Quasi‐shear waves in inhomogeneous weakly anisotropic media by the quasi‐isotropic approach: A model study

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 308-319 ◽  
Author(s):  
Ivan Pšenčík ◽  
Joe A. Dellinger
Keyword(s):  
Order Approximation ◽  
Anisotropic Media ◽  
Ray Method ◽  
Weak Anisotropy ◽  
S Waves ◽  
Model Study ◽  
Arbitrary Strength ◽  
Isotropic Media ◽  
Synthetic Models ◽  
Ray Methods

In inhomogeneous isotropic regions, S-waves can be modeled using the ray method for isotropic media. In inhomogeneous strongly anisotropic regions, the independently propagating qS1- and qS2-waves can similarly be modeled using the ray method for anisotropic media. The latter method does not work properly in inhomogenous weakly anisotropic regions, however, where the split qS-waves couple. The zeroth‐order approximation of the quasi‐isotropic (QI) approach was designed for just such inhomogeneous weakly anisotropic media, for which neither the ray method for isotropic nor anisotropic media applies. We test the ranges of validity of these three methods using two simple synthetic models. Our results show that the QI approach more than spans the gap between the ray methods: it can be used in isotropic regions (where it reduces to the ray method for isotropic media), in regions of weak anisotropy (where the ray method for anisotropic media does not work properly), and even in regions of moderately strong anisotropy (in which the qS-waves decouple and thus could be modeled using the ray method for anisotropic media). A modeling program that switches between these three methods as necessary should be valid for arbitrary‐strength anisotropy.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

First-order ray tracing for qP waves in inhomogeneous, weakly anisotropic media

Geophysics ◽  
2005 ◽  
Vol 70 (6) ◽  
pp. D65-D75 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Ray Tracing ◽  
Anisotropic Media ◽  
Order Accuracy ◽  
Weak Anisotropy ◽  
Higher Symmetry ◽  
First Order ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Order Quantities ◽  
Relative Errors

We propose approximate ray-tracing equations for qP-waves propagating in smooth, inhomogeneous, weakly anisotropic media. For their derivation, we use perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. The proposed ray-tracing equations and corresponding traveltimes are of the first order. Accuracy of the traveltimes can be increased by calculating a secondorder correction along first-order rays. The first-order ray-tracing equations for qP-waves propagating in a general weakly anisotropic medium depend on only 15 weak-anisotropy parameters (generalization of Thomsen’s parameters). The equations are thus considerably simpler than the exact ray-tracing equations. For higher-symmetry anisotropic media the equations differ only slightly from equations for isotropic media. They can thus substitute for the traditional isotropic ray tracers used in seismic processing. For vanishing anisotropy, the first-order ray-tracing equations reduce to standard, exact ray-tracing equations for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of calculated traveltimes on inhomogeneity of the medium. For anisotropy of about 8%, considered in the examples presented, the relative errors of the traveltimes, including the second-order correction, are well under 0.05%; for anisotropy of about 20%, they do not exceed 0.3%.

Polarization, phase velocity, and NMO velocity of qP-waves in arbitrary weakly anisotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1754-1766 ◽  
Author(s):  
Ivan Pšenčk ◽  
Dirk Gajewski
Keyword(s):  
Phase Velocity ◽  
Polarization Vector ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Natural Generalization ◽  
Weak Anisotropy ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Approximate Formulas

We present approximate formulas for the qP-wave phase velocity, polarization vector, and normal moveout velocity in an arbitrary weakly anisotropic medium obtained with first‐order perturbation theory. All these quantities are expressed in terms of weak anisotropy (WA) parameters, which represent a natural generalization of parameters introduced by Thomsen. The formulas presented and the WA parameters have properties of Thomsen’s formulas and parameters: (1) the approximate equations are considerably simpler than exact equations for qP-waves, (2) the WA parameters are nondimensional quantities, and (3) in isotropic media, the WA parameters are zero and the corresponding equations reduce to equations for isotropic media. In contrast to Thomsen’s parameters, the WA parameters are related linearly to the density normalized elastic parameters. For the transversely isotropic media with vertical axis of symmetry, the equations presented and the WA parameters reduce to the equations and linearized parameters of Thomsen. The accuracy of the formulas presented is tested on two examples of anisotropic media with relatively strong anisotropy: on a transversely isotropic medium with the horizontal axis of symmetry and on a medium with triclinic anisotropy. Although anisotropy is rather strong, the approximate formulas presented yield satisfactory results.

