Modeling seismic waves in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (11) ◽  
pp. 1745-1749
Author(s):  
Pin Yan ◽  
Qiaodeng He
Keyword(s):  
Fourier Transformation ◽  
Seismic Waves ◽  
Transversely Isotropic ◽  
Eigenvalues And Eigenvectors ◽  
Transversely Isotropic Medium ◽  
Inverse Fourier Transformation ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Eigen Decomposition ◽  
D Model

Seismic waves in anisotropic media are more complex than in isotropic media. Here we derive the propagating matrices for seismic waves in 2-D transversely isotropic medium (TIM). With eigen‐decomposition, eigenvalues and eigenvectors are given in analytical forms, therefore, calculation of propagators are simple and accurate. For a 2-D model of layered media, we compute the seismic responses to an impulse in the f-k domain, and then do a 2-D inverse Fourier transformation. Clear qP and qSV waves can be recognized from the resultant sections.

Moveout approximations for P- and SV-waves in dip-constrained transversely isotropic media

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. C53-C59 ◽  
Author(s):  
Véronique Farra ◽  
Ivan Pšenčík
Keyword(s):  
Transversely Isotropic ◽  
Arbitrary Orientation ◽  
Transversely Isotropic Medium ◽  
Weak Anisotropy ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Dip Angle ◽  
P And Sv Waves ◽  
Axes Of Symmetry

We generalize the P- and SV-wave moveout formulas obtained for transversely isotropic media with vertical axes of symmetry (VTI), based on the weak-anisotropy approximation. We generalize them for 3D dip-constrained transversely isotropic (DTI) media. A DTI medium is a transversely isotropic medium whose axis of symmetry is perpendicular to a dipping reflector. The formulas are derived in the plane defined by the source-receiver line and the normal to the reflector. In this configuration, they can be easily obtained from the corresponding VTI formulas. It is only necessary to replace the expression for the normalized offset by the expression containing the apparent dip angle. The final results apply to general 3D situations, in which the plane reflector may have arbitrary orientation, and the source and the receiver may be situated arbitrarily in the DTI medium. The accuracy of the proposed formulas is tested on models with varying dip of the reflector, and for several orientations of the horizontal source-receiver line with respect to the dipping reflector.

scholarly journals Dip‐moveout error in transversely isotropic media with linear velocity variation in depth

Geophysics ◽  
1993 ◽  
Vol 58 (10) ◽  
pp. 1442-1453 ◽  
Author(s):  
Ken L. Larner
Keyword(s):  
Homogeneous Model ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Velocity Variation ◽  
Transversely Isotropic Medium ◽  
Velocity Gradients ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axis Of Symmetry ◽  
Homogeneous Media

Levin modeled the moveout, within common‐midpoint (CMP) gathers, of reflections from plane‐dipping reflectors beneath homogeneous, transversely isotropic media. For some media, when the axis of symmetry for the anisotropy was vertical, he found departures in stacking velocity from predictions based upon the familiar cosine‐of‐dip correction for isotropic media. Here, I do similar tests, again with transversely isotropic models with vertical axis of symmetry, but now allowing the medium velocity to vary linearly with depth. Results for the same four anisotropic media studied by Levin show behavior of dip‐corrected stacking velocity with reflector dip that, for all velocity gradients considered, differs little from that for the counterpart homogeneous media. As with isotropic media, traveltimes in an inhomogeneous, transversely isotropic medium can be modeled adequately with a homogeneous model with vertical velocity equal to the vertical rms velocity of the inhomogeneous medium. In practice, dip‐moveout (DMO) is based on the assumption that either the medium is homogeneous or its velocity varies with depth, but in both cases isotropy is assumed. It turns out that for only one of the transversely isotropic media considered here—shale‐limestone—would v(z) DMO fail to give an adequate correction within CMP gathers. For the shale‐limestone, fortuitously the constant‐velocity DMO gives a better moveout correction than does the v(z) DMO.

Traveltime computation in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (2) ◽  
pp. 272-281 ◽  
Author(s):  
Eduardo L. Faria ◽  
Paul L. Stoffa
Keyword(s):  
Heterogeneous Media ◽  
Grid Point ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Prestack Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
First Arrival ◽  
Grid Points ◽  
Arbitrary Velocity

An approach for calculating first‐arrival traveltimes in a transversely isotropic medium is developed and has the advantage of avoiding shadow zones while still being computationally fast. Also, it works with an arbitrary velocity grid that may have discontinuities. The method is based on Fermat’s principle. The traveltime for each point in the grid is calculated several times using previously calculated traveltimes at surrounding grid points until the minimum time is found. Different ranges of propagation angle are covered in each traveltime calculation such that at the end of the process all propagation angles are covered. This guarantees that the first‐arrival traveltime for a specific grid point is correctly calculated. The resulting algorithm is fully vectorizable. The method is robust and can accurately determine first‐arrival traveltimes in heterogeneous media. Traveltimes are compared to finite‐difference modeling of transversely isotropic media and are found to be in excellent agreement. An application to prestack migration is used to illustrate the usefulness of the method.

