Acoustic approximations for processing in transversely isotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 623-631 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Elastic Model ◽  
S Wave ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Two Parameters ◽  
Vti Media

When transversely isotropic (VTI) media with vertical symmetry axes are characterized using the zero‐dip normal moveout (NMO) velocity [[Formula: see text]] and the anisotropy parameter ηinstead of Thomsen’s parameters, time‐related processing [moveout correction, dip moveout (DMO), and time migration] become nearly independent of the vertical P- and S-wave velocities ([Formula: see text] and [Formula: see text], respectively). The independence on [Formula: see text] and [Formula: see text] is well within the limits of seismic accuracy, even for relatively strong anisotropy. The dependency on [Formula: see text] and [Formula: see text] reduces even further as the ratio [Formula: see text] decreases. In fact, for [Formula: see text], all time‐related processing depends exactly on only [Formula: see text] and η. This fortunate dependence on two parameters is demonstrated here through analytical derivations of time‐related processing equations in terms of [Formula: see text] and η. The time‐migration dispersion relation, the NMO velocity for dipping events, and the ray‐tracing equations extracted by setting [Formula: see text] (i.e., by considering VTI as acoustic) not only depend solely on [Formula: see text] and η but are much simpler than the counterpart expressions for elastic media. Errors attributed to this use of the acoustic assumption are small and may be neglected. Therefore, as in isotropic media, the acoustic model arising from setting [Formula: see text], although not exactly true for VTI media, can serve as a useful approximation to the elastic model for the kinematics of P-wave data. This approximation can boost the efficiency of imaging and DMO programs for VTI media as well as simplify their description.

scholarly journals The offset‐midpoint traveltime pyramid in transversely isotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1316-1325 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Fourth Power ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Equivalent Equation ◽  
Two Parameters ◽  
Vti Media

Prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an offset‐midpoint traveltime equation, Cheop’s pyramid equivalent equation for VTI media. The derivation of such an equation for VTI media requires approximations that pertain to high frequency and weak anisotropy. Yet the resultant offset‐midpoint traveltime equation for VTI media is highly accurate for even strong anisotropy. It is also strictly dependent on two parameters: NMO velocity and the anisotropy parameter, η. It reduces to the exact offset‐midpoint traveltime equation for isotropic media when η = 0. In vertically inhomogeneous media, the NMO velocity and η parameters in the offset‐midpoint traveltime equation are replaced by their effective values: the velocity is replaced by the rms velocity and η is given by a more complicated equation that includes summation of the fourth power of velocity.

Velocity analysis using nonhyperbolic moveout in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1839-1854 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Vertical Heterogeneity ◽  
Isotropic Media ◽  
D Values ◽  
Two Parameters ◽  
Vti Media ◽  
Ti Media

P‐wave reflections from horizontal interfaces in transversely isotropic (TI) media have nonhyperbolic moveout. It has been shown that such moveout as well as all time‐related processing in TI media with a vertical symmetry axis (VTI media) depends on only two parameters, [Formula: see text] and η. These two parameters can be estimated from the dip‐moveout behavior of P‐wave surface seismic data. Alternatively, one could use the nonhyperbolic moveout for parameter estimation. The quality of resulting estimates depends largely on the departure of the moveout from hyperbolic and its sensitivity to the estimated parameters. The size of the nonhyperbolic moveout in TI media is dependent primarily on the anisotropy parameter η. An “effective” version of this parameter provides a useful measure of the nonhyperbolic moveout even in v(z) isotropic media. Moreover, effective η, [Formula: see text], is used to show that the nonhyperbolic moveout associated with typical TI media (e.g., shales, with η ≃ 0.1) is larger than that associated with typical v(z) isotropic media. The departure of the moveout from hyperbolic is increased when typical anisotropy is combined with vertical heterogeneity. Larger offset‐to‐depth ratios (X/D) provide more nonhyperbolic information and, therefore, increased stability and resolution in the inversion for [Formula: see text]. The X/D values (e.g., X/D > 1.5) needed for obtaining stability and resolution are within conventional acquisition limits, especially for shallow targets. Although estimation of η using nonhyperbolic moveouts is not as stable as using the dip‐moveout method of Alkhalifah and Tsvankin, particularly in the absence of large offsets, it does offer some flexibility. It can be applied in the absence of dipping reflectors and also may be used to estimate lateral η variations. Application of the nonhyperbolic inversion to data from offshore Africa demonstrates its usefulness, especially in estimating lateral and vertical variations in η.

