Reflection coefficients in attenuative anisotropic media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB193-WB202 ◽  
Author(s):  
Jyoti Behura ◽  
Ilya Tsvankin
Keyword(s):  
Plane Wave ◽  
Wave Reflection ◽  
Symmetry Plane ◽  
Anisotropic Media ◽  
P Wave ◽  
Substantial Contribution ◽  
Reflection Coefficients ◽  
Slowness Vector ◽  
Anisotropy Parameters ◽  
Attenuation Anisotropy

Such reservoir rocks as tar sands are characterized by significant attenuation and, in some cases, attenuation anisotropy. Most existing attenuation studies are focused on plane-wave attenuation coefficients, which determine the amplitude decay along the raypath of seismic waves. Here we study the influence of attenuation on PP- and PS-wave reflection coefficients for anisotropic media with the main emphasis on transversely isotropic models with a vertical symmetry axis (VTI). Concise analytic solutions obtained by linearizing the exact plane-wave reflection coefficients are verified by numerical modeling. To make a substantial contribution to reflection coefficients, attenuation must be strong, with the quality factor [Formula: see text] not exceeding 10. For such highly attenuative media, it is also necessary to take attenuation anisotropy into account if the magnitude of the Thomsen-styleattenuation-anisotropy parameters is relatively large. In general, the linearized reflection coefficients in attenuative media include velocity-anisotropy parameters but have almost “isotropic” dependence on attenuation. Our formalism also helps evaluate the influence of the inhomogeneity angle (the angle between the real and imaginary parts of the slowness vector) on the reflection coefficients. A nonzero inhomogeneity angle of the incident wave introduces additional terms into the PP- and PS-wave reflection coefficients, which makes conventional amplitude-variation-with-offset (AVO) analysis inadequate for strongly attenuative media. For instance, an incident P-wave with a nonzero inhomogeneity angle generates a mode-converted PS-wave at normal incidence, even if both half-spaces have a horizontal symmetry plane. The developed linearized solutions can be used in AVO inversion for highly attenuative (e.g., gas-sand and heavy-oil) reservoirs.

Variation of P-wave reflectivity with offset and azimuth in anisotropic media

Geophysics ◽  
1998 ◽  
Vol 63 (3) ◽  
pp. 935-947 ◽  
Author(s):  
Andreas Rüger
Keyword(s):  
Reflection Coefficient ◽  
Shear Wave ◽  
Wave Reflection ◽  
Symmetry Plane ◽  
Anisotropic Media ◽  
Shear Wave Splitting ◽  
P Wave ◽  
Reflection Coefficients ◽  
Wave Splitting ◽  
Splitting Parameter

P-wave amplitudes may be sensitive even to relatively weak anisotropy of rock mass. Recent results on symmetry‐plane P-wave reflection coefficients in azimuthally anisotropic media are extended to observations at arbitrary azimuth, large incidence angles, and lower symmetry systems. The approximate P-wave reflection coefficient in transversely isotropic media with a horizontal axis of symmetry (HTI) (typical for a system of parallel vertical cracks embedded in an isotropic matrix) shows that the amplitude versus offset (AVO) gradient varies as a function of the squared cosine of the azimuthal angle. This change can be inverted for the symmetry‐plane directions and a combination of the shear‐wave splitting parameter γ and the anisotropy coefficient [Formula: see text]. The reflection coefficient study is also extended to media of orthorhombic symmetry that are believed to be more realistic models of fractured reservoirs. The study shows the orthorhombic and HTI reflection coefficients are very similar and the azimuthal variation in the orthorhombic P-wave reflection response is a function of the shear‐wave splitting parameter γ and two anisotropy parameters describing P-wave anisotropy for near‐vertical propagation in the symmetry planes. The simple relationships between the reflection amplitudes and anisotropic coefficients given here can be regarded as helpful rules of thumb in quickly evaluating the importance of anisotropy in a particular play, integrating results of NMO and shear‐wave‐splitting analyses, planning data acquisition, and guiding more advanced numerical amplitude‐inversion procedures.

