AccScience Publishing / JSE / Article
Submit to JSE
Cite this article
2
Views
Journal Browser
Volume | Year
Issue
Search
Forthcoming Issue
Current Issue
View All
News and Announcements
View All
ARTICLE

Seismic data reconstruction via complex shearlet transform and block coordinate relaxation

JICHENG LIU,  YA GU,  YONGXIN CHOU
Show Less
School of Electrical & Automatic Engineering, Changshu Institute of Technology. Changshu 215500, P.R. China,
JSE 2019, 28(4), 307–332;
Submitted: 9 June 2025 | Revised: 9 June 2025 | Accepted: 9 June 2025 | Published: 9 June 2025
© 2025 by the Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
Download PDF
Cite
XML
Abstract

Liu, J., Gu, Y. and Chou, Y.X., 2019. Seismic data reconstruction via complex shearlet transform and block coordinate relaxation. Journal of Seismic Exploration, 28: 307-332. Due to practical and economic limitations, real seismic data is not densely sampled in all coordinates, which will affect the subsequent seismic data processing steps, such as migration, surface-related multiple elimination and inversion. Therefore, it is necessary to reconstruct the incomplete seismic data. This paper explains the application of the complex shearlet transform to seismic data reconstruction. With a Z; constraint and the Block Coordinate Relaxation (BCR) method, performance of the complex shearlet-based, real shearlet-based and well-accepted curvelet-based reconstruction are compared in terms of recovered f-k spectrum and signal to noise ratio (SNR). We also discuss the shift invariance of the complex shearlet transform and compare the performances of the BCR and the widely used Projection Onto Convex Sets (POCS) method. The numerical experiments on synthetic and real data with different under-sampling rates demonstrate the validity of the proposed method, especially for the case of large amounts of traces missing.

Keywords
seismic data reconstruction
complex shearlet transform
curvelet transform
block coordinate relaxation
References
  1. Abma, R. and Kabir, N., 2006. 3D interpolation of irregular data with a POCS algorithm.
  2. Geophysics, 71(6): E91-E97.
  3. Alliney, S. and Ruzinsky, S.A., 1994. An algorithm for the minimization of mixed /,
  4. and /, norm with application to Bayesian estimation. IEEE Transact. Signal
  5. Process., 42: 618-627
  6. Bishop, Y.M.M., Fienberg, S.E. and Holland, P.W., 1975. Discrete Multivariate Analysis:
  7. Theory and Practice. MIT Press, Cambridge, MA.
  8. Bruce, A., Sardy, S. and Tseng P., 1998. Block coordinate relaxation methods for
  9. nonparametric signal de-noising. Proc. SPIE - The International Society for Optical
  10. Engineering, 3391: 75-86.
  11. Cao, J., Wang, Y., Yang, C. and Zhao, J., 2011. A review on restoration of seismic
  12. wavefields based on regularization and compressive sensing. Inverse Probl. Sci.
  13. Engineer., 19: 679-704.
  14. Cao, J., Wang, Y. and Yang, C., 2012. Seismic data restoration based on compressive
  15. sensing using the regularization and zero-norm sparse optimization. Chin. J.
  16. Geophys., 55: 596-607.
  17. Cao, J. and Wang, Y., 2014. Seismic data restoration with a fast Li norm trust region
  18. method. J. Geophys. Engineer., 11: 045010.
  19. Cao, J. Wang, Y. and Wang, B., 2015a. Accelerating seismic interpolation with a
  20. gradient projection method based on tight frame property of curvelet. Explor.
  21. Geophys., 43: 253-260.
  22. Cao, J. and Zhao, J., 2015b. 3D seismic interpolation with a low redundancy, fast curvelet
  23. transform. J. Seismic Explor., 24: 121-134.
  24. Cao, J. and Wang, B., 2015c. An improved projection onto convex sets methods for
  25. simultaneous interpolation and denoising. Chin. J. Geophys. (in Chinese), 58:
  26. 2935-2947.
  27. Candés, E. and Donoho, D.L., 2004. New tight frames of curvelets and optimal
  28. representation of objects with C singularities. Comm. Pure Appl. Math., 57:
  29. 219-266.
  30. Chemingui, N. and Biondi, B., 2002. Seismic data reconstruction by inversion to common
  31. offset. Geophysics, 67: 1575-1585.
