Linearized quantities in transversely isotropic media

2004 ◽  
Vol 41 (3) ◽  
pp. 349-354 ◽  
Author(s):  
P F Daley ◽  
L R Lines
Keyword(s):  
Energy Surface ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Slowness Surface ◽  
Transversely Isotropic Media ◽  
Alternative Characterization ◽  
Isotropic Media ◽  
Ellipsoid Of Revolution ◽  
Two Parameters ◽  
Ti Media

This paper advocates a modified parameterization for transversely isotropic (TI) media in seismology. Transversely isotropic media may be parameterized by the reference quasi-compressional (qP) and quasi-shear (qSV) velocities, α and β, together with the two parameters, ε and δ. These last two variables account for the deviation of the coupled qP and qSV modes of wave propagation from the isotropic case. The dimensionless quantity ε is a measure of the ellipticity of the qP wavefront. The "strange" parameter δ has been employed as a measure of deviation of the qP wavefront or slowness surface from an ellipsoid of revolution and also of the qSV wavefront or slowness surface from a sphere. As the parameter δ has been described as "conceptually inaccessible"; it is logical to determine an alternative characterization in physically realizable quantities. As in the earlier literature, the defining parameter for the degree of ellipticity of the qP slowness or energy surface in a TI medium is chosen here to be ε. In addition, the deviation of both the qP and qSV surfaces from the degenerate ellipsoidal case will be specified here as σ. An examination of the linearized phase (wavefront normal) velocities, and the linearized PP and SVSV reflection coefficients at an interface between two TI media is undertaken to emphasize the merits of this modified parameterization. All formulae used here have appeared in one form or another in the cited literature. In this paper, a reformulation of existing equations as functions of the two independent variables, (ε, σ), is presented. This results in a more physically realizable context for the linearized formulae.

Velocity analysis using nonhyperbolic moveout in transversely isotropic media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1839-1854 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Vertical Heterogeneity ◽  
Isotropic Media ◽  
D Values ◽  
Two Parameters ◽  
Vti Media ◽  
Ti Media

P‐wave reflections from horizontal interfaces in transversely isotropic (TI) media have nonhyperbolic moveout. It has been shown that such moveout as well as all time‐related processing in TI media with a vertical symmetry axis (VTI media) depends on only two parameters, [Formula: see text] and η. These two parameters can be estimated from the dip‐moveout behavior of P‐wave surface seismic data. Alternatively, one could use the nonhyperbolic moveout for parameter estimation. The quality of resulting estimates depends largely on the departure of the moveout from hyperbolic and its sensitivity to the estimated parameters. The size of the nonhyperbolic moveout in TI media is dependent primarily on the anisotropy parameter η. An “effective” version of this parameter provides a useful measure of the nonhyperbolic moveout even in v(z) isotropic media. Moreover, effective η, [Formula: see text], is used to show that the nonhyperbolic moveout associated with typical TI media (e.g., shales, with η ≃ 0.1) is larger than that associated with typical v(z) isotropic media. The departure of the moveout from hyperbolic is increased when typical anisotropy is combined with vertical heterogeneity. Larger offset‐to‐depth ratios (X/D) provide more nonhyperbolic information and, therefore, increased stability and resolution in the inversion for [Formula: see text]. The X/D values (e.g., X/D > 1.5) needed for obtaining stability and resolution are within conventional acquisition limits, especially for shallow targets. Although estimation of η using nonhyperbolic moveouts is not as stable as using the dip‐moveout method of Alkhalifah and Tsvankin, particularly in the absence of large offsets, it does offer some flexibility. It can be applied in the absence of dipping reflectors and also may be used to estimate lateral η variations. Application of the nonhyperbolic inversion to data from offshore Africa demonstrates its usefulness, especially in estimating lateral and vertical variations in η.

scholarly journals The offset‐midpoint traveltime pyramid in transversely isotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1316-1325 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Fourth Power ◽  
Weak Anisotropy ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Equivalent Equation ◽  
Two Parameters ◽  
Vti Media

Prestack Kirchhoff time migration for transversely isotropic media with a vertical symmetry axis (VTI media) is implemented using an offset‐midpoint traveltime equation, Cheop’s pyramid equivalent equation for VTI media. The derivation of such an equation for VTI media requires approximations that pertain to high frequency and weak anisotropy. Yet the resultant offset‐midpoint traveltime equation for VTI media is highly accurate for even strong anisotropy. It is also strictly dependent on two parameters: NMO velocity and the anisotropy parameter, η. It reduces to the exact offset‐midpoint traveltime equation for isotropic media when η = 0. In vertically inhomogeneous media, the NMO velocity and η parameters in the offset‐midpoint traveltime equation are replaced by their effective values: the velocity is replaced by the rms velocity and η is given by a more complicated equation that includes summation of the fourth power of velocity.

