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.

A convenient expression for the NMO velocity function in terms of ray parameter

Geophysics ◽  
1998 ◽  
Vol 63 (1) ◽  
pp. 275-278 ◽  
Author(s):  
Jack K. Cohen
Keyword(s):  
Phase Velocity ◽  
Phase Angle ◽  
Transversely Isotropic ◽  
P Waves ◽  
Velocity Function ◽  
Analysis Methodology ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Vti Media ◽  
Ray Parameter

Tsvankin (1995) derived an expression for the normal‐moveout (NMO) velocity from horizontal or dipping reflectors that is valid in symmetry planes of anisotropic media. This equation is written in terms of the phase velocity and its derivatives with respect to the phase angle with vertical. The phase angle, however, is not a seismic observable, while the ray parameter p is. Indeed, Alkhalifah and Tsvankin (1995) pointed out the importance of representing the NMO velocity function as a function of ray parameter and obtained a number of significant results via an indirect use of this variable. Subsequently, Cohen (1997) directly transformed the Tsvankin expression for the moveout velocity obtaining an expression involving the phase velocity expressed as a function of ray parameter. Applying this result to the normal moveout of P-waves in transversely isotropic media with a vertical symmetry axis (VTI media), he was able to get various approximations for the ray‐parameter form of the moveout function and thus give additional analytic insight into the success of the velocity‐analysis methodology Alkhalifah and Tsvankin.

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.

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.

Approximate dispersion relations for qP-qSV-waves in transversely isotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (3) ◽  
pp. 919-933 ◽  
Author(s):  
Michael A. Schoenberg ◽  
Maarten V. de Hoop
Keyword(s):  
Wave Propagation ◽  
Order Approximation ◽  
Transverse Isotropy ◽  
Slight Modification ◽  
Transversely Isotropic ◽  
P Wave ◽  
Slowness Surface ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
First Order Approximation

To decouple qP and qSV sheets of the slowness surface of a transversely isotropic (TI) medium, a sequence of rational approximations to the solution of the dispersion relation of a TI medium is introduced. Originally conceived to allow isotropic P-wave processing schemes to be generalized to encompass the case of qP-waves in transverse isotropy, the sequence of approximations was found to be applicable to qSV-wave processing as well, although a higher order of approximation is necessary for qSV-waves than for qP-waves to yield the same accuracy. The zeroth‐order approximation, about which all other approximations are taken, is that of elliptical TI, which contains the correct values of slowness and its derivative along and perpendicular to the medium’s axis of symmetry. Successive orders of approximation yield the correct values of successive orders of derivatives in these directions, thereby forcing the approximation into increasingly better fit at the intervening oblique angles. Practically, the first‐order approximation for qP-wave propagation and the second‐order approximation for qSV-wave propagation yield sufficiently accurate results for the typical transverse isotropy found in geological settings. After only slight modification to existing programs, the rational approximation allows for ray tracing, (f-k) domain migration, and split‐step Fourier migration in TI media—with little more difficulty than that encountered presently with such algorithms in isotropic media.

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 Dip‐moveout processing by Fourier transform in anisotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1260-1269 ◽  
Author(s):  
John E. Anderson ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Computing Time ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
P Wave ◽  
Surface Reflection ◽  
Anisotropic Models ◽  
Reflection Data ◽  
Isotropic Media ◽  
Vti Media

Conventional dip‐moveout (DMO) processing is designed for isotropic media and cannot handle angle‐dependent velocity. We show that Hale's isotropic DMO algorithm remains valid for elliptical anisotropy but may lead to serious errors for nonelliptical models, even if velocity anisotropy is moderate. Here, Hale's constant‐velocity DMO method is extended to anisotropic media. The DMO operator, to be applied to common‐offset data corrected for normal moveout (NMO), is based on the analytic expression for dip‐dependent NMO velocity given by Tsvankin. Since DMO correction in anisotropic media requires knowledge of the velocity field, it should be preceded by an inversion procedure designed to obtain the normal‐moveout velocity as a function of ray parameter. For transversely isotropic models with a vertical symmetry axis (VTI media), P‐wave NMO velocity depends on a single anisotropic coefficient (η) that can be determined from surface reflection data. Impulse responses and synthetic examples for typical VTI media demonstrate the accuracy and efficiency of this DMO technique. Once the inversion step has been completed, the NMO-DMO sequence does not take any more computing time than the genetic Hale method in isotropic media. Our DMO operator is not limited to vertical transverse isotropy as it can be applied in the same fashion in symmetry planes of more complicated anisotropic models such as orthorhombic.

