Advertisement

Studia Geophysica et Geodaetica

, Volume 63, Issue 4, pp 569–583 | Cite as

Simultaneous interpolation and denoising based on a modified thresholding method

  • Jingjie CaoEmail author
  • Shangxu Wang
  • Wenquan Liang
Article
First Online:
  • 17 Downloads

Abstract

Seismic interpolation can provide complete data for some multichannel processing techniques such as time lapse imaging and wave equation migration. However, field seismic data often contains random noise and noisy data interpolation is a challenging task. A traditional method applies interpolation and denoising separately, but this needs two workflows. Simultaneous interpolation and denoising combines interpolation and denoising in one workflow and can also get acceptable results. Most existing interpolation methods can only recover missing traces but fail to attenuate noise in sampled traces. In this study, a novel thresholding strategy is proposed to remove the noise in the sampled traces and meanwhile recover missing traces during interpolation. For each iteration, the residual is multiplied by a weighting factor and then added to the iterative solution, after which the sum in the transformed domain is calculated using the thresholding operation to update the iterative solution. To ensure that the interpolation and denoising results are robust, the exponential method was chosen to reduce the threshold values in small quantities. The curvelet transform was used as sparse representation and three interpolation methods were chosen as benchmarks. Three numerical tests results proved the effectiveness of the proposed method on removing noise in the sampled traces when the minimum threshold values are correctly chosen.

Keywords

interpolation sparsity seismic inversion noise thresholding method 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors wish to thank F. Mahmoudian and three anonymous reviewers for greatly improving this manuscript. We thank the authors of CurveLab for providing access to their curvelet transform codes. This work was funded by National Natural Science Foundation of China (41674114, 41974116 and 41704120), Postdoctoral Research Foundation of China (2016M600171 and 2017T100137), Hundreds of Outstanding Innovative Talent Support Program for Colleges in Hebei Province (III) under grant number SLRC2017024, and Natural Science Foundation of Hebei Province under grant number D2017403027.

