8. From Transverse Isotropy to Arbitrary Anisotropy for qP-Waves in Weakly Anisotropic Media

2001 ◽  
pp. 145-162 ◽  
Author(s):  
Patrick N. J. Rasolofosaon
Keyword(s):  
Transverse Isotropy ◽  
Anisotropic Media ◽  
Arbitrary Anisotropy

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

The effect of transverse isotropy on isotropic traveltime inversion of vertical seismic profile data—A modeling study

Geophysics ◽  
1990 ◽  
Vol 55 (9) ◽  
pp. 1235-1241 ◽  
Author(s):  
Jan Douma
Keyword(s):  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Seismic Profile ◽  
Isotropic Layer ◽  
Low Velocity ◽  
Traveltime Inversion ◽  
Axis Of Symmetry ◽  
Transversely Isotropic Layer

Traveltime inversion of multioffset VSP data can be used to determine the depths of the interfaces in layered media. Many inversion schemes, however, assume isotropy and consequently may introduce erroneous structures for anisotropic media. Synthetic traveltime data are computed for layered anisotropic media and inverted assuming isotropic layers. Only the interfaces between these layers are inverted. For a medium consisting of a horizontal isotropic low‐velocity layer on top of a transversely isotropic layer with a horizontal axis of symmetry (e.g., anisotropy due to aligned vertical cracks), 2-D isotropic inversion results in an anticline. For a given axis of symmetry the form of this anticline depends on the azimuth of the source‐borehole direction. The inversion result is a syncline (in 3-D a “bowl” structure), regardless of the azimuth of the source‐borehole direction for a vertical axis of symmetry of the transversely isotropic layer (e.g., anisotropy due to horizontal bedding).

Accurate simulations of pure quasi-P-waves in complex anisotropic media

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T341-T348 ◽  
Author(s):  
Sheng Xu ◽  
Hongbo Zhou
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Differential Equations ◽  
Transverse Isotropy ◽  
Anisotropic Media ◽  
Second Order ◽  
P Wave ◽  
Pseudodifferential Equation ◽  
Reverse Time ◽  
Time Migration

Reverse time migration (RTM) in complex anisotropic media requires calculation of the propagation of a single-mode wave, the quasi-P-wave. This was conventionally realized by solving a [Formula: see text] system of second-order partial differential equations. The implementation of this [Formula: see text] system required at least twice the computational resources as compared with the acoustic wave equation. The S-waves, an introduced auxiliary function in this system, were treated as artifacts in the RTM. Furthermore, the [Formula: see text] system suffered numerical stability problems at the places in which abrupt changes of symmetric axis of anisotropy exist, which brings more challenges to real data implementation. On the other hand, the Alkhalifah’s equation, which governs the pure quasi-P-wave propagation, was hard to solve because it was a pseudodifferential equation. We proposed a pure quasi-P-wave equation that can be easily implemented within current imaging framework. Our new equation was obtained by decomposing the original pseudodifferential operator into two numerical solvable operators: a differential operator and a scalar operator. The combination of these two operators yielded an accurate phase of quasi-P-wave propagation. Our solution was S-wave free and numerically stable for very complicated models. Because only one equation was required to resolve numerically, the new proposed scheme was more efficient than those conventional schemes that solve the [Formula: see text] second-order differential equations system. For tilted transverse isotropy (TTI) RTM implementation, the required increase of numerical cost was minimal, and we could expect at least a factor of two of improvement of efficiency. We showed the effectiveness and robustness of our method with numerical examples with simple and very complicated TTI models, the SEG Advanced Modeling (SEAM) model. Further extension of our approach to orthorhombic anisotropy or tilted orthorhombic anisotropy was straightforward.

Traveltime and conversion‐point computations and parameter estimation in layered, anisotropic media by τ‐p transform

Geophysics ◽  
2003 ◽  
Vol 68 (1) ◽  
pp. 210-224 ◽  
Author(s):  
Mirko van der Baan ◽  
J.‐Michael Kendall
Keyword(s):  
Ray Tracing ◽  
Transverse Isotropy ◽  
Anisotropic Media ◽  
Forward Modeling ◽  
Seismic Wave Propagation ◽  
Pure Mode ◽  
Fast Method ◽  
Conversion Point ◽  
Arbitrary Strength ◽  
Taylor Series Expansions

