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.

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.

Elastic constants of transversely isotropic media from constrained aperture traveltimes

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 658-667 ◽  
Author(s):  
Reinaldo J. Michelena
Keyword(s):  
Elastic Constants ◽  
Heterogeneous Media ◽  
Transversely Isotropic ◽  
System Of Equations ◽  
Sh Waves ◽  
Velocity Models ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
P And Sv Waves ◽  
Homogeneous Media

The elastic constants that control P‐ and SV‐wave propagation in a transversely isotropic media can be estimated by using P‐ and SV‐wave traveltimes from either crosswell or VSP geometries. The procedure consists of two steps. First, elliptical velocity models are used to fit the traveltimes near one axis. The result is four elliptical parameters that represent direct and normal moveout velocities near the chosen axis for P‐ and SV‐waves. Second, the elliptical parameters are used to solve a system of four equations and four unknown elastic constants. The system of equations is solved analytically, yielding simple expressions for the elastic constants as a function of direct‐ and normal‐moveout velocities. For SH‐waves, the estimation of the corresponding elastic constants is easier because the phase velocity is already elliptical. The procedure, introduced for homogeneous media, is generalized to heterogeneous media by using tomographic techniques.

Analysis of reflection traveltimes in 3-D transversely isotropic heterogeneous media

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1884-1895 ◽  
Author(s):  
Vladimir Y. Grechka ◽  
George A. McMechan
Keyword(s):  
Ray Tracing ◽  
Heterogeneous Media ◽  
Chebyshev Approximation ◽  
Transversely Isotropic ◽  
Singular Value ◽  
Common Source ◽  
Model Parameters ◽  
Value Analysis ◽  
Frechet Derivatives ◽  
Fréchet Derivatives

A two‐point ray‐tracing technique for rays reflected from irregular, but smooth, interfaces in 3-D transversely isotropic heterogeneous media is developed. The method is based on Chebyshev parameterization of curved segments of the reflected rays, of the reflectors, and of the velocity and anisotropy distributions in the model. Chebyshev approximation also can describe the reflection traveltime surfaces to compress traveltime data by replacing them with coefficients of the corresponding Chebyshev series. The advantage of the proposed parameterization is that it gives traveltime as an explicit function of the model parameters. This explicitly provides the Frechét derivatives of the traveltime with respect to the model parameters. The Frechét derivatives are used in two ways. First, a two‐term Taylor series is constructed to relate variations in the model parameters to the corresponding perturbations in the traveltimes. This makes it possible, based on the results of a single ray tracing in a relatively simple model, to predict traveltimes for a range of more complicated models, without any additional ray tracing. Second, singular‐value decomposition of the Frechét matrix determines the influence of various model parameters on common‐source and common‐midpoint traveltimes. The singular‐value analysis shows that common‐source traveltimes depend mainly on the reflector position and shape. The common‐midpoint traveltimes also contain additional information about lateral velocity heterogeneity and anisotropy. However, both of these parameters affect the traveltimes in similar ways and so usually cannot be determined separately.

scholarly journals NMO‐velocity surfaces and Dix‐type formulas in anisotropic heterogeneous media

Geophysics ◽  
2002 ◽  
Vol 67 (3) ◽  
pp. 939-951 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Cross Sections ◽  
Heterogeneous Media ◽  
Transversely Isotropic ◽  
Radius Vector ◽  
P Wave ◽  
Reflection Data ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Anisotropic Layers ◽  
Zero Offset

Reflection moveout of pure modes recorded on conventional‐length spreads is described by a normal‐moveout (NMO) velocity that depends on the orientation of the common‐midpoint (CMP) line. Here, we introduce the concept of NMO‐velocity surfaces, which are obtained by plotting the NMO velocity as the radius‐vector along all possible directions in 3‐D space, and use it to develop Dix‐type averaging and differentiation algorithms in anisotropic heterogeneous media. The intersection of the NMO‐velocity surface with the horizontal plane represents the NMO ellipse that can be estimated from wide‐azimuth reflection data. We demonstrate that the NMO ellipse and conventional‐spread moveout as a whole can be modeled by Dix‐type averaging of specifically oriented cross‐sections of the NMO‐velocity surfaces along the zero‐offset reflection raypath. This formalism is particularly simple to implement for a stack of homogeneous anisotropic layers separated by plane dipping boundaries. Since our method involves computing just a single (zero‐offset) ray for a given reflection event, it can be efficiently used in anisotropic stacking‐velocity tomography. Application of the Dix‐type averaging to layered transversely isotropic media with a vertical symmetry axis (VTI) shows that the presence of dipping interfaces above the reflector makes the P‐wave NMO ellipse dependent on the vertical velocity and anisotropic coefficients ε and δ. In contrast, P‐wave moveout in VTI models with a horizontally layered overburden is fully controlled by the NMO velocity of horizontal events and the Alkhalifah‐Tsvankin coefficient η ≈ ε − δ. Hence, in some laterally heterogeneous, layered VTI models P‐wave reflection data may provide enough information for anisotropic depth processing.

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