEE Signal Processing Letters, vol.10, no.7, pp.211-214,2003.
[42]L. Wang, H. Liu, Curvelet-based Noise Attenuation in Prestack Seismic Data, [J]IEEE computer society,2008.
[43]Trad, D., T.J. Ulrych and M.D. Sacchi, Latest views of the sparse Radon transform: Geophysics,68,386-399,2003
[44]Zwartjes, P. M., and A. Gisolf, Fourier reconstruction with sparse inversion:Geophys. Prosp.,55, no.2,199-221,2007
[45]Hennenfent, G, F. J. Herrmann, and R. Neelamani,2005, Sparseness-constrained seismic deconvolution with Curvelets:Presented at the CSEG National Convention.
[46]Yarham, C., U. Boeniger, and F. Herrmann,2006, Curvelet-based ground roll removal: Presented at the SEG International Exposition and 76th Annual Meeting.
[47]M.L. Brady. A fast discrete approximation algorithm for the Radon transform. SIAM J. Comput.,27(1):107-119,1998
[48]Oz Yilmaz, Seismic Data Processing (SEG Investigations in Geophysics N 2.) 2000.
[49]Averbuch, A., Coifman, R., Donoho, D. L., Elad, M., and Israeli, M.2006. Fast and accurate polar Fourier transform. Appl. Comput. Harmon. Anal.21.
[50]Donoho, D., Maleki, A., and Shaharam, M.2006. Wavelab 850. http://www-stat.stanford.edu/-wavelab.
[51]Duijndam, A. J. W. and Schonewille, M. A.1999. Nonuniform fast Fourier transform. Geophysics 64,539.
[52]Dutt, A. and Rokhlin, V.1993. Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Stat. Comput.14,1368.
[53]Frigo, M. and Johnson, S. G. 2005b. FFTW, C subroutine library, http://www.fftw.org.
[54]K. Guo, W-Q. Lim, D. Labate, G. Weiss and E. Wilson, The theory of wavelets with composite dilations, Boston (2005).
[2]G Hennenfent. Sampling and reconstruction of seismic wavefields in the curvelet domain, Ph. D. Thesis, BRITISH COLUMBIA,2008.
[3]张军华,吕宁,田连玉,等.地震资料去噪方法技术综合评价[J].地球物学进展.2006,21(2)：546-553.
[4]S. Mallat. A wavelet tour of signal processing, Academic Press, San Diego, CA,1998.
[5]D.L. Donoho and I.M. Johnstone, "Minimax Estimation via Wavelet Shrinkage," Annals of Statistics,1998,26(3):879-921.
[6]Candes, E. J., Ridgelets:theory and applications, Ph.D. Thesis, Statistics, Stanford 1998.
[7]Candes E J, Donoho D L. Curvelets:A surprisingly effective nonadaptive representation of objects with edges[M]. TN:Vanderbilt University Press,1999.
[8]仝中飞.Curvelet阈值迭代法在地震数据去噪和插值中的应用研究[D].长春：吉林大学,2009.
[9]付燕.人工地震信号去噪方法研究[D].西安：西北工业大学博士学位论文.2002.
[10]Candes E J, Demanet L, Donoho D L, et al. Fast discrete curvelet transforms[R]. Applied and computational mathematics, California Institute of Technology,2005:1-43.
[11]Herrmann F J, Verschuur E. Curvelet domain multiple elimination with sparseness constraints[C]. Society of Exploration Geophysicists, Expanded Abstracts, p1333-1336,2004.
[12]G. Hennenfent and F. J.Herrmann. Seismic denoising with nonuniformly sampled curvelets, IEEE Computing in Science and Engineering, vol.8, pp.16-25,2006.
[13]Herrmann F J, Deli Wang, G Hennenfent, et al. Seismic data processing with curvelets: A multiscale and nonlinear approach[C]. Society of Exploration Geophysicists, Expanded Abstracts, p2220-2224,2007.
[14]R. Neelamani, A. Baumstein, D. Gillard, M. Hadidi, W. Soroka, Coherent and random noise attenuation using the curvelet transform, the leading edge,27 (2),240-248 (2008).
[15]H. Chauris, J. Ma, Seismic imaging in the curvelet domain:achievements and perspectives,71st EAGE conference and Exhibition, Amsterdam, Netherlands,8-11 June 2009.
[16]H. Douma, M. de Hoop, Leading-order seismic imaging using curvelets, Geophysics,
72(6),S231-S248(2007).
[17]H. Chauris, T. Nguyen, Seismic demigration/migration in the curvelet domain, Geophysics,73 (2), S35-S46 (2008).
[18]F. Herrmann, D. Wang, D. Verschuur, Adaptive curvelet-domain primary-multiple separation, Geophysics,73 (3), A17-A21 (2008).
[19]E. Candes, J. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math.,59,1207-1233 (2005).
[20]D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (4),1289-1306(2006).
[21]Richard Baraniuk, A Lecture on Compressive Sensing. To appear in IEEE Signal Processing Magazine, July 2007.