  32. Chen, S., Donoho, D.L. and Saunders, M., 1996. Atomic decomposition by basis pursuit
  33. Tech. Report, Stanford University.
  34. Easley, G., Labate, D. and Lim, W.Q., 2008. Sparse directional image representations
  35. using the discrete shearlet transform. Appl. Computat. Harmon. Analys., 25:
  36. 25-46.
  37. Feng, F., Wang, Z., Liu, C. and Wang, D., 2016. Seismic data reconstruction based on
  38. sparse constraint in the shearlet domain. Geophys. Prosp. Petrol., 55: 682-691.
  39. Fomel, S., 2003. Seismic reflection data interpolation with differential offset and shot
  40. continuation. Geophysics, 68: 733-744.
  41. Gao, J., Chen, X., Li, J., Liu, G.-C. and Ma, J., 2010. Irregular seismic data reconstruction
  42. based on exponential threshold model of POCS method. Appl. Geophys. 72:
  43. 229-238.
  44. Gao, J., Sacchi, M. and Chen, X., 2013. A fast reduced-rank interpolation method for
  45. prestack seismic volumes that depend on four spatial dimensions. Geophysics,
  46. 78(1): V21-V30.
  47. Genzel, M. and Kutyniok, G., 2014. Asymptotic analysis of inpainting via universal
  48. shearlet systems. SIAM J. Imag. Sci., 7: 2301-2339.
  49. Glenn, R.E., Demetrio, L. and Flavia, C., 2009. Shearlet-based total variation diffusion
  50. for denoising. IEEE Transact. Image Process., 18: 260-268.
  51. Guo, J.Y., Zhou, X.Y. and Yu, S.P., 1996. Iso-interval trace interpolation in F-X domain.
  52. OGP, 31: 28-34.
  53. Guo, K., Kutyniok, G. and Labate, D., 2006. Sparse multidimensional representations
  54. using anisotropic dilation und shear operators. Wavelets and Splines, Athens, GA.
  55. TN: 189-201.
  56. Hennenfent, G. and Herrmann, F.J., 2008. Simply denoise: wavefield reconstruction via
  57. jittered under-sampling. Geophysics, 73(3): V19-V28.
  58. Herrmann, F.J. and Hennenfent, G., 2008. Non-parametric seismic data recovery with
  59. curvelet frames. Geophys. J. Internat., 173: 233-248.
  60. Kong, D. and Peng, Z., 2015. Seismic random noise attenuation using shearlet and total
  61. generalized variation. J. Geophys. Engineer., 12: 1024-1035.
  62. Kutyniok, G. and Labate, D., 2007. Optimally sparse multidimensional representation
  63. using shearlets. SIAM J. Mathemat. Analys., 39: 298-318.
  64. Kutyniok, G. and Labate, D., 2009. Resolution of the wavefront set using continuous
  65. shearlets. Transact. Am. Mathemat. Soc., 361: 2719-2754.
  66. Kutyniok, G. and Labate, D., 2012. Introduction to Shearlets: Multiscale Analysis for
  67. Multivariate Data. Springer Verlag, New York.
  68. Kutyniok, G., Shahram, M. and Zhuang, X.S., 2012. ShearLab: A rational design of a
  69. digital parabolic scaling algorithm. SIAM J. Imag. Sci., 5: 1291-1332.
  70. Kutyniok, G. and Petersen, P., 2015. Classification of edges using compactly supported
  71. shearlets. Appl. Comput. Harmon. Anal., 42: 245-293.
  72. Kutyniok, G., Lim, W.Q. and Reisenhofer, R., 2016. Shearlab 3D: faithful digital shearlet
  73. transforms based on compactly supported shearlets. ACM Transact. Mathemat.
  74. Softw., 42(5): 1-42.
  75. Lim, W.Q., 2010. The discrete shearlets transform: a new directional transform and
  76. compactly supported shearlets transform. IEEE Transact. Image Process., 19:
  77. 1166-1180.
  78. Liu, J.C., Chou, Y.X. and Zhu, J.J., 2018. Interpolating seismic data via the POCS
  79. method based on shearlet transform. J. Geophys. Engineer., 15: 852-876.
  80. Liu, Y., Zhang, P. and Liu, C., 2017. Seismic data interpolation using generalised
  81. velocity-dependent seislet transform. Geophys. Prosp. 65(S1): 82-93.
  82. Ma, J.W., 2013. Three-dimensional irregular seismic data reconstruction via low-rank
  83. matrix completion. Geophysics, 78(5): V181-V192.