Anisotropy influences many aspects of seismic wave propagation and, therefore, has implications for conventional processing schemes. It also holds information about the nature of the medium. To estimate anisotropy, we need both forward modeling and inversion tools. Forward modeling in anisotropic media is generally done by ray tracing. We present a new and fast method using the τ‐p transform to calculate exact reflection‐moveout curves in stratified, laterally homogeneous, anisotropic media for all pure‐mode and converted phases which requires no conventional ray tracing. Moreover, we obtain the common conversion points for both P‐SV and P‐SH converted waves. Results are exact for arbitrary strength of anisotropy in both HTI and VTI media (transverse isotropy with a horizontal or vertical symmetry axis, respectively). Since inversion for anisotropic parameters is a highly nonunique problem, we also develop expressions describing the phase velocities that require only a reduced number of parameters for both types of anisotropy. Nevertheless, resulting predictions for traveltimes and conversion points are generally more accurate than those obtained using the conventional Taylor‐series expansions. In addition, the reduced‐parameter expressions are also able to handle kinks or cusps in the SV traveltime curves for either VTI or HTI symmetry.

Theory and numerical simulation of fluid-pressure diffusion in anisotropic porous media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. N31-N39 ◽  
Author(s):  
José M. Carcione ◽  
Davide Gei
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Transverse Isotropy ◽  
Fluid Pressure ◽  
Pseudospectral Method ◽  
Anisotropic Media ◽  
Quality Factors ◽  
Anisotropic Porous Media ◽  
Fourier Pseudospectral Method ◽  
Hydrocarbon Reservoir

The physics of fluid diffusion in anisotropic media was studied, based on Biot’s theory of poroelasticity and using wave propagation concepts. Diffusion and elastic strain can be uncoupled fully, being a good approximation in many situations. We have used a correction to the stiffness of the rock under conditions of transverse isotropy and uniaxial strain to model borehole conditions. The concepts of phase, group, and energy velocities were analyzed to describe the location of the diffusion front, and the attenuation and quality factors were obtained to quantify the amplitude decay. We have found that the location of the front is described correctly by the energy velocity. The Green’s function in anisotropic media can be obtained by applying a change of coordinates to the isotropic solution. We have simulated the diffusion in inhomogeneous media using a time-domain spectral explicit scheme and the staggered Fourier pseudospectral method to compute the spatial derivatives. The method is based on a spectral Chebychev expansion of the evolution operator of the system. The scheme allows the solution of linear periodic parabolic equations, having accuracy within the machine precision, in time and in space. The results match the analytic solution obtained from the Green’s function. The performance of the algorithm is confirmed in the case of a pressure field generated by a fluid-injection source in a hydrocarbon reservoir where the properties vary fractally.

Shear-wave group-velocity surfaces in low-symmetry anisotropic media

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. C1-C7 ◽  
Author(s):  
Vladimir Grechka
Keyword(s):  
Group Velocity ◽  
Body Wave ◽  
Transverse Isotropy ◽  
Shear Waves ◽  
Anisotropic Media ◽  
The Body ◽  
Velocity Models ◽  
Phase And Group Velocities ◽  
Low Symmetry ◽  
Group Velocities

Shear waves excited by natural sources constitute a significant part of useful energy recorded in downhole microseismic surveys. In rocks, such as fractured shales, exhibiting symmetries lower than transverse isotropy (TI), the shear wavefronts are always multivalued in certain directions, potentially complicating the data processing and analysis. This paper discusses a basic tool — the computation of the phase and group velocities of all waves propagating along a given ray — that intends to facilitate the understanding of geometries of the shear wavefronts in homogeneous anisotropic media. With this tool, arbitrarily complex group-velocity surfaces can be conveniently analyzed, providing insights into possible challenges to be faced when processing shear waves in anisotropic velocity models that have symmetries lower than TI. Among those challenges are complicated multipathing and the presence of cones of directions, known as internal refraction cones, in which no fast shear waves propagate and the entire shear portion of the body-wave seismic data consists of several branches of the slow shear wavefronts.

An expanding-wavefront method for solving the eikonal equations in general anisotropic media

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. T129-T135 ◽  
Author(s):  
Yue Wang ◽  
Tamas Nemeth ◽  
Robert T. Langan
Keyword(s):  
Group Velocity ◽  
Eikonal Equation ◽  
Anisotropic Media ◽  
Unconditionally Stable ◽  
Eikonal Equations ◽  
Arbitrary Anisotropy ◽  
Arbitrary Shapes

We present a procedure that solves the eikonal equation for general anisotropic media. It allows one to incorporate arbitrary shapes and types of anisotropic formations. We use an expanding-wavefront scheme and explicit tracking of group velocity propagation directions to choose the causal upgrade stencils for computing traveltime. The method is of first degree in accuracy and is unconditionally stable. The relative traveltime errors are controlled mainly by initial wavefront formation and by the choice of the update stencils. We illustrate the method’s ability to generate smoothly changing qP traveltimes in models of arbitrary anisotropy.

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).

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