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Volume 65, Issue 6
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

We present a new inversion method to estimate, from prestack seismic data, blocky P‐ and S‐wave velocity and density images and the associated sparse reflectivity levels. The method uses the three‐term Aki and Richards approximation to linearise the seismic inversion problem. To this end, we adopt a weighted mixed ‐norm that promotes structured forms of sparsity, thus leading to blocky solutions in time. In addition, our algorithm incorporates a covariance or scale matrix to simultaneously constrain P‐ and S‐wave velocities and density. This information is obtained by nearby well‐log data. We also include a term containing a low‐frequency background model. The mixed norm leads to a convex objective function that can be minimised using proximal algorithms. In particular, we use the fast iterative shrinkage‐thresholding algorithm. A key advantage of this algorithm is that it only requires matrix–vector multiplications and no direct matrix inversion. The latter makes our algorithm numerically stable, easy to apply, and economical in terms of computational cost. Tests on synthetic and field data show that the proposed method, contrarily to conventional ‐ or ‐norm regularised solutions, is able to provide consistent blocky and/or sparse estimators of P‐ and S‐wave velocities and density from a noisy and limited number of observations.

© 2017 European Association of Geoscientists & Engineers © 2017 European Association of Geoscientists & Engineers
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2017-02-20
2024-04-23

