Numerical simulation of seismic wave propagation in viscoelastic-anisotropic media using frequency-independent Q wave equation

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. WA1-WA10 ◽  
Author(s):  
Tieyuan Zhu
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Inhomogeneous Medium ◽  
Seismic Anisotropy ◽  
Numerical Solutions ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Theoretical Formula ◽  
Frequency Independent

Seismic anisotropy is the fundamental phenomenon of wave propagation in the earth’s interior. Numerical modeling of wave behavior is critical for exploration and global seismology studies. The full elastic (anisotropy) wave equation is often used to model the complexity of velocity anisotropy, but it ignores attenuation anisotropy. I have presented a time-domain displacement-stress formulation of the anisotropic-viscoelastic wave equation, which holds for arbitrarily anisotropic velocity and attenuation [Formula: see text]. The frequency-independent [Formula: see text] model is considered in the seismic frequency band; thus, anisotropic attenuation is mathematically expressed by way of fractional time derivatives, which are solved using the truncated Grünwald-Letnikov approximation. I evaluate the accuracy of numerical solutions in a homogeneous transversely isotropic (TI) medium by comparing with theoretical [Formula: see text] and [Formula: see text] values calculated from the Christoffel equation. Numerical modeling results show that the anisotropic attenuation is angle dependent and significantly different from the isotropic attenuation. In synthetic examples, I have proved its generality and feasibility by modeling wave propagation in a 2D TI inhomogeneous medium and a 3D orthorhombic inhomogeneous medium.

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T121-T131 ◽  
Author(s):  
Tieyuan Zhu ◽  
Tong Bai
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Seismic Attenuation ◽  
Fourier Pseudospectral Method ◽  
Frequency Independent ◽  
Theoretical Predictions

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

On the construction and efficiency of staggered numerical differentiators for the wave equation

Geophysics ◽  
1990 ◽  
Vol 55 (1) ◽  
pp. 107-110 ◽  
Author(s):  
M. Kindelan ◽  
A. Kamel ◽  
P. Sguazzero
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Computational Efficiency ◽  
Differential Operators ◽  
High Order

Finite‐difference (FD) techniques have established themselves as viable tools for the numerical modeling of wave propagation. The accuracy and the computational efficiency of numerical modeling can be enhanced by using high‐order spatial differential operators (Dablain,1986).

A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. A1-A6 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Numerical Modeling ◽  
Wave Equation ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Computational Domain ◽  
Perfect Absorption ◽  
Transition Area ◽  
Efficient Scheme ◽  
Grid Points ◽  
The One

We propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations. This scheme divides the computational domain into boundary, transition, and inner areas. The wavefields within the inner and boundary areas are computed by the wave equation and the one-way wave equation, respectively. The wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one-way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone. The results from our finite-difference numerical modeling tests of the 2D acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area. We obtain equally good performance using pseudospectral and finite-element modeling with the same scheme. Our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption.

A separable strong-anisotropy approximation for pure qP-wave propagation in transversely isotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. C337-C354 ◽  
Author(s):  
Jörg Schleicher ◽  
Jessé C. Costa
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Dispersion Relation ◽  
Transversely Isotropic ◽  
Wave Dispersion ◽  
Low Rank ◽  
S Wave ◽  
Finite Difference Approximations ◽  
Transversely Isotropic Media ◽  
Isotropic Media

The wave equation can be tailored to describe wave propagation in vertical-symmetry axis transversely isotropic (VTI) media. The qP- and qS-wave eikonal equations derived from the VTI wave equation indicate that in the pseudoacoustic approximation, their dispersion relations degenerate into a single one. Therefore, when using this dispersion relation for wave simulation, for instance, by means of finite-difference approximations, both events are generated. To avoid the occurrence of the pseudo-S-wave, the qP-wave dispersion relation alone needs to be approximated. This can be done with or without the pseudoacoustic approximation. A Padé expansion of the exact qP-wave dispersion relation leads to a very good approximation. Our implementation of a separable version of this equation in the mixed space-wavenumber domain permits it to be compared with a low-rank solution of the exact qP-wave dispersion relation. Our numerical experiments showed that this approximation can provide highly accurate wavefields, even in strongly anisotropic inhomogeneous media.

