A spectral scheme for wave propagation simulation in 3-D elastic‐anisotropic media

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1593-1607 ◽  
Author(s):  
José M. Carcione ◽  
Dan Kosloff ◽  
Alfred Behle ◽  
Geza Seriani
Keyword(s):  
Wave Propagation ◽  
Fourier Method ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Computational Effort ◽  
Second Order ◽  
Rapid Expansion ◽  
Integration Algorithm ◽  
Fourier Pseudospectral Method ◽  
Band Limited

This work presents a new scheme for wave propagation simulation in three‐dimensional (3-D) elastic-anisotropic media. The algorithm is based on the rapid expansion method (REM) as a time integration algorithm, and the Fourier pseudospectral method for computation of the spatial derivatives. The REM expands the evolution operator of the second‐order wave equation in terms of Chebychev polynomials, constituting an optimal series expansion with exponential convergence. The modeling allows arbitrary elastic coefficients and density in lateral and vertical directions. Numerical methods which are based on finite‐difference techniques (in time and space) are not efficient when applied to realistic 3-D models, since they require considerable computer memory and time to obtain accurate results. On the other hand, the Fourier method permits a significant reduction of the working space, and the REM algorithm gives machine accuracy with half the computational effort as the usual second-order temporal differencing scheme. The new algorithm provides spectral accuracy for band limited wave propagation with no temporal or spatial dispersion. Hence, the combination REM/Fourier method could be considered at present the fastest and the most accurate of all the algorithms based on spectral methods in terms of efficiency of computer time. The technique is successfully tested with the analytical solution in the symmetry axis of a 3-D homogeneous transversely isotropic solid. Computed radiation patterns in clay shale and sandstone show the characteristics predicted by the theory. A final example computes synthetic seismograms showing the effects of shear‐wave splitting of a model where an azimuthally anisotropic cracked shale layer is inside a transversely isotropic sandstone.

Efficient modeling of wave propagation in a vertical transversely isotropic attenuative medium based on fractional Laplacian

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T121-T131 ◽  
Author(s):  
Tieyuan Zhu ◽  
Tong Bai
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fractional Laplacian ◽  
Numerical Solutions ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Seismic Attenuation ◽  
Fourier Pseudospectral Method ◽  
Frequency Independent ◽  
Theoretical Predictions

To efficiently simulate wave propagation in a vertical transversely isotropic (VTI) attenuative medium, we have developed a viscoelastic VTI wave equation based on fractional Laplacian operators under the assumption of weak attenuation ([Formula: see text]), where the frequency-independent [Formula: see text] model is used to mathematically represent seismic attenuation. These operators that are nonlocal in space can be efficiently computed using the Fourier pseudospectral method. We evaluated the accuracy of numerical solutions in a homogeneous transversely isotropic medium by comparing with theoretical predictions and numerical solutions by an existing viscoelastic-anisotropic wave equation based on fractional time derivatives. To accurately handle heterogeneous [Formula: see text], we select several [Formula: see text] values to compute their corresponding fractional Laplacians in the wavenumber domain and interpolate other fractional Laplacians in space. We determined its feasibility by modeling wave propagation in a 2D heterogeneous attenuative VTI medium. We concluded that the new wave equation is able to improve the efficiency of wave simulation in viscoelastic-VTI media by several orders and still maintain accuracy.

Forbidden directions for inhomogeneous pure shear waves in dissipative anisotropic media

Geophysics ◽  
1995 ◽  
Vol 60 (2) ◽  
pp. 522-530 ◽  
Author(s):  
José M. Carcione ◽  
Fabio Cavallini
Keyword(s):  
Wave Propagation ◽  
Pure Shear ◽  
Plane Waves ◽  
Symmetry Plane ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Material Interfaces ◽  
Inhomogeneous Waves ◽  
Propagation Properties ◽  
The Waves

In this work we investigate the wave‐propagation properties of pure shear, inhomogeneous, viscoelastic plane waves in the symmetry plane of a monoclinic medium. In terms of seismic propagation, the problem is to describe SH‐waves traveling through a fractured transversely isotropic formation where we assume that the waves are inhomogeneous with amplitudes varying across surfaces of constant phase. This assumption is widely supported by theoretical and experimental evidence. The results are presented in terms of polar diagrams of the quality factor, attenuation, slowness, and energy velocity curves. Inhomogeneous waves are more anisotropic and dissipative than homogeneous viscoelastic plane waves, for which the wavenumber and attenuation directions coincide. Moreover, the theory predicts, beyond a given degree of inhomogeneity, the existence of “stop bands” where there is no wave propagation. This phenomenon does not occur in dissipative isotropic and elastic anisotropic media. The combination of anelasticity and anisotropy activates these bands. They exist even in very weakly anisotropic and quasi‐elastic materials; only a finite value of Q is required. Weaker anisotropy does not affect the width of the bands, but increases the threshold of inhomogeneity above which they appear; moreover, near the threshold, lower attenuation implies narrower bands. A numerical simulation suggests that, in the absence of material interfaces or heterogeneities, the wavefield is mainly composed of homogeneous waves.

