Effective and efficient approaches for calculating seismic ray velocity and attenuation in viscoelastic anisotropic media

Geophysics ◽  
2020 ◽  
Vol 86 (1) ◽  
pp. C19-C35
Author(s):  
Jianlu Wu ◽  
Bing Zhou ◽  
Xingwang Li ◽  
Youcef Bouzidi
Keyword(s):  
Velocity Vector ◽  
Hamiltonian Function ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Quality Factors ◽  
Approximate Methods ◽  
Slowness Vector ◽  
Homogeneous Complex ◽  
Isotropic Media ◽  
Complex Ray

In viscoelastic anisotropic media, the elastic moduli, slowness vector, phase, and ray velocity are all complex-valued quantities in the frequency domain. Solving the complex eikonal equation becomes computationally complex and time-consuming. We have developed two approximate methods to effectively calculate the ray velocity vector, attenuation, and quality factor in viscoelastic transversely isotropic media with a vertical symmetry axis (VTI) and in orthorhombic (ORT) anisotropy. The first method is based on the perturbation theory (PER) under the assumption of a homogeneous complex ray vector, which is obtained by applying the elastic background and viscoelastic perturbations to the real and imaginary components of the modulus tensor, respectively. The perturbations of the slowness vectors of the three wave modes (qP, qSV, and qSH) are determined through the vanishing Hamiltonian function. The second method is derived by applying a real slowness direction (RSD) to the inhomogeneous complex slowness vector and then approximately calculating the complex ray velocity vector with the condition of the homogeneous complex vector. The numerical results verify that the two approaches can produce accurate ray velocity vector, attenuation, and quality factors of the qP-wave in viscoelastic VTI and ORT media. The RSD method can yield high accuracies of ray velocity for the qSV- and qSH-wave in viscoelastic VTI models even at triplication of the qSV wavefronts, as well as qS1 and qS2 in a weak ORT medium ([Formula: see text] > 20), except for near the cusp of the qS1 wavefronts (errors approximately 6%) where the PER has more than 10% error.

Calculation of slowness vectors from ray directions for qP-, qSV-, and qSH-waves in tilted transversely isotropic media

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. C153-C160 ◽  
Author(s):  
Lijiao Zhang ◽  
Bing Zhou
Keyword(s):  
Ray Tracing ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Seismic Wave Propagation ◽  
Previous Method ◽  
New Methods ◽  
Slowness Vector ◽  
Isotropic Media ◽  
Tti Media ◽  
Wave Modes

Kinematic ray tracing is an effective way to simulate the seismic wave propagation in isotropic and anisotropic media. It is essential to know the ray velocity when tracing seismic rays. But in anisotropic media, the ray velocity is a function of the direction of the slowness vector instead of the ray direction and it often deviates from the phase velocity. In this case, it causes a critical problem for ray tracing, which is how to calculate the ray velocity from a known ray direction. If we could calculate the phase slowness vector from ray directions, the ray velocity could be computed. We have evaluated a previous method in the first place. Then, we developed two new methods to solve two existing problems of the previous method: (1) It leads to complex and multiple solutions of the slowness vector and (2) it mixes up the qP- and qSV-wave modes. Our first method solves the two problems by applying eigenvalues to separate the wave modes and decrease the two unknowns ([Formula: see text] and [Formula: see text]) to only one unknown in two equations. Our second method is based on the general relationship between the slowness and ray-velocity vectors and shows that only one unknown is involved in one equation for tilted transversely isotropic (TTI) media. After obtaining the slowness vector, the ray velocity can be computed easily. A 2D model is designed to test the feasibility of the new methods. Using the results for the model, we found that the presented approaches were applicable for ray tracing in TTI media.