Kirchhoff time migration for transversely isotropic media: An application to Trinidad data

Geophysics ◽  
2006 ◽  
Vol 71 (1) ◽  
pp. S29-S35 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Data Set ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Migration Method ◽  
Vti Media ◽  
Vertical Inhomogeneity

Using a newly developed nonhyperbolic offset-mid-point traveltime equation for prestack Kirchhoff time migration, instead of the conventional double-square-root (DSR) equation, results in overall better images from anisotropic data. Specifically, prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an analytical offset-midpoint traveltime equation that represents the equivalent of Cheop's pyramid for VTI media. It includes higher-order terms necessary to better handle anisotropy as well as vertical inhomogeneity. Application of this enhanced Kirchhoff time-migration method to the anisotropic Marmousi data set demonstrates the effectiveness of the approach. Further application of the method to field data from Trinidad results in sharper reflectivity images of the subsurface, with the faults better focused and positioned than with images obtained using isotropic methods. The superiority of the anisotropic time migration is evident in the flatness of the image gathers.

Analytic study of the effective parameters for determination of the NMO velocity function in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1855-1866 ◽  
Author(s):  
Jack K. Cohen
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Velocity Function ◽  
Transversely Isotropic Media ◽  
Wave Phase Velocity ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Vti Media ◽  
Ray Parameter

In their studies of transversely isotropic media with a vertical symmetry axis (VTI media), Alkhalifah and Tsvankin observed that, to a high numerical accuracy, the normal moveout (NMO) velocity for dipping reflectors as a function of ray parameter p depends mainly on just two parameters, each of which can be determined from surface P‐wave observations. They substantiated this result by using the weak‐anisotropy approximation and exploited it to develop a time‐domain processing sequence that takes into account vertical transverse isotropy. In this study, the two‐parameter Alkhalifah‐Tsvankin result was further examined analytically. It was found that although there is (as these authors already observed) some dependence on the remaining parameters of the problem, this dependence is weak, especially in the practically important regimes of weak to moderately strong transverse isotropy and small ray parameter. In each of these regimes, an analytic solution is derived for the anisotropy parameter η required for time‐domain P‐wave imaging in VTI media. In the case of elliptical anisotropy (η = 0), NMO velocity expressed through p is fully controlled just by the zero‐dip NMO velocity—one of the Alkhalifah‐ Tsvankin parameters. The two‐parameter representation of NMO velocity also was shown to be exact in another limit—that of the zero shear‐wave vertical velociy. The analytic results derived here are based on new representations for both the P‐wave phase velocity and normal moveout velocity in terms of the ray parameter, with explicit expressions given for the cases of vanishing onaxis shear speed, weak to moderate transverse isotropy, and small to moderate ray parameter. Using these formulas, I have rederived and, in some cases, extended in a uniform manner various results of Tsvankin, Alkhalifah, and others. Examples include second‐order expansions in the anisotropy parameters for both the P‐wave phase‐velocity function and NMO‐velocity function, as well as expansions in powers of the ray parameter for both of these functions. I have checked these expansions against the corresponding exact functions for several choices of the anisotropy parameters.

The effects of transverse isotropy on reflection amplitude versus offset

Geophysics ◽  
1987 ◽  
Vol 52 (4) ◽  
pp. 564-567 ◽  
Author(s):  
J. Wright
Keyword(s):  
Wave Velocity ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
S Wave ◽  
Reflection Amplitude ◽  
Amplitude Versus Offset ◽  
Transversely Isotropic Media ◽  
P Wave Velocity ◽  
Isotropic Media

Studies have shown that elastic properties of materials such as shale and chalk are anisotropic. With the increasing emphasis on extraction of lithology and fluid content from changes in reflection amplitude with shot‐to‐group offset, one needs to know the effects of anisotropy on reflectivity. Since anisotropy means that velocity depends upon the direction of propagation, this angular dependence of velocity is expected to influence reflectivity changes with offset. These effects might be particularly evident in deltaic sand‐shale sequences since measurements have shown that the P-wave velocity of shales in the horizontal direction can be 20 percent higher than the vertical P-wave velocity. To investigate this behavior, a computer program was written to find the P- and S-wave reflectivities at an interface between two transversely isotropic media with the axis of symmetry perpendicular to the interface. Models for shale‐chalk and shale‐sand P-wave reflectivities were analyzed.

