1887
Volume 48, Issue 4
  • ISSN: 0812-3985
  • E-ISSN: 1834-7533

Abstract

[

This paper introduces a low redundancy curvelet transform which can reduce the redundancy to 10 for three dimensional data and simultaneously interpolate and denoise. Numerical experiments on synthetic and field data demonstrate that the low redundancy transform can provide reliable results while at the same time improving the computational efficiency.

,

The simultaneous seismic interpolation and denoising problem can be solved as a sparse inversion problem by using the sparseness of seismic data in a transformed domain as the information, where the properties of the sparse transform will significantly influence the numerical results and computational efficiency. Curvelet transform has nearly optimal sparse expression for seismic data, thus seismic signal processing based on this transform tends to result in preferable results. However, the redundancy of this transform can be 24–32 for three dimensional data which is not computationally cost efficient. This paper introduces a low redundancy curvelet transform to simultaneously interpolate and denoise. The redundancy of the proposed transform can be reduced to 10 for three dimensional data, and this property will improve the computational efficiency of the curvelet transform-based signal processing. The iterative soft thresholding method was chosen to solve the sparse inversion problem. Some practical principles on how to choose the regularisation parameters are discussed, due to the crucial nature of the regularisation parameter for simultaneous interpolation and denoising. Numerical experiments on synthetic and field data demonstrate that the low redundancy transform can provide reliable results while at the same time improving the computational efficiency.

]
ASEG
Loading

Article metrics loading...