Normal moveout from dipping reflectors in anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 268-284 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Symmetry Axis ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Weak Anisotropy ◽  
Anisotropic Models ◽  
Transversely Isotropic Media ◽  
Wide Range ◽  
Isotropic Media ◽  
The Difference

Description of reflection moveout from dipping interfaces is important in developing seismic processing methods for anisotropic media, as well as in the inversion of reflection data. Here, I present a concise analytic expression for normal‐moveout (NMO) velocities valid for a wide range of homogeneous anisotropic models including transverse isotropy with a tilted in‐plane symmetry axis and symmetry planes in orthorhombic media. In transversely isotropic media, NMO velocity for quasi‐P‐waves may deviate substantially from the isotropic cosine‐of‐dip dependence used in conventional constant‐velocity dip‐moveout (DMO) algorithms. However, numerical studies of NMO velocities have revealed no apparent correlation between the conventional measures of anisotropy and errors in the cosine‐of‐dip DMO correction (“DMO errors”). The analytic treatment developed here shows that for transverse isotropy with a vertical symmetry axis, the magnitude of DMO errors is dependent primarily on the difference between Thomsen parameters ε and δ. For the most common case, ε − δ > 0, the cosine‐of‐dip–corrected moveout velocity remains significantly larger than the moveout velocity for a horizontal reflector. DMO errors at a dip of 45 degrees may exceed 20–25 percent, even for weak anisotropy. By comparing analytically derived NMO velocities with moveout velocities calculated on finite spreads, I analyze anisotropy‐induced deviations from hyperbolic moveout for dipping reflectors. For transversely isotropic media with a vertical velocity gradient and typical (positive) values of the difference ε − δ, inhomogeneity tends to reduce (sometimes significantly) the influence of anisotropy on the dip dependence of moveout velocity.

scholarly journals Transformation to zero offset in transversely isotropic media

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 947-963 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Ray Tracing ◽  
Impulse Response ◽  
Homogeneous Medium ◽  
Anisotropy Parameter ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Amplitude Distribution ◽  
Isotropic Media ◽  
Zero Offset ◽  
Vertical Inhomogeneity

Nearly all dip‐moveout correction (DMO) implementations to date assume isotropic homogeneous media. Usually, this has been acceptable considering the tremendous cost savings of homogeneous isotropic DMO and considering the difficulty of obtaining the anisotropy parameters required for effective implementation. In the presence of typical anisotropy, however, ignoring the anisotropy can yield inadequate results. Since anisotropy may introduce large deviations from hyperbolic moveout, accurate transformation to zero‐offset in anisotropic media should address such nonhyperbolic moveout behavior of reflections. Artley and Hale’s v(z) ray‐tracing‐based DMO, developed for isotropic media, provides an attractive approach to treating such problems. By using a ray‐tracing procedure crafted for anisotropic media, I modify some aspects of their DMO so that it can work for v(z) anisotropic media. DMO impulse responses in typical transversely isotropic (TI) models (such as those associated with shales) deviate substantially from the familiar elliptical shape associated with responses in homogeneous isotropic media (to the extent that triplications arise even where the medium is homogeneous). Such deviations can exceed those caused by vertical inhomogeneity, thus emphasizing the importance of taking anisotropy into account in DMO processing. For isotropic or elliptically anisotropic media, the impulse response is an ellipse; but as the key anisotropy parameter η varies, the shape of the response differs substantially from elliptical. For typical η > 0, the impulse response in TI media tends to broaden compared to the response in an isotropic homogeneous medium, a behavior opposite to that encountered in typical v(z) isotropic media, where the response tends to be squeezed. Furthermore, the amplitude distribution along the DMO operator differs significantly from that for isotropic media. Application of this anisotropic DMO to data from offshore Africa resulted in a considerably better alignment of reflections from horizontal and dipping reflectors in common‐midpoint gather than that obtained using an isotropic DMO. Even the presence of vertical inhomogeneity in this medium could not eliminate the importance of considering the shale‐induced anisotropy.