Rational interpolation of qP-traveltimes for semblance-based anisotropy estimation in layered VTI media

Geophysics ◽  
2008 ◽  
Vol 73 (4) ◽  
pp. D53-D62 ◽  
Author(s):  
Huub Douma ◽  
Mirko van der Baan
Keyword(s):  
Parameter Estimation ◽  
Random Noise ◽  
Anisotropy Parameter ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Rational Interpolation ◽  
Transversely Isotropic Media ◽  
Interpolation Procedure ◽  
Isotropic Media ◽  
Vti Media

The [Formula: see text] domain is the natural domain for anisotropy parameter estimation in horizontally layered media. The need to transform the data to the [Formula: see text] domain or to pick traveltimes in the [Formula: see text] domain is, however, a practical disadvantage. To overcome this, we combine [Formula: see text]-derived traveltimes and offsets in horizontally layered transversely isotropic media with a vertical symmetry axis (VTI) with a rational interpolation procedure applied in the [Formula: see text] domain. This combination results in an accurate and efficient [Formula: see text]-based semblance analysis for anisotropy parameter estimation from the moveout of qP-waves in horizontally layered VTI media. The semblance analysis is applied to the moveout to search directly for the interval values of the relevant parameters. To achieve this, the method is applied in a layer-stripping fashion. We demonstrate the method using synthetic data examples and show that it is robust in the presence of random noise and moderate statics.

An effective anisotropy parameter in transversely isotropic media

Geophysics ◽  
1987 ◽  
Vol 52 (12) ◽  
pp. 1654-1664 ◽  
Author(s):  
N. C. Banik
Keyword(s):  
Transverse Isotropy ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
Reflection Seismic ◽  
The North ◽  
Reflection Amplitude ◽  
Transversely Isotropic Media ◽  
Isotropic Media

An interesting physical meaning is presented for the anisotropy parameter δ, previously introduced by Thomsen to describe weak anisotropy in transversely isotropic media. Roughly, δ is the difference between the P-wave and SV-wave anisotropies of the medium. The observed systematic depth errors in the North Sea are reexamined in view of the new interpretation of the moveout velocity through δ. The changes in δ at an interface adequately describe the effects of transverse isotropy on the P-wave reflection amplitude, The reflection coefficient expression is linearized in terms of changes in elastic parameters. The linearized expression clearly shows that it is the variation of δ at the interface that gives the anisotropic effects at small incidence angles. Thus, δ effectively describes both the moveout velocity and the reflection amplitude variation, two very important pieces of information in reflection seismic prospecting, in the presence of transverse isotropy.

scholarly journals An acoustic eikonal equation for attenuating transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. C9-C20 ◽  
Author(s):  
Qi Hao ◽  
Tariq Alkhalifah
Keyword(s):  
Eikonal Equation ◽  
Wave Attenuation ◽  
Transversely Isotropic ◽  
S Wave ◽  
P Waves ◽  
The Earth ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Vti Media ◽  
Complex Valued

