1887
  1. Home
  2. A-Z Publications
  3. Geophysical Prospecting
  4. Volume 71, Issue 8
  5. Article
Volume 71, Issue 8
  • E-ISSN: 1365-2478

Abstract

Abstract

Tensors have been proposed to represent pre‐stack seismic data, particularly for seismic data denoising and reconstruction. They naturally permit describing multi‐dimensional seismic signals that depend on time (or frequency) and source and receiver coordinates. A tensor representation aims to preserve the information embedded in the multi‐linear array in a reduced space. Such a representation is part of many algorithms for seismic data reconstruction via tensor completion methodologies. We investigate and apply one particular tensor tree representation to seismic data reconstruction. The Tensor Tree decomposition methodology permits decomposing a high‐order tensor into third‐order tensors. The technique relies on the truncated singular value decomposition, which branches the tensors into low‐dimensional tensors. As a benefit, the tensor tree allows us to reorganize the tensor into a matrix that demands the least singular values to have an optimal low‐rank approximation. We have developed an algorithm that uses the tensor tree for data reconstruction in an iterative optimization scheme and directly compared it to the parallel matrix factorization. At first, we demonstrated the proposed methodology results with five‐dimensional synthetic shot data and then moved forward with five‐dimensional field data, where we analysed it both pre and post‐stack. The tensor tree performs well in reconstructing both synthetic and field data with high fidelity, at the same level as the well‐established parallel matrix factorization.

© 2023 European Association of Geoscientists & Engineers. © 2023 European Association of Geoscientists & Engineers.
Loading

Article metrics loading...