Amplitude analysis of isotropic P-wave reflections

Geophysics ◽  
2006 ◽  
Vol 71 (6) ◽  
pp. C93-C103 ◽  
Author(s):  
Mirko van der Baan ◽  
Dirk Smit
Keyword(s):  
Plane Wave ◽  
Wave Reflection ◽  
P Wave ◽  
Background Information ◽  
Head Wave ◽  
Reflection Coefficients ◽  
Plane Wave Decomposition ◽  
Free Parameters ◽  
Time Offset ◽  
Wave Decomposition

The analysis of amplitude variation with offset (AVO) of seismic reflections is a very popular tool for detecting gas sands. It is assumed in AVO, however, that plane-wave reflection coefficients can be used directly to analyze amplitudes measured in the time-offset domain. This is not true for near-critical angles of reflection. Plane-wave reflection coefficients incorporate the contribution of the head wave. A plane-wave decomposition such as a proper [Formula: see text] transform must be applied to the seismic data for accurate analysis of reflection coefficients near critical angles. Amplitudes after plane-wave decomposition are related directly to the plane-wave reflection coefficients; geometric-spreading corrections are no longer required, and polarization effects of P-P reflections recorded on the [Formula: see text]-component are also removed. Conventional, linearized expressions for the isotropic P-P-wave reflection coefficient depend on contrasts in three parameters, and they require background information about average P-wave/S-wave velocity ratios. We derive a new reduced-parameter expression that depends only on two free parameters without loss of accuracy. No extra prior parameter information is needed either. The reduction in free parameters is achieved by explicitly incorporating P-wave moveout information. A new AVO strategy is developed that requires moveout analysis of three reflections: the target horizon, the reflections directly above and below the target horizon, and the amplitudes of the target horizon. The new AVO expression can be used in the time-offset domain for precritical arrivals and in the [Formula: see text] domain for precritical and critical reflections.

Plane-wave attenuation anisotropy in orthorhombic media

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. D9-D19 ◽  
Author(s):  
Yaping Zhu ◽  
Ilya Tsvankin
Keyword(s):  
Attenuation Coefficient ◽  
Wave Attenuation ◽  
Fractured Reservoirs ◽  
The Body ◽  
P Wave ◽  
Attenuation Coefficients ◽  
Velocity Anisotropy ◽  
Homogeneous Wave ◽  
Anisotropy Parameters ◽  
Attenuation Anisotropy

Orthorhombic models are often used in the interpretation of azimuthally varying seismic signatures recorded over fractured reservoirs. Here, we develop an analytic framework for describing the attenuation coefficients in orthorhombic media with orthorhombic attenuation (i.e., the symmetry of both the real and imaginary parts of the stiffness tensor is identical) under the assumption of homogeneous wave propagation. The analogous form of the Christoffel equation in the symmetry planes of orthorhombic and VTI (transversely isotropic with a vertical symmetry axis) media helps to obtain the symmetry-plane attenuation coefficients by adapting the existing VTI equations. To take full advantage of this equivalence with transverse isotropy, we introduce a parameter set similar to the VTI attenuation-anisotropy parameters [Formula: see text], [Formula: see text], and [Formula: see text]. This notation, based on the same principle as Tsvankin’s velocity-anisotropy parameters for orthorhombic media, leads to concise linearized equations for thesymmetry-plane attenuation coefficients of all three modes (P, [Formula: see text], and [Formula: see text]).The attenuation-anisotropy parameters also allow us to simplify the P-wave attenuation coefficient [Formula: see text] outside the symmetry planes under the assumptions of small attenuation and weak velocity and attenuation anisotropy. The approximate coefficient [Formula: see text] has the same form as the linearized P-wave phase-velocity function, with the velocity parameters [Formula: see text] and [Formula: see text] replaced by the attenuation parameters [Formula: see text] and [Formula: see text]. The exact attenuation coefficient, however, also depends on the velocity-anisotropy parameters, while the body-wave velocities are almost unperturbed by the presence of attenuation. The reduction in the number of parameters responsible for the P-wave attenuation and the simple approximation for the coefficient [Formula: see text] provide a basis for inverting P-wave attenuation measurements from orthorhombic media. The attenuation processing must be preceded by anisotropic velocity analysis that can be performed (in the absence of pronounced velocity dispersion) using existing algorithms for nonattenuative media.