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References

  1. AkiK. and RichardsP.1980. Quantitative seismology: theory and methods, W.H. Freeman and Co.
     [Google Scholar]
  2. AlemieW.M.2010. Regularization of the AVO inverse problem by means of a multivariate Cauchy probability distribution . Master's thesis, The University of Alberta, Canada.
  3. AlemieW.M. and SacchiM.D.2011. High‐resolution three‐term AVO inversion by means of a trivariate Cauchy probability distribution. Geophysics76(3), R43–R55.
     [Google Scholar]
  4. BabacanS.D., NakajimaS. and DoM.N.2014. Bayesian group‐sparse modeling and variational inference. IEEE Transactions on Signal Processing62(11), 2906–2921.
     [Google Scholar]
  5. BachF., JenattonR., MairalJ. and ObozinskiG.2012. Optimization with sparsity‐inducing penalties. Foundations and Trends® in Machine Learning4(1), 1–106.
     [Google Scholar]
  6. BayesT.1763. An essay towards solving a problem in the doctrine of chances. By the late Rev. Mr. Bayes, F. R. S. communicated by Mr. Price, in a letter to John Canton, A. M. F. R. S. Philosophical Transactions of the Royal Society53, 370–418.
     [Google Scholar]
  7. BeckA. and TeboulleM.2009. A fast iterative shrinkage‐thresholding algorithm for linear inverse problems. SIAM Journal of Imaging Sciences2, 183–202.
     [Google Scholar]
  8. BortfeldR.1961. Approximations to the reflection and transmission coefficients of plane longitudinal and transverse waves. Geophysical Prospecting9(4), 485–502.
     [Google Scholar]
  9. BulandA. and OmreH.2003. Bayesian linearized AVO inversion. Geophysics68(1), 185–198.
     [Google Scholar]
  10. CastagnaJ., BatzleM. and EastwoodR.1985. Relationships between compressional and shear‐wave velocities in clastic silicate rocks. Geophysics50, 551–570.
     [Google Scholar]
  11. CastagnaJ.P., SwanH.W. and FosterD.J.1998. Framework for AVO gradient and intercept interpretation. Geophysics63(3), 948–956.
     [Google Scholar]
  12. ChopraS. and CastagnaJ.2007. Introduction to this special section – AVO. The Leading Edge26, 1506–1507.
     [Google Scholar]
  13. CookeD. and SchneiderW.1983. Generalized linear inversion of reflection seismic data. Geopphysics46(6), 665–676.
     [Google Scholar]
  14. DaubechiesI., DefriseM. and MolC.D.2004. An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics57, 1413–1457.
     [Google Scholar]
  15. DebeyeH.W.J. and van RielP.1990. L p ‐norm deconvolution. Geophysical Prospecting38, 381–403.
     [Google Scholar]
  16. DowntonJ.E.2005. Seismic parameter estimation from AVO inversion . PhD thesis, University of Calgary, Canada.
  17. DowntonJ.E. and LinesL.R.2003. High‐resolution AVO analysis before NMO. 2003 SEG annual meeting, Dallas, USA, Expanded Abstracts, 219–222.
  18. FarquharsonC.G. and OldenburgD.W.2004. A comparison of automatic techniques for estimating the regularization parameter in non‐linear inverse problems. Geophysical Journal International156, 411–425.
     [Google Scholar]
  19. FattiJ.L., SmithG.C., VailP.J., StraussP.J. and LevittP.R.1994. Detection of gas in sandstone reservoirs using AVO analysis: a 3‐D seismic case history using the Geostack technique. Geophysics59(9), 1362–1376.
     [Google Scholar]
  20. FigueiredoM.A.T., NowakR.D. and WrightS.J.2007. Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE Journal of Selected Topics in Signal Processing1, 586–597.
     [Google Scholar]
  21. GardnerG., GardnerL. and GregoryA.1974. Formation velocity and density: the diagnostic basics for stratigraphic traps. Geophysics74, 770–780.
     [Google Scholar]
  22. GramfortA., KowalskiM. and HämäläinenM.2012. Mixed‐norm estimates for the M/EEG inverse problem using accelerated gradient methods. Physics in Medicine and Biology57(7), 1937–1961.
     [Google Scholar]
  23. GramfortA., StrohmeierD., HaueisenJ., HämäläinenM. and KowalskiM.2013. Time–frequency mixed‐norm estimates: sparse M/EEG imaging with non‐stationary source activations. NeuroImage70, 410–422.
     [Google Scholar]
  24. HennenfentG., van denBerg E., FriedlanderM.P. and HermannF.J.2008. New insights into one‐norm solvers from the Pareto curve. Geophysics73(4), 23–26.
     [Google Scholar]
  25. IkelleL.T. and AmundsenL.2005. Introduction to Petroleum Seismology. Investigations in Geophysics, Society of Exploration Geophysicists.
     [Google Scholar]
  26. JensåsI.Ø.2008. A blockyness constraint for seismic AVA inversion . Master's thesis, Norwegian University of Science and Technology, Norway.
  27. KhajehnejadM.A., XuW., AvestimehrA.S. and HassibiB.2011. Analyzing weighted minimization for sparse recovery with nonuniform sparse models. IEEE Transactions on Signal Processing59(5), 1985–2001.
     [Google Scholar]
