An anisotropic acoustic wave equation for modeling and migration in 2D TTI media

Zhou, Hongbo; Bloor, Robert; Zhang, Guanquan

doi:10.1190/1.2369913

Cited by 114 publications

(54 citation statements)

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“…temporal frequency ω a is expressed by an algebraic equation of the sixth order. This is a consequence of the fact that in elements a 11 , a 22 , a 33 of matrix (7) quadric frequency ω a remains.…”

Section: Dispersion Relationsmentioning

confidence: 99%

The pseudo-acoustic equations of the scalar wavefield in anisotropic media

Kostecki

Żuławiński

2015

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The pseudo-acoustic equations of the scalar wavefield in anisotropic mediaIn this paper we present a new formulation of scalar pseudo-acoustic wave equation in arbitrary anisotropic media TI (Transverse Isotropy) type for 2D and 3D cases. These equations, based on precise dispersion relation, determine eigenvalue -temporal frequency as the function of wavenumber and anisotropy parameters. Here we present a few snapshots obtained for different signals by one-way equation.Key words: forward modelling, TI, Transverse Isotropy, pseudo-acoustic equation, one-way equation, dispersion relation.Pseudoakustyczne równanie skalarnego pola falowego w ośrodkach anizotropowych W artykule przedstawiono nowe sformułowanie skalarnych pseudospektralnych równań falowych w dowolnych ośrodkach anizotropowych typu TI (Transverse Isotropy) dla przypadków 2D i 3D. Równania bazują na dyspersyjnych relacjach wyznaczających wartości własne -częstotliwości w funkcji liczb falowych i parametrów anizotropii na podstawie pełnego systemu równań akustycznych. Zaprezentowano kilka przypadków propagacji falowej określonej jednostronnym równaniem w formie migawkowych zdjęć dla różnych typów sygnału.Słowa kluczowe: anizotropia, poprzeczna izotropia (TI), modelowanie, jednostronne równanie falowe, relacja dyspersyjna.Forward modelling of seismic waves in anisotropic media should be executed by solving a full system of elastic wave equations using numerical techniques such as finite-difference, spectral and pseudo-spectral method [3,7,9,10,11,17,19,20,23]. Due to the very time-consuming calculation system of modelling three component acoustic field and insufficient data concerning elastic parameters, in practical seismic research the tendency to simplify the theory and to limit oneself to the considerations of a few models of anisotropy can be noted. Alkhalifah T. [1, 2] introduced for P (compressional) wave fourth-order pseudo-acoustic scalar equations in the space-time domain. In this equation for vertical transverse isotropy (VTI) shear wave velocity V S = 0 is assumed. Alkhalifah's approximation was an inspiration for many studies [4-6, 21, 22].In this paper, we present pseudo-acoustic wave equations, first and second order, versus time in the wavenumbers domain for transverse isotropic media (TI). Some numerical examples were presented as snapshots of propagating wavefields.

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Section: Dispersion Relationsmentioning

confidence: 99%

The pseudo-acoustic equations of the scalar wavefield in anisotropic media

Kostecki

Żuławiński

2015

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“…The commonly used method to do this is to square both sides of the equation after moving the square root term to one side. For example, Alkhalifah's AWE for VTI media [1] , Zhang et al and Zhou et al's AWEs [3,5] for TTI media are derived from Eq. (7) based on this squaring-two-sides method.…”

Section: Approximation Of the Phase Velocity Formulamentioning

confidence: 99%

“…Du et al [4] discussed AWE migration for TTI media using a pseudo-spectral method. Zhou et al [5] divided Zhang et al's AWE [3] into two coupled second-order partial differential equations. Wu and Liu [6,7] solved the AWE for VTI media in frequency-space domain.…”

Section: Introductionmentioning

confidence: 99%

“…Though Alkhalifah [1] presented a method to deal with the shear wave artifacts problem by changing the media around the source to be isotropic, it has great limitations because the method changes the parameters of the media on the one hand and is hard to deal with shear waves coming from different sorts of scattered waves on the other hand. Consequently, Klie and Toro [3] derived a pure qP-wave equation for media with any strength of anisotropy and Du et al [5] used Taylor series expansion to derive a pure qP-wave equation for TTI media.…”

