The optimized expansion based low-rank method for wavefield extrapolation

Spectral methods are fast becoming an indispensable tool for wavefield extrapolation, especially in anisotropic media because it tends to be dispersion and artifact free as well as highly accurate when solving the wave equation. However, for inhomogeneous media, we face difficulties in dealing with the mixed space-wavenumber domain extrapolation operator efficiently. To solve this problem, we evaluated an optimized expansion method that can approximate this operator with a low-rank variable separation representation. The rank defines the number of inverse Fourier transforms for each time extrapolation step, and thus, the lower the rank, the faster the extrapolation. The method uses optimization instead of matrix decomposition to find the optimal wavenumbers and velocities needed to approximate the full operator with its explicit low-rank representation. As a result, we obtain lower rank representations compared with the standard low-rank method within reasonable accuracy and thus cheaper extrapolations. Additional bounds set on the range of propagated wavenumbers to adhere to the physical wave limits yield unconditionally stable extrapolations regardless of the time step. An application on the BP model provided superior results compared to those obtained using the decomposition approach. For transversely isotopic media, because we used the pure P-wave dispersion relation, we obtained solutions that were free of the shear wave artifacts, and the algorithm does not require that [Formula: see text]. In addition, the required rank for the optimization approach to obtain high accuracy in anisotropic media was lower than that obtained by the decomposition approach, and thus, it was more efficient. A reverse time migration result for the BP tilted transverse isotropy model using this method as a wave propagator demonstrated the ability of the algorithm.

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

Geophysics ◽

10.1190/geo2014-0242.1 ◽

2014 ◽

Vol 79 (6) ◽

pp. T341-T348 ◽

Cited By ~ 50

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.

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A hybrid finite-difference/low-rank solution to anisotropy acoustic wave equations

Geophysics ◽

10.1190/geo2018-0333.1 ◽

2019 ◽

Vol 84 (2) ◽

pp. T83-T91 ◽

Cited By ~ 2

Author(s):

Zhen-Dong Zhang ◽

Tariq Alkhalifah ◽

Zedong Wu

Keyword(s):

Finite Difference ◽

Pseudodifferential Operator ◽

Fourier Transforms ◽

Pseudodifferential Operators ◽

Correction Term ◽

P Wave ◽

Low Rank ◽

Low Rank Approximation ◽

Time Step ◽

Wavenumber Domain

P-wave extrapolation in anisotropic media suffers from SV-wave artifacts and computational dependency on the complexity of anisotropy. The anisotropic pseudodifferential wave equation cannot be solved using an efficient time-domain finite-difference (FD) scheme directly. The wavenumber domain allows us to handle pseudodifferential operators accurately; however, it requires either smoothly varying media or more computational resources. In the limit of elliptical anisotropy, the pseudodifferential operator reduces to a conventional operator. Therefore, we have developed a hybrid-domain solution that includes a space-domain FD solver for the elliptical anisotropic part of the anisotropic operator and a wavenumber-domain low-rank scheme to solve the pseudodifferential part. Thus, we split the original pseudodifferential operator into a second-order differentiable background and a pseudodifferential correction term. The background equation is solved using the efficient FD scheme, and the correction term is approximated by the low-rank approximation. As a result, the correction wavefield is independent of the velocity model, and, thus, it has a reduced rank compared with the full operator. The total computation cost of our method includes the cost of solving a spatial FD time-step update plus several fast Fourier transforms related to the rank. The accuracy of our method is of the order of the FD scheme. Applications to a simple homogeneous tilted transverse isotropic (TTI) medium and modified BP TTI models demonstrate the effectiveness of the approach.

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

Geophysics ◽

10.1190/1.1443974 ◽

1996 ◽

Vol 61 (2) ◽

pp. 467-483 ◽

Cited By ~ 152

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).

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Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media

Geophysics ◽

10.1190/geo2015-0184.1 ◽

2016 ◽

Vol 81 (2) ◽

pp. T63-T77 ◽

Cited By ~ 26

Author(s):

Jiubing Cheng ◽

Tariq Alkhalifah ◽

Zedong Wu ◽

Peng Zou ◽

Chenlong Wang

Keyword(s):