/content/journals/10.1111/1365-2478.13374
2023-09-22
2025-07-11

  • From This Site

    /content/journals/10.1111/1365-2478.13374
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. Abma, R. & Kabir, N. (2006) 3D interpolation of irregular data with a POCS algorithm. Geophysics, 71, E91–E97.
     [Google Scholar]
  2. Baraniuk, R.G. & Steeghs, P. (2017) Compressive sensing: a new approach to seismic data acquisition. The Leading Edge, 36, 642–645.
     [Google Scholar]
  3. Berry, M.W., Dumais, S.T. & O'Brien, G.W. (1995) Using linear algebra for intelligent information retrieval. SIAM Review, 37, 573–595.
     [Google Scholar]
  4. Carozzi, F. & Sacchi, M.D. (2021) Interpolated multichannel singular spectrum analysis: a reconstruction method that honours true trace coordinates. Geophysics, 86, V55–V70.
     [Google Scholar]
  5. Carozzi, F. & Sacchi, M.D.S. (2017) 5D seismic reconstruction via parallel matrix factorization with randomized QR decomposition. In 2017 SEG International Exposition and Annual Meeting. Houston, TX: Society of Exploration Geophysicists, pp. 4251–4256.
     [Google Scholar]
  6. Cavalcante, Q. & Porsani, M.J. (2022) Low‐rank seismic data reconstruction and denoising by CUR matrix decompositions. Geophysical Prospecting, 70, 362–376.
     [Google Scholar]
  7. Chen, Y., Zhang, D., Jin, Z., Chen, X., Zu, S., Huang, W. & Gan, S. (2016) Simultaneous denoising and reconstruction of 5‐D seismic data via damped rank‐reduction method. Geophysical Journal International, 206, 1695–1717.
     [Google Scholar]
  8. Chiron, L., van Agthoven, M.A., Kieffer, B., Rolando, C. & Delsuc, M.‐A. (2014) Efficient denoising algorithms for large experimental datasets and their applications in Fourier transform ion cyclotron resonance mass spectrometry. Proceedings of the National Academy of Sciences, 111, 1385–1390.
     [Google Scholar]
  9. Drineas, P., Mahoney, M.W. & Muthukrishnan, S. (2006) Sampling algorithms for L 2 regression and applications. In Proceedings of the Seventeenth Annual ACM‐SIAM Symposium on Discrete Algorithm. New York, ACM: pp. 1127–1136.
     [Google Scholar]
  10. Duijndam, A. & Schonewille, M. (1999) Nonuniform fast Fourier transform. Geophysics, 64, 539–551.
     [Google Scholar]
  11. Duijndam, A., Schonewille, M. & Hindriks, C. (1999) Reconstruction of band‐limited signals, irregularly sampled along one spatial direction. Geophysics, 64, 524–538.
     [Google Scholar]
  12. Eckart, C. & Young, G. (1936) The approximation of one matrix by another of lower rank. Psychometrika, 1, 211–218.
     [Google Scholar]
  13. Ely, G., Aeron, S., Hao, N. & Kilmer, M.E. (2015) 5D seismic data completion and denoising using a novel class of tensor decompositions. Geophysics, 80, V83–V95.
     [Google Scholar]
  14. Freire, S.L. & Ulrych, T.J. (1988) Application of singular value decomposition to vertical seismic profiling. Geophysics, 53, 778–785.
     [Google Scholar]
  15. Gao, J., Sacchi, M.D. & Chen, X. (2013) A fast reduced‐rank interpolation method for prestack seismic volumes that depend on four spatial dimensions. Geophysics, 78, V21–V30.
     [Google Scholar]
  16. Halko, N., Martinsson, P.‐G. & Tropp, J.A. (2011) Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53, 217–288.
     [Google Scholar]
  17. Hennenfent, G., Fenelon, L. & Herrmann, F.J. (2010) Nonequispaced curvelet transform for seismic data reconstruction: a sparsity‐promoting approach. Geophysics, 75, WB203–WB210.
     [Google Scholar]
  18. Hennenfent, G. & Herrmann, F.J. (2008) Simply denoise: wavefield reconstruction via jittered undersampling. Geophysics, 73, V19–V28.
     [Google Scholar]
  19. Herrmann, F.J. (2010) Randomized sampling and sparsity: Getting more information from fewer samples. Geophysics, 75, WB173–WB187.
     [Google Scholar]
  20. Herrmann, F.J., Friedlander, M.P. & Yilmaz, O. (2012) Fighting the curse of dimensionality: Compressive sensing in exploration seismology. IEEE Signal Processing Magazine, 29, 88–100.
     [Google Scholar]
  21. Jermyn, A.S. (2019) Efficient tree decomposition of high‐rank tensors. Journal of Computational Physics, 377, 142–154.
     [Google Scholar]
  22. Kolda, T.G. & Bader, B.W. (2009) Tensor decompositions and applications. SIAM Review, 51, 455–500.
     [Google Scholar]
  23. Koren, Y. (2009) The Bellkor solution to the Netflix grand prize. Netflix Prize Documentation, 81, 1–10.
     [Google Scholar]
  24. Kreimer, N. & Sacchi, M.D. (2012) A tensor higher‐order singular value decomposition for prestack seismic data noise reduction and interpolation. Geophysics, 77, V113–V122.
     [Google Scholar]
  25. Kreimer, N., Stanton, A. & Sacchi, M.D. (2013) Tensor completion based on nuclear norm minimization for 5D seismic data reconstruction. Geophysics, 78, V273–V284.
     [Google Scholar]