  28. KowalskiM. and TorrésaniB.2009. Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients. Signal, Image and Video Processing3(3), 251–264.
     [Google Scholar]
  29. KowalskiM., SiedenburgK. and DörflerM.2013. Social sparsity! Neighborhood systems enrich structured shrinkage operators. IEEE Transactions on Signal Processing61(10), 2498–2511.
     [Google Scholar]
  30. LarsonR. and EdwardsB.H.1999. Elementary Linear Algebra, 4th edn. Houghton Mifflin Company.
     [Google Scholar]
  31. LiY., DowntonJ. and XuY.2007. Practical aspects of AVO modeling. The Leading Edge26(3), 295–311.
     [Google Scholar]
  32. MalinvernoA. and BriggsV.A.2004. Expanded uncertainty quantification in inverse problems: hierarchical Bayes and empirical Bayes. Geophysics69(4), 1005–1016.
     [Google Scholar]
  33. MisraS. and SacchiM.D.2008. Global optimization with model‐space preconditioning: application to AVO inversion. Geophysics73(5), R71–R82.
     [Google Scholar]
  34. OstranderW.J.1984. Plane‐wave reflection coefficients for gas sands at nonnormal angles of incidence. Geophysics49, 1637–1648.
     [Google Scholar]
  35. PérezD.O. and VelisD.R.2011. Sparse‐spike AVO/AVA attributes from prestack data. 2011 SEG annual meeting, San Antonio, USA, Expanded Abstracts, 340–344.
  36. PérezD.O., VelisD.R. and SacchiM.D.2013. High‐resolution prestack seismic inversion using a hybrid FISTA least‐squares strategy. Geophysics78(5), R185–R195.
     [Google Scholar]
  37. PérezD.O., VelisD.R. and SacchiM.D.2014. Blocky inversion of prestack seismic data using mixed‐norms. 2014 SEG annual meeting, Denver, USA, Expanded Abstracts, 3106–3111.
  38. RodriguezI.V., BonarD. and SacchiM.2012. Microseismic data denoising using a 3C group sparsity constrained time‐frequency transform. Geophysics77(2), V21–V29.
     [Google Scholar]
  39. SacchiM.D., UlrychT.J. and WalkerC.J.1998. Interpolation and extrapolation using a high‐resolution discrete Fourier transform. IEEE Transactions on Signal Processing, 31–38.
     [Google Scholar]
  40. ShueyR.1985. A simplification of the Zoeppritz equations. Geophysics50(4), 609–614.
     [Google Scholar]
  41. SiviaD.S.1996. Data Analysis: A Bayesian Tutorial, 2nd edn. Oxford, UK: Oxford University Press.
     [Google Scholar]
  42. SmithG.C. and GidlowM.2000. A comparison of the fluid factor with λ and μ in AVO analysis. 2000 SEG annual meeting, Calgary, Canada, Expanded Abstracts, 122–125.
  43. TarantolaA.1987. Inverse Problem Theory, Methods for Data Fitting and Model Parameter Estimation. Elsevier Science Publishing Co.
     [Google Scholar]
  44. TheuneU., JensåsI.Ø. and EidsvikJ.2010. Analysis of prior models for a blocky inversion of seismic AVA data. Geophysics75(3), C25–C35.
     [Google Scholar]
  45. UlrychT., SacchiM. and WoodburyA.2001. A Bayes tour of inversion: a tutorial. Geophysics66(1), 55–69.
     [Google Scholar]
  46. UlrychT.J. and SacchiM.D.2005. Information‐based Inversion and Processing with Applications. Elsevier.
     [Google Scholar]
  47. van den BergE. and FriedlanderM.2008. Probing the Pareto frontier for basis pursuit solutions. SIAM Journal of Scientific Computing31(2), 890–912.
     [Google Scholar]
  48. VelisD.R.2003. Estimating the distribution of primary reflection coefficients. Geophysics68(4), 1417–1422.
     [Google Scholar]
  49. VelisD.R.2008. Stochastic sparse‐spike deconvolution. Geophysics73, R1–R9.
     [Google Scholar]
  50. WaldenA. and HoskenJ.1986. The nature of the non‐Gaussianity of primary reflection coefficients and its significance for deconvolution. Geophysical Prospecting34, 1038–1066.
     [Google Scholar]
  51. YilmazO.2001. Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data . Investigations in Geophysics, Society of Exploration Geophysicists.
  52. YuanM. and LinY.2006. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society, Series B68, 49–67.
     [Google Scholar]
  53. ZhangR., SenM. and SrinivasanS.2013. A prestack basis pursuit seismic inversion. Geophysics78(1), R1–R11.
     [Google Scholar]
  54. ZoeppritzK.1919. Über Reflexion and Durchgang seismischer Wellen durch Unstetigkeits‐flächen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu GöttingenK1, 66–84.
     [Google Scholar]
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Three‐term inversion of prestack seismic data using a weighted l2, 1 mixed norm
Geophysical Prospecting 65, 1477 (2017); https://doi.org/10.1111/1365-2478.12500
/content/journals/10.1111/1365-2478.12500
/content/journals/10.1111/1365-2478.12500
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  • Article Type: Research Article
Keyword(s): Inverse problem; Parameter estimation; Seismic

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