scholarly journals Numerical Analysis of the Electrical Potential Calculation in a Transversely Isotropic Media

2014 ◽  
Vol 2 (1-2) ◽  
Author(s):  
Sri Mardiyati
Keyword(s):  
Electrical Conductivity ◽  
Finite Element ◽  
Numerical Analysis ◽  
Electrical Potential ◽  
Transversely Isotropic ◽  
Element Formulation ◽  
Transversely Isotropic Medium ◽  
Infinite Elements ◽  
Transversely Isotropic Media ◽  
Isotropic Media

The electrical potential due to a point source of current supplied at the surface of a transversely isotropic medium is calculated using a finite element formulation. The finite and infinite elements are applied to calculate the potential for arbitrary electrical conductivity profiles. The accuracy of the scheme is checked against results obtainable using Chave's algorithm.

Aberration‐free image for SH reflection in transversely isotropic media

Geophysics ◽  
1987 ◽  
Vol 52 (11) ◽  
pp. 1563-1565 ◽  
Author(s):  
J. M. Blair ◽  
J. Korringa
Keyword(s):  
Group Velocity ◽  
Isotropic Medium ◽  
Homogeneous Medium ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Elastic Symmetry ◽  
Source Point ◽  
Slowness Vector ◽  
Transversely Isotropic Media ◽  
Isotropic Media

This note is intended formulate and prove a theorem about shear (S) waves in a transversely isotropic medium for which we have found no reference in the literature. The theorem states the following: SH waves emanating from a point source in a homogeneous transversely isotropic medium are reflected from a planar interface between the transversely isotropic medium and another homogeneous medium in such a way that they define a reflective image that is free of aberrations, regardless of the relative orientation of the elastic symmetry axis and the interface. It is an image for rays in the direction of the group velocity vectors, not the slowness vectors. The image is located on a line through the source point in the direction of the group velocity of a wave for which the slowness vector is perpendicular to the interface. The distance, measured along this line, of the image behind the interface is equal to that of the source point in front. An analogous theorem for slowness vectors exists only for isotropic media, where it is trivial and coincides with the above.

Crosswell tomographic estimation of elastic constants in heterogeneous transversely isotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (3) ◽  
pp. 774-783 ◽  
Author(s):  
Reinaldo J. Michelena ◽  
Jerry M. Harris ◽  
Francis Muir
Keyword(s):  
Elastic Constants ◽  
Isotropic Medium ◽  
Field Data ◽  
Transversely Isotropic ◽  
Transversely Isotropic Medium ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Axes Of Symmetry

The procedure to estimate elastic constants of a transversely isotropic medium from limited‐aperture traveltimes has two steps. First, P‐ and SV‐wave traveltimes are fitted with elliptical velocity functions around one of the axes of symmetry. Second, the parameters that describe the elliptical velocity functions are transformed analytically into elastic constants. When the medium is heterogeneous, the process of fitting the traveltimes with elliptical velocity functions is performed tomographically, and the transformation to elastic constants is performed locally at each position in space. Crosswell synthetic and field data examples show that the procedure is accurate as long as the data aperture is constrained as follows: it should not be too large otherwise the elliptical approximation may not be adequate, and it should not be too small because the tomographic estimation of elliptical velocities fails, even if the medium is actually isotropic.

Approximate polarization of plane waves in a medium having weak transverse isotropy

Geophysics ◽  
1994 ◽  
Vol 59 (10) ◽  
pp. 1605-1612 ◽  
Author(s):  
Björn E. Rommel
Keyword(s):  
Isotropic Medium ◽  
Transverse Isotropy ◽  
Full Range ◽  
Velocity Ratio ◽  
Plane Waves ◽  
Transversely Isotropic ◽  
Seismic Exploration ◽  
Transversely Isotropic Medium ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Many real rocks and sediments relevant to seismic exploration can be described by elastic, transversely isotropic media. The properties of plane waves propagating in a transversely isotropic medium can be given in an analytically exact form. Here the polarization is recast into a comprehensive form that includes Daley and Hron’s normalization and Helbig’s full range of elastic constants. But these formulas are rather lengthy and do not easily reveal the features caused by anisotropy. Hence Thomsen suggested an approximation scheme for weak transverse isotropy. His derivation of the approximate polarization, however, is based on a property that is not suitable to measure small differences between an isotropic and a weakly transversely isotropic medium. Therefore the approximation of the polarization is recast. The corrected approximation does show a dependence on weak transverse isotropy. This feature can be viewed as an additional rotation of the polarization with respect to the wavenormal. It depends on the anisotropy as well as the inverse velocity ratio. An approximate condition of pure polarization, which occurs in certain directions, is also obtained. The corrected approximation results in a better match of the approximate polarization with the exact one, providing the assumption of weak transverse isotropy is met. When comparing the additional rotation with the deviation of the (observable) ray direction from the wavenormal, the qSV‐wave indicates transverse isotropy most clearly.

Estimating SV‐wave stacking velocities for transversely isotropic solids

Geophysics ◽  
1991 ◽  
Vol 56 (10) ◽  
pp. 1596-1602 ◽  
Author(s):  
Patricia A. Berge
Keyword(s):  
Shear Wave ◽  
Isotropic Medium ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Transversely Isotropic Medium ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Snell’S Law ◽  
Snell's Law

Conventional seismic experiments can record converted shear waves in anisotropic media, but the shear‐wave stacking velocities pose a problem when processing and interpreting the data. Methods used to find shear‐wave stacking velocities in isotropic media will not always provide good estimates in anisotropic media. Although isotropic methods often can be used to estimate shear‐wave stacking velocities in transversely isotropic media with vertical symmetry axes, the estimations fail for some transversely isotropic media even though the anisotropy is weak. For a given anisotropic medium, the shear‐wave stacking velocity can be estimated using isotropic methods if the isotropic Snell’s law approximates the anisotropic Snell’s law and if the shear wavefront is smooth enough near the vertical axis to be fit with an ellipse. Most of the 15 transversely isotropic media examined in this paper met these conditions for short reflection spreads and small ray angles. Any transversely isotropic medium will meet these conditions if the transverse isotropy is weak and caused by thin subhorizontal layering. For three of the media examined, the anisotropy was weak but the shear wave-fronts were not even approximately elliptical near the vertical axis. Thus, isotropic methods provided poor estimates of the shear‐wave stacking velocities. These results confirm that for any given transversely isotropic medium, it is possible to determine whether or not shear‐wave stacking velocities can be estimated using isotropic velocity analysis.

Azimuthal variations in scattering amplitudes induced by transverse isotropy

Geophysics ◽  
1991 ◽  
Vol 56 (10) ◽  
pp. 1584-1595 ◽  
Author(s):  
M. A. Pelissier ◽  
Anna Thomas‐Betts ◽  
Peter D. Vestergaard
Keyword(s):  
Seismic Waves ◽  
Symmetry Axis ◽  
Mode Conversion ◽  
Transverse Isotropy ◽  
Full Range ◽  
Transversely Isotropic ◽  
Anisotropy Axis ◽  
Scattering Coefficients ◽  
Transversely Isotropic Media ◽  
Isotropic Media

The study of amplitude variations of reflected and transmitted seismic waves due to anistropy has received considerable attention in recent years, but most investigations have concentrated on the effect of transverse isotropy with the symmetry axis either vertical or horizontal. The published results on the whole tend to exclude mode conversions. Amplitudes of all reflected and transmitted wave modes are addressed for [Formula: see text]-waves incident on boundaries between isotropic and transversely isotropic media, the symmetry axis of which is oriented at 45 degrees to the interface. The results cover the full range of incidence angles and all “acquisition azimuth” in the plane of the interface. When the anistropy axis is not normal to the interface, the scattering coefficients are shown to be highly dependent on the azimuth. The pattern of azimuthal variation is especially complicated in the case of mode conversion, and scattering coefficient profiles that are 180 degrees apart are not the same. This has the implication that source‐receiver interchangeability does not hold and could have serious consequences to amplitude studies in split‐spread surveys. Both the [Formula: see text] and [Formula: see text] reflections show strong azimuthal variations, dependent on both the dip and the strike of the anisotropy axis. It may be possible to recover shown that the scattered amplitude patterns are dominantly controlled by the value of the elastic modulus [Formula: see text].

Improved geometric-spreading approximation in layered transversely isotropic media

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. D85-D95 ◽  
Author(s):  
Alexey Stovas ◽  
Bjorn Ursin
Keyword(s):  
Single Layer ◽  
Transversely Isotropic ◽  
Velocity Analysis ◽  
Transversely Isotropic Medium ◽  
Numerical Tests ◽  
Transversely Isotropic Media ◽  
Geometric Spreading ◽  
Isotropic Media ◽  
Direct Approximation ◽  
Acoustic Approximation

A new approximation (direct approximation) for the relative geometric spreading of qP-waves in a layered transversely isotropic medium uses three traveltime parameters: two-way vertical traveltime, normal-moveout velocity, and the heterogeneity coefficient. These traveltime parameters, which can be estimated in velocity analysis, enter the parameters of geometric-spreading approximation. The new approximation is based on the acoustic approximation for a single-layer vertical transversely isotropic (VTI) medium with coefficients defined from Taylor expansion and asymptotic behavior at infinite offset. The acoustic approximation is used to reduce the number of parameters and to make the expression for the direct geometric-spreading approximation similar to a well-known traveltime approximation. The same traveltime parameters also allow for an independent approximation of the radiation pattern. Two numerical tests (single-layer and multilayered VTI models) prove that the new approximation is more accurate than standard approximations based on shifted hyperbola or two rational approximations.

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