Reflection moveout approximations for P-waves in a moderately anisotropic homogeneous tilted transverse isotropy layer

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. C175-C185 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
3D Space ◽  
Axis Of Symmetry ◽  
Relative Errors ◽  
Symmetry Tests ◽  
Ti Media

We have developed approximate nonhyperbolic P-wave moveout formulas applicable to weakly or moderately anisotropic media of arbitrary anisotropy symmetry and orientation. Instead of the commonly used Taylor expansion of the square of the reflection traveltime in terms of the square of the offset, we expand the square of the reflection traveltime in terms of weak-anisotropy (WA) parameters. No acoustic approximation is used. We specify the formulas designed for anisotropy of arbitrary symmetry for the transversely isotropic (TI) media with the axis of symmetry oriented arbitrarily in the 3D space. Resulting formulas depend on three P-wave WA parameters specifying the TI symmetry and two angles specifying the orientation of the axis of symmetry. Tests of the accuracy of the more accurate of the approximate formulas indicate that maximum relative errors do not exceed 0.3% or 2.5% for weak or moderate P-wave anisotropy, respectively.

Quartic moveout coefficient: 3D description and application to tilted TI media

Geophysics ◽  
2003 ◽  
Vol 68 (5) ◽  
pp. 1600-1610 ◽  
Author(s):  
Andres Pech ◽  
Ilya Tsvankin ◽  
Vladimir Grechka
Keyword(s):  
Heterogeneous Media ◽  
Symmetry Axis ◽  
High Sensitivity ◽  
Transversely Isotropic ◽  
Seismic Inversion ◽  
P Wave ◽  
Individual Values ◽  
Weak Anisotropy ◽  
Essential Information ◽  
Ti Media

Nonhyperbolic (long‐spread) moveout provides essential information for a number of seismic inversion/processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4 that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection‐point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4 can be evaluated during the tracing of the zero‐offset ray, so long‐spread moveout can be modeled without time‐consuming multioffset, multiazimuth ray tracing. The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P‐waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt ν of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak‐anisotropy approximation to analyze the dependence of A4 on the anisotropic parameters. The linearized expression for A4 is proportional to the anellipticity coefficient η ≈ ε − δ and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips. The azimuthal variation of the quartic coefficient is governed by the tilt ν and reflector dip φ and has a much more complicated character than the NMO–velocity ellipse. For example, if the symmetry axis is vertical (VTI media, ν = 0) and the dip φ < 30°, A4 goes to zero on two lines with different azimuths where it changes sign. If the symmetry axis is orthogonal to the reflector (this model is typical for thrust‐and‐fold belts), the strike‐line quartic coefficient is defined by the well‐known expression for a horizontal VTI layer (i.e., it is independent of dip), while the dip‐line A4 is proportional to cos4 φ and rapidly decreases with dip. The high sensitivity of the quartic moveout coefficient to the parameter η and the tilt of the symmetry axis can be exploited in the inversion of wide‐azimuth, long‐spread P‐wave data for the parameters of TI media.

Analytic study of the geometrical spreading of P-waves in a layered transversely isotropic medium with a vertical symmetry axis

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1305-1315 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Isotropic Medium ◽  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Analytical Formula ◽  
Transversely Isotropic ◽  
P Wave ◽  
Geometrical Spreading ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Special Cases

An analytical formula for geometrical spreading is derived for a horizontally layered transversely isotropic medium with a vertical symmetry axis (VTI). With this expression, geometrical spreading can be determined using only the anisotropy parameters in the first layer, the traveltime derivatives, and the source‐receiver offset. Explicit, numerically feasible expressions for geometrical spreading are obtained for special cases of transverse isotropy (weak anisotropy and elliptic anisotropy). Geometrical spreading can be calculated for transversly isotropic (TI) media by using picked traveltimes of primary nonhyperbolic P-wave reflections without having to know the actual parameters in the deeper subsurface; no ray tracing is needed. Synthetic examples verify the algorithm and show that it is numerically feasible for calculation of geometrical spreading. For media with a few (4–5) layers, relative errors in the computed geometrical spreading remain less than 0.5% for offset/depth ratios less than 1.0. Errors that change with offset are attributed to inaccuracy in the expression used for nonhyberbolic moveout. Geometrical spreading is most sensitive to errors in NMO velocity, followed by errors in zero‐offset reflection time, followed by errors in anisotropy of the surface layer. New relations between group and phase velocities and between group and phase angles are shown in appendices.

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.

