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.

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).

Q least-squares reverse time migration based on the first-order viscoacoustic quasidifferential equations

Geophysics ◽  
2021 ◽  
pp. 1-65
Author(s):  
Yingming Qu ◽  
Yixin Wang ◽  
Zhenchun Li ◽  
Chang Liu
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Surface Topography ◽  
Density Perturbation ◽  
Second Order ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Order ◽  
Time Migration ◽  
Grid Scheme

Seismic wave attenuation caused by subsurface viscoelasticity reduces the quality of migration and the reliability of interpretation. A variety of Q-compensated migration methods have been developed based on the second-order viscoacoustic quasidifferential equations. However, these second-order wave-equation-based methods are difficult to handle with density perturbation and surface topography. In addition, the staggered grid scheme, which has an advantage over the collocated grid scheme because of its reduced numerical dispersion and enhanced stability, works in first-order wave-equation-based methods. We have developed a Q least-squares reverse time migration method based on the first-order viscoacoustic quasidifferential equations by deriving Q-compensated forward-propagated operators, Q-compensated adjoint operators, and Q-attenuated Born modeling operators. Besides, our method using curvilinear grids is available even when the attenuating medium has surface topography and can conduct Q-compensated migration with density perturbation. The results of numerical tests on two synthetic and a field data sets indicate that our method improves the imaging quality with iterations and produces better imaging results with clearer structures, higher signal-to-noise ratio, higher resolution, and more balanced amplitude by correcting the energy loss and phase distortion caused by Q attenuation. It also suppresses the scattering and diffracted noise caused by the surface topography.

3-D prestack migration in anisotropic media

Geophysics ◽  
1993 ◽  
Vol 58 (1) ◽  
pp. 79-90 ◽  
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.

Depth imaging with multiples

Geophysics ◽  
2001 ◽  
Vol 66 (1) ◽  
pp. 246-255 ◽  
Author(s):  
Oong K. Youn ◽  
Hua‐wei Zhou
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Ray Tracing ◽  
Scalar Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Prestack Depth Migration ◽  
Depth Imaging ◽  
Depth Migration ◽  
Time Migration

Depth imaging with multiples is a prestack depth migration method that uses multiples as the signal for more accurate boundary mapping and amplitude recovery. The idea is partially related to model‐based multiple‐suppression techniques and reverse‐time depth migration. Conventional reverse