scholarly journals Reflection moveout approximation for a P-SV wave in a moderately anisotropic homogeneous vertical transverse isotropic layer

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. C75-C83 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Numerical Solution ◽  
Taylor Series Expansion ◽  
Pure Mode ◽  
Isotropic Layer ◽  
Weak Anisotropy ◽  
Reflected Waves ◽  
Quartic Equation ◽  
S Waves ◽  
Conversion Point ◽  
Quartic Term

A description of the subsurface is incomplete without the use of S-waves. Use of converted waves is one way to involve S-waves. We have developed and tested an approximate formula for the reflection moveout of a wave converted at a horizontal reflector underlying a homogeneous transversely isotropic layer with the vertical axis of symmetry. For its derivation, we use the weak-anisotropy approximation; i.e., we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. Traveltimes are calculated along reference rays of converted reflected waves in a reference isotropic medium. This requires the determination of the point of reflection (the conversion point) of the reference ray, at which the conversion occurs. This can be done either by a numerical solution of a quartic equation or by using a simple approximate solution. Presented tests indicate that the accuracy of the proposed moveout formula is comparable with the accuracy of formulas derived in a weak-anisotropy approximation for pure-mode reflected waves. Specifically, the tests indicate that the maximum relative traveltime errors are well below 1% for models with P- and SV-wave anisotropy of approximately 10% and less than 2% for models with P- and SV-wave anisotropy of 25% and 12%, respectively. For isotropic media, the use of the conversion point obtained by numerical solution of the quartic equation yields exact results. The approximate moveout formula is used for the derivation of approximate expressions for the two-way zero-offset traveltime, the normal moveout velocity and the quartic term of the Taylor series expansion of the squared traveltime.

New Principles of the Substructure Development in Metal Materials under Plastic Deformation, Revealed by Advanced X-Ray Methods

2004 ◽  
Vol 443-444 ◽  
pp. 259-262 ◽  
Author(s):  
Yuriy Perlovich ◽  
Margarita Isaenkova
Keyword(s):  
Wide Spectrum ◽  
Residual Deformation ◽  
Ray Method ◽  
Computer Data ◽  
Rolled Material ◽  
X Ray ◽  
Pole Figures ◽  
Metal Materials ◽  
First Time ◽  
Ray Methods

The substructure inhomogeneity of real textured metal materials was studied by use of the X-ray method of Generalized Pole Figures and the computer data treatment. Main regularities of substructure inhomogeneity were revealed for the first time. Substructure conditions of grains in rolled material form an extremely wide spectrum and vary by passing from texture maxima to texture minima, where residual deformation effects are most significant. The distribution of residual elastic microstrains in the orientational space of rolled material shows the distinct system.

Ray Method for Volume Waves in Anisotropic Media

Elastic Waves ◽  
2018 ◽  
pp. 191-205
Author(s):  
Vassily M. Babich ◽  
Aleksei P. Kiselev
Keyword(s):  
Anisotropic Media ◽  
Ray Method

scholarly journals Transformation to zero offset in transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 947-963 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Ray Tracing ◽  
Impulse Response ◽  
Homogeneous Medium ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Amplitude Distribution ◽  
Isotropic Media ◽  
Zero Offset ◽  
Vertical Inhomogeneity

Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.

Reflection and transmission responses in a layered transversely isotropic medium with horizontal symmetry axis

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. C143-C157 ◽  
Author(s):  
Song Jin ◽  
Alexey Stovas
Keyword(s):  
Good Accuracy ◽  
Symmetry Axis ◽  
Local Property ◽  
Transversely Isotropic ◽  
Weak Anisotropy ◽  
Reflection And Transmission ◽  
Numerical Tests ◽  
Transversely Isotropic Media ◽  
Horizontal Symmetry ◽  
Isotropic Media

Seismic wave reflection and transmission (R/T) responses characterize the subsurface local property, and the widely spread anisotropy has considerable influences even at small incident angles. We have considered layered transversely isotropic media with horizontal symmetry axes (HTI), and the symmetry axes were not restricted to be aligned. With the assumption of weak contrast across the interface, linear approximations for R/T coefficients normalized by vertical energy flux are derived based on a simple layered HTI model. We also obtain the approximation with the isotropic background medium under an additional weak anisotropy assumption. Numerical tests illustrate the good accuracy of the approximations compared with the exact results.

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