Effects of transverse isotropy on P‐wave AVO for gas sands

Geophysics ◽  
1993 ◽  
Vol 58 (6) ◽  
pp. 883-888 ◽  
Author(s):  
Ki Young Kim ◽  
Keith H. Wrolstad ◽  
Fred Aminzadeh
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
P Wave ◽  
Amplitude Variation ◽  
Amplitude Variation With Offset ◽  
Velocity Anisotropy ◽  
Special Cases ◽  
Class 1 ◽  
Zero Offset ◽  
Ti Media

Velocity anisotropy should be taken into account when analyzing the amplitude variation with offset (AVO) response of gas sands encased in shales. The anisotropic effects on the AVO of gas sands in transversely isotropic (TI) media are reviewed. Reflection coefficients in TI media are computed using a planewave formula based on ray theory. We present results of modeling special cases of exploration interest having positive reflectivity, near‐zero reflectivity, and negative reflectivity. The AVO reflectivity in anisotropic media can be decomposed into two parts; one for isotropy and the other for anisotropy. Zero‐offset reflectivity and Poisson’s ratio contrast are the most significant parameters for the isotropic component while the δ difference (Δδ) between shale and gas sand is the most important factor for the anisotropic component. For typical values of Tl anisotropy in shale (positive δ and ε), both δ difference (Δδ) and ε difference (Δε) amplify AVO effects. For small angles of incidence, Δδ plays an important role in AVO while Δε dominates for large angles of incidence. For typical values of δ and ε, the effects of anisotropy in shale are: (1) a more rapid increase in AVO for Class 3 and Class 2 gas sands, (2) a more rapid decrease in AVO for Class 1 gas sands, and (3) a shift in the offset of polarity reversal for some Class 1 and Class 2 gas sands.

Geometrical spreading of P-waves in horizontally layered, azimuthally anisotropic media

Geophysics ◽  
2005 ◽  
Vol 70 (5) ◽  
pp. D43-D53 ◽  
Author(s):  
Xiaoxia Xu ◽  
Ilya Tsvankin ◽  
Andrés Pech
Keyword(s):  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Azimuthal Velocity ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Geometrical Spreading ◽  
Weak Anisotropy ◽  
P Waves ◽  
Anisotropic Models ◽  
Wide Range

For processing and inverting reflection data, it is convenient to represent geometrical spreading through the reflection traveltime measured at the earth's surface. Such expressions are particularly important for azimuthally anisotropic models in which variations of geometrical spreading with both offset and azimuth can significantly distort the results of wide-azimuth amplitude-variation-with-offset (AVO) analysis. Here, we present an equation for relative geometrical spreading in laterally homogeneous, arbitrarily anisotropic media as a simple function of the spatial derivatives of reflection traveltimes. By employing the Tsvankin-Thomsen nonhyperbolic moveout equation, the spreading is represented through the moveout coefficients, which can be estimated from surface seismic data. This formulation is then applied to P-wave reflections in an orthorhombic layer to evaluate the distortions of the geometrical spreading caused by both polar and azimuthal anisotropy. The relative geometrical spreading of P-waves in homogeneous orthorhombic media is controlled by five parameters that are also responsible for time processing. The weak-anisotropy approximation, verified by numerical tests, shows that azimuthal velocity variations contribute significantly to geometrical spreading, and the existing equations for transversely isotropic media with a vertical symmetry axis (VTI) cannot be applied even in the vertical symmetry planes. The shape of the azimuthally varying spreading factor is close to an ellipse for offsets smaller than the reflector depth but becomes more complicated for larger offset-to-depth ratios. The overall magnitude of the azimuthal variation of the geometrical spreading for the moderately anisotropic model used in the tests exceeds 25% for a wide range of offsets. While the methodology developed here is helpful in modeling and analyzing anisotropic geometrical spreading, its main practical application is in correcting the wide-azimuth AVO signature for the influence of the anisotropic overburden.

scholarly journals Velocity analysis for transversely isotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (5) ◽  
pp. 1550-1566 ◽  
Author(s):  
Tariq Alkhalifah ◽  
Ilya Tsvankin
Keyword(s):  
Transversely Isotropic ◽  
Anisotropic Media ◽  
P Wave ◽  
Velocity Analysis ◽  
Analytic Equation ◽  
Weak Anisotropy ◽  
Time Processing ◽  
Time Migration ◽  
Ti Media ◽  
Ray Parameter

The main difficulty in extending seismic processing to anisotropic media is the recovery of anisotropic velocity fields from surface reflection data. We suggest carrying out velocity analysis for transversely isotropic (TI) media by inverting the dependence of P‐wave moveout velocities on the ray parameter. The inversion technique is based on the exact analytic equation for the normal‐moveout (NMO) velocity for dipping reflectors in anisotropic media. We show that P‐wave NMO velocity for dipping reflectors in homogeneous TI media with a vertical symmetry axis depends just on the zero‐dip value [Formula: see text] and a new effective parameter η that reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy. Our inversion procedure makes it possible to obtain η and reconstruct the NMO velocity as a function of ray parameter using moveout velocities for two different dips. Moreover, [Formula: see text] and η determine not only the NMO velocity, but also long‐spread (nonhyperbolic) P‐wave moveout for horizontal reflectors and the time‐migration impulse response. This means that inversion of dip‐moveout information allows one to perform all time‐processing steps in TI media using only surface P‐wave data. For elliptical anisotropy (ε = δ), isotropic time‐processing methods remain entirely valid. We show the performance of our velocity‐analysis method not only on synthetic, but also on field data from offshore Africa. Accurate time‐to‐depth conversion, however, requires that the vertical velocity [Formula: see text] be resolved independently. Unfortunately, it cannot be done using P‐wave surface moveout data alone, no matter how many dips are available. In some cases [Formula: see text] is known (e.g., from check shots or well logs); then the anisotropy parameters ε and δ can be found by inverting two P‐wave NMO velocities corresponding to a horizontal and a dipping reflector. If no well information is available, all three parameters ([Formula: see text], ε, and δ) can be obtained by combining our inversion results with shear‐wave information, such as the P‐SV or SV‐SV wave NMO velocities for a horizontal reflector. Generalization of the single‐layer NMO equation to layered anisotropic media with a dipping reflector provides a basis for extending anisotropic velocity analysis to vertically inhomogeneous media. We demonstrate how the influence of a stratified anisotropic overburden on moveout velocity can be stripped through a Dix‐type differentiation procedure.

