1887
Volume 49, Issue 4
  • ISSN: 0812-3985
  • E-ISSN: 1834-7533

Abstract

[

Reverse-time migration can be formulated in a least-squares inversion framework. This is referred to as least-squares reverse-time migration, which attempts to find an optimal model of the reflectors that fits the observed data in a least-squares sense. Based on different representations of the model, different formulas of the forward modelling for least-squares reverse-time migration can be derived. In this paper, we derive two different formulas. One formula is to recover the impedance-perturbation-related images based on Born approximation. The other is to invert the reflectivity-related images based on Kirchhoff approximation. The theoretical analysis unveils there is an i difference between the two formulas. Consequently, the seismic image using the two formulas has different shape/phase: the one based on Born approximation produces anti-symmetric images; the other based on Kirchhoff approximation gives symmetric images. Two numerical examples demonstrate the similarities and differences between the two formulas.

,

We derive two formulas of forward modelling for least-squares reverse-time migration based on Born and Kirchhoff approximations. Analysis unveils an i difference exists between the two formulas. Consequently, their seismic images have different shape/phase: the Born approximation produces anti-symmetric images while the Kirchhoff approximation gives symmetric images. Numerical examples demonstrate these features between the two formulas.

]
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2018-08-01
2026-01-18

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References

  1. Barzilai J. Borwein J. M. 1988 Two-point step size gradient methods: IMA Journal of Numerical Analysis 8 141 148 10.1093/imanum/8.1.141
    https://doi.org/10.1093/imanum/8.1.141 [Google Scholar]
  2. Baysal E. Kosloff D. D. Sherwood J. W. C. 1983 Reverse time migration: Geophysics 48 1514 1524 10.1190/1.1441434
    https://doi.org/10.1190/1.1441434 [Google Scholar]
  3. Bednar J. B. 2005 A brief history of seismic migration: Geophysics 70 3MJ 20MJ 10.1190/1.1926579
    https://doi.org/10.1190/1.1926579 [Google Scholar]
  4. Bleistein N. Zhang Y. Xu S. Zhang G. Gray S. H. 2005 Migration/inversion: think image point coordinates, process in acquisition surface coordinates: Inverse Problems 21 1715 1744 10.1088/0266‑5611/21/5/013
    https://doi.org/10.1088/0266-5611/21/5/013 [Google Scholar]
  5. Bube K. P. Langan R. T. 2008 A continuation approach to regularization of ill-posed problems with application to crosswell traveltime tomography: Geophysics 73 VE337 VE351 10.1190/1.2969460
    https://doi.org/10.1190/1.2969460 [Google Scholar]
  6. Claerbout J. F. 1971 Toward a unified theory of reflector mapping: Geophysics 36 467 481 10.1190/1.1440185
    https://doi.org/10.1190/1.1440185 [Google Scholar]
  7. Claerbout J. Doherty S. 1972 Downward continuation of moveout-corrected seismograms: Geophysics 37 741 768 10.1190/1.1440298
    https://doi.org/10.1190/1.1440298 [Google Scholar]
  8. Dai W. Fowler P. Schuster G. T. 2012 Multi-source least-squares reverse time migration: Geophysical Prospecting 60 681 695 10.1111/j.1365‑2478.2012.01092.x
    https://doi.org/10.1111/j.1365-2478.2012.01092.x [Google Scholar]
  9. Dong, S., Suh, S., and Wang, B., 2013, Practical aspects of least-squares reverse time migration: 75th Conference and Exhibition, EAGE, Expanded Abstracts, We P09 02.
  10. Gazdag J. 1978 Wave equation migration with the phase-shift method: Geophysics 43 1342 1351 10.1190/1.1440899
