THREE‐TERM TAYLOR SERIES FOR t2 ‐ x2‐CURVES OF P‐ AND S‐WAVES OVER LAYERED TRANSVERSELY ISOTROPIC GROUND*

Hake, H.; Helbig, Klaus; Mesdag, C.S.

doi:10.1111/j.1365-2478.1984.tb00742.x

Cited by 117 publications

(93 citation statements)

References 14 publications

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“…For example, in the case of isotropy or elliptical anisotropy, the reflection moveout is hyperbolic (A 4 = 0). For VTI symmetry, equation (6) reduces to the known azimuthally independent quartic coefficient A 4 given by Hake et al (1984) and Tsvankin and Thomsen (1994). In the case of a horizontal HTI layer, with a horizontal symmetry axis parallel to x 1 of the (x 1 ,x 3 ) plane, the components A …”

Section: Nhmo Coefficient Amentioning

confidence: 99%

“…On conventional-length spreads (spreadlength-to-depth ratio ≤ 1), the hyperbolic moveout equation (1) can be expected to provide an adequate description of the moveout, but the NMO velocity should be averaged over the stack of layers. In isotropic and VTI media, this averaging is performed by means of the conventional isotropic Dix (1955) equation (Hake et al, 1984). Furthermore, Alkhalifah and Tsvankin (1995) showed that the Dix equation remains valid in symmetry planes of any anisotropic medium, if the interval NMO velocities are evaluated at the ray-parameter value of the zerooffset ray.…”

Section: Moveout In Multilayered Mediamentioning

confidence: 99%

“…The parameter A 4 for pure modes in horizontally layered VTI media was given by Hake et al (1984) and represented in a more compact form by Tsvankin and Thomsen (1994). Due to the influence of the x 4 term, the quartic equation (2) becomes divergent with increasing offset and can be replaced by a more accurate nonhyperbolic moveout equation developed by Tsvankin and Thomsen (1994):…”

Section: Analytic Approximations Of Reflection Moveoutmentioning

confidence: 99%

“…Recent studies and case histories (Lynn et al, 1996;Corrigan et al, 1996) have shown that wave propagation signatures, including reflection moveout and amplitude-variation-with-offset (AVO), are sensitive to the presence of azimuthal anisotropy. Hake et al (1984) derived the quartic Taylor series term A 4 of t 2 − x 2 reflection-moveout curves for pure modes in TI media with a vertical axis of symmetry. Tsvankin and Thomsen (1994) recasted the quartic term of Hake et al (1984) in a more compact form using Thomsen's (1986) notation.…”

Section: Introductionmentioning

confidence: 99%

“…Hake et al (1984) derived the quartic Taylor series term A 4 of t 2 − x 2 reflection-moveout curves for pure modes in TI media with a vertical axis of symmetry. Tsvankin and Thomsen (1994) recasted the quartic term of Hake et al (1984) in a more compact form using Thomsen's (1986) notation. They also introduced a normalization factor for the quartic term that ensures the convergence of the Taylor series traveltime expansion at infinitely large horizontal offsets for VTI media.…”

Section: Introductionmentioning

confidence: 99%

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Nonhyperbolic Reflection Moveout for Orthorhombic Media

Al-Dajani¹,

Toksöz²

1998

60th EAGE Conference and Exhibition

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Reflection moveout in azimuthally anisotropic media is not only azimuthally dependent but it is also nonhyperbolic. As a result, the conventional hyperbolic normal moveout (NMO) equation parameterized by the exact NMO (stacking) velocity loses accuracy with increasing offset (i.e., spreadlength). This is true even for a single-homogeneous azimuthally anisotropic layer. The most common azimuthally anisotropic models used to describe fractured media are the horizontal transverse isotropy (HTI) and the orthorhombic (ORT).Here, we introduce an analytic representation for the quartic coefficient of the Taylor's series expansion of the two-way traveltime for pure mode reflection (i.e., no conversion) in arbitrary anisotropic media with arbitrary strength of anisotropy. In addition, we present an analytic expression for the long-spread (large-offset) nonhyperbolic reflection moveout (NHMO). In this study, special attention is given to Pwave propagation in orthorhombic media with horizontal interfaces. The quartic coefficient, in general, has a relatively simple form, especially for shear wave propagation. The reflection moveout for each shear-wave mode in a homogeneous orthorhombic medium is purely hyperbolic in the direction normal to the polarization. In addition, the nonhyperbolic portion of the moveout for shear-wave propagation reaches its maximum along the polarization direction, and it decreases rapidly away from the direction of polarization. Hence, the anisotropy-induced nonhyperbolic reflection moveout for shear-wave propagation is significant in the vicinity of the polarization directions.In multilayered azimuthally anisotropic media, the NMO (stacking) velocity and the quartic moveout coefficient can be calculated with good accuracy using Dix-type averaging (e.g., the known averaging equations for VTI media). The interval NMO velocities and the interval quartic coefficients, however, are azimuthally dependent. This allows us to extend the nonhyperbolic moveout (NHMO) equation, originally designed for VTI media, to more general horizontally stratified azimuthally anisotropic media. Numerical examples from reflection moveout in orthorhombic media, the focus of this paper, show that this NHMO equation accurately describes the azimuthally-dependent P -wave reflection traveltimes, even on spreadlengths twice as large as the reflector depth. This work provides analytic insight into the behavior of nonhyperbolic moveout, and it has important applications in modeling and inversion of reflection moveout in azimuthally anisotropic media.

