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.

First-order ray tracing for qP waves in inhomogeneous, weakly anisotropic media

Geophysics ◽  
2005 ◽  
Vol 70 (6) ◽  
pp. D65-D75 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Ray Tracing ◽  
Anisotropic Media ◽  
Order Accuracy ◽  
Weak Anisotropy ◽  
Higher Symmetry ◽  
First Order ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Order Quantities ◽  
Relative Errors

We propose approximate ray-tracing equations for qP-waves propagating in smooth, inhomogeneous, weakly anisotropic media. For their derivation, we use perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. The proposed ray-tracing equations and corresponding traveltimes are of the first order. Accuracy of the traveltimes can be increased by calculating a secondorder correction along first-order rays. The first-order ray-tracing equations for qP-waves propagating in a general weakly anisotropic medium depend on only 15 weak-anisotropy parameters (generalization of Thomsen’s parameters). The equations are thus considerably simpler than the exact ray-tracing equations. For higher-symmetry anisotropic media the equations differ only slightly from equations for isotropic media. They can thus substitute for the traditional isotropic ray tracers used in seismic processing. For vanishing anisotropy, the first-order ray-tracing equations reduce to standard, exact ray-tracing equations for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of calculated traveltimes on inhomogeneity of the medium. For anisotropy of about 8%, considered in the examples presented, the relative errors of the traveltimes, including the second-order correction, are well under 0.05%; for anisotropy of about 20%, they do not exceed 0.3%.

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 Anisotropic P-wave traveltime tomography implementing Thomsen's weak approximation in TOMO3D

2019 ◽  
Author(s):  
Adrià Meléndez ◽  
Clara Estela Jiménez ◽  
Valentí Sallarès ◽  
César R. Ranero
Keyword(s):  
P Wave ◽  
Weak Approximation ◽  
Weak Anisotropy ◽  
Traveltime Tomography ◽  
Simultaneous Inversion ◽  
Advantages And Disadvantages ◽  
Parameter Combination ◽  
P Wave Velocity ◽  
Rough Approximation ◽  
Vti Media

Abstract. We present the implementation of Thomsen's weak anisotropy approximation for VTI media within TOMO3D, our code for 2-D and 3-D joint refraction and reflection traveltime tomographic inversion. In addition to the inversion of seismic P-wave velocity and reflector depth, the code can now retrieve models of the Thomsen's parameters δ and ε. Here we test this new implementation following four different strategies on a canonical synthetic experiment. First, we study the sensitivity of traveltimes to the presence of a 25 % anomaly in each of the parameters. Next, we invert for two combinations of parameters, (v, δ, ε) and (v, δ, v⟂), following two inversion strategies, simultaneous and sequential, and compare the results to study their performances and discuss their advantages and disadvantages. Simultaneous inversion is the preferred strategy and the parameter combination (v, δ, ε) produces the best overall results. The only advantage of the parameter combination (v, δ, v⟂) is a better recovery of the magnitude of v. In each case we derive the fourth parameter from the equation relating ε, v⟂ and v. Recovery of v, ε and v⟂ is satisfactory whereas δ proves to be impossible to recover even in the most favorable scenario. However, this does not hinder the recovery of the other parameters, and we show that it is still possible to obtain a rough approximation of δ distribution in the medium by sampling a reasonable range of homogeneous initial δ models and averaging the final δ models that are satisfactory in terms of data fit.

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.

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.

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.

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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