    https://doi.org/10.1190/1.1440899 [Google Scholar]
  11. Jaramillo H. H. Bleistein N. 1999 The link of Kirchhoff migration and demigration to Kirchhoff and Born modeling: Geophysics 64 1793 1805 10.1190/1.1444685
    https://doi.org/10.1190/1.1444685 [Google Scholar]
  12. Kaplan S. T. Routh P. S. Sacchi M. D. 2010 Derivation of forward and adjoint operators for least-squares shot-profile split-step migration: Geophysics 75 S225 S235 10.1190/1.3506146
    https://doi.org/10.1190/1.3506146 [Google Scholar]
  13. Khalil, A., Sun, J., Zhang, Y., and Poole, G., 2013, RTM noise attenuation and image enhancement using time-shift gathers: 83rd Annual International Meeting, SEG, Expanded Abstracts, 1–5.
  14. Khaniani H. Bancroft J. C. von Lunen E. 2016 Iterative multiparameter waveform inversion of precritical reflection data using prestack time Kirchhoff approximation: Geophysics 81 R15 R27 10.1190/geo2014‑0560.1
    https://doi.org/10.1190/geo2014-0560.1 [Google Scholar]
  15. Kühl, H., and Sacchi, M. D., 2001, Generalized least-squares DSR migration using a common angle imaging condition: 71st Annual International Meeting, SEG, Expanded Abstracts, 1025–1028.
  16. Kühl H. Sacchi M. D. 2003 Least-squares wave-equation migration for AVP/AVA inversion: Geophysics 68 262 273 10.1190/1.1543212
    https://doi.org/10.1190/1.1543212 [Google Scholar]
  17. Lancaster, S., and Whitcombe, D., 2000, Fast‐track ‘coloured’ inversion: 70th Annual International Meeting, SEG, Expanded Abstracts, 1572–1575.
  18. Liu Y. Sun H. Chang X. 2005 Least-squares wave-path migration: Geophysical Prospecting 53 811 816 10.1111/j.1365‑2478.2005.00505.x
    https://doi.org/10.1111/j.1365-2478.2005.00505.x [Google Scholar]
  19. McMechan G. A. 1983 Migration by extrapolation of time-dependent of time-dependent boundary values: Geophysical Prospecting 31 413 420 10.1111/j.1365‑2478.1983.tb01060.x
    https://doi.org/10.1111/j.1365-2478.1983.tb01060.x [Google Scholar]
  20. Nemeth T. Wu C. Schuster G. T. 1999 Least-squares migration of incomplete reflection data: Geophysics 64 208 221 10.1190/1.1444517
    https://doi.org/10.1190/1.1444517 [Google Scholar]
  21. Nocedal J. 1980 Updating quasi-Newton matrices with limited storage: Mathematics of Computation 35 773 782 10.1090/S0025‑5718‑1980‑0572855‑7
    https://doi.org/10.1090/S0025-5718-1980-0572855-7 [Google Scholar]
  22. Sava P. C. Fomel S. 2003 Angle-domain common-image gathers by wavefield continuation methods: Geophysics 68 1065 1074 10.1190/1.1581078
    https://doi.org/10.1190/1.1581078 [Google Scholar]
  23. Scales J. A. 1987 Tomographic inversion via the conjugate gradient method: Geophysics 52 179 185 10.1190/1.1442293
    https://doi.org/10.1190/1.1442293 [Google Scholar]
  24. Schneider W. A. 1978 Integral formulation for migration in two and three dimensions: Geophysics 43 49 76 10.1190/1.1440828
    https://doi.org/10.1190/1.1440828 [Google Scholar]
  25. Stolt R. H. 1978 Migration by Fourier transform: Geophysics 43 23 48 10.1190/1.1440826
    https://doi.org/10.1190/1.1440826 [Google Scholar]
  26. Tarantola A 1984 Inversion of seismic reflection data in the acoustic approximation: Geophysics 49 1259 1266 doi:10.1190/1.1441754
    https://doi.org/10.1190/1.1441754 [Google Scholar]
  27. Tu N. Herrmann F. J. 2015 Fast imaging with surface-related multiples by sparse inversion: Geophysical Journal International 201 304 317 10.1093/gji/ggv020
    https://doi.org/10.1093/gji/ggv020 [Google Scholar]