Section: Introductionmentioning

confidence: 99%

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An Approximate Acoustic Wave Equation for VTI Media

Huang

Zhu

Liu

2011

Chinese J of Geophysics

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Acoustic approximation is an effective way to simplify modeling and migration of quasi‐P waves in the presence of anisotropy. To generate a new acoustic wave equation for VTI media (transversely isotropic media with a vertical symmetric axis) from the accurate phase velocity expression, we make simplification to the square root term by an undetermined coefficients method to get an approximate phase velocity formula, from which we derive a second‐order temporal and fourth‐order spatial partial differential equation through inverse Fourier transform. Quantitative computations show that the phase velocity specified by the new acoustic wave equation is accurate when Thomsen parameter ε is equal to δ (elliptical anisotropy) and the errors of the phase velocity become bigger as the difference between ε and δ gets bigger. The maximum relative error of the phase velocity is 0.13% in the case of ε = 0.1 and δ = 0.2, and becomes –1.65% in the case of ε = 0.8 and δ = 0.3. The shear‐wave‐free snapshots from finite difference modeling demonstrate that the new acoustic wave equation is a pure quasi‐P wave equation.

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“…In addition to the ability to handle extreme velocity complexity, many present-day RTM algorithms can handle anisotropy media such as vertical transverse isotropy (VTI) and tilted transverse isotropy (TTI) (Zhou et al, 2006a(Zhou et al, , 2006bDuveneck et al, 2008;Fletcher et al, 2008;Zhang and Zhang, 2008;Zhang and Zhang, 2009). On real data imaging examples, TTI RTM has given the best images in a complex Gulf of Mexico wide-azimuth survey (Huang et al, 2009), although the velocity models for TTI migration were simplified as structurally conformable transverse isotropy (STI), which requires the symmetry axis to be consistent with the normal vectors of reflectors (Audebert et al, 2006).…”

Section: Introductionmentioning

confidence: 98%

3D angle gathers from reverse time migration

Xu¹,

Zhang²,

Tang³

2011

GEOPHYSICS

232

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Common-image gathers are an important output of prestack depth migration. They provide information needed for velocity model building and amplitude and phase information for subsurface attribute interpretation. Conventionally, common-image gathers are computed using Kirchhoff migration on commonoffset/azimuth data volumes. When geologic structures are complex and strong contrasts exist in the velocity model, the complicated wave behaviors will create migration artifacts in the image gathers. As long as the gather output traces are indexed by any surface attribute, such as source location, receiver location, or surface plane-wave direction, they suffer from the migration artifacts caused by multiple raypaths. These problems have been addressed in a significant amount of work, resulting in common-image gathers computed in the reflection angle domain, whose traces are indexed by the subsurface reflection angle and/or the subsurface azimuth angle.Most of these efforts have concentrated on Kirchhoff and oneway wave-equation migration methods. For reverse time migration, subsurface angle gathers can be produced using the same approach as that used for one-way wave-equation migration. However, these approaches need to be revisited when producing high-quality subsurface angle gathers in three dimensions (reflection angle/azimuth angle), especially for wide-azimuth data. We have developed a method for obtaining 3D subsurface reflection angle/azimuth angle common-image gathers specifically for the amplitude-preserved reverse time migration. The method builds image gathers with a highdimensional convolution of wavefields in the wavenumber domain. We have found a windowed antileakage Fourier transform method that leads to an efficient and practical implementation. This approach has generated high-resolution angle-domain gathers on synthetic 2.5D data and 3D wideazimuth real data.

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An anisotropic acoustic wave equation for modeling and migration in 2D TTI media

Cited by 114 publications

References 10 publications

The pseudo-acoustic equations of the scalar wavefield in anisotropic media

The pseudo-acoustic equations of the scalar wavefield in anisotropic media

An Approximate Acoustic Wave Equation for VTI Media

3D angle gathers from reverse time migration

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