Migration of P‐wave reflection data in transversely isotropic media

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 591-596 ◽  
Author(s):  
Suhas Phadke ◽  
S. Kapotas ◽  
N. Dai ◽  
Ernest R. Kanasewich
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Limited Range ◽  
Horizontal Velocity ◽  
P Wave ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Wave propagation in transversely isotropic media is governed by the horizontal and vertical wave velocities. The quasi‐P(qP) wavefront is not an ellipse; therefore, the propagation cannot be described by the wave equation appropriate for elliptically anisotropic media. However, for a limited range of angles from the vertical, the dispersion relation for qP‐waves can be approximated by an ellipse. The horizontal velocity necessary for this approximation is different from the true horizontal velocity and depends upon the physical properties of the media. In the method described here, seismic data is migrated using a 45-degree wave equation for elliptically anisotropic media with the horizontal velocity determined by comparing the 45-degree elliptical dispersion relation and the quasi‐P‐dispersion relation. The method is demonstrated for some synthetic data sets.

The migrator’s equation for anisotropic media

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1429-1434 ◽  
Author(s):  
N. F. Uren ◽  
G. H. F. Gardner ◽  
J. A. McDonald
Keyword(s):  
Physical Model ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Vertical Axis ◽  
Original Structure ◽  
Model Data ◽  
P Waves ◽  
Reflection Seismic ◽  
Isotropic Media ◽  
Zero Offset

The migrator’s equation, which gives the relationship between real and apparent dips on a reflector in zero‐offset reflection seismic sections, may be readily implemented in one step with a frequency‐domain migration algorithm for homogeneous media. Huygens’ principle is used to derive a similar relationship for anisotropic media where velocities are directionally dependent. The anisotropic form of the migrator’s equation is applicable to both elliptically and nonelliptically anisotropic media. Transversely isotropic media are used to demonstrate the performance of an f-k implementation of the migrator’s equation for anisotropic media. In such a medium SH-waves are elliptically anisotropic, while P-waves are nonelliptically anisotropic. Numerical model data and physical model data demonstrate the performance of the algorithm, in each case recovering the original structure. Isotropic and anisotropic migration of anisotropic physical model data are compared experimentally, where the anisotropic velocity function of the medium has a vertical axis of symmetry. Only when anisotropic migration is used is the original structure recovered.

Refraction across an angular unconformity between nonparallel TI media

Geophysics ◽  
2005 ◽  
Vol 70 (2) ◽  
pp. D19-D28 ◽  
Author(s):  
John Hodgkinson ◽  
R. James Brown
Keyword(s):  
Vertical Plane ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Demonstration Program ◽  
P Waves ◽  
Angular Unconformity ◽  
Isotropic Media ◽  
The Media ◽  
P And Sv Waves ◽  
Ti Media

Assuming elliptical wavefronts, we reformulate refraction theory for transversely isotropic (TI) media based on the use of the auxiliary angle, α, which is intermediate between the phase angle, θ, and the group angle, ϕ. When considering the application of stretching to transform elliptically anisotropic media into isotropic media, the auxiliary angle is a natural one to use because both θ and ϕ → α under such stretching. Our present formulation for TI media makes the assumption of elliptical anisotropy, which is valid generally for SH-waves but only as a special case for P-and SV-waves, where, in the SV case, the only possible ellipses are circles. Nevertheless, the theory has useful applications for P-waves over limited ranges of propagation direction (e.g., in the short-spread approximation). Our formulation provides explicit results for all angles of incidence and for what we term an angular unconformity between two TI media, that is, for all orientations of the axes of symmetry for each of the media, and for all orientations of the interface, assuming these two axes and the interface normal to lie in the same vertical plane. Our conclusions have been verified by showing that the phase angles and phase velocities of the incident and refracted waves obey Snell's law across the interface. We also demonstrate, using auxiliary angles, that the description of refraction between elliptically anisotropic media by stretching the media to make them isotropic, then applying isotropic refraction, is also valid for our general angular-unconformity case. However, both stretching (1D) and either scaling (2D) or shearing must be applied correctly and separately to the two media. The refraction algorithm developed from this theory and another developed by Byun in terms of phase-velocity theory are currently the only published noniterative algorithms known to us for refraction across an angular unconformity where the axes of anisotropy are parallel neither to each other nor to the interface. Based on this theory, we have developed a demonstration program, AUXDEMOC, that computes the refracted-ray angles for any combination of parameters by the two equivalent methods: (1) anisotropic refraction and (2) stretching plus isotropic refraction. This program can be downloaded from http://www.crewes.org/under Free Software.