P‐wave signatures and notation for transversely isotropic media: An overview

Geophysics ◽  
1996 ◽  
Vol 61 (2) ◽  
pp. 467-483 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Wave Propagation ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Seismic Inversion ◽  
P Wave ◽  
Symmetry Direction ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Progress in seismic inversion and processing in anisotropic media depends on our ability to relate different seismic signatures to the anisotropic parameters. While the conventional notation (stiffness coefficients) is suitable for forward modeling, it is inconvenient in developing analytic insight into the influence of anisotropy on wave propagation. Here, a consistent description of P‐wave signatures in transversely isotropic (TI) media with arbitrary strength of the anisotropy is given in terms of Thomsen notation. The influence of transverse isotropy on P‐wave propagation is shown to be practically independent of the vertical S‐wave velocity [Formula: see text], even in models with strong velocity variations. Therefore, the contribution of transverse isotropy to P‐wave kinematic and dynamic signatures is controlled by just two anisotropic parameters, ε and δ, with the vertical velocity [Formula: see text] being a scaling coefficient in homogeneous models. The distortions of reflection moveouts and amplitudes are not necessarily correlated with the magnitude of velocity anisotropy. The influence of transverse isotropy on P‐wave normal‐moveout (NMO) velocity in a horizontally layered medium, on small‐angle reflection coefficient, and on point‐force radiation in the symmetry direction is entirely determined by the parameter δ. Another group of signatures of interest in reflection seisimology—the dip‐dependence of NMO velocity, magnitude of nonhyperbolic moveout, time‐migration impulse response, and the radiation pattern near vertical—is dependent on both anisotropic parameters (ε and δ) and is primarily governed by the difference between ε and δ. Since P‐wave signatures are so sensitive to the value of ε − δ, application of the elliptical‐anisotropy approximation (ε = δ) in P‐wave processing may lead to significant errors. Many analytic expressions given in the paper remain valid in transversely isotropic media with a tilted symmetry axis. Moreover, the equation for NMO velocity from dipping reflectors, as well as the nonhyperbolic moveout equation, can be used in symmetry planes of any anisotropic media (e.g., orthorhombic).

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.

Inversion of ultrasonic data for transversely isotropic media

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. C1-C7 ◽  
Author(s):  
Yevhen Kovalyshen ◽  
Joel Sarout ◽  
Jeremie Dautriat
Keyword(s):  
Numerical Algorithm ◽  
Seismic Data ◽  
Symmetry Axis ◽  
Rock Sample ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Wave Velocities ◽  
Transversely Isotropic Media ◽  
Isotropic Media

We have developed a new numerical algorithm for inversion of ultrasonic data in transversely isotropic media. This algorithm is able to determine from the measured P-wave velocities the orientation of the symmetry axis of a rock sample and the Thomsen’s parameters, only assuming transverse isotropy. The inversion of ultrasonic data acquired on natural and potentially heterogeneous shale samples produced reasonable results. In addition, the algorithm was successfully tested on ultrasonic data acquired on synthetic samples with predefined orientations of the symmetry axis. An additional outcome of the algorithm is a simple approximation of Thomsen’s formulation, which can be effectively used for interpretation of seismic data in transversely isotropic media.

Fowler DMO and time migration for transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 835-845 ◽  
Author(s):  
John Anderson ◽  
Tariq Alkhalifah ◽  
Ilya Tsvankin
Keyword(s):  
Transverse Isotropy ◽  
Computing Time ◽  
Transversely Isotropic ◽  
P Wave ◽  
Effective Parameters ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Zero Offset

The main advantage of Fowler’s dip‐moveout (DMO) method is the ability to perform velocity analysis along with the DMO removal. This feature of Fowler DMO becomes even more attractive in anisotropic media, where imaging methods are hampered by the difficulty in reconstructing the velocity field from surface data. We have devised a Fowler‐type DMO algorithm for transversely isotropic media using the analytic expression for normal‐moveout velocity. The parameter‐estimation procedure is based on the results of Alkhalifah and Tsvankin showing that in transversely isotropic media with a vertical axis of symmetry (VTI) the P‐wave normal‐moveout (NMO) velocity as a function of ray parameter can be described fully by just two coefficients: the zero‐dip NMO velocity [Formula: see text] and the anisotropic parameter η (η reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy). In this extension of Fowler DMO, resampling in the frequency‐wavenumber domain makes it possible to obtain the values of [Formula: see text] and η by inspecting zero‐offset (stacked) panels for different pairs of the two parameters. Since most of the computing time is spent on generating constant‐velocity stacks, the added computational effort caused by the presence of anisotropy is relatively minor. Synthetic and field‐data examples demonstrate that the isotropic Fowler DMO technique fails to generate an accurate zero‐offset section and to obtain the zero‐dip NMO velocity for nonelliptical VTI models. In contrast, this anisotropic algorithm allows one to find the values of the parameters [Formula: see text] and η (sufficient to perform time migration as well) and to correct for the influence of transverse isotropy in the DMO processing. When combined with poststack F-K Stolt migration, this method represents a complete inversion‐processing sequence capable of recovering the effective parameters of transversely isotropic media and producing migrated images for the best‐fit homogeneous anisotropic model.

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