Elastic Waves ◽

Vector Fields ◽

Integral Operators ◽

Transversely Isotropic ◽

Anisotropic Media ◽

Low Rank ◽

Elastic Displacement ◽

Low Rank Approximation ◽

Time Step ◽

Domain Integral

In elastic imaging, the extrapolated vector fields are decoupled into pure wave modes, such that the imaging condition produces interpretable images. Conventionally, mode decoupling in anisotropic media is costly because the operators involved are dependent on the velocity, and thus they are not stationary. We have developed an efficient pseudospectral approach to directly extrapolate the decoupled elastic waves using low-rank approximate mixed-domain integral operators on the basis of the elastic displacement wave equation. We have applied [Formula: see text]-space adjustment to the pseudospectral solution to allow for a relatively large extrapolation time step. The low-rank approximation was, thus, applied to the spectral operators that simultaneously extrapolate and decompose the elastic wavefields. Synthetic examples on transversely isotropic and orthorhombic models showed that our approach has the potential to efficiently and accurately simulate the propagations of the decoupled quasi-P and quasi-S modes as well as the total wavefields for elastic wave modeling, imaging, and inversion.

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3-D prestack migration in anisotropic media

Geophysics ◽

10.1190/1.1443353 ◽

1993 ◽

Vol 58 (1) ◽

pp. 79-90 ◽

Cited By ~ 16

Author(s):

Zhengxin Dong ◽

George A. McMechan

Keyword(s):

A Priori ◽

Three Dimensional ◽

Synthetic Data ◽

Anisotropic Media ◽

P Wave ◽

Common Source ◽

Reverse Time ◽

Reverse Time Migration ◽

Prestack Migration ◽

Time Migration

A three‐dimensional (3-D) prestack reverse‐time migration algorithm for common‐source P‐wave data from anisotropic media is developed and illustrated by application to synthetic data. Both extrapolation of the data and computation of the excitation‐time imaging condition are implemented using a second‐order finite‐ difference solution of the 3-D anisotropic scalar‐wave equation. Poorly focused, distorted images are obtained if data from anisotropic media are migrated using isotropic extrapolation; well focused, clear images are obtained using anisotropic extrapolation. A priori estimation of the 3-D anisotropic velocity distribution is required. Zones of anomalous, directionally dependent reflectivity associated with anisotropic fracture zones are detectable in both the 3-D common‐ source data and the corresponding migrated images.

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Feasibility of nonhyperbolic moveout inversion in transversely isotropic media

Geophysics ◽

10.1190/1.1444407 ◽

1998 ◽

Vol 63 (3) ◽

pp. 957-969 ◽

Cited By ~ 78

Author(s):

Vladimir Grechka ◽

Ilya Tsvankin

Keyword(s):

Vertical Velocity ◽

Transverse Isotropy ◽

Horizontal Velocity ◽

P Wave ◽

Vacuum Field ◽

Percentage Error ◽

Correlated Errors ◽

Data Set ◽

Time Migration ◽

Quartic Term

Inversion of reflection traveltimes in anisotropic media can provide estimates of anisotropic coefficients required for seismic processing and lithology discrimination. Nonhyperbolic P-wave moveout for transverse isotropy with a vertical symmetry axis (VTI media) is controlled by the parameter η (or, alternatively, by the horizontal velocity Vhor), which is also responsible for the influence of anisotropy on all time‐processing steps, including dip‐moveout (DMO) correction and time migration. Here, we recast the nonhyperbolic moveout equation, originally developed by Tsvankin and Thomsen, as a function of Vhor and normal‐moveout (NMO) velocity Vnmo and introduce a correction factor in the denominator that increases the accuracy at intermediate offsets. Then we apply this equation to obtain Vhor and η from nonhyperbolic semblance analysis on long common midpoint (CMP) spreads and study the accuracy and stability of the inversion procedure. Our error analysis shows that the horizontal velocity becomes relatively well‐constrained by reflection traveltimes if the spreadlength exceeds twice the reflector depth. There is, however, a certain degree of tradeoff between Vhor and Vnmo caused by the interplay between the quadratic and quartic term of the moveout series. Since the errors in Vhor and Vnmo have opposite signs, the absolute error in the parameter η (which depends on the ratio Vhor/Vnmo) turns out to be at least two times bigger than the percentage error in Vhor. Therefore, the inverted value of η is highly sensitive to small correlated errors in reflection traveltimes, with moveout distortions on the order of 3–4 ms leading to errors in η up to ±0.1—even in the simplest model of a single VTI layer. Similar conclusions apply to vertically inhomogeneous media, in which the interval horizontal velocity can be obtained with an accuracy often comparable to that of the NMO velocity, while the interval values of η are distorted by the tradeoff between Vhor and Vnmo that gets amplified by the Dix‐type differentiation procedure. We applied nonhyperbolic semblance analysis to a walkaway VSP data set acquired at Vacuum field, New Mexico, and obtained a significant value of η = 0.19 indicative of nonnegligible anisotropy in this area. Then we combined moveout inversion results with the known vertical velocity to resolve the anisotropic coefficients ε and δ. However, in agreement with our modeling results, η estimation was significantly compounded by the scatter in the measured traveltimes. Certain instability in η inversion has no influence on the results of anisotropic poststack time migration because all kinematically equivalent models obtained from nonhyperbolic moveout give an adequate description of long‐spread reflection traveltimes. Also, inversion of nonhyperbolic moveout provides a relatively accurate horizontal‐velocity function that can be combined with the vertical velocity (if it is available) to estimate the anisotropic coefficient ε. However, η represents a valuable lithology indicator that can be obtained from surface P-wave data. Therefore, for purposes of lithology discrimination, it is preferable to find η by means of the more stable DMO method of Alkhalifah and Tsvankin.