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.

Accurate simulations of pure quasi-P-waves in complex anisotropic media

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T341-T348 ◽  
Author(s):  
Sheng Xu ◽  
Hongbo Zhou
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Differential Equations ◽  
Transverse Isotropy ◽  
Anisotropic Media ◽  
Second Order ◽  
P Wave ◽  
Pseudodifferential Equation ◽  
Reverse Time ◽  
Time Migration

Reverse time migration (RTM) in complex anisotropic media requires calculation of the propagation of a single-mode wave, the quasi-P-wave. This was conventionally realized by solving a [Formula: see text] system of second-order partial differential equations. The implementation of this [Formula: see text] system required at least twice the computational resources as compared with the acoustic wave equation. The S-waves, an introduced auxiliary function in this system, were treated as artifacts in the RTM. Furthermore, the [Formula: see text] system suffered numerical stability problems at the places in which abrupt changes of symmetric axis of anisotropy exist, which brings more challenges to real data implementation. On the other hand, the Alkhalifah’s equation, which governs the pure quasi-P-wave propagation, was hard to solve because it was a pseudodifferential equation. We proposed a pure quasi-P-wave equation that can be easily implemented within current imaging framework. Our new equation was obtained by decomposing the original pseudodifferential operator into two numerical solvable operators: a differential operator and a scalar operator. The combination of these two operators yielded an accurate phase of quasi-P-wave propagation. Our solution was S-wave free and numerically stable for very complicated models. Because only one equation was required to resolve numerically, the new proposed scheme was more efficient than those conventional schemes that solve the [Formula: see text] second-order differential equations system. For tilted transverse isotropy (TTI) RTM implementation, the required increase of numerical cost was minimal, and we could expect at least a factor of two of improvement of efficiency. We showed the effectiveness and robustness of our method with numerical examples with simple and very complicated TTI models, the SEG Advanced Modeling (SEAM) model. Further extension of our approach to orthorhombic anisotropy or tilted orthorhombic anisotropy was straightforward.

Simulation of anisotropic wave propagation based upon a spectral element method

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1251-1260 ◽  
Author(s):  
Dimitri Komatitsch ◽  
Christophe Barnes ◽  
Jeroen Tromp
Keyword(s):  
Wave Propagation ◽  
Spectral Element Method ◽  
Mass Matrix ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Spectral Element ◽  
Weak Formulation ◽  
Element Mesh ◽  
Element Method

We introduce a numerical approach for modeling elastic wave propagation in 2-D and 3-D fully anisotropic media based upon a spectral element method. The technique solves a weak formulation of the wave equation, which is discretized using a high‐order polynomial representation on a finite element mesh. For isotropic media, the spectral element method is known for its high degree of accuracy, its ability to handle complex model geometries, and its low computational cost. We show that the method can be extended to fully anisotropic media. The mass matrix obtained is diagonal by construction, which leads to a very efficient fully explicit solver. We demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2-D transversely isotropic test case, and by comparing its predictions against those based upon a finite difference method for a 2-D heterogeneous, anisotropic medium. We show its generality and its flexibility by modeling wave propagation in a 3-D transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid.

P‐wave signatures and notation for transversely isotropic media: An overview

Geophysics ◽  
1996 ◽  
Vol 61 (2) ◽  
pp. 467-483 ◽  
Author(s):  
Ilya Tsvankin
Keyword(s):  
Wave Propagation ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Seismic Inversion ◽  
P Wave ◽  
Symmetry Direction ◽  
Time Migration ◽  
Transversely Isotropic Media ◽  
Isotropic Media

Progress in seismic inversion and processing in anisotropic media depends on our ability to relate different seismic signatures to the anisotropic parameters. While the conventional notation (stiffness coefficients) is suitable for forward modeling, it is inconvenient in developing analytic insight into the influence of anisotropy on wave propagation. Here, a consistent description of P‐wave signatures in transversely isotropic (TI) media with arbitrary strength of the anisotropy is given in terms of Thomsen notation. The influence of transverse isotropy on P‐wave propagation is shown to be practically independent of the vertical S‐wave velocity [Formula: see text], even in models with strong velocity variations. Therefore, the contribution of transverse isotropy to P‐wave kinematic and dynamic signatures is controlled by just two anisotropic parameters, ε and δ, with the vertical velocity [Formula: see text] being a scaling coefficient in homogeneous models. The distortions of reflection moveouts and amplitudes are not necessarily correlated with the magnitude of velocity anisotropy. The influence of transverse isotropy on P‐wave normal‐moveout (NMO) velocity in a horizontally layered medium, on small‐angle reflection coefficient, and on point‐force radiation in the symmetry direction is entirely determined by the parameter δ. Another group of signatures of interest in reflection seisimology—the dip‐dependence of NMO velocity, magnitude of nonhyperbolic moveout, time‐migration impulse response, and the radiation pattern near vertical—is dependent on both anisotropic parameters (ε and δ) and is primarily governed by the difference between ε and δ. Since P‐wave signatures are so sensitive to the value of ε − δ, application of the elliptical‐anisotropy approximation (ε = δ) in P‐wave processing may lead to significant errors. Many analytic expressions given in the paper remain valid in transversely isotropic media with a tilted symmetry axis. Moreover, the equation for NMO velocity from dipping reflectors, as well as the nonhyperbolic moveout equation, can be used in symmetry planes of any anisotropic media (e.g., orthorhombic).