/content/journals/10.1071/EG15097
2017-12-01
2026-01-23

  • From This Site

    /content/journals/10.1071/EG15097
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. Berkhout A. J. Verschuur D. J. 1997 Estimation of multiple scattering by iterative inversion, part I: theoretical considerations: Geophysics 62 1586 1595 10.1190/1.1444261
    https://doi.org/10.1190/1.1444261 [Google Scholar]
  2. Candes, E., and Donoho, D., 2000, Curvelets – a surprisingly effective nonadaptive representation for objects with edges: Vanderbilt University Press.
  3. Candes E. Donoho D. 2004 New tight frames of curvelets and optimal representation of objects with piecewise C2 singularities: Communications on Pure and Applied Mathematics 57 219 266 10.1002/cpa.10116
    https://doi.org/10.1002/cpa.10116 [Google Scholar]
  4. Candes E. J. Demanet L. Donoho D. Ying L. 2006 Fast discrete curvelet transforms: SIAM Journal on Multiscale Modeling and Simulation 5 861 899 10.1137/05064182X
    https://doi.org/10.1137/05064182X [Google Scholar]
  5. Cao J. J. Wang B. F. 2015 An improved projection onto convex sets method for simultaneous interpolation and denoising: Chinese Journal of Geophysics 58 2935 2947 [in Chinese]
    [Google Scholar]
  6. Cao J. J. Zhao J. T. 2015 3D seismic interpolation with a low redundancy, fast curvelet transform: Journal of Seismic Exploration 24 121 134
    [Google Scholar]
  7. Cao J. J. Wang Y. F. Zhao J. T. Yang C. C. 2011 A review on restoration of seismic wavefields based on regularization and compressive sensing: Inverse Problems in Science and Engineering 19 679 704 10.1080/17415977.2011.576342
    https://doi.org/10.1080/17415977.2011.576342 [Google Scholar]
  8. Cao J. J. Wang Y. F. Yang C. C. 2012 Seismic data restoration based on compressive sensing using the regularization and zero-norm sparse optimization: Chinese Journal of Geophysics 55 239 251 10.1002/cjg2.1718
    https://doi.org/10.1002/cjg2.1718 [Google Scholar]
  9. Cao J. J. Wang Y. F. Wang B. F. 2015 a Accelerating seismic interpolation with a gradient projection method based on tight frame property of curvelet: Exploration Geophysics 46 253 260 10.1071/EG14016
    https://doi.org/10.1071/EG14016 [Google Scholar]
  10. Cao J. J. Zhao J. T. Hu Z. Y. 2015 b 3D seismic denoising based on a low-redundancy curvelet transform: Journal of Geophysics and Engineering 12 566 576 10.1088/1742‑2132/12/4/566
    https://doi.org/10.1088/1742-2132/12/4/566 [Google Scholar]
  11. Chauris H. Nguyen T. 2008 Seismic demigration/migration in the curvelet domain: Geophysics 73 S35 S46 10.1190/1.2831933
    https://doi.org/10.1190/1.2831933 [Google Scholar]
  12. Chen S. Donoho D. Saunders M. 1998 Atomic decomposition by basis pursuit: SIAM Journal on Scientific Computing 20 33 61 10.1137/S1064827596304010
    https://doi.org/10.1137/S1064827596304010 [Google Scholar]
  13. Daubechies I. Defrise M. Mol C. D. 2004 An iterative thresholding algorithm for linear inverse problems with a sparsity constraint: Communications on Pure and Applied Mathematics 57 1413 1457 10.1002/cpa.20042
    https://doi.org/10.1002/cpa.20042 [Google Scholar]
  14. Fomel S. Liu Y. 2010 Seislet transform and seislet frame: Geophysics 75 V25 V38 10.1190/1.3380591
    https://doi.org/10.1190/1.3380591 [Google Scholar]
  15. Gao J. J. Stanton A. Naghizadeh M. Sacchi M. D. Chen X. H. 2013 Convergence improvement and noise attenuation considerations for beyond alias projection onto convex sets reconstruction: Geophysical Prospecting 61 138 151 10.1111/j.1365‑2478.2012.01103.x
    https://doi.org/10.1111/j.1365-2478.2012.01103.x [Google Scholar]
  16. Geng Y. Wu R. S. Gao J. H. 2012 Dreamlet compression of seismic data: Chinese Journal of Geophysics 55 2705 2715 [in Chinese]
    [Google Scholar]
  17. Hennenfent G. Herrmann F. 2006 Seismic denoising with non-uniformly sampled curvelets: Computing in Science & Engineering 8 16 25 10.1109/MCSE.2006.49
    https://doi.org/10.1109/MCSE.2006.49 [Google Scholar]
  18. Herrmann F. Hennenfent G. 2008 Non-parametric seismic data recovery with curvelet frames: Geophysical Journal International 173 233 248 10.1111/j.1365‑246X.2007.03698.x
    https://doi.org/10.1111/j.1365-246X.2007.03698.x [Google Scholar]
  19. Herrmann F. Moghaddam P. Stolk C. 2008 a Sparsity- and continuity-promoting seismic image recovery with curvelet frames: Applied and Computational Harmonic Analysis 24 150 173 10.1016/j.acha.2007.06.007
    https://doi.org/10.1016/j.acha.2007.06.007 [Google Scholar]
  20. Herrmann F. Wang D. Hennenfent G. Moghaddam P. 2008 b Curvelet-based seismic data processing: a multiscale and nonlinear approach: Geophysics 73 A1 A5 10.1190/1.2799517
    https://doi.org/10.1190/1.2799517 [Google Scholar]
  21. Kreimer N. Sacchi M. D. 2012 A tensor higher-order singular value decomposition for prestack seismic data reduction and interpolation: Geophysics 77 V113 V122 10.1190/geo2011‑0399.1
