Elastic impedance in weakly anisotropic media

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. D73-D83 ◽  
Author(s):  
Jorge L. Martins
Keyword(s):  
Seismic Anisotropy ◽  
Anisotropic Media ◽  
P Wave ◽  
Weak Anisotropy ◽  
Numerical Tests ◽  
Pp Wave ◽  
Elastic Impedance ◽  
Phase Angles ◽  
Reflection Formula ◽  
Suitable Approximation

The original formulation for the P-wave elastic impedance (EI) equation ignores seismic anisotropy. Incorporation of anisotropy effects into the EI formula requires a suitable approximation for reflection coefficients. In order to derive an anisotropic EI equation, this paper uses an approximation for PP-wave reflection [Formula: see text] coefficients which holds for weak-contrast interfaces separating weakly anisotropic media of arbitrary symmetry. Inserting the chosen [Formula: see text] coefficient approximation into the original formalism provides an anisotropic EI formula, which is written as a product of two terms: a modified version for the isotropic EI equation and a correction because of weak anisotropy. The latter term shows dependence of the anisotropic EI formula on the so-called weak anisotropy (WA) parameters, on a reference isotropic medium, and on the azimuthal and incident phase angles. Numerical tests show the performance of the EI formula in calculating anisotropic [Formula: see text] coefficients and in constructing azimuthal far-offset EI logs. Since EI allows applying poststack algorithms without modification, an inversion methodology can be designed for investigating anisotropy in sedimentary formations.

Practical concept of traveltime inversion of simulated P-wave vertical seismic profile data in weak to moderate arbitrary anisotropy

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. C107-C123
Author(s):  
Ivan Pšenčík ◽  
Bohuslav Růžek ◽  
Petr Jílek
Keyword(s):  
Waveform Inversion ◽  
Anisotropic Media ◽  
Compressional Wave ◽  
Seismic Profile ◽  
P Wave ◽  
Source Distribution ◽  
S Wave ◽  
Vertical Seismic Profile ◽  
Weak Anisotropy ◽  
Traveltime Inversion

We have developed a practical concept of compressional wave (P-wave) traveltime inversion in weakly to moderately anisotropic media of arbitrary symmetry and orientation. The concept provides sufficient freedom to explain and reproduce observed anisotropic seismic signatures to a high degree of accuracy. The key to this concept is the proposed P-wave anisotropy parameterization (A-parameters) that, together with the use of the weak-anisotropy approximation, leads to a significantly simplified theory. Here, as an example, we use a simple and transparent formula relating P-wave traveltimes to 15 P-wave A-parameters describing anisotropy of arbitrary symmetry. The formula is used in the inversion scheme, which does not require any a priori information about anisotropy symmetry and its orientation, and it is applicable to weak and moderate anisotropy. As the first step, we test applicability of the proposed scheme on a blind inversion of synthetic P-wave traveltimes generated in vertical seismic profile experiments in homogeneous models. Three models of varying anisotropy are used: tilted orthorhombic and triclinic models of moderate anisotropy (approximately 10%) and an orthorhombic model of strong anisotropy (>25%) with a horizontal plane of symmetry. In all cases, the inversion yields the complete set of 15 P-wave A-parameters, which make reconstruction of corresponding phase-velocity surfaces possible with high accuracy. The inversion scheme is robust with respect to noise and the source distribution pattern. Its quality depends on the angular illumination of the medium; we determine how the absence of nearly horizontal propagation directions affects inversion accuracy. The results of the inversion are applicable, for example, in migration or as a starting model for inversion methods, such as full-waveform inversion, if a model refinement is desired. A similar procedure could be designed for the inversion of S-wave traveltimes in anisotropic media of arbitrary symmetry.

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.

Cross‐borehole tomography in anisotropic media

Geophysics ◽  
1992 ◽  
Vol 57 (9) ◽  
pp. 1194-1198 ◽  
Author(s):  
Philip Carrion ◽  
Jesse Costa ◽  
Jose E. Ferrer Pinheiro ◽  
Michael Schoenberg
Keyword(s):  
Ray Tracing ◽  
Anisotropic Media ◽  
P Wave ◽  
Weak Anisotropy ◽  
Tomographic Inversion ◽  
Traveltime Tomography ◽  
Accurate Imaging

Anisotropy has significant effect on traveltime cross‐borehole tomography. Even relatively weak anisotropy cannot be ignored if accurate velocity estimates are desired, since isotropic traveltime tomography treats anisotropy as inhomogeneity. Traveltime data in our examples were synthetically generated by a ray‐tracing code for anisotropic media, and the computed quasi‐P‐wave traveltimes were subsequently inverted using the “dual tomography” technique (Carrion, 1991). The results of the tomographic inversion show typical artifacts due to the anisotropy, and that accurate imaging is impossible without taking the anisotropy into account.