Seismic-wave attenuation is an important component of describing wave propagation. Certain regions, such as gas clouds inside the earth, exert highly localized attenuation. In fact, the anisotropic nature of the earth induces anisotropic attenuation because the quasi P-wave dispersion effect should be profound along the symmetry direction. We have developed a 2D acoustic eikonal equation governing the complex-valued traveltime of quasi P-waves in attenuating, transversely isotropic media with a vertical-symmetry axis (VTI). This equation is derived under the assumption that the complex-valued traveltime of quasi P-waves in attenuating VTI media are independent of the S-wave velocity parameter [Formula: see text] in Thomsen’s notation and the S-wave attenuation coefficient [Formula: see text] in Zhu and Tsvankin’s notation. We combine perturbation theory and Shanks transform to develop practical approximations to the acoustic attenuating eikonal equation, capable of admitting an analytical description of the attenuation in homogeneous media. For a horizontal-attenuating VTI layer, we also derive the nonhyperbolic approximations for the real and imaginary parts of the complex-valued reflection traveltime. These equations reveal that (1) the quasi SV-wave velocity and the corresponding quasi SV-wave attenuation coefficient given as part of Thomsen-type notation barely affect the ray velocity and ray attenuation of quasi P-waves in attenuating VTI media; (2) combining the perturbation method and Shanks transform provides an accurate analytic eikonal solution for homogeneous attenuating VTI media; (3) for a horizontal attenuating VTI layer with weak attenuation, the real part of the complex-valued reflection traveltime may still be described by the existing nonhyperbolic approximations developed for nonattenuating VTI media, and the imaginary part of the complex-valued reflection traveltime still has the shape of nonhyperbolic curves. In addition, we have evaluated the possible extension of the proposed eikonal equation to realistic attenuating media, an alternative perturbation solution to the proposed eikonal equation, and the feasibility of applying the proposed nonhyperbolic equation for the imaginary part of the complex-valued traveltime to invert for interval attenuation parameters.

Scalar and vector imaging based on wave mode decoupling for elastic reverse time migration in isotropic and transversely isotropic media

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. S383-S398 ◽  
Author(s):  
Chenlong Wang ◽  
Jiubing Cheng ◽  
Børge Arntsen
Keyword(s):  
Computational Cost ◽  
Transversely Isotropic ◽  
Polarity Reversal ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Wave Modes

Recording P- and S-wave modes acquires more information related to rock properties of the earth’s interior. Elastic migration, as a part of multicomponent seismic data processing, potentially offers a great improvement over conventional acoustic migration to create a spatial image of some medium properties. In the framework of elastic reverse time migration, we have developed new scalar and vector imaging conditions assisted by efficient polarization-based mode decoupling to avoid crosstalk among the different wave modes for isotropic and transversely isotropic media. For the scalar imaging, we corrected polarity reversal of zero-lag PS images using the local angular attributes on the fly of angle-domain imaging. For the vector imaging, we naturally used the polarization information in the decoupled single-mode vector fields to automatically avoid the polarity reversal and to estimate the local angular attributes for angle-domain imaging. Examples of increasing complexity in 2D and 3D cases found that the proposed approaches can be used to obtain a physically interpretable image and angle-domain common-image gather at an acceptable computational cost. Decoupling and imaging the 3D S-waves involves some complexity, which has not been addressed in the literature. For this reason, we also attempted at illustrating the physical contents of the two separated S-wave modes and their contribution to seismic full-wave imaging.

The joint nonhyperbolic moveout inversion of PP and PS data in VTI media

Geophysics ◽  
2002 ◽  
Vol 67 (6) ◽  
pp. 1929-1932 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Vertical Velocity ◽  
Single Layer ◽  
Transversely Isotropic ◽  
Accurate Estimate ◽  
P Wave ◽  
S Wave ◽  
P Waves ◽  
Layer Cake ◽  
Two Parameters ◽  
Vti Media

Nonhyperbolic moveout of P‐waves in horizontally layered transversely isotropic media with a vertical symmetry axis (VTI) can be used to estimate the anellipticity coefficient η in addition to the NMO velocity Vnmo,P. Those two parameters are sufficient for time processing of P‐wave data (despite a certain instability in the inversion for η), but they do not constrain the vertical velocity VP0 and the depth scale of the model. It has been suggested in the literature that this ambiguity in the depth‐domain velocity analysis for layer‐cake VTI media can be resolved by combining long‐spread reflection traveltimes of P‐waves and mode‐converted PSV‐waves. Here, we show that reflection traveltimes of horizontal PSV events help to determine the ratio of the P‐ and S‐wave vertical velocities and the NMO velocity of SV‐waves, and they give a more accurate estimate of η. However, nonhyperbolic moveout of PSV‐waves turns out to be mostly controlled by wide‐angle P‐wave traveltimes and does not provide independent information for the inversion. As a result, even for a single‐layer model and uncommonly large offsets, traveltimes of P‐ and PSV‐waves cannot be inverted for the vertical velocity and anisotropic parameters ε and δ. To reconstruct the horizontally layered VTI model from surface data, it is necessary to combine long‐spread traveltimes of pure P and SV reflections.

Sign in / Sign up

Export Citation Format

Share Document