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Inversion of azimuthally dependent NMO velocity in transversely isotropic media with a tilted axis of symmetry

Geophysics ◽

10.1190/1.1444714 ◽

2000 ◽

Vol 65 (1) ◽

pp. 232-246 ◽

Cited By ~ 27

Author(s):

Vladimir Grechka ◽

Ilya Tsvankin

Keyword(s):

Wave Velocity ◽

Symmetry Axis ◽

Transverse Isotropy ◽

Transversely Isotropic ◽

P Wave ◽

Model Parameters ◽

Symmetry Direction ◽

Time Migration ◽

Inversion Procedure ◽

Tti Media

Just as the transversely isotropic model with a vertical symmetry axis (VTI media) is typical for describing horizontally layered sediments, transverse isotropy with a tilted symmetry axis (TTI) describes dipping TI layers (such as tilted shale beds near salt domes) or crack systems. P-wave kinematic signatures in TTI media are controlled by the velocity [Formula: see text] in the symmetry direction, Thomsen’s anisotropic coefficients ε and δ, and the orientation (tilt ν and azimuth β) of the symmetry axis. Here, we show that all five parameters can be obtained from azimuthally varying P-wave NMO velocities measured for two reflectors with different dips and/or azimuths (one of the reflectors can be horizontal). The shear‐wave velocity [Formula: see text] in the symmetry direction, which has negligible influence on P-wave kinematic signatures, can be found only from the moveout of shear waves. Using the exact NMO equation, we examine the propagation of errors in observed moveout velocities into estimated values of the anisotropic parameters and establish the necessary conditions for a stable inversion procedure. Since the azimuthal variation of the NMO velocity is elliptical, each reflection event provides us with up to three constraints on the model parameters. Generally, the five parameters responsible for P-wave velocity can be obtained from two P-wave NMO ellipses, but the feasibility of the moveout inversion strongly depends on the tilt ν. If the symmetry axis is close to vertical (small ν), the P-wave NMO ellipse is largely governed by the NMO velocity from a horizontal reflector Vnmo(0) and the anellipticity coefficient η. Although for mild tilts the medium parameters cannot be determined separately, the NMO-velocity inversion provides enough information for building TTI models suitable for time processing (NMO, DMO, time migration). If the tilt of the symmetry axis exceeds 30°–40° (e.g., the symmetry axis can be horizontal), it is possible to find all P-wave kinematic parameters and construct the anisotropic model in depth. Another condition required for a stable parameter estimate is that the medium be sufficiently different from elliptical (i.e., ε cannot be close to δ). This limitation, however, can be overcome by including the SV-wave NMO ellipse from a horizontal reflector in the inversion procedure. While most of the analysis is carried out for a single layer, we also extend the inversion algorithm to vertically heterogeneous TTI media above a dipping reflector using the generalized Dix equation. A synthetic example for a strongly anisotropic, stratified TTI medium demonstrates a high accuracy of the inversion (subject to the above limitations).