Time step n-tupling for wave equations

Geophysics ◽  
2017 ◽  
Vol 82 (6) ◽  
pp. T249-T254 ◽  
Author(s):  
Lasse Amundsen ◽  
Ørjan Pedersen
Keyword(s):  
Discrete Time ◽  
Wave Equations ◽  
Fourier Method ◽  
Computational Effort ◽  
Second Order ◽  
Time Step ◽  
Time Stepping ◽  
Previous Time ◽  
Time Integrators ◽  
New Time

We have constructed novel temporal discretizations for wave equations. We first select an explicit time integrator that is of second order, leading to classic time marching schemes in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. Then, we determine how the time step can be doubled, tripled, or generally, [Formula: see text]-tupled, producing a new time-stepping method in which the next value of the wavefield at the discrete time [Formula: see text] is computed from current values known at time [Formula: see text] and the previous time [Formula: see text]. In-between time values of the wavefield are eliminated. Using the Fourier method to calculate space derivatives, the new time integrators allow larger stable time steps than traditional time integrators; however, like the Lax-Wendroff procedure, they require more computational effort per time step. Because the new schemes are developed from the classic second-order time-stepping scheme, they will have the same properties, except the Courant-Friedrichs-Lewy stability condition, which becomes relaxed by the factor [Formula: see text] compared with the classic scheme. As an example, we determine the method for solving scalar wave propagation in which doubling the time step is 15% faster than a Lax-Wendroff correction scheme of the same spatial order because it can increase the time step by [Formula: see text] only.

Reflection and transmission coefficients for seismic waves in ellipsoidally anisotropic media

Geophysics ◽  
1979 ◽  
Vol 44 (1) ◽  
pp. 27-38 ◽  
Author(s):  
P. F. Daley ◽  
F. Hron
Keyword(s):  
Wave Propagation ◽  
Anisotropic Medium ◽  
Transverse Isotropy ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
The Earth ◽  
Special Case

The deficiency of an isotropic model of the earth in the explanation of observed traveltime phenomena has led to the mathematical investigation of elastic wave propagation in anisotropic media. A type of anisotropy dealt with in the literature (Potsma, 1955; Cerveny and Psencik, 1972; and Vlaar, 1968) is uniaxial anisotropy or transverse isotropy. A special case of transverse isotropy which assumes the wavefronts to be ellipsoids of revolution has been used by Cholet and Richard (1954) and Richards (1960) in accounting for the observed traveltimes at Berraine in the Sahara and in the foothills of Western Canada. The kinematics of this problem have been treated in a number of papers, the most notable being Gassmann (1964). However, to appreciate fully the effect of anisotropy, the dynamics of the problem must be explored. Assuming a model of the earth consisting of plane transversely isotropic layers with the axes of anisotropy perpendicular to the interfaces, a prime requisite for obtaining amplitude distance curves or synthetic seismograms is the calculation of reflection and transmission coefficients at the interfaces. In this paper the special case of ellipsoidal anisotropy will be considered. That the quasi‐shear SV wavefront is forced to be spherical by this assumption is unfortunate, but it is instructive to investigate this simple anisotropic model as it incorporates many features inherent to wave propagation in a more general anisotropic medium. A brief outline of the theory for wave propagation in an ellipsoidally anisotropic medium is given and the analytic expressions for the reflection and transmission coefficients are listed. A complete derivation of reflection and transmission coefficients in transversely isotropic media can be found in Daley and Hron (1977). Finally, all 24 possible reflection and transmission coefficients and surface conversion coefficients are displayed for varying degrees of anisotropy.

Stability Conditions for Wave Simulation in 3-D Anisotropic Media with the Pseudospectral Method

2012 ◽  
Vol 12 (3) ◽  
pp. 703-720 ◽  
Author(s):  
Wensheng Zhang
Keyword(s):  
Inverse Problem ◽  
Wave Propagation ◽  
Pseudospectral Method ◽  
Transversely Isotropic ◽  
Anisotropic Media ◽  
Stability Conditions ◽  
Wave Simulation ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
D Wave

AbstractSimulation of elastic wave propagation has important applications in many areas such as inverse problem and geophysical exploration. In this paper, stability conditions for wave simulation in 3-D anisotropic media with the pseudospectral method are investigated. They can be expressed explicitly by elasticity constants which are easy to be applied in computations. The 3-D wave simulation for two typical anisotropic media, transversely isotropic media and orthorhombic media, are carried out. The results demonstrate some satisfactory behaviors of the pseudospectral method.

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