1887
Volume 64, Issue 2
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

This paper introduces an efficiency improvement to the sparse‐grid geometric sampling methodology for assessing uncertainty in non‐linear geophysical inverse problems. Traditional sparse‐grid geometric sampling works by sampling in a reduced‐dimension parameter space bounded by a feasible polytope, e.g., a generalization of a polygon to dimension above two. The feasible polytope is approximated by a hypercube. When the polytope is very irregular, the hypercube can be a poor approximation leading to computational inefficiency in sampling. We show how the polytope can be regularized using a rotation and scaling based on principal component analysis. This simple regularization helps to increase the efficiency of the sampling and by extension the computational complexity of the uncertainty solution. We demonstrate this on two synthetic 1D examples related to controlled‐source electromagnetic and amplitude versus offset inversion. The results show an improvement of about 50% in the performance of the proposed methodology when compared with the traditional one. However, as the amplitude versus offset example shows, the differences in the efficiency of the proposed methodology are very likely to be dependent on the shape and complexity of the original polytope. However, it is necessary to pursue further investigations on the regularization of the original polytope in order to fully understand when a simple regularization step based on rotation and scaling is enough.

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/content/journals/10.1111/1365-2478.12286
2015-06-29
2024-03-29
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References

  1. AzevedoL., NunesR., SoaresA. and NetoG.S.2013. Stochastic Seismic AVO Inversion. 75th EAGE Conference & Exhibition, London, U.K., June 2013.
    [Google Scholar]
  2. BoschM., MukerjiT. and GonzálezE.F.2010. Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: a review. Geophysics75(5), 75A165–75A176.
    [Google Scholar]
  3. BulandA. and OmreH.2003. Bayesian wavelet estimation from seismic and well data. Geophysics68(6), 2000–2009.
    [Google Scholar]
  4. DoyenP.M.2007. Seismic Reservoir Characterization. EAGE.
    [Google Scholar]
  5. Fernández MartínezJ.L., NazariS., Fernández MuñizZ. and RectorJ.W.2010a. Simulating models using geological bases, well logs and seismic attributes. 80th SEG Meeting, Denver, CO, USA.
    [Google Scholar]
  6. Fernández MartínezJ.L., TompkinsM.J., MukerjiT. and AlumbaughD.L.2010b. Geometric sampling: an approach to uncertainty in high dimensional spaces. In: Advances in Intelligent and Soft Computing: Combining Soft Computing and StatisticalMethods in Data Analysis, pp. 247–254 (eds C.Borgelt , G.González‐Rodríguez , W.Trutschnig , M.Asunción Lubiano , M. ÁngelesGil , P.Grzegorzewski and O.Hryniewicz ). Springer.
    [Google Scholar]
  7. GanapathysubramanianB. and ZabarasN.2007. Modeling diffusion in random heterogeneous media: data‐driven models, stochastic collocation and the variational multiscale method. Journal of Computational Physics226(1), 326–353.
    [Google Scholar]
  8. GranaD. and Della RossaE.2010. Probabilistic Petrophysical‐properties estimation integrating statistical rock physics with seismic inversion. Geophysics75(3), O21–O37.
    [Google Scholar]
  9. HortaA. and SoaresA.2010. Direct sequential co‐simulation with joint probability distributions. Mathematicial Geosciences42(3), 269–292.
    [Google Scholar]
  10. JafarpourB. and McLaughlinD.B.2009. Reservoir characterization with discrete cosine transform. Part 1: parameterization. SPE Journal14, 182–188.
    [Google Scholar]
  11. JolliffeI.T.2002. Principal Component Analysis, 2nd Edn.Springer.
    [Google Scholar]
  12. KamounF., FouratiW. and BouhlelM.S.2004. Comparative survey of the DCT and wavelet transforms for image compression. Journal of Testing and Evaluation34(6).
    [Google Scholar]
  13. NunesR., SoaresA., SchwederskyG., DillonL., GuerreiroL., CaetanoH.et al. 2012. Geostatistical Inversion of Prestack Seismic Data. In: Proceedings of the Ninth International Geostatistics Congress, Oslo, Norway, pp. 1–8.
    [Google Scholar]
  14. SambridgeM.1999. Geophysical inversion with a neighbourhood algorithm–I. searching a parameter space. Geophysical Journal International138(2), 479–494.
    [Google Scholar]
  15. ScalesJ.A. and TenorioL.2001. Prior information and uncertainty in inverse problems. Geophysics66(2), 389–397.
    [Google Scholar]
  16. SenM.K. and StoffaP.L.1991. Nonlinear one dimensional seismic waveform inversion using simulated annealing. Geophysics56(10), 1624–1638.
    [Google Scholar]
  17. ShueyR.T.1985. A simplification of the Zoeppritz equations. Geophysics50(4), 609–614.
    [Google Scholar]
  18. SmolyakS.1963. Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Mathematics4, 240–243.
    [Google Scholar]
  19. SoaresA.2001. Direct sequential simulation and cosimulation. Mathematical Geology33(8), 911–926.
    [Google Scholar]
  20. SoaresA., DietJ.D. and GuerreiroL.2007. Stochastic Inversion with a Global Perturbation Method. In: EAGE Petroleum Geostatistics.
    [Google Scholar]
  21. TarantolaA.2005. Inverse Problem Theory. SIAM.
    [Google Scholar]
  22. TompkinsM.J. and AlumbaughD.L.2002, A transversely isotropic 1D electromagnetic inversion scheme requiring minimal a priori information. 72nd SEG Annual International Meeting, Salt Lake City, USA, Expanded Abstracts, 676–679.
    [Google Scholar]
  23. TompkinsM.J., Fernández MartínezJ.L. and Fernández MuñizZ.2011a. Marine electromagnetic inverse solution appraisal and uncertainty using model‐derived basis functions and sparse geometric sampling. Geophysical Prospecting59(4), 947–965.
    [Google Scholar]
  24. TompkinsM.J., Fernández MartínezJ.L., AlumbaughD.L. and MukerjiT.2011b. Scalable uncertainty estimation for nonlinear inverse problems using parameter reduction, constraint mapping, and geometric sampling: marine controlled‐source. Geophysics76(4), 263–281.
    [Google Scholar]
  25. TompkinsM.J., Fernandez MartinezJ.L. and Fernandez MunizZ.2013. Comparison of sparse‐grid geometric and random sampling methods in nonlinear inverse solution uncertainty estimation. Geophysical Prospecting61, 28–41.
    [Google Scholar]
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