AVO in transversely Isotropic media—An overview

Geophysics ◽  
1994 ◽  
Vol 59 (5) ◽  
pp. 775-781 ◽  
Author(s):  
J. P. Blangy
Keyword(s):  
Transversely Isotropic ◽  
Shear Velocity ◽  
Velocity Shear ◽  
Type I ◽  
Amplitude Variation With Offset ◽  
Type Iii ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Compressional Velocity ◽  
Ti Media

The amplitude variation with offset (AVO) responses of elastic transversely isotropic media are sensitive to contrasts in both of Thomsen’s anisotropic parameters δ and ε. The equation describing P-P reflections indicates that the smaller the contrasts in isotropic properties (compressional velocity, shear velocity, and density) and the larger the contrasts in δ and ε across an interface of reflection, the greater the effects of anisotropy on the AVO signature. Contrasts in δ are most important under small‐to‐medium angles of incidence as previously described by Banik (1987), while contrasts in ε can have a strong influence on amplitudes for the larger angles of incidence commonly encountered in exploration (20 degrees and beyond). Consequently, using Rutherford and Williams’ AVO classification of isotropic gas sands, type I gas sands overlain by a transversely isotropic (TI) shale exhibit a larger decrease in AVO than if the shale had been isotropic, and type III gas sands overlain by a transversely isotropic (TI) shale exhibit a larger increase in AVO than if the shale had been isotropic. Furthermore, it is possible for a “type III” isotropic water sand to exhibit an “unexpected) increase in AVO if the overlying shale is sufficiently anisotropic. More quantitative AVO interpretations in TI media require considerations of viscoelastic TI in addition to elastic TI and lead to complicated integrated earth models. However, when elastic and viscoelastic TI have the same axis of symmetry in a shale overlying an isotropic sand, both elastic and viscoelastic TI affect the overall AVO response in the same direction by constructively increasing/decreasing the isotropic component of the AVO response. Continued efforts in this area will lead to more realistic reservoir models and hopefully answer some of the unexplained pitfalls in AVO interpretation.

scholarly journals Traveltime approximations for transversely isotropic media with an inhomogeneous background

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA31-WA42 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Symmetry Axis ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Transversely Isotropic Media ◽  
Higher Order Terms ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Series Type ◽  
Ti Media ◽  
Taylor’S Series

A transversely isotropic (TI) model with a tilted symmetry axis is regarded as one of the most effective approximations to the Earth subsurface, especially for imaging purposes. However, we commonly utilize this model by setting the axis of symmetry normal to the reflector. This assumption may be accurate in many places, but deviations from this assumption will cause errors in the wavefield description. Using perturbation theory and Taylor’s series, I expand the solutions of the eikonal equation for 2D TI media with respect to the independent parameter [Formula: see text], the angle the tilt of the axis of symmetry makes with the vertical, in a generally inhomogeneous TI background with a vertical axis of symmetry. I do an additional expansion in terms of the independent (anellipticity) parameter [Formula: see text] in a generally inhomogeneous elliptically anisotropic background medium. These new TI traveltime solutions are given by expansions in [Formula: see text] and [Formula: see text] with coefficients extracted from solving linear first-order partial differential equations. Pade approximations are used to enhance the accuracy of the representation by predicting the behavior of the higher-order terms of the expansion. A simplification of the expansion for homogenous media provides nonhyperbolic moveout descriptions of the traveltime for TI models that are more accurate than other recently derived approximations. In addition, for 3D media, I develop traveltime approximations using Taylor’s series type of expansions in the azimuth of the axis of symmetry. The coefficients of all these expansions can also provide us with the medium sensitivity gradients (Jacobian) for nonlinear tomographic-based inversion for the tilt in the symmetry axis.

scholarly journals Mapping moveout approximations in TI media

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. C19-C26 ◽  
Author(s):  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Transversely Isotropic ◽  
Vertical Axis ◽  
Accurate Approximation ◽  
Seismic Modeling ◽  
Effective Parameters ◽  
Transversely Isotropic Media ◽  
Vertical Heterogeneity ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Ti Media

Moveout approximations play a very important role in seismic modeling, inversion, and scanning for parameters in complex media. We developed a scheme to map one-way moveout approximations for transversely isotropic media with a vertical axis of symmetry (VTI), which is widely available, to the tilted case (TTI) by introducing the effective tilt angle. As a result, we obtained highly accurate TTI moveout equations analogous with their VTI counterparts. Our analysis showed that the most accurate approximation is obtained from the mapping of generalized approximation. The new moveout approximations allow for, as the examples demonstrate, accurate description of moveout in the TTI case even for vertical heterogeneity. The proposed moveout approximations can be easily used for inversion in a layered TTI medium because the parameters of these approximations explicitly depend on corresponding effective parameters in a layered VTI medium.