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Section: Nhmo Coefficient Amentioning

confidence: 99%

Section: Moveout In Multilayered Mediamentioning

confidence: 99%

Section: Analytic Approximations Of Reflection Moveoutmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonhyperbolic Reflection Moveout for Orthorhombic Media

Al-Dajani¹,

Toksöz²

1998

60th EAGE Conference and Exhibition

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The Characteristic of Quartic Moveout Coefficient of P‐Wave in the TI Homogeneous Media with an Arbitrary Orientation Symmetric‐Axis

Chen¹,

Chen²

2008

Chinese J of Geophysics

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Deviations of any pure mode (non‐converted) wave reflection traveltimes from a hyperbola, called the nonhyperbolic or quartic moveout, need to be corrected properly in processing long‐spread anisotropic seismic data. Based on our exact analytic expression for the quartic moveout coefficient (A4) of pure mode (non‐converted) wave reflection on a horizontal interface in the TI media with an arbitrarily orientated symmetry axis (ATI) in space, we numerically study the characteristics of the P‐wave A4 with the CMP survey line azimuth variation. The forward modeling results show that the characteristics of the A4 are related not only to the anisotropy parameters of the TI, but also the spatial orientation of the TI. The variation characteristic of the A4 is determined by the anisotropy parameters and tilt angle of the TI symmetric‐axis, and the azimuth of the TI symmetric‐axis results in the azimuth symmetry of the A4 with the survey line variation. Our numerical results will benefit the explanation of the anisotropic seismic data and the parameter inversion.

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Analytic Approximation of Quartic Moveout Coefficient in the TI Media with an Arbitrary Spatial Orientation

Chen¹,

Chen²

2009

Chinese J of Geophysics

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Nonhyperbolic (far‐offset) moveout provides essential information for seismic modeling and inversion studies, particularly for parameter estimation in anisotropic media. Based on the exact analytic solution of the quartic moveout coefficient A4 for a horizontal interface in the TI media with an arbitrary spatial orientation of symmetry axis (ATI), we further discuss our derived approximate analytic expression of A4, in order to study the difference between the approximate and exact solution of A4 when the azimuth of CMP survey line varies in the ATI media. This research may provide valuable information for the analytic analysis of nonhyperbolic traveltimeoffset and anisotropy parameter inversion by using approximate solution of A4 in the ATI media. Combined with real rock data, the modeling results show that the distinction between the approximate and exact solution of the P‐wave A4 exists not only in the value and sign of A4, but also prominently in the azimuthal signature of A4. For the strongly anisotropic media, the approximate solution has notable errors comparing with the exact solution. At the same time, there is little difference between the approximate and exact solution of A4 when the dip angle of the symmetry axis is between 75°~80° and the anisotropic parameters satisfy 0 <ε−δ < 0.15 and |δ| < 0.20, in such cases we can utilize the approximate solution of A4 for making anisotropy observation, explanation, and parameter inversion.

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THREE‐TERM TAYLOR SERIES FOR t² ‐ x²‐CURVES OF P‐ AND S‐WAVES OVER LAYERED TRANSVERSELY ISOTROPIC GROUND*

Cited by 117 publications

References 14 publications

Nonhyperbolic Reflection Moveout for Orthorhombic Media

Nonhyperbolic Reflection Moveout for Orthorhombic Media

The Characteristic of Quartic Moveout Coefficient of P‐Wave in the TI Homogeneous Media with an Arbitrary Orientation Symmetric‐Axis

Analytic Approximation of Quartic Moveout Coefficient in the TI Media with an Arbitrary Spatial Orientation

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