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Fast algorithms for elastic-wave-mode separation and vector decomposition using low-rank approximation for anisotropic media

Geophysics ◽

10.1190/geo2014-0032.1 ◽

2014 ◽

Vol 79 (4) ◽

pp. C97-C110 ◽

Cited By ~ 61

Author(s):

Jiubing Cheng ◽

Sergey Fomel

Keyword(s):

Elastic Wave ◽

Fast Algorithms ◽

Anisotropic Media ◽

Wave Mode ◽

Low Rank ◽

Fourier Integral Operators ◽

Reverse Time ◽

Time Migration ◽

Mode Separation ◽

Vector Decomposition

Wave-mode separation and vector decomposition are significantly more expensive than wavefield extrapolation and are the computational bottleneck for elastic reverse time migration in heterogeneous anisotropic media. We have expressed elastic-wave-mode separation and vector decomposition for anisotropic media as space-wavenumber-domain operations in the form of Fourier integral operators and developed fast algorithms for their implementation using their low-rank approximations. Synthetic data generated from 2D and 3D models demonstrated that these methods are accurate and efficient.

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Elastic imaging with exact wavefield extrapolation for application to ocean-bottom 4C seismic data

Geophysics ◽

10.1190/geo2013-0152.1 ◽

2013 ◽

Vol 78 (6) ◽

pp. S265-S284 ◽

Cited By ~ 41

Author(s):

Matteo Ravasi ◽

Andrew Curtis

Keyword(s):

Model Simulation ◽

P Wave ◽

Solid Boundary ◽

Central Component ◽

Ocean Bottom ◽

Reverse Time ◽

Reverse Time Migration ◽

Time Migration ◽

True Time ◽

Wavefield Extrapolation

A central component of imaging methods is receiver-side wavefield backpropagation or extrapolation in which the wavefield from a physical source scattered at any point in the subsurface is estimated from data recorded by receivers located near or at the Earth’s surface. Elastic reverse-time migration usually accomplishes wavefield extrapolation by simultaneous reversed-time ‘injection’ of the particle displacements (or velocities) recorded at each receiver location into a wavefield modeling code. Here, we formulate an exact integral expression based on reciprocity theory that uses a combination of velocity-stress recordings and quadrupole-dipole backpropagating sources, rather than the commonly used approximate formula involving only particle velocity data and dipole backpropagating sources. The latter approximation results in two types of nonphysical waves in the scattered wavefield estimate: First, each arrival contained in the data is injected upward and downward rather than unidirectionally as in the true time-reversed experiment; second, all injected energy emits compressional and shear propagating modes in the model simulation (e.g., if a recorded P-wave is injected, both P and S propagating waves result). These artifacts vanish if the exact wavefield extrapolation integral is used. Finally, we show that such a formula is suitable for extrapolation of ocean-bottom 4C data: Due to the fluid-solid boundary conditions at the seabed, the data recorded in standard surveys are sufficient to perform backpropagation using the exact equations. Synthetic examples provide numerical evidence of the importance of correcting such errors.

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Simulating propagation of separated wave modes in general anisotropic media, Part I: qP-wave propagators

Geophysics ◽

10.1190/geo2012-0504.1 ◽

2014 ◽

Vol 79 (1) ◽

pp. C1-C18 ◽

Cited By ~ 20

Author(s):

Jiubing Cheng ◽

Wei Kang

Keyword(s):

Wave Equation ◽

Anisotropic Media ◽

Pure Mode ◽

Reverse Time ◽

S Wave ◽

Time Step ◽

Flexible Tool ◽

Mode Wave ◽

Time Migration ◽

Wave Modes

Wave propagation in an anisotropic medium is inherently described by elastic wave equations with P- and S-wave modes intrinsically coupled. We present an approach to simulate propagation of separated wave modes for forward modeling, migration, waveform inversion, and other applications in general anisotropic media. The proposed approach consists of two cascaded computational steps. First, we simulate equivalent elastic anisotropic wavefields with a minimal second-order coupled system (that we call here a pseudo-pure-mode wave equation), which describes propagation of all wave modes with a partial wave mode separation. Such a system for qP-wave is derived from the inverse Fourier transform of the Christoffel equation after a similarity transformation, which aims to project the original vector displacement wavefields onto isotropic references of the polarization directions of qP-waves. It accurately describes the kinematics of all wave modes and enhances qP-waves when the pseudo-pure-mode wavefield components are summed. The second step is a filtering to further project the pseudo-pure-mode wavefields onto the polarization directions of qP-waves so that residual qS-wave energy is removed and scalar qP-wave fields are accurately separated at each time step during wavefield extrapolation. As demonstrated in the numerical examples, pseudo-pure-mode wave equation plus correction of projection deviation provides a robust and flexible tool for simulating propagation of separated wave modes in transversely isotropic and orthorhombic media. The synthetic example of a Hess VTI model shows that the pseudo-pure-mode qP-wave equation can be used in prestack reverse-time migration applications.

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