Wave propagation in an initially stressed transversely isotropic thermoelastic half-space

2014 ◽  
Vol 23 (5-6) ◽  
pp. 185-190 ◽  
Author(s):  
Raj Rani Gupta ◽  
M.S. Saroa
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Boundary Conditions ◽  
Half Space ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Type Ii ◽  
Transversely Isotropic Medium ◽  
Temperature Distributions ◽  
Thermoelasticity Theory

AbstractThe present paper deals with the study of reflection waves in an initially stressed transversely isotropic medium, in the context of Green and Naghdi (GN) thermoelasticity theory type II and III. The components of displacement, stresses and temperature distributions are determined through the solution of the wave equation by imposing the appropriate boundary conditions. Numerically simulated results are plotted graphically with respect to frequency in order to show the effect of anisotropy.

A physical model of shear‐wave propagation in a transversely isotropic solid

Geophysics ◽  
1994 ◽  
Vol 59 (3) ◽  
pp. 484-487 ◽  
Author(s):  
Chih‐Hsiung Chang ◽  
G. H. F. Gardner ◽  
John A. McDonald
Keyword(s):  
Wave Propagation ◽  
Physical Model ◽  
Shear Wave ◽  
Seismic Anisotropy ◽  
Sedimentary Rocks ◽  
Physical Property ◽  
Transversely Isotropic ◽  
Velocity Variation ◽  
Common Phenomenon ◽  
Isotropic Solid

It is now understood that seismic anisotropy is a comparatively common phenomenon in sedimentary layers. The elastic properties of most sedimentary rocks have been shown to be anisotropic. (Anisotropy means that the physical property of the material is a function of the measuring direction). Seismologists are generally concerned with velocity variation with the direction of propagation.

Laboratory results for the features of body‐wave propagation in a transversely isotropic media

Geophysics ◽  
2001 ◽  
Vol 66 (6) ◽  
pp. 1921-1924 ◽  
Author(s):  
Young‐Fo Chang ◽  
Chih‐Hsiung Chang
Keyword(s):  
Wave Propagation ◽  
Shear Wave ◽  
Elastic Anisotropy ◽  
Body Wave ◽  
Transversely Isotropic ◽  
Sedimentary Basins ◽  
Seismic Profiles ◽  
Vertical Seismic Profiles ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Much of the earth’s crust appears to have some degree of elastic anisotropy (Crampin, 1981; Crampin and Lovell, 1991; Helbig, 1993). The phenomena of elastic wave propagation in anisotropic media are more complex than those in isotropic media. Shear‐wave propagation in an orthorhombic physical model is most complex when the direction of the wave is close to the neighborhood of the cusp on the group velocity surfaces (Brown et al., 1991). The first identification of singularities in wave propagation through sedimentary basins occurred in the examination of shear‐wave splitting in multioffset vertical seismic profiles (VSPs) at a borehole site in the Paris Basin (Bush and Crampin, 1991), where large variations in shear‐wave polarizations in propagation directions close to point singularities were observed. Computation of synthetic seismograms for layer sequences showed that the shear‐wave polarizations and amplitudes were irregular near point singularities (Crampin, 1991).

Reverse time migration in tilted transversely isotropic (TTI) media

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA179-WCA187 ◽  
Author(s):  
Robin P. Fletcher ◽  
Xiang Du ◽  
Paul J. Fowler
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
Transversely Isotropic ◽  
Shear Velocity ◽  
Vertical Axis ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Axis Of Symmetry ◽  
Tti Media

Reverse time migration (RTM) exhibits great advantages over other imaging methods because it is based on computing numerical solutions to a two-way wave equation. It does not suffer from dip limitation like one-way downward continuation techniques do, thus enabling overturned reflections to be imaged. As well as correctly handling multipathing, RTM has the potential to image internal multiples when the boundaries responsible for generating the multiples are present in the model. In isotropic media, one can use a scalar acoustic wave equation for RTM of pressure data. In anisotropic media, P- and SV-waves are coupled together so, formally, elastic wave equations must be used for RTM. A new wave equation for P-waves is proposed in tilted transversely isotropic (TTI) media that can be solved as part of an acoustic anisotropic RTM algorithm, using standard explicit finite differencing. If the shear velocity along the axis of symmetry is set to zero, stable numerical solutions can be computed for media with a vertical axis of symmetry and [Formula: see text] not less than [Formula: see text]. In TTI media with rapid variations in the direction of the axis of symmetry, setting the shear velocity along the axis of symmetry to zero can cause numerical solutions to become unstable. A solution to this problem is proposed that involves using a small amount of nonzero shear velocity. The amount of shear velocity added is chosen to remove triplications from the SV wavefront and to minimize the anisotropic term of the SV reflection coefficient. We show modeling and high-quality RTM results in complex TTI media using this equation.

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