Anisotropic parameters and P‐wave velocity for orthorhombic media

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1292-1309 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Fractured Reservoirs ◽  
Transversely Isotropic ◽  
Azimuthal Velocity ◽  
Analytic Solutions ◽  
Seismic Inversion ◽  
P Wave ◽  
Weak Anisotropy ◽  
P Waves ◽  
Velocity Anisotropy ◽  
Wide Range

Although orthorhombic (or orthotropic) symmetry is believed to be common for fractured reservoirs, the difficulties in dealing with nine independent elastic constants have precluded this model from being used in seismology. A notation introduced in this work is designed to help make seismic inversion and processing for orthorhombic media more practical by simplifying the description of a wide range of seismic signatures. Taking advantage of the fact that the Christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic (TI) media, we can replace the stiffness coefficients by two vertical (P and S) velocities and seven dimensionless parameters that represent an extension of Thomsen's anisotropy coefficients to orthorhombic models. By design, this notation provides a uniform description of anisotropic media with both orthorhombic and TI symmetry. The dimensionless anisotropic parameters introduced here preserve all attractive features of Thomsen notation in treating wave propagation and performing 2-D processing in the symmetry planes of orthorhombic media. The new notation has proved useful in describing seismic signatures outside the symmetry planes as well, especially for P‐waves. Linearization of P‐wave phase velocity in the anisotropic coefficients leads to a concise weak‐anisotropy approximation that provides good accuracy even for models with pronounced polar and azimuthal velocity variations. This approximation can be used efficiently to build analytic solutions for various seismic signatures. One of the most important advantages of the new notation is the reduction in the number of parameters responsible for P‐wave velocities and traveltimes. All kinematic signatures of P‐waves in orthorhombic media depend on just the vertical velocity [Formula: see text] and five anisotropic parameters, with [Formula: see text] serving as a scaling coefficient in homogeneous media. This conclusion, which holds even for orthorhombic models with strong velocity anisotropy, provides an analytic basis for application of P‐wave traveltime inversion and data processing algorithms in orthorhombic media.

Transverse isotropy versus lateral heterogeneity in the inversion of P-wave reflection traveltimes

Geophysics ◽  
1998 ◽  
Vol 63 (1) ◽  
pp. 204-212 ◽  
Author(s):  
Vladimir Y. Grechka
Keyword(s):  
Wave Reflection ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Numerical Differentiation ◽  
Linear Functions ◽  
P Wave ◽  
Lateral Heterogeneity ◽  
Anisotropic Parameter ◽  
Single Plane ◽  
Ti Media

Nonelliptic transverse isotropy may cause pronounced nonhyperbolic moveout of long‐spread P-wave reflection data. Lateral heterogeneity may alter the moveout in much the same way, and one can expect that a given P-wave reflection moveout may be interpreted equally well in terms of parameters of homogeneous transversely isotropic (TI) or laterally heterogeneous (LH) isotropic models. Here, the common‐midpoint (CMP) moveout of a P-wave reflected from a horizontal interface beneath a weakly laterally heterogeneous medium that is also weakly transversely isotropic is represented analytically in the form similar to that in homogeneous TI media. Both the normal‐moveout (NMO) velocity and the quartic moveout coefficient contain derivatives of the zero‐ offset traveltime t0 and the NMO velocity Vnmo with respect to the lateral coordinate. Despite the presence of heterogeneity, nonhyperbolic velocity analysis can be performed in the same way as in homogeneous TI models. If all parameters of the medium are linear functions of the lateral coordinate, heterogeneity does not influence the results of inversion for the anisotropic parameter η. However, to find η in the case of general lateral heterogeneity, the second derivative of Vnmo and the fourth derivative of t0 are needed. Since these high‐order derivatives are calculated from the data measured at discrete points by numerical differentiation, stability of η estimation is further reduced as compared to that in homogeneous TI media. Consequently, the trade‐off between anisotropy and heterogeneity significantly complicates the inversion of P-wave reflection traveltimes, even in the simplest model of a single plane layer.

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.

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.

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