References

  1. Abma R. and Kabir N., 2006. 3D interpolation of irregular data with a POCS algorithm. Geophysics, 71, E91–E97.CrossRefGoogle Scholar
  2. Candes E., Demanet L., Donoho D. and Ying L., 2006. Fast discrete curvelet transforms. Multiscale Model. Simul., 5, 861–899.CrossRefGoogle Scholar
  3. Cao J. and Wang B., 2015. An improved projection onto convex sets methods for simultaneous interpolation and denoising. Chinese J. Geophys., 58, 2935–2947 (in Chinese).Google Scholar
  4. Cao J., Wang Y. and Yang C., 2012. Seismic data restoration based on compressive sensing using the regularization and zero-norm sparse optimization. Chinese J. Geophys., 55, 596–607 (in Chinese).CrossRefGoogle Scholar
  5. Cao J. and Zhao J., 2017. Simultaneous seismic interpolation and denoising based on sparse inversion with a 3D low redundancy curvelet transform. Explor. Geophys., 48, 422–429.CrossRefGoogle Scholar
  6. Chen Y., Zhang D., Jin Z., Chen X.H., Zu S.H., Huang W.L. and Gan S.W., 2016. Simultaneous denoising and reconstruction of 5-D seismic data via damped rank-reduction method. Geophys. J. Int., 206, 1695–1717.CrossRefGoogle Scholar
  7. Daubechies I., Defrise M. and Mol C., 2004. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pur. Appl. Math., 57, 1413–1457.CrossRefGoogle Scholar
  8. Fomel S., 2003. Seismic reflection data interpolation with differential offset and shot continuation. Geophysics, 68, 733–744.CrossRefGoogle Scholar
  9. Fomel S. and Liu Y., 2010. Seislet transform and seislet frame. Geophysics, 75, V25–V38.CrossRefGoogle Scholar
  10. Gao J., Stanton A., Naghizadeh M., Naghizadeh M., Sacchi M. and Chen X., 2013. Convergence improvement and noise attenuation considerations for beyond alias projection onto convex sets reconstruction. Geophys. Prospect., 61, 138–151.CrossRefGoogle Scholar
  11. Gao J., Stanton A. and Sacchi M., 2015. Parallel matrix factorization algorithm and its application to 5D seismic reconstruction and denoising. Geophysics, 80, V173–V187.CrossRefGoogle Scholar
  12. Gong X., Yu S. and Wang S., 2016. Prestack seismic data regularization using a time-variant anisotropic Radon transform. J. Geophys. Eng., 13, 462–469.CrossRefGoogle Scholar
  13. Herrmann F. and Hennenfent G., 2008. Non-parametric seismic data recovery with curvelet frames. Geophys. J. Int., 173, 233–248.CrossRefGoogle Scholar
  14. Herrmann P., Mojesky T., Magasan M. and Hugonnet P., 2000. De-aliased, high-resolution Radon transforms. SEG Technical Program Expanded 2000, 1953?1956.Google Scholar
  15. Kreimer N. and Sacchi M., 2012. A tensor higher-order singular value decomposition for pre-stack simultaneous noise reduction and interpolation. Geophysics, 77, V113–V122.CrossRefGoogle Scholar
  16. Liang J., Ma J. and Zhang X., 2014. Seismic data restoration via data-driven tight frame. Geophysics, 79, V65–V74.CrossRefGoogle Scholar
  17. Ma M., Zhang R. and Yuan, S. Y., 2018. Multichannel impedance inversion for nonstationary seismic data based on the modified alternating direction method of multipliers. Geophysics, 84, A1–A6.CrossRefGoogle Scholar
  18. Mosher C., Li C., Morley L., Ji Y., Janiszewski F., Olson R. and Brewer, J., 2014. Increasing the efficiency of seismic data acquisition via compressive sensing. The Leading Edge, 33, 386–388.CrossRefGoogle Scholar
  19. Oropeza V. and Sacchi M., 2011. Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis. Geophysics, 76, V25–V32.CrossRefGoogle Scholar
  20. Parsani M., 1999. Seismic trace interpolation using half-step prediction filters. Geophysics, 64, 1461–1467.CrossRefGoogle Scholar
  21. Ronen J., 1987. Wave-equation trace interpolation. Geophysics, 52, 973–984.CrossRefGoogle Scholar
  22. Spitz S., 1991. Seismic trace interpolation in the f-x domain. Geophysics, 56, 785–794.CrossRefGoogle Scholar
  23. Smith P., Scott I. and Traylen T., 2012. Simultaneous time-lapse binning and regularization for 4D data. Extended Abstract. 74th EAGE Conference and Exhibition incorporating EUROPEC 2012, DOI: 10.3997/2214-4609.20148240.Google Scholar
  24. Sternfels R., Viguier G., Gondoin R. and Le Meur D., 2015. Multidimensional simultaneous random plus erratic noise attenuation and interpolation for seismic data by joint low-rank and sparse inversion. Geophysics, 80, WD129–WD141.CrossRefGoogle Scholar
  25. Wang B, Wu R., Chen X. and Li J., 2015. Simultaneous seismic data interpolation and denoising with a new adaptive method based on dreamlet transform. Geophys. J. Int., 201, 1182–1194.CrossRefGoogle Scholar
  26. Wang B., Chen X., Li J. and Cao J., 2016. An improved weighted projection onto convex sets methods for seismic data interpolation and denoising. IEEE J. Sel. Top. Appl. Earth Observ. Remote Sens., 9, 228–235.CrossRefGoogle Scholar
  27. Wang Y., Cao J. and Yang C., 2011. Recovery of seismic wavefields based on compressive sensing by an 1 l-norm constrained trust region method and the piecewise random subsampling. Geophys. J. Int., 187, 199–213.CrossRefGoogle Scholar
  28. Xu S., Zhang Y., Pham D. and Lambaré G., 2005. Antileakage Fourier transform for seismic data regularization. Geophysics, 70, Z87–Z93.CrossRefGoogle Scholar
  29. Yang P., Gao J. and Chen W., 2013. On alaysis-based two-step interpolation methods for randomly sampled seismic data. Comput. Geosci., 51, 449–461.CrossRefGoogle Scholar
  30. Yao G. and Jakubowicz H., 2016. Least-squares reverse-time migration in a matrix-based formulation. Geophys. Prospect., 64, 611–621.CrossRefGoogle Scholar
  31. Yao G., Silva N., Warner M. and Kalinicheva T., 2018. Separation of migration and tomography modes of full-waveform inversion in the plane wave domain. J. Geophys. Res.-Solid Earth, 123, 1486–1501.CrossRefGoogle Scholar
  32. Yao G., Wu D. and Debens H., 2016. Adaptive finite difference for seismic wavefield modelling in acoustic media. Sci. Rep., 6, 30302, DOI: 10.1038/srep30302.CrossRefGoogle Scholar
  33. Yuan S., Su Y., Wang T., Wang J. and Wang S., 2019a. Geosteering phase attributes: A new detector for the discontinuities of seismic images. IEEE Geosci. Remote Sens. Lett., 16, 145–149.CrossRefGoogle Scholar
  34. Yuan S., Wang S., Luo Y., Wei W. and Wang G., 2019b. Impedance inversion by using the lowfrequency full-waveform inversion result as a priori model. Geophysics, 84, R149–R164.CrossRefGoogle Scholar
  35. Zhang D., Chen Y., Huang W. and Gan S., 2016. Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data. J. Geophys. Eng., 13, 704–720.CrossRefGoogle Scholar
  36. Zhang D., Zhou Y., Chen H., Chen W., Zu S., and Chen Y., 2017. Hybrid rank-sparsity constraint model for simultaneous reconstruction and denoising of 3D seismic data. Geophysics, 82, V351–V367.CrossRefGoogle Scholar
  37. Zhang H. and Chen X., 2013. Seismic data reconstruction based on jittered sampling and curvelet transform. Chinese J. Geophys., 56, 1637–1649 (in Chinese).Google Scholar
  38. Zhou Y., Liu Z. and Zhang Z., 2015. Seismic signal reconstruction under the morphological component analysis frame-work combined with DCT and curvelet dictionary. Geophys. Prospect. Petrol., 54, 560–568.Google Scholar
  39. Zhou Y., Wang L. and Pu Q., 2014. Seismic data reconstruction based on K-SVD dictionary learning under compressive sensing framework. Oil Geophys. Prospect., 49, 652–660.Google Scholar

Copyright information

© Inst. Geophys. CAS, Prague 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Petroleum Resources and ProspectingChina University of Petroleum (Beijing)BeijingChina
  2. 2.Hebei GEO UniversityShijiazhuang, HebeiChina
  3. 3.Longyan UniversityLongyan, FujianChina

Personalised recommendations

Cite article

Buy options