Approximate dispersion relations for qP-qSV-waves in transversely isotropic media

Geophysics ◽  
2000 ◽  
Vol 65 (3) ◽  
pp. 919-933 ◽  
Author(s):  
Michael A. Schoenberg ◽  
Maarten V. de Hoop
Keyword(s):  
Wave Propagation ◽  
Order Approximation ◽  
Transverse Isotropy ◽  
Slight Modification ◽  
Transversely Isotropic ◽  
P Wave ◽  
Slowness Surface ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
First Order Approximation

To decouple qP and qSV sheets of the slowness surface of a transversely isotropic (TI) medium, a sequence of rational approximations to the solution of the dispersion relation of a TI medium is introduced. Originally conceived to allow isotropic P-wave processing schemes to be generalized to encompass the case of qP-waves in transverse isotropy, the sequence of approximations was found to be applicable to qSV-wave processing as well, although a higher order of approximation is necessary for qSV-waves than for qP-waves to yield the same accuracy. The zeroth‐order approximation, about which all other approximations are taken, is that of elliptical TI, which contains the correct values of slowness and its derivative along and perpendicular to the medium’s axis of symmetry. Successive orders of approximation yield the correct values of successive orders of derivatives in these directions, thereby forcing the approximation into increasingly better fit at the intervening oblique angles. Practically, the first‐order approximation for qP-wave propagation and the second‐order approximation for qSV-wave propagation yield sufficiently accurate results for the typical transverse isotropy found in geological settings. After only slight modification to existing programs, the rational approximation allows for ray tracing, (f-k) domain migration, and split‐step Fourier migration in TI media—with little more difficulty than that encountered presently with such algorithms in isotropic media.

Scalar generalized‐screen algorithms in transversely isotropic media with a vertical symmetry axis

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1538-1550 ◽  
Author(s):  
Jéro⁁me H. Le Rousseau ◽  
Maarten V. de Hoop
Keyword(s):  
Symmetry Axis ◽  
Complex Structure ◽  
Numerical Algorithms ◽  
Transversely Isotropic ◽  
Isotropic Case ◽  
Polarization Angle ◽  
Screen Method ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
The One

The scalar generalized‐screen method in isotropic media is extended here to transversely isotropic media with a vertical symmetry axis (VTI). Although wave propagation in a transversely isotropic medium is essentially elastic, we use an equivalent “acoustic” system of equations for the qP‐waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of “mild” anisotropy. The enhanced accuracy of the generalized‐screen method as compared to the split‐step Fourier methods allows the extension to VTI media. The generalized‐screen expansion of the one‐way propagator follows closely the method used in the isotropic case. The medium is defined in terms of a background and a perturbation. The generalized‐screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation. We cast the theory into numerical algorithms, and assess the accuracy of the generalized‐screen method in a particular VTI medium with complex structure (the BP Amoco Valhall model) in which multipathing is significant.

Velocity analysis in transversely isotropic media

Geophysics ◽  
1980 ◽  
Vol 45 (6) ◽  
pp. 1094-1095 ◽  
Author(s):  
A. L. Lucas ◽  
P. N. S. O’Brien ◽  
J. H. Thomas
Keyword(s):  
Transversely Isotropic ◽  
Vertical Axis ◽  
Two Dimensions ◽  
P Wave ◽  
Velocity Analysis ◽  
Wave Surface ◽  
Common Depth Point ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Ellipsoid Of Revolution

In transversely isotropic media, the moveout velocity obtained from common‐depth‐point (CDP) analysis may be significantly different from the horizontal velocity of the pseudo‐P wave. In Levin’s (1978) paper, he discusses, among other things, the problem of velocity determination in a medium in which the pseudo‐P wave surface produced by a point source is an ellipsoid of revolution. He points out that one would expect many sedimentary rocks to be transversely isotropic with a vertical axis of symmetry. In his Appendix he proves that an ellipse (using two dimensions for convenience) is one possible shape for the wave surface in such a medium. He also shows, as have others, that in this case CDP velocity analysis measures the velocity of horizontal propagation.

Fast simulation of triaxial borehole induction measurements acquired in axially symmetrical and transversely isotropic media

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. E233-E249 ◽  
Author(s):  
Gong Li Wang ◽  
Carlos Torres-Verdín ◽  
Stan Gianzero
Keyword(s):  
Transversely Isotropic ◽  
Simulation Method ◽  
Fourier Series Expansion ◽  
Radial Dependence ◽  
Transversely Isotropic Media ◽  
Hankel Functions ◽  
Isotropic Media ◽  
Em Induction ◽  
Ti Media ◽  
3D Problem

We introduce and successfully test an efficient method to simulate triaxial borehole electromagnetic (EM) induction measurements acquired in axially symmetrical and transversely isotropic (TI) media. The method uses a Fourier series expansion to express the azimuthal dependence of EM fields and the source term whereby the essentially 3D problem collapses to a series of independent 2D problems. Each 2D problem is solved with a semianalytic method that uses normalized Bessel functions and normalized Hankel functions to express the radial dependence of EM fields, thereby improving numerical stability. In addition, use is made of amplitude and slope basis functions to describe the longitudinal dependence of EM fields to avoid grid refinement in the vicinity of horizontal formation boundaries. For validation, we compare the new simulation method to two 1D analytic methods in horizontally and radially layered formations, and to one 3D finite-difference method (3DFD) in multilayered formations that include borehole and invasion zones. Numerical results indicate that the method is accurate in formations with high conductivity contrasts compared to 1D methods and is more than ten times more efficient than the 3DFD method in multilayer formations.

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