Approximations to a generalized real ray-tracing method for heterogeneous anisotropic viscoelastic media

Geophysics ◽  
2021 ◽  
pp. 1-108
Author(s):  
Jianlu Wu ◽  
Bing Zhou ◽  
youcef bouzidi ◽  
Xingwang Li
Keyword(s):  
Ray Tracing ◽  
Shear Waves ◽  
Transversely Isotropic ◽  
Vertical Axis ◽  
Real Space ◽  
Quality Factors ◽  
Ray Tracing Method ◽  
The Real ◽  
Complex Ray ◽  
Tracing Method

The real raytracing approach leads to an effective solution in the real space domain using a homogenous ray velocity vector. However, it fails to yield solutions for quasi-shear waves, which suffer triplication of the wavefronts. To address this challenging problem, a generalized real ray-tracing method and its new approximations are presented to solve the complex ray equation. The numerical results show that the generalized ray-tracing method is superior to the real ray-tracing method in the presence of triplications of the quasi-shear waves in the computation of ray velocity, ray attenuation, and ray quality factors, as well as the reflection and transmission coefficients in viscoelastic anisotropic media. Based on the assumptions of the real slowness direction and real polarization vectors, two new approximations of the generalized real ray-tracing method are developed for directly computing the homogeneous complex ray velocity vectors of three wave modes (qP, qS1, and qS2). These approximations significantly improve the computational efficiency by avoiding the iterative process required by the generalized real ray-tracing method that is inherited from the real ray-tracing method. The computational accuracies are verified through transversely isotropic models and orthorhombic models with different strengths of attenuation and anisotropy. The incorporation of the new approximation into the shortest-path method turned out to be an efficient and accurate method for seismic ray tracing in heterogeneous viscoelastic and transversely isotropic media with a vertical axis of symmetry, even in the presence of strong attenuation and anisotropy.

Gaussian beam depth migration for anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (5) ◽  
pp. 1474-1484 ◽  
Author(s):  
Tariq Alkhalifah
Keyword(s):  
Gaussian Beam ◽  
Ray Tracing ◽  
Anisotropy Parameter ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Inhomogeneous Media ◽  
Computational Effort ◽  
Isotropic Media ◽  
Gaussian Beam Migration

Gaussian beam migration (GBM), as it is implemented today, efficiently handles isotropic inhomogeneous media. The approach is based on the solution of the wave equation in ray‐centered coordinates. Here, I extend the method to work for 2-D migration in generally anisotropic inhomogeneous media. Extension of the Gaussian‐beam method from isotropic to anisotropic media involves modification of the kinematics and dynamics in the required ray tracing. While the accuracy of the paraxial expansion for anisotropic media is comparable to that for isotropic media, ray tracing in anisotropic media is much slower than that in isotropic media. However, because ray tracing is just a small portion of the computation in GBM, the increased computational effort in general anisotropic GBM is typically only about 40%. Application of this method to synthetic examples shows successful migration in inhomogeneous, transversely isotropic media for reflector dips up to and beyond 90°. Further applications to synthetic data of layered anisotropic media show the importance of applying the proper smoothing to the velocity field used in the migration. Also, tests with synthetic data show that the quality of anisotropic migration of steep events in a medium with velocity increasing with depth is much more sensitive to the Thomsen anisotropy parameter ε than to the parameter δ. Thus, a good estimate of ε is needed to apply anisotropic migration with confidence.

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.

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