Weak-anisotropy moveout approximations for P-waves in homogeneous layers of monoclinic or higher anisotropy symmetries

Geophysics ◽  
2016 ◽  
Vol 81 (2) ◽  
pp. C17-C37 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík ◽  
Petr Jílek
Keyword(s):  
Seismic Data ◽  
Symmetry Plane ◽  
Anisotropic Media ◽  
P Wave ◽  
Homogeneous Layer ◽  
Stiffness Tensor ◽  
Weak Anisotropy ◽  
P Waves ◽  
Order Of Magnitude ◽  
Better Than

We have used so-called weak anisotropy (WA) parameterization as an alternative to the parameterization of generally anisotropic media by a stiffness tensor. WA parameters consist of linear combinations of normalized stiffness-tensor elements controlling various seismic signatures; hence, they are theoretically extractable from seismic data. They are dimensionless and can be designed to have the same order of magnitude. WA parameters, similarly to Thomsen-type parameters, have a clear physical interpretation. They are, however, applicable to anisotropy of any symmetry, strength, and orientation. They are defined in coordinate systems independent of the symmetry elements of the studied media. Expressions using WA parameters naturally simplify as the anisotropy becomes weaker or as the anisotropy symmetry increases. We expect that, due to these useful properties, WA parameterization can potentially provide a framework for seismic data processing in generally anisotropic media. Using the WA parameterization, we have derived and tested approximate P-wave moveout formulas for a homogeneous layer of up-to-monoclinic symmetry, underlain by a horizontal reflector coinciding with a symmetry plane. The derived traveltime formulas represent an expansion of the traveltime with respect to (small) WA parameters. For the comparison with standard moveout formulas, we expressed ours in the common form of nonhyperbolic moveout, containing normal moveout velocity and a quartic coefficient as functions of the WA parameters. The accuracy of our formulas depends strongly on the deviation of ray- and phase-velocity directions (controlled by the deviation of the ray and phase velocities). The errors do not generally increase with increasing offset, nor do they increase with decreasing anisotropy symmetry. The accuracy of our formulas is comparable with, or better than, the accuracy of commonly used formulas.

Quasi‐shear waves in inhomogeneous weakly anisotropic media by the quasi‐isotropic approach: A model study

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 308-319 ◽  
Author(s):  
Ivan Pšenčík ◽  
Joe A. Dellinger
Keyword(s):  
Order Approximation ◽  
Anisotropic Media ◽  
Ray Method ◽  
Weak Anisotropy ◽  
S Waves ◽  
Model Study ◽  
Arbitrary Strength ◽  
Isotropic Media ◽  
Synthetic Models ◽  
Ray Methods

In inhomogeneous isotropic regions, S-waves can be modeled using the ray method for isotropic media. In inhomogeneous strongly anisotropic regions, the independently propagating qS1- and qS2-waves can similarly be modeled using the ray method for anisotropic media. The latter method does not work properly in inhomogenous weakly anisotropic regions, however, where the split qS-waves couple. The zeroth‐order approximation of the quasi‐isotropic (QI) approach was designed for just such inhomogeneous weakly anisotropic media, for which neither the ray method for isotropic nor anisotropic media applies. We test the ranges of validity of these three methods using two simple synthetic models. Our results show that the QI approach more than spans the gap between the ray methods: it can be used in isotropic regions (where it reduces to the ray method for isotropic media), in regions of weak anisotropy (where the ray method for anisotropic media does not work properly), and even in regions of moderately strong anisotropy (in which the qS-waves decouple and thus could be modeled using the ray method for anisotropic media). A modeling program that switches between these three methods as necessary should be valid for arbitrary‐strength anisotropy.

Enabling isotropic reflectivity modeling and inversion in anisotropic media

2021 ◽  
Vol 40 (4) ◽  
pp. 267-276
Author(s):  
Peter Mesdag ◽  
Leonardo Quevedo ◽  
Cătălin Tănase
Keyword(s):  
Synthetic Data ◽  
Anisotropic Media ◽  
Forward Modeling ◽  
Seismic Inversion ◽  
Stratified Medium ◽  
Effective Parameters ◽  
Elastic Parameters ◽  
Pp Wave ◽  
Anisotropic Elastic ◽  
And Inversion

Exploration and development of unconventional reservoirs, where fractures and in-situ stresses play a key role, call for improved characterization workflows. Here, we expand on a previously proposed method that makes use of standard isotropic modeling and inversion techniques in anisotropic media. Based on approximations for PP-wave reflection coefficients in orthorhombic media, we build a set of transforms that map the isotropic elastic parameters used in prestack inversion into effective anisotropic elastic parameters. When used in isotropic forward modeling and inversion, these effective parameters accurately mimic the anisotropic reflectivity behavior of the seismic data, thus closing the loop between well-log data and seismic inversion results in the anisotropic case. We show that modeling and inversion of orthorhombic anisotropic media can be achieved by superimposing effective elastic parameters describing the behavior of a horizontally stratified medium and a set of parallel vertical fractures. The process of sequential forward modeling and postinversion analysis is exemplified using synthetic data.

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.

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.

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