1887
Volume 65, Issue S1
  • E-ISSN: 1365-2478

Abstract

ABSTRACT

In this paper, we present a methodology to perform geophysical inversion of large‐scale linear systems via a covariance‐free orthogonal transformation: the discrete cosine transform. The methodology consists of compressing the matrix of the linear system as a digital image and using the interesting properties of orthogonal transformations to define an approximation of the Moore–Penrose pseudo‐inverse. This methodology is also highly scalable since the model reduction achieved by these techniques increases with the number of parameters of the linear system involved due to the high correlation needed for these parameters to accomplish very detailed forward predictions and allows for a very fast computation of the inverse problem solution. We show the application of this methodology to a simple synthetic two‐dimensional gravimetric problem for different dimensionalities and different levels of white Gaussian noise and to a synthetic linear system whose system matrix has been generated via geostatistical simulation to produce a random field with a given spatial correlation. The numerical results show that the discrete cosine transform pseudo‐inverse outperforms the classical least‐squares techniques, mainly in the presence of noise, since the solutions that are obtained are more stable and fit the observed data with the lowest root‐mean‐square error. Besides, we show that model reduction is a very effective way of parameter regularisation when the conditioning of the reduced discrete cosine transform matrix is taken into account. We finally show its application to the inversion of a real gravity profile in the Atacama Desert (north Chile) obtaining very successful results in this non‐linear inverse problem. The methodology presented here has a general character and can be applied to solve any linear and non‐linear inverse problems (through linearisation) arising in technology and, particularly, in geophysics, independently of the geophysical model discretisation and dimensionality. Nevertheless, the results shown in this paper are better in the case of ill‐conditioned inverse problems for which the matrix compression is more efficient. In that sense, a natural extension of this methodology would be its application to the set of normal equations.

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2017-12-26
2024-03-29
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References

