Finite-difference frequency-domain modeling of viscoacoustic wave propagation in 2D tilted transversely isotropic (TTI) media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. T75-T95 ◽  
Author(s):  
Stéphane Operto ◽  
Jean Virieux ◽  
A. Ribodetti ◽  
J. E. Anderson
Keyword(s):  
Finite Difference ◽  
Frequency Domain ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Wave Mode ◽  
Absorbing Boundary Conditions ◽  
P Wave ◽  
Domain Modeling ◽  
S Wave ◽  
Absorbing Boundary

A 2D finite-difference, frequency-domain method was developed for modeling viscoacoustic seismic waves in transversely isotropic media with a tilted symmetry axis. The medium is parameterized by the P-wave velocity on the symmetry axis, the density, the attenuation factor, Thomsen’s anisotropic parameters [Formula: see text] and [Formula: see text], and the tilt angle. The finite-difference discretization relies on a parsimonious mixed-grid approach that designs accurate yet spatially compact stencils. The system of linear equations resulting from discretizing the time-harmonic wave equation is solved with a parallel direct solver that computes monochromatic wavefields efficiently for many sources. Dispersion analysis shows that four grid points per P-wavelength provide sufficiently accurate solutions in homogeneous media. The absorbing boundary conditions are perfectly matched layers (PMLs). The kinematic and dynamic accuracy of the method wasassessed with several synthetic examples which illustrate the propagation of S-waves excited at the source or at seismic discontinuities when [Formula: see text]. In frequency-domain modeling with absorbing boundary conditions, the unstable S-wave mode is not excited when [Formula: see text], allowing stable simulations of the P-wave mode for such anisotropic media. Some S-wave instabilities are seen in the PMLs when the symmetry axis is tilted and [Formula: see text]. These instabilities are consistent with previous theoretical analyses of PMLs in anisotropic media; they are removed if the grid interval is matched to the P-wavelength that leads to dispersive S-waves. Comparisons between seismograms computed with the frequency-domain acoustic TTI method and a finite-difference, time-domain method for the vertical transversely isotropic elastic equation show good agreement for weak to moderate anisotropy. This suggests the method can be used as a forward problem for viscoacoustic anisotropic full-waveform inversion.

Azimuthal AVO for tilted TI medium

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. C21-C33 ◽  
Author(s):  
Hongwei Wang ◽  
Suping Peng ◽  
Wenfeng Du
Keyword(s):  
Isotropic Medium ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Transverse Waves ◽  
Propagation Angle ◽  
S Waves ◽  
The Difference ◽  
Dip Angle

With the incident P-wave, we derive approximate formulas for amplitudes and polarizations of waves reflected from and transmitted through a planar, horizontal boundary between an overlying isotropic medium and an underlying tilted transversely isotropic (TTI) medium assuming that the directions of the phase and group velocities are consistent. Provided that the velocities in the isotropic medium are equal to the velocities along the symmetry axis direction, we derive the relational expression between the propagation angle in the TTI medium and the propagation angle in the hypothetical isotropic medium, under the condition that the horizontal slowness is the same, and then we update the approximate formula of the polarization in the TTI medium. Provided that the slow and fast transverse waves (qS and SH) are generated simultaneously in the anisotropic interface, we linearize for a six-order Zoeppritz equation, derive the azimuthal formula of longitudinal and S-waves, and determine their detailed expressions within the symmetry axis plane. According to the derived azimuthal AVO formula, we establish medium models, compare the derived AVO with the precision, and obtain the following conclusions: (1) The dip angle for the symmetry axis with respect to the vertical may have a sufficiently large impact on AVO, and the vertical longitudinal wave can generate an S-wave. (2) For the derived AVO formula, within the symmetry axis plane, the fitting effect of the approximate and exact formulas is good; however, within the other incident planes, taking the azimuth angle 45° as an example, the approximation is suitable for the large impedance contrast if the anisotropic parameters are set properly. (3) The error between the approximation and precision is mainly caused by the difference between the reflected and transmitted angles, the velocities’ derivation with respect to azimuth, and the division of approximation into isotropic and anisotropic parts.

A finite-difference approach for solving pure quasi-P-wave equations in transversely isotropic and orthorhombic media

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. C161-C172 ◽  
Author(s):  
Xueyan Li ◽  
Hejun Zhu
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Differential Operators ◽  
Transversely Isotropic ◽  
Coupled Systems ◽  
P Wave ◽  
Finite Difference Schemes ◽  
S Wave ◽  
P Waves ◽  
Pseudo Differential Operators

Starting from the dispersion relation and setting S-wave velocity along symmetry axes to zero, pseudoacoustic-wave equations have been proposed to describe the kinematics of acoustic wavefields in transversely isotropic (TI) and orthorhombic media. To date, the numerical stability of the pseudoacoustic-wave equations has been improved by developing coupled systems of wave equations; however, most simulations still suffer from S-wave artifacts that are the fundamental solutions of the fourth- and sixth-order partial differential equations. Pure quasi-P-wave equations accurately describe the traveltimes of P-waves in TI and orthorhombic media and are free of S-wave artifacts. However, it is difficult to directly solve the pure quasi-P-wave equations using conventional finite-difference schemes due to the presence of pseudo-differential operators. We approximated these pseudo-differential operators by algebraic expressions, whose coefficients can be determined by minimizing differences between the true and approximated values of the pseudo-differential operators in the wavenumber domain. The derived new coupled systems involve modified acoustic-wave equations and a Poisson’s equation that can be solved by conventional finite-difference stencils and fast Poisson’s solver. Several 2D and 3D numerical examples demonstrate that the simulations based on the new systems are free of S-wave artifacts and have correct kinematics of quasi-P-waves in TI and orthorhombic media.