    https://doi.org/10.1190/geo2011-0399.1 [Google Scholar]
  22. Kumar, V., and Herrmann, F., 2008, Deconvolution with curvelet-domain sparsity: SEG Technical Program, Expanded Abstracts, 1996–2000.
  23. Kumar V. Oueity J. Clowes R. M. Herrmann F. 2011 Enhancing crustal reflection data through curvelet denoising: Tectonophysics 508 106 116 10.1016/j.tecto.2010.07.017
    https://doi.org/10.1016/j.tecto.2010.07.017 [Google Scholar]
  24. Lin T. Herrmann F. 2013 Robust estimation of primaries by sparse inversion via one-norm minimization: Geophysics 78 R133 R150 10.1190/geo2012‑0097.1
    https://doi.org/10.1190/geo2012-0097.1 [Google Scholar]
  25. Liu B. Sacchi M. D. 2004 Minimum weighted norm interpolation of seismic records: Geophysics 69 1560 1568 10.1190/1.1836829
    https://doi.org/10.1190/1.1836829 [Google Scholar]
  26. Ma J. W. 2013 Three-dimensional irregular seismic data reconstruction via low-rank matrix completion: Geophysics 78 V181 V192 10.1190/geo2012‑0465.1
    https://doi.org/10.1190/geo2012-0465.1 [Google Scholar]
  27. Mansour H. Herrmann F. Yilmaz O. 2013 Improved wavefield reconstruction from randomized sampling via weighted one-norm minimization: Geophysics 78 V193 V206 10.1190/geo2012‑0383.1
    https://doi.org/10.1190/geo2012-0383.1 [Google Scholar]
  28. Montefusco L. B. Papi S. 2003 A parameter selection method for wavelet shrinkage denoising: BIT Numerical Mathematics 43 611 626 10.1023/B:BITN.0000007055.60934.b7
    https://doi.org/10.1023/B:BITN.0000007055.60934.b7 [Google Scholar]
  29. Neelamani R. Baumstein A. Gillard D. Hadidi M. Soroka W. 2008 Coherent and random noise attenuation using the curvelet transform: The Leading Edge 27 240 248 10.1190/1.2840373
    https://doi.org/10.1190/1.2840373 [Google Scholar]
  30. Oropeza V. Sacchi M. D. 2011 Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis: Geophysics 76 V25 V32 10.1190/1.3552706
    https://doi.org/10.1190/1.3552706 [Google Scholar]
  31. Ronen J. 1987 Wave-equation trace interpolation: Geophysics 52 973 984 10.1190/1.1442366
    https://doi.org/10.1190/1.1442366 [Google Scholar]
  32. Sacchi M. Liu B. 2005 Minimum weighted norm wavefield reconstruction for AVA imaging: Geophysical Prospecting 53 787 801 10.1111/j.1365‑2478.2005.00503.x
    https://doi.org/10.1111/j.1365-2478.2005.00503.x [Google Scholar]
  33. Spitz S. 1991 Seismic trace interpolation in the f-x domain: Geophysics 56 785 794 10.1190/1.1443096
    https://doi.org/10.1190/1.1443096 [Google Scholar]
  34. Stolt R. H. 2002 Seismic data mapping and reconstruction: Geophysics 67 890 908 10.1190/1.1484532
    https://doi.org/10.1190/1.1484532 [Google Scholar]
  35. Symes W. W. 2007 Reverse time migration with optimal checkpointing: Geophysics 72 SM213 SM221 10.1190/1.2742686
    https://doi.org/10.1190/1.2742686 [Google Scholar]
  36. Trad D. Ulrych T. Sacchi M. 2002 Accurate interpolation with high-resolution time-variant Radon transforms: Geophysics 67 644 656 10.1190/1.1468626
    https://doi.org/10.1190/1.1468626 [Google Scholar]
  37. van den Berg E. Friedlander M. P. 2009 Probing the Pareto frontier for basis pursuit solutions: SIAM Journal on Scientific Computing 31 890 912 10.1137/080714488
    https://doi.org/10.1137/080714488 [Google Scholar]
  38. Wang Y. F. Cao J. J. Yang C. C. 2011 Recovery of seismic wavefields based on compressive sensing by an l 1-norm constrained trust region method and the piecewise random subsampling: Geophysical Journal International 187 199 213 10.1111/j.1365‑246X.2011.05130.x
    https://doi.org/10.1111/j.1365-246X.2011.05130.x [Google Scholar]
  39. Woiselle A. Starck J. Fadili J. 2011 3D data denoising and inpainting with the low-redundancy fast curvelet transform: Journal of Mathematical Imaging and Vision 39 121 139 10.1007/s10851‑010‑0231‑5
    https://doi.org/10.1007/s10851-010-0231-5 [Google Scholar]
  40. Xiao, T. Y., Yu, S. G., and Wang, Y. F., 2003, Numerical methods of inverse problems [in Chinese]: Science Press (Beijing).
  41. Xu S. Zhang Y. Lambaré G. 2010 Antileakage Fourier transform for seismic data regularization in higher dimensions: Geophysics 75 WB113 WB120 10.1190/1.3507248
    https://doi.org/10.1190/1.3507248 [Google Scholar]
  42. Zhang R. F. Ulrych T. 2003 Physical wavelet frame denoising: Geophysics 68 225 231 10.1190/1.1543209
    https://doi.org/10.1190/1.1543209 [Google Scholar]
  43. Zwartjes, P., 2005, Fourier reconstruction with sparse inversion: Ph.D. thesis, Delft University of Technology.
/content/journals/10.1071/EG15097
Loading
Simultaneous seismic interpolation and denoising based on sparse inversion with a 3D low redundancy curvelet transform
Exploration Geophysics 48, 422 (2017); https://doi.org/10.1071/EG15097
/content/journals/10.1071/EG15097
/content/journals/10.1071/EG15097
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): curvelet transform, inverse problems, sparse inversion, wavefield interpolation.

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error