[22]Candes E J, Wakin M B, An Introduction To Compressive Sampling. IEEE Signal Processing Magazine,March 2008
[23]F. J. Herrmann and G. Hennenfent. Non-parametric seismic data recovery with curvelet frames:Geophysical Journal International,2008.
[24]E. Candes, D. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise singularities, Comm. Pure Appl. Math.,57,219-266 (2004).
[25]E. Le Pennec and S.Mallat, Sparse Geometrical Image Approximation with Bandelets, IEEE Transaction on Image Processing,2004.
[26]孙延奎.小波分析及其应用[M].北京：机械工业出版社,2005：31-36，157-158
[27]王建英,尹忠科,张春梅.信号与图像的稀疏分解及其初步应用[M].成都：西南交通大学出版社,2006：49-139.
[28]Mallat S. and Zhang Z. Matching pursuit with time-frequency dictionaries. IEEE Trans. On Signal Processing,1993,41(12):3397-3415.
[29]Bergeau F., and Mallat S. Matching pursuit of images. Proceedings of IEEE-SP, 1994:330-333.
[30]冈萨雷斯等著.数字图像处理[M].阮秋琦等译.第2版.北京：电子工业出版社2005：375-379
[31]E. Candes, D. Donoho, Continuous curvelet transform:Ⅰ. Resolution of the wavefront set, Appl. Comput. Harmon. Anal.,19,162-197 (2003).
[32]E. Candes, D. Donoho, Continuous curvelet transform:Ⅱ. Discretization and frames, Appl. Comput. Harmon. Anal.,19,198-222 (2003).
[33]J. Ma, G Plonka, A review of curvelets and recent applications, IEEE Sig. Proc.Mag, submitted, (2008)
[34]G Hennenfent and F. J. Herrmann Irregular sampling:from aliasing to noise:Presented at the EAGE 69th Conference & Exhibition,2007b.
[35]G Hennenfent and F. J., Random sampling:new insights into the reconstruction of coarsely-sampled wavefields:Presented at the SEG International Exposition and 77th Annual Meeting.2007c.
[36]H. Shan, J. Ma, H. Yang, Comparisons of wavelets, contourlets, and curvelets for seismic denoising, J. Applied Geophysics,2009.
[37]Brian Russell, Dan Hampson, Joong Chun. Noise elimination and the radon transform, Part1[J].The Leading Edge,1990,9(11):18-23.
[38]Hennenfent G, Herrmann F J. Simply denoise:Wavefield reconstruction via jitred undersampling[J].Geophysics,2008,73(3):V19-V28.
[39]S. Tong, L. Wang, Marine Linear Noise Suppression in the Curvelet Domain, [J]IEEE computer society,2009.
[40]L.Shan, J. Fu, Curvelet Transform and its Application in Seismic Data Denoising, [J]IEEE computer society,2009.
[41]G. Y. Chen and T. D. Bui, Multiwavelet Denoising using Neighbouring Coefficients, IEEE Signal Processing Letters, vol.10, no.7, pp.211-214,2003.
[42]L. Wang, H. Liu, Curvelet-based Noise Attenuation in Prestack Seismic Data, [J]IEEE computer society,2008.
[43]Trad, D., T.J. Ulrych and M.D. Sacchi, Latest views of the sparse Radon transform: Geophysics,68,386-399,2003
[44]Zwartjes, P. M., and A. Gisolf, Fourier reconstruction with sparse inversion:Geophys. Prosp.,55, no.2,199-221,2007
[45]Hennenfent, G, F. J. Herrmann, and R. Neelamani,2005, Sparseness-constrained seismic deconvolution with Curvelets:Presented at the CSEG National Convention.
[46]Yarham, C., U. Boeniger, and F. Herrmann,2006, Curvelet-based ground roll removal: Presented at the SEG International Exposition and 76th Annual Meeting.
[47]M.L. Brady. A fast discrete approximation algorithm for the Radon transform. SIAM J. Comput.,27(1):107-119,1998
[48]Oz Yilmaz, Seismic Data Processing (SEG Investigations in Geophysics N 2.) 2000.
[49]Averbuch, A., Coifman, R., Donoho, D. L., Elad, M., and Israeli, M.2006. Fast and accurate polar Fourier transform. Appl. Comput. Harmon. Anal.21.
[50]Donoho, D., Maleki, A., and Shaharam, M.2006. Wavelab 850. http://www-stat.stanford.edu/-wavelab.
[51]Duijndam, A. J. W. and Schonewille, M. A.1999. Nonuniform fast Fourier transform. Geophysics 64,539.
[52]Dutt, A. and Rokhlin, V.1993. Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Stat. Comput.14,1368.
[53]Frigo, M. and Johnson, S. G. 2005b. FFTW, C subroutine library, http://www.fftw.org.
[54]K. Guo, W-Q. Lim, D. Labate, G. Weiss and E. Wilson, The theory of wavelets with composite dilations, Boston (2005).