scholarly journals New acoustic approximation for transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2019 ◽  
Vol 85 (1) ◽  
pp. C1-C12 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Tariq Alkhalifah ◽  
Hitoshi Mikada
Keyword(s):  
Wave Propagation ◽  
Seismic Data ◽  
Eikonal Equation ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Seismic Data Processing ◽  
S Wave ◽  
P Waves ◽  
Acoustic Approximation

Seismic data processing in the elastic anisotropic model is complicated due to multiparameter dependency. Approximations to the P-wave kinematics are necessary for practical purposes. The acoustic approximation for P-waves in a transversely isotropic medium with a vertical symmetry axis (VTI) simplifies the description of wave propagation in elastic media, and as a result, it is widely adopted in seismic data processing and analysis. However, finite-difference implementations of that approximation are plagued with S-wave artifacts. Specifically, the resulting wavefield also includes artificial diamond-shaped S-waves resulting in a redundant signal for many applications that require pure P-wave data. To derive a totally S-wave-free acoustic approximation, we have developed a new acoustic approximation for pure P-waves that is totally free of S-wave artifacts in the homogeneous VTI model. To keep the S-wave velocity equal to zero, we formulate the vertical S-wave velocity to be a function of the model parameters, rather than setting it to zero. Then, the corresponding P-wave phase and group velocities for the new acoustic approximation are derived. For this new acoustic approximation, the kinematics is described by a new eikonal equation for pure P-wave propagation, which defines the new vertical slowness for the P-waves. The corresponding perturbation-based approximation for our new eikonal equation is used to compare the new equation with the original acoustic eikonal. The accuracy of our new P-wave acoustic approximation is tested on numerical examples for homogeneous and multilayered VTI models. We find that the accuracy of our new acoustic approximation is as good as the original one for the phase velocity, group velocity, and the kinematic parameters such as vertical slowness, traveltime, and relative geometric spreading. Therefore, the S-wave-free acoustic approximation could be further applied in seismic processing that requires pure P-wave data.

Geometrical spreading in a layered transversely isotropic medium with vertical symmetry axis

Geophysics ◽  
2003 ◽  
Vol 68 (6) ◽  
pp. 2082-2091 ◽  
Author(s):  
Bjørn Ursin ◽  
Ketil Hokstad
Keyword(s):  
Ray Tracing ◽  
Seismic Data ◽  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Analytical Approximation ◽  
Geometrical Spreading ◽  
S Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
Amplitude Versus

Compensation for geometrical spreading is important in prestack Kirchhoff migration and in amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of seismic data. We present equations for the relative geometrical spreading of reflected and transmitted P‐ and S‐wave in horizontally layered transversely isotropic media with vertical symmetry axis (VTI). We show that relatively simple expressions are obtained when the geometrical spreading is expressed in terms of group velocities. In weakly anisotropic media, we obtain simple expressions also in terms of phase velocities. Also, we derive analytical equations for geometrical spreading based on the nonhyperbolic traveltime formula of Tsvankin and Thomsen, such that the geometrical spreading can be expressed in terms of the parameters used in time processing of seismic data. Comparison with numerical ray tracing demonstrates that the weak anisotropy approximation to geometrical spreading is accurate for P‐waves. It is less accurate for SV‐waves, but has qualitatively the correct form. For P waves, the nonhyperbolic equation for geometrical spreading compares favorably with ray‐tracing results for offset‐depth ratios less than five. For SV‐waves, the analytical approximation is accurate only at small offsets, and breaks down at offset‐depth ratios less than unity. The numerical results are in agreement with the range of validity for the nonhyperbolic traveltime equations.

scholarly journals The offset-midpoint traveltime pyramid in 3D transversely isotropic media with a horizontal symmetry axis

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T51-T62 ◽  
Author(s):  
Qi Hao ◽  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
P Wave ◽  
Analytic Representation ◽  
Velocity Inversion ◽  
Transversely Isotropic Media ◽  
Horizontal Symmetry ◽  
Time Domains ◽  
Isotropic Media ◽  
Hti Media

Analytic representation of the offset-midpoint traveltime equation for anisotropy is very important for prestack Kirchhoff migration and velocity inversion in anisotropic media. For transversely isotropic media with a vertical symmetry axis, the offset-midpoint traveltime resembles the shape of a Cheops’ pyramid. This is also valid for homogeneous 3D transversely isotropic media with a horizontal symmetry axis (HTI). We extended the offset-midpoint traveltime pyramid to the case of homogeneous 3D HTI. Under the assumption of weak anellipticity of HTI media, we derived an analytic representation of the P-wave traveltime equation and used Shanks transformation to improve the accuracy of horizontal and vertical slownesses. The traveltime pyramid was derived in the depth and time domains. Numerical examples confirmed the accuracy of the proposed approximation for the traveltime function in 3D HTI media.

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