  1. AsterR.C., BorchersB. and ThurberC.H.2005. Parameter Estimation and Inverse Problems. Elsevier Academic Press.
    [Google Scholar]
  2. BarbosaV.C.F. and SilvaJ.B.C.1994. Generalized compact gravity inversion. Geophysics59(1), 57–68.
    [Google Scholar]
  3. ChakravarthiV., RamammaB. and ReddyT.V.2013. Gravity anomaly modeling of sedimentary basins by means of multiple structures and exponential density contrast‐depth variations: a space domain approach. Journal of the Geological Society of India82, 561–569.
    [Google Scholar]
  4. ChenK.2005. Matrix Preconditioning Techniques and Applications. Cambridge University Press.
    [Google Scholar]
  5. DadashpourM., RwechunguraR.W. and KleppeJ.2011. Fast reservoir parameter estimation by using effect of principal components sensitivities and discrete cosine transform. SPE‐141913‐MS.
  6. DoiT., HayanoS. and SaitoY.1997. Wavelet solution of the inverse source problems. IEEE Transactions on Magnetics33(2), 1935–1938.
    [Google Scholar]
  7. DonohoD.L., JohnstoneI., KerkyacharianG. and PicardD.1996. Density estimation by wavelet thresholding. Annals of Statistics24(2), 508–539.
    [Google Scholar]
  8. EnglH.W., HankeM. and NeubauerA.1996. Regularization of Inverse Problems. Boston, MA: Kluwer Academic Publishers.
    [Google Scholar]
  9. Fernández‐MartínezJ.L., Fernández‐MuñizZ. and TompkinsM.J.2012. On the topography of the cost functional in linear and nonlinear inverse problems. Geophysics77(1), W1–W15.
    [Google Scholar]
  10. Fernández‐MartínezJ.L., Fernández‐MuñizZ., PalleroJ.L.G. and Pedruelo‐GonzálezL.M.2013. From Bayes to Tarantola: new insights to understand uncertainty in inverse problems. Journal of Applied Geophysics98, 62–72.
    [Google Scholar]
  11. Fernández‐MuñizZ., Fernández‐MartínezJ.L., SrinivasanS. and MukerjiT.2015. Comparative analysis of the solution of linear continuous inverse problems using different basis expansions. Journal of Applied Geophysics113, 92–102.
    [Google Scholar]
  12. GabaldaG., BonvalotS. and HipkinR.2003. CG3TOOL: an interactive computer program for Scintrex CG‐3M gravity data processing. Computers & Geosciences29(2), 155–171.
    [Google Scholar]
  13. GabaldaG., NalpasT. and BonvalotS.2005. The base of the Atacama Gravels Formation (26°S, Northern Chile): first results from gravity data. 6th International Symposium on Andean Geodynamics (ISAG 2005), Barcelona, Spain, Extended Abstracts, 286–289. Paris, France: Institut de Recherche Pour le Développement.
  14. GillP.R., MurrayW. and WrightM.H.1982. The Levenberg‐Marquardt method. In: Practical Optimization. Emerald Group Publishing Limited.
    [Google Scholar]
  15. GonzálezR.C. and WoodsR.E.2007. Digital Image Processing, 3rd edn.Prentice Hall.
    [Google Scholar]
  16. GoovaertsP.1997. Geostatistics for Natural Resource Evaluation. Oxford University Press.
    [Google Scholar]
  17. GroetschC.W.1999. Inverse Problems: Activities for Undergraduates. The Mathematical Association of America.
    [Google Scholar]
  18. HansenP.C.1994. Regularizations tools. A MATLAB package for analysis and solution of discrete ill‐posed problems. Numerical Algorithms6(1), 1–35.
    [Google Scholar]
  19. HansenP.C.1998. Rank‐deficient and discrete ill‐posed problems. In: Numerical Aspects of Linear Inversion, Society for Industrial and Applied Mathematics. Philadelphia, PA.
    [Google Scholar]
  20. HansenP.C.2010. Discrete Inverse Problems: Insight and Algorithms. Society for Industrial and Applied Mathematics.
    [Google Scholar]
  21. HansenP.C., PereyraV. and SchererG.2013. Least Squares Data Fitting With Applications. Johns Hopkins University Press.
    [Google Scholar]
  22. IsacksB.L.1988. Uplift of the Central Andean Plateau and bending of the Bolivian Orocline. Journal of Geophysical Research93 (B4), 3211–3231.
    [Google Scholar]
  23. JafarpourB. and Mac LaughlinD.B.2009. Reservoir characterization with the discrete cosine transform. SPE Journal14(1), 182–201.
    [Google Scholar]
  24. KhayamS.A.2003. Seminar 1 – The Discrete Cosine Transform (DCT): Theory and Application. ECE 802‐602: Information Theory and Coding. Michigan State University.
    [Google Scholar]
  25. MenkeW.1989. Geophysical Data Analysis: Discrete Inverse Theory. Elsevier Science.
    [Google Scholar]
  26. MortimerC.1973. The Cenozoic history of the southern Atacama Desert, Chile. Journal of the Geological Society129(5), 505–526.
    [Google Scholar]
  27. PalleroJ.L.G., Fernández‐MartínezJ.L., BonvalotS. and FudymO.2015. Gravity inversion and uncertainty assessment of basement relief via particle swarm optimization. Journal of Applied Geophysics116, 80–191.
    [Google Scholar]
  28. RaidA.M., KhedrW.M., El‐dosukyM.A. and AhmedW.2014. JPEG image compression using discrete cosine transform ‐ a survey. International Journal of Computer Science & Engineering Survey5, 2.
    [Google Scholar]
  29. RaoK.R. and YipP.1990. Discrete Cosine Transform: Algorithms, Advantages, Applications. San Diego, CA: Academic Press Professional, Inc.
    [Google Scholar]
  30. SayoodK.2003. Introduction to Date Compression, 2nd edn.Morgan Kaufmann Publisher.
    [Google Scholar]
  31. SilvaJ.B.C., TeixeiraW.A. and BarbosaV.C.F.2009. Gravity data as a tool for landfill study. Environmental Geology57, 749–757.
    [Google Scholar]
  32. TarantolaA.2005. Inverse Problem Theory and Model Parameter Estimation. Society for Industrial and Applied Mathematics.
  33. TompkinsM.J., Fernández‐MartínezJ.L. and Fernández‐MuñizZ.2011. Marine electromagnetic inverse solution appraisal and uncertainty using model‐derived basis functions and sparse geometric sampling. Geophysical Prospecting59, 947–965.
    [Google Scholar]
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