  28. Versteeg R. J. 1993 Sensitivity of prestack depth migration to the velocity model: Geophysics 58 873 882 10.1190/1.1443471
    https://doi.org/10.1190/1.1443471 [Google Scholar]
  29. Virieux J. Operto S. 2009 An overview of full-waveform inversion in exploration geophysics: Geophysics 74 WCC1 WCC26 10.1190/1.3238367
    https://doi.org/10.1190/1.3238367 [Google Scholar]
  30. Wang Y. Liang W. Nashed Z. Li X. Liang G. Yang C. 2014 Seismic modeling by optimizing regularized staggered-grid finite-difference operators using a time-space-domain dispersion-relationship-preserving method: Geophysics 79 T277 T285 10.1190/geo2014‑0078.1
    https://doi.org/10.1190/geo2014-0078.1 [Google Scholar]
  31. Whitmore, N. D., 1983, Iterative depth migration by backward time propagation: 53rd Annual International Meeting, SEG, Expanded Abstracts, 382–385.
  32. Wu Z. Alkhalifah T. 2015 Simultaneous inversion of the background velocity and the perturbation in full-waveform inversion: Geophysics 80 R317 R329 10.1190/geo2014‑0365.1
    https://doi.org/10.1190/geo2014-0365.1 [Google Scholar]
  33. Wu D. Yao G. Cao J. Wang Y. 2016 Least-squares RTM with L1 norm regularisation: Journal of Geophysics and Engineering 13 666 673 10.1088/1742‑2132/13/5/666
    https://doi.org/10.1088/1742-2132/13/5/666 [Google Scholar]
  34. Xu S. Zhang Y. Tang B. 2011 3D angle gathers from reverse time migration: Geophysics 76 S77 S92 10.1190/1.3536527
    https://doi.org/10.1190/1.3536527 [Google Scholar]
  35. Yao, G., and Jakubowicz, H., 2012, Least-squares reverse-time migration: 74th Conference and Exhibition, EAGE, Expanded Abstracts, X043.
  36. Yao G. Jakubowicz H. 2016 Least-squares reverse-time migration in a matrix-based formulation: Geophysical Prospecting 64 611 621 10.1111/1365‑2478.12305
    https://doi.org/10.1111/1365-2478.12305 [Google Scholar]
  37. Yao G. Wu D. 2015 Least-squares reverse-time migration for reflectivity imaging: Science China: Earth Sciences 58 1982 1992 10.1007/s11430‑015‑5143‑1
    https://doi.org/10.1007/s11430-015-5143-1 [Google Scholar]
  38. Yao G. Wu D. Debens H. A. 2016 Adaptive finite difference for seismic wavefield modelling in acoustic media: Scientific Reports 6 30302 10.1038/srep30302
    https://doi.org/10.1038/srep30302 [Google Scholar]
  39. Yilmaz, O., 2001, Seismic data analysis: processing, inversion, and interpretation of seismic data: Society of Exploration Geophysicists.
  40. Zeng, C., Dong, S., Mao, J., and Wang, B., 2014, Broadband least-squares reverse time migration for complex structure imaging: 84th Annual International Meeting, SEG, Expanded Abstracts, 3715–3719.
  41. Zhang, Y., Sun, J., and Gray, S., 2007, Reverse-time migration: amplitude and implementation issues: 77th Annual International Meeting, SEG, Expanded Abstracts, 2145–2149.
  42. Zhang Y. Duan L. Xie Y. 2015 A stable and practical implementation of least-squares reverse time migration: Geophysics 80 V23 V31 10.1190/geo2013‑0461.1
    https://doi.org/10.1190/geo2013-0461.1 [Google Scholar]
/content/journals/10.1071/EG16157
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Forward modelling formulas for least-squares reverse-time migration
Exploration Geophysics 49, 506 (2018); https://doi.org/10.1071/EG16157
/content/journals/10.1071/EG16157
/content/journals/10.1071/EG16157
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  • Article Type: Research Article
Keyword(s): Born approximation; forward modelling; Kirchhoff approximation; least-squares inversion; least-squares reverse-time migration; reverse-time migration

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