Explicit analytic expression for normal moveout from horizontal and dipping reflectors in weakly anisotropic media of arbitrary symmetry type

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1294-1304 ◽  
Author(s):  
P. N. J. Rasolofosaon
Keyword(s):  
Wave Velocity ◽  
Symmetry Plane ◽  
Vertical Plane ◽  
Anisotropic Media ◽  
P Wave ◽  
Symmetry Type ◽  
Seismic Reflection Data ◽  
General Symmetry ◽  
P Wave Velocity ◽  
Analytic Expression

When processing and inverting seismic reflection data, the NMO velocity must be correctly described, taking into account realistic situations such as the presence of anisotropy and dipping reflectors. Some dip‐moveout (DMO) algorithms have been developed, such as Tsvankin’s analytic formula. It describes the anisotropy‐induced distortions in the classical isotropic cosine of dip dependence of the NMO velocity. However, it is restricted to the vertical symmetry planes of anisotropic media, so the technique is unsuitable for the azimuthal inspection of sedimentary rocks, either with horizontal bedding and vertical fractures or with dipping bedding but no fractures. However, under the weak anisotropy approximation the deviations of the rays from a vertical plane can be neglected for the traveltimes computation, and the equation can still be applicable. Based on this approach, an explicit analytic expression for the P-wave NMO velocity in the presence of horizontal or dipping reflectors in media exhibiting the most general symmetry type (triclinic) is obtained in this work. If the medium exhibits a horizontal symmetry plane, the concise DMO equations are formally identical to Tsvankin’s except that the parameters δ and ε are not constant but depend on the azimuth ψ Physically, δ(ψ) is the deviation from the vertical P-wave velocity of the P-wave NMO velocity for a horizontal reflector normalized by the vertical P-wave velocity for the azimuth ψ. The function ε(ψ) has the same definition as δ(ψ) except that the P-wave NMO velocity is replaced by the horizontal P-wave velocity. Both depend linearly on (1) new dimensionless anisotropy parameters and (2) generalizing to arbitrary symmetry the transversely isotropic parameters δ and ε. In the most general symmetry case (triclinic), an additional term to the DMO formula is necessary. The numerical examples, based on experimental data in rocks, show two things. First, the magnitude of the DMO errors induced by anisotropy depends primarily on the absolute value of ε(ψ) − δ(ψ) and not on the individual values of ε(ψ) and δ(ψ), which is a direct consequence of the similarity between Tsvankin’s equation and the equation presented here. Second, the anisotropy‐induced DMO correction can be significant even in the presence of moderate anisotropy and can exhibit complex azimuthal dependence.

QVOA analysis: P-wave attenuation anisotropy for fracture characterization

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. C37-C48 ◽  
Author(s):  
Tatiana Chichinina ◽  
Vladimir Sabinin ◽  
Gerardo Ronquillo-Jarillo
Keyword(s):  
Wave Reflection ◽  
Wave Attenuation ◽  
Fractured Reservoirs ◽  
Anisotropy Parameter ◽  
P Wave ◽  
Fracture Parameters ◽  
Fractured Medium ◽  
Seismic Quality Factor ◽  
Pp Wave ◽  
Attenuation Anisotropy

This paper investigates [Formula: see text]-anisotropy for characterizing fractured reservoirs — specifically, the variation of the seismic quality factor [Formula: see text] versus offset and azimuth (QVOA). We derive an analytical expression for P-wave attenuation in a transversely isotropic medium with horizontal symmetry axis (HTI) and provide a method (QVOA) for estimating fracture direction from azimuthally varying [Formula: see text] in PP-wave reflection data. The QVOA formula is similar to Rüger’s approximation for PP-wave reflection coefficients, the theoretical basis for amplitude variation with angle offset (AVOA) analysis. The technique for QVOA analysis is similar to azimuthal AVO analysis. We introduce two new seismic attributes: [Formula: see text] versus offset (QVO) gradient and intercept. QVO gradient inversion not only indicates fracture orientation but also characterizes [Formula: see text]-anisotropy. We relate the [Formula: see text]-anisotropy parameter [Formula: see text] to fractured-medium parameters and invert the QVO gradient to estimate [Formula: see text]. The attenuation parameter [Formula: see text] and Thomsen-style anisotropy parameter [Formula: see text] are found to be interdependent. The attenuation anisotropy magnitude strongly depends on the host rock’s [Formula: see text] parameter, whereas the dependence on fracture parameters is weak. This complicates the QVO gradient inversion for the fracture parameters. This result is independent of the attenuation mechanism. To illustrate the QVOA method in synthetic data, we use Hudson’s first-order effective-medium model of a dissipative fractured reservoir with fluid flow between aligned cracks and random pores as a possible mechanism for P-wave attenuation.

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