Solving 1-D inverse problems by Chebyshev polynomial expansion

Vladimir Y. Grechka, George A. McMechan, Vitaly A. Volovodenko

    Research output: Contribution to journalArticle

    5 Citations (Scopus)

    Abstract

    Seismic wave propagation described by differential equations with variable coefficients may be solved by the Chebyshev polynomial expansion method (CPEM). This method approximates a model and forward solutions by orthonormal Chebyshev polynomials. The CPEM provides an alternative to the usual formulation. In CPEM, the model is approximated globally and the forward solutions can explicitly depend on the parameters of the model. The former ensures that a smooth model is produced by inversion. The latter produces partial derivatives of the forward solutions directly with respect to the model parameters, which streamlines the inversion and also gives a quantitative tool for determining the feasibility of inversion in the presence of noise (by singular value decomposition). Two examples of inversion demonstrate the potential of the CPEM. The first is nonlinear inversion for velocity and density using borehole SH-wave data. The second is linear inversion for interval velocities from rms velocity data. Estimation of interval velocity from stacking velocity is illustrated using field data from the Gulf of Mexico.

    Original languageEnglish
    Pages (from-to)1758-1768
    Number of pages11
    JournalGeophysics
    Volume61
    Issue number6
    Publication statusPublished - Nov 1996

    Fingerprint

    inverse problem
    Inverse problems
    polynomials
    Polynomials
    inversions
    expansion
    Seismic waves
    intervals
    SH waves
    Singular value decomposition
    Gulf of Mexico
    SH-wave
    Boreholes
    Wave propagation
    seismic waves
    boreholes
    stacking
    Differential equations
    seismic wave
    wave propagation

    ASJC Scopus subject areas

    • Geochemistry and Petrology
    • Geophysics

    Cite this

    Grechka, V. Y., McMechan, G. A., & Volovodenko, V. A. (1996). Solving 1-D inverse problems by Chebyshev polynomial expansion. Geophysics, 61(6), 1758-1768.

    Solving 1-D inverse problems by Chebyshev polynomial expansion. / Grechka, Vladimir Y.; McMechan, George A.; Volovodenko, Vitaly A.

    In: Geophysics, Vol. 61, No. 6, 11.1996, p. 1758-1768.

    Research output: Contribution to journalArticle

    Grechka, VY, McMechan, GA & Volovodenko, VA 1996, 'Solving 1-D inverse problems by Chebyshev polynomial expansion', Geophysics, vol. 61, no. 6, pp. 1758-1768.
    Grechka VY, McMechan GA, Volovodenko VA. Solving 1-D inverse problems by Chebyshev polynomial expansion. Geophysics. 1996 Nov;61(6):1758-1768.
    Grechka, Vladimir Y. ; McMechan, George A. ; Volovodenko, Vitaly A. / Solving 1-D inverse problems by Chebyshev polynomial expansion. In: Geophysics. 1996 ; Vol. 61, No. 6. pp. 1758-1768.
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