The Gabor transform, pseudodifferential operators, and seismic deconvolution

Margrave, Gary F.; Gibson, Peter C.; Grossman, J. P.; Henley, David C.; Iliescu, Victor; Lamoureux, Michael P.

doi:10.3233/ica-2005-12104

Cited by 41 publications

(24 citation statements)

References 11 publications

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“…They have applications in mathematics and engineering in a wide variety of areas, including sampling theory [1], operator theory [36], harmonic analysis [45], nonlinear sparse approximation [23,29], pseudodifferential operators [34], wavelet theory [20], wireless communications [53,54,40], data transmission with erasures [27,10], filter banks [11], signal processing [28,9], image processing [14], geophysics [47], and quantum computing [24].…”

Section: Introductionmentioning

confidence: 99%

Density, Overcompleteness, and Localization of Frames. I. Theory

Bălan

Casazza

Heil

et al. 2006

J Fourier Anal Appl

136

231

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ABSTRACT. Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two frames F = {f i } i∈I and E = {e j } j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E via a map a : I → G. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of F , the relative measure of E -both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements -and the density of the set a(I) in G.Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results.The notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented.Math Subject Classifications. 42C15, 46C99.

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Section: Introductionmentioning

confidence: 99%

Density, Overcompleteness, and Localization of Frames. I. Theory

Bălan

Casazza

Heil

et al. 2006

J Fourier Anal Appl

136

231

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“…Techniques using spectral method for estimation of quality Q factors from surface seismic common midpoint gathers (surface seismic reflection data) were presented in [11,40]. A Gabor deconvolution algorithm to correct reflection seismograms for the effects of elastic attenuation and source signature has been developed in [27,28]. An extensive comparison of different techniques of the frequency-dependent attenuation of seismic waves from VSP data has been investigated in [33,35] and numerical comparison of seismic planewave dispersion and attenuation models has been reported in [34].…”

Section: Literature Reviewmentioning

confidence: 99%

Rational approximation for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media

et al. 2010

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The article presents a numerical inversion method for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media using reconstruction of relaxation spectrum from measured or computed complex velocity or modulus of the medium. Mathematically, the problem is formulated as an inverse problem for reconstruction of relaxation spectrum in the analytic Stieltjes representation of the complex modulus using rational approximation. A rational (Padé) approximation to the relaxation spectrum is derived from a constrained least squares minimization problem with regularization. The recovered stress-strain relaxation spectrum is applied to numerical calculation of frequency-dependent Q factor and frequency-dependent phase velocity for known analytical models of a standard linear viscoelastic solid (Zener) model as well as a nearly constant-Q model which has a continuous spectrum. Numerical results for these analytic models show good agreement between theoretical and predicted values and demonstrate the validity of the algorithm. The proposed method can be used for evaluating relaxation mechanisms in seismic wavefield simulation of viscoelastic media. The constructed lower order Padé approximation can be used for determination of the internal memory variables in time-domain finite difference numerical simulation of viscoelastic wave propagation.

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“…The editor coordinating the review of this manuscript and approving it for publication was Alejandro Frery. [10], Gabor deconvolution to enhance seismic resolution [11] [12] [13], and multichannel blind deconvolution which exploits the lateral coherence among the seismic traces [14] [15] [16].…”

Section: Introductionmentioning

confidence: 99%

Seismic Resolution Enhancement by Frequency-Dependent Wavelet Scaling

Chen

Wang

2018

IEEE Geosci. Remote Sensing Lett.

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Abstract-When seismic waves propagate through the Earth, their high-frequency energy is absorbed by subsurface viscoelastic media. Seismic wavelet appears to be stretched out, as it is dominated by low-frequency components. In order to enhance seismic resolution, we propose here a wavelet compression method that utilizes the scale characteristic in the Fourier transform. The novelty of the scheme is a frequency-dependent scaling that extends the amplitude spectrum to both high-and low-frequency axes simultaneously. This is for the first time to make this frequency-dependent proposal, instead of a constant scaling scheme in the classic Fourier theory. It compresses seismic wavelet in the time domain, and also simplifies the wavelet form effectively. This frequency-dependent scaling scheme leads to a transferring filter that is applicable to seismic field data. It results in an improvement in data resolution and in the ability of thin-layer identification, which will facilitate further seismic inversion and reservoir characterization.Index Terms-Seismic data processing, signal processing, resolution enhancement.

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The Gabor transform, pseudodifferential operators, and seismic deconvolution

Cited by 41 publications

References 11 publications

Density, Overcompleteness, and Localization of Frames. I. Theory

Density, Overcompleteness, and Localization of Frames. I. Theory

Rational approximation for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media

Seismic Resolution Enhancement by Frequency-Dependent Wavelet Scaling

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