1887
Volume 65, Issue 4
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

Seismic field data are often irregularly or coarsely sampled in space due to acquisition limits. However, complete and regular data need to be acquired in most conventional seismic processing and imaging algorithms. We have developed a fast joint curvelet‐domain seismic data reconstruction method by sparsity‐promoting inversion based on compressive sensing. We have made an attempt to seek a sparse representation of incomplete seismic data by curvelet coefficients and solve sparsity‐promoting problems through an iterative thresholding process to reconstruct the missing data. In conventional iterative thresholding algorithms, the updated reconstruction result of each iteration is obtained by adding the gradient to the previous result and thresholding it. The algorithm is stable and accurate but always requires sufficient iterations. The linearised Bregman method can accelerate the convergence by replacing the previous result with that before thresholding, thus promoting the effective coefficients added to the result. The method is faster than conventional one, but it can cause artefacts near the missing traces while reconstructing small‐amplitude coefficients because some coefficients in the unthresholded results wrongly represent the residual of the data. The key process in the joint curvelet‐domain reconstruction method is that we use both the previous results of the conventional method and the linearised Bregman method to stabilise the reconstruction quality and accelerate the recovery for a while. The acceleration rate is controlled through weighting to adjust the contribution of the acceleration term and the stable term. A fierce acceleration could be performed for the recovery of comparatively small gaps, whereas a mild acceleration is more appropriate when the incomplete data has a large gap of high‐amplitude events. Finally, we carry out a fast and stable recovery using the trade‐off algorithm. Synthetic and field data tests verified that the joint curvelet‐domain reconstruction method can effectively and quickly reconstruct seismic data with missing traces.

Loading

Article metrics loading...

/content/journals/10.1111/1365-2478.12455
2016-09-23
2020-01-29
Loading full text...

Full text loading...

References

  1. BeckA. and FigueiredoM.2009. A fast iterative shrinkage–thresholding algorithm for linear inverse problems. Siam Journal on Imaging Sciences2(1), 183–202.
    [Google Scholar]
  2. Bioucas‐DiasJ.M. and FigueiredoM.A.T.2007. A new TwIST: two‐step iterative shrinkage/thresholding algorithms for image restoration. IEEE Transactions on Image Processing16(12), 2992–3004.
    [Google Scholar]
  3. BlumensathT. and DaviesM.E.2008. Iterative thresholding for sparse approximations. Journal of Fourier Analysis and Applications14(5–6), 629–654.
    [Google Scholar]
  4. BregmanL.M.1967. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Computational Mathematics and Mathematical Physics7(3), 200–217.
    [Google Scholar]
  5. CandèsE.J.2008. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Mathematique346(9–10), 589–592.
    [Google Scholar]
  6. CandèsE.J., DemanetL., DonohoD.L. and YingL.2006a. Fast discrete curvelet transforms. Society for Industrial and Applied Mathematics Multiscale Modeling and Simulation5(3), 861–899.
    [Google Scholar]
  7. CandèsE.J. and DonohoD.L.2005a. Continuous curvelet transform: I. resolution of the wavefront set. Applied and Computational Harmonic Analysis19(2), 162–197.
    [Google Scholar]
  8. CandèsE.J. and DonohoD.L.2005b. Continuous curvelet transform: II. discretization and frames. Applied and Computational Harmonic Analysis19(2), 198–222.
    [Google Scholar]
  9. CandèsE.J., RombergJ. and TaoT.2006b. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory52(2), 489–509.
    [Google Scholar]
  10. CandèsE.J., RombergJ. and TaoT.2006c. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics59(8), 1207–1223.
    [Google Scholar]
  11. CandèsE.J. and TaoT.2006. Near optimal signal recovery from random projections: universal encoding strategies. IEEE Transactions on Information Theory52(12), 5406–5425.
    [Google Scholar]
  12. ChenS.B., DonohoD.L. and SaundersM.A.1994. Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing20(1), 33–61.
    [Google Scholar]
  13. CormenT.H., LeisersonC.E., RivestR.L. and SteinC.2001. Chapter 16: Greedy algorithms. In: Introduction to Algorithms (ed. T.H.Cormen ). The MIT Press.
    [Google Scholar]
  14. DaubechisI., DefriseM. and De MolC.2004. An iterative thresholding algorithm for linear inver problems with a sparsity constraint. Communications on Pure and Applied Mathematics57(11), 1413–1457.
    [Google Scholar]
  15. DonohoD.L.2006. Compressed sensing. IEEE Transactions on Information Theory52(4), 1289–1306.
    [Google Scholar]
  16. DonohoD.L, TsaigY., DroriI. and StarckJ.‐L.2012. Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit. IEEE Transactions on Information Theory58(2), 1094–1121.
    [Google Scholar]
  17. HerrmannF.J. and HennenfentG.2008. Non‐parametric seismic data recovery with Curvelet frames. Geophysical Journal of International173(1), 233–248.
    [Google Scholar]
  18. LandweberL.1951. An iteration formula for Fredholm integral equations of the first kind. American Journal of Mathematics73(3), 615–624.
    [Google Scholar]
  19. MaJ.W.2011. Improved iterative curvelet thresholding for compressed sensing. IEEE Transactions on Instrumentation and Measurement60(1), 126–136.
    [Google Scholar]
  20. MallatS.2008. A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press.
    [Google Scholar]
  21. MallatS.G. and ZhangZ.1993. Matching pursuits with time–frequency dictionaries. IEEE Transactions on Signal Processing41(12), 3397–3415.
    [Google Scholar]
  22. OsherS., MaoY., DongB. and YinW.2008. Fast Linearized Bregman Iteration for Compressed Sensing and Sparse Denoising , UCLA CAM Report (08‐37).
    [Google Scholar]
  23. PatiY.C., RezaiifarR. and KrishnaprasadP.S.1993. Orthogonal matching pursuit: recursive function approximation with application to wavelet decomposition. Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA.
  24. TroppJ.A. and GilbertA.C.2007. Signal recovery from random measurements via orthogonal matching pursuit. Transactions of Information Theory53(12), 4655–4666.
    [Google Scholar]
  25. TrussellH.J. and CivanlarM.R.1985. The Landweber iteration and projection onto convex sets. IEEE Transactions on Acoustics, Speech, and Signal Processing33, 1632–1634.
    [Google Scholar]
  26. TuN. and HerrmannF.J.2015. Fast imaging with surface‐related multiples by sparse inversion. Geophysical Journal International201(1), 304–317.
    [Google Scholar]
  27. YinW.2010. Analysis and generalizations of the linearized Bregman method. SIAM Journal of Imaging Science3(4), 856–877.
    [Google Scholar]
  28. YinW., OsherS., GoldfarbD. and DarbonJ.2008. Bregman iterative algorithms for l 1 ‐minimization with applications to compressed sensing, SIAM Journal of Imaging Science1(1), 143–168.
    [Google Scholar]
http://instance.metastore.ingenta.com/content/journals/10.1111/1365-2478.12455
Loading
/content/journals/10.1111/1365-2478.12455
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): Curvelet transform , Seismic data reconstruction and Sparsity inversion
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