A constant-Q time-domain wave equation using the fractional Laplacian

Tieyuan Zhu, Jerry M. Harris

Research output: Contribution to journalConference article

1 Citation (Scopus)

Abstract

We present a constant-Q time-domain wave equation. It is derived from Kjartansson's constant-Q constitutive stressstrain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second order temporal derivative and two fractional Laplacian operators, models attenuation and dispersion effects. The temporal derivative is solved by the staggered-grid finite-difference approach. The fractional Laplacian is easily calculated in the spatial frequency domain using say a Fourier pseudo-spectral implementation. The advantage of using our fractional Laplacian formulation over the traditional fractional time derivative approach is the avoidance of storing the time history of variables and thus more economic in computational costs. In numerical simulations, we incorporate PML (perfectly matched layer) absorbing boundaries. Furthermore, we verify the accuracy of numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, McDonal's measurements in the Pierre shale, and results from 2-D viscoacoustic analytical modeling for the homogeneous Pierre shale. We then generalize our rigorous formulation for viscoacoustic waves in homogeneous media to an approximate equation for heterogeneous media. We demonstrate the applicability and efficiency of our constant-Q time-domain approach wave equation in a heterogeneous medium with highly atttenuative formation.

Original languageEnglish (US)
Pages (from-to)3417-3422
Number of pages6
JournalSEG Technical Program Expanded Abstracts
Volume32
DOIs
StatePublished - Jan 1 2013
EventSEG Houston 2013 Annual Meeting, SEG 2013 - Houston, United States
Duration: Sep 22 2011Sep 27 2011

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wave equation
Wave equations
wave equations
Shale
Derivatives
heterogeneous medium
shale
attenuation
absorbing boundary
formulations
perfectly matched layers
homogeneous medium
avoidance
conservation equations
Conservation
Momentum
finite difference method
economics
momentum
Economics

All Science Journal Classification (ASJC) codes

  • Geotechnical Engineering and Engineering Geology
  • Geophysics

Cite this

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title = "A constant-Q time-domain wave equation using the fractional Laplacian",
abstract = "We present a constant-Q time-domain wave equation. It is derived from Kjartansson's constant-Q constitutive stressstrain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second order temporal derivative and two fractional Laplacian operators, models attenuation and dispersion effects. The temporal derivative is solved by the staggered-grid finite-difference approach. The fractional Laplacian is easily calculated in the spatial frequency domain using say a Fourier pseudo-spectral implementation. The advantage of using our fractional Laplacian formulation over the traditional fractional time derivative approach is the avoidance of storing the time history of variables and thus more economic in computational costs. In numerical simulations, we incorporate PML (perfectly matched layer) absorbing boundaries. Furthermore, we verify the accuracy of numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, McDonal's measurements in the Pierre shale, and results from 2-D viscoacoustic analytical modeling for the homogeneous Pierre shale. We then generalize our rigorous formulation for viscoacoustic waves in homogeneous media to an approximate equation for heterogeneous media. We demonstrate the applicability and efficiency of our constant-Q time-domain approach wave equation in a heterogeneous medium with highly atttenuative formation.",
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A constant-Q time-domain wave equation using the fractional Laplacian. / Zhu, Tieyuan; Harris, Jerry M.

In: SEG Technical Program Expanded Abstracts, Vol. 32, 01.01.2013, p. 3417-3422.

Research output: Contribution to journalConference article

TY - JOUR

T1 - A constant-Q time-domain wave equation using the fractional Laplacian

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AU - Harris, Jerry M.

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N2 - We present a constant-Q time-domain wave equation. It is derived from Kjartansson's constant-Q constitutive stressstrain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second order temporal derivative and two fractional Laplacian operators, models attenuation and dispersion effects. The temporal derivative is solved by the staggered-grid finite-difference approach. The fractional Laplacian is easily calculated in the spatial frequency domain using say a Fourier pseudo-spectral implementation. The advantage of using our fractional Laplacian formulation over the traditional fractional time derivative approach is the avoidance of storing the time history of variables and thus more economic in computational costs. In numerical simulations, we incorporate PML (perfectly matched layer) absorbing boundaries. Furthermore, we verify the accuracy of numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, McDonal's measurements in the Pierre shale, and results from 2-D viscoacoustic analytical modeling for the homogeneous Pierre shale. We then generalize our rigorous formulation for viscoacoustic waves in homogeneous media to an approximate equation for heterogeneous media. We demonstrate the applicability and efficiency of our constant-Q time-domain approach wave equation in a heterogeneous medium with highly atttenuative formation.

AB - We present a constant-Q time-domain wave equation. It is derived from Kjartansson's constant-Q constitutive stressstrain relation in combination with the mass and momentum conservation equations. Our wave equation, expressed by a second order temporal derivative and two fractional Laplacian operators, models attenuation and dispersion effects. The temporal derivative is solved by the staggered-grid finite-difference approach. The fractional Laplacian is easily calculated in the spatial frequency domain using say a Fourier pseudo-spectral implementation. The advantage of using our fractional Laplacian formulation over the traditional fractional time derivative approach is the avoidance of storing the time history of variables and thus more economic in computational costs. In numerical simulations, we incorporate PML (perfectly matched layer) absorbing boundaries. Furthermore, we verify the accuracy of numerical results through comparisons with theoretical constant-Q attenuation and dispersion solutions, McDonal's measurements in the Pierre shale, and results from 2-D viscoacoustic analytical modeling for the homogeneous Pierre shale. We then generalize our rigorous formulation for viscoacoustic waves in homogeneous media to an approximate equation for heterogeneous media. We demonstrate the applicability and efficiency of our constant-Q time-domain approach wave equation in a heterogeneous medium with highly atttenuative formation.

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