  26. Kumar, R., Da Silva, C., Akalin, O., Aravkin, A.Y., Mansour, H., Recht, B. & Herrmann, F.J. (2015) Efficient matrix completion for seismic data reconstruction. Geophysics, 80, V97–V114.
     [Google Scholar]
  27. Liberty, E., Woolfe, F., Martinsson, P.‐G., Rokhlin, V. & Tygert, M. (2007) Randomized algorithms for the low‐rank approximation of matrices. Proceedings of the National Academy of Sciences, 104, 20167–20172.
     [Google Scholar]
  28. Lin, L.‐F., Hsu, H.‐H., Liu, C.‐S., Chao, K.‐H., Ko, C.‐C., Chiu, S.‐D., Hsieh, H.‐S., Ma, Y.‐F. & Chen, S.‐C. (2019) Marine 3D seismic volumes from 2D seismic survey with large streamer feathering. Marine Geophysical Research, 40, 619–633.
     [Google Scholar]
  29. Liu, B. & Sacchi, M.D. (2004) Minimum weighted norm interpolation of seismic records. Geophysics, 69, 1560–1568.
     [Google Scholar]
  30. Liu, Y., Long, Z. & Zhu, C. (2018) Image completion using low tensor tree rank and total variation minimization. IEEE Transactions on Multimedia, 21, 338–350.
     [Google Scholar]
  31. Lorentsen, H.S. (2019) The effect of streamer feathering on seismic data. Master's thesis, NTNU.
  32. Manenti, R. & Sacchi, M. (2022) Analysis of rank‐reduction methods for filtering and interpolation. SEG Technical Program Expanded Abstracts. Houston, TX: Society of Exploration Geophysicists, pp. 2952–2956.
     [Google Scholar]
  33. Maurer, H., Curtis, A. & Boerner, D.E. (2010) Recent advances in optimized geophysical survey design. Geophysics, 75, 75A177–75A194.
     [Google Scholar]
  34. Naghizadeh, M. & Innanen, K.A. (2011) Seismic data interpolation using a fast generalized Fourier transform. Geophysics, 76, V1–V10.
     [Google Scholar]
  35. Naghizadeh, M. & Sacchi, M.D. (2010) Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data. Geophysics, 75, WB189–WB202.
     [Google Scholar]
  36. Nakatani, N. & Chan, G. K.‐L. (2013) Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm. The Journal of Chemical Physics, 138, 134113.
     [Google Scholar]
  37. Oropeza, V. & Sacchi, M. (2011) Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis. Geophysics, 76, V25–V32.
     [Google Scholar]
  38. Oropeza, V.E. & Sacchi, M.D. (2010) A randomized SVD for multichannel singular spectrum analysis (MSSA) noise attenuation. In SEG Technical Program Expanded Abstracts 2010 (pp. 3539–3544). Houston, TX: Society of Exploration Geophysicists.
     [Google Scholar]
  39. Rokhlin, V., Szlam, A. & Tygert, M. (2010) A randomized algorithm for principal component analysis. SIAM Journal on Matrix Analysis and Applications, 31, 1100–1124.
     [Google Scholar]
  40. Sacchi, M.D. et al. (2009) FX singular spectrum analysis. In CSPG CSEG CWLS Convention. Calgary, Alberta: Canadian Society of Petroleum Geologists, pp. 392–395.
  41. Sacchi, M.D., Ulrych, T.J. & Walker, C.J. (1998) Interpolation and extrapolation using a high‐resolution discrete Fourier transform. IEEE Transactions on Signal Processing, 46, 31–38.
     [Google Scholar]
  42. Trad, D. (2009) Five‐dimensional interpolation: recovering from acquisition constraints. Geophysics, 74, V123–V132.
     [Google Scholar]
  43. Trad, D. (2014) Five‐dimensional interpolation: new directions and challenges. CSEG Recorder, 39(3), 1–14.
     [Google Scholar]
  44. Trickett, S. & Burroughs, L. (2009) Prestack rank‐reducing noise suppression: Theory. In SEG Technical Program Expanded Abstracts 2009. Houston, TX: Society of Exploration Geophysicists, pp. 3332–3336.
  45. Vidal, G. (2008) Class of quantum many‐body states that can be efficiently simulated. Physical Review Letters, 101, 110501.
     [Google Scholar]
  46. Xu, S., Zhang, Y., Pham, D. & Lambaré, G. (2005) Antileakage Fourier transform for seismic data regularization. Geophysics, 70, V87–V95.
     [Google Scholar]
  47. Yilmaz, Ö. (2001) Seismic data analysis: Processing, inversion, and interpretation of seismic data. Houston, TX: Society of Exploration Geophysicists.
     [Google Scholar]
  48. Zwartjes, P. & Gisolf, A. (2007) Fourier reconstruction with sparse inversion. Geophysical Prospecting, 55, 199–221.
     [Google Scholar]
  49. Zwartjes, P. & Sacchi, M. (2007) Fourier reconstruction of nonuniformly sampled, aliased seismic data. Geophysics, 72, V21–V32.
     [Google Scholar]
/content/journals/10.1111/1365-2478.13374
Loading
Tensor tree decomposition as a rank‐reduction method for pre‐stack interpolation
Geophysical Prospecting 71, 1404 (2023); https://doi.org/10.1111/1365-2478.13374
/content/journals/10.1111/1365-2478.13374
/content/journals/10.1111/1365-2478.13374
Loading

Data & Media loading...

  • Article Type: Research Article
Keyword(s): 5D interpolation; data processing; data regularization; noise; signal processing

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