  84. Minh, N.D. and Vetterli, M., 2005. The contourlet transform: An efficient directional
  85. multiresolution image representation. IEEE Transact. Image Proc., 14: 2091-2106.
  86. Naghizadeh, M. and Sacchi, M. D. 2007 Multistep autogressive reconstruction of seismic
  87. records Geophysics 72(6) V111-V118.
  88. Naghizadeh, M. and Sacchi, M.D., 2009. F-x adaptive seismic-trace interpolation.
  89. Geophysics, 74(1): V9-V16.
  90. Naghizadeh, M. and Innanen, K.A., 2011. Seismic data interpolation using a fast
  91. generalized Fourier transform Geophysics 76(1) V1-V10.
  92. Nurul Kabir, M.M. and ,Verschuur, D.J., 1995. Restoration of missing offsets by
  93. parabolic Radon transform. Geophys. Prosp., 43: 347-368.
  94. Ozkan, M., Tekalp, M. and Sezan, M., 1994. POCS-based restoration of space-varying
  95. blurred images. IEEE Transact. Image Process., 3: 450-454.
  96. Patti, A., Sezan, M. and Tekalp, A., 1997. Superresolution video reconstruction with
  97. arbitrary sampling lattices and nonzereo aperture time. IEEE Transact. Image
  98. Process., 6: 1064-1076.
  99. Pein, A., Loock, S., Plonka, G. and Salditt, T., 2016. Using sparsity information for
  100. iterative phase retrieval in x-ray propagation imaging. Opt. Express, 24:
  101. 8332-8343.
  102. Pejoski, S., Kafedziski, V. and Gleich, D., 2015. Compressed sensing MRI using discrete
  103. nonseperable shearlet transform and FISTA. IEEE Sign. Process. Lett., 22:
  104. 1566-1570.
  105. Porsani, M., 1999. Seismic trace interpolation using half-step prediction filters.
  106. Geophysics, 64: 1461-1467.
  107. Reisenhofer, R., Kiefer, J. and King, E.J., 2016. Shearlet-based detection of flame fronts.
  108. Experim. Fluids, 57: 41.
  109. Ronen, J., 1987. Wave equation trace interpolation. Geophysics, 52: 973-984.
  110. Sheng, Y., Demetrio, L., Glenn, R.E. and Hamid, K., 2009. A shearlet approach to edge
  111. analysis and detection. IEEE Transact. Image Process., 18: 929-941.
  112. Spitz, S., 1991. Seismic trace interpolation in the F-X domain. Geophysics, 56: 785-794.
  113. Starck, J.L., Elad, M. and Donoho, D.L., 2005. Image decomposition via the combination
  114. of sparse representation and a variational approach. IEEE Transact. Image
  115. Process., 14: 1570-1582.
  116. Wang, B., Wu, R., Geng, Y. and Chen, X., 2014. Dreamlet-based interpolation using
  117. POCS method. J. Appl. Geophys., 109: 256-265.
  118. Wang, B., Wu, R., Chen, X. and Li, J., 2015. Simultaneous seismic data interpolation and
  119. denoising with a new adaptive method based on dreamlet transform. Geophys. J.
  120. Internat., 201: 1180-1192.
  121. Wang, B., 2016. An Efficient POCS Interpolation method in the frequency-space domain.
  122. IEEE Geosci. Remote Sens. Lett., 13: 1384-1387.
  123. Wang, B., Chen, X., Li, J. and Cao, J., 2016. An improved weighted Projection Onto
  124. Convex Sets method for seismic data interpolation and denoising. IEEE J. Select.
  125. Topics Appl. Earth Observat. Remote Sens., 9: 228-235.
  126. Wang, Q., 2010. The discrete shearlet transform: a new directional transform and
  127. compactly supported shearlet frames. IEEE Transact., 19: 1166-1180.
  128. Yang, P., Gao, J. and Chen, W., 2012. Curvelet-based POCS interpolation of
  129. nonuniformly sampled seismic records. J. Appl. Geophys., 79: 90-99.
  130. Zhou, X., 1997. Seismic trace interpolation in F-X domain. Oil Geophys. Prosp., 32:
  131. 154-162.
  132. Zhang, L., Han, L.G., Xu, D.X., Li, Y. and Li, H., 2017. Seismic data reconstruction with
  133. Shearlet transform based on compressed sensing technology. OGP, 52: 220-225.
Share
Back to top
Journal of Seismic Exploration, Electronic ISSN: 0963-0651 Print ISSN: 0963-0651, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept