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 3-D two‐point ray tracing for heterogeneous, weakly transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1883-1894 ◽  
Author(s):  
Vladimir Y. Grechka ◽  
George A. McMechan
Keyword(s):  
Ray Tracing ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Heterogeneous Media ◽  
Chebyshev Approximation ◽  
Transversely Isotropic ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Derivatives Of

A two‐point ray‐tracing technique for 3-D smoothly heterogeneous, weakly transversely isotropic media is based on Fermat’s principle and takes advantage of global Chebyshev approximation of both the model and curved rays. This approximation gives explicit relations for derivatives of traveltime with respect to ray parameters and allows use of the rapidly converging conjugate gradient method to compute traveltimes. The method is fast because, for most smoothly heterogeneous media, approximation of rays by only a few polynomials and a few conjugate gradient iterations provide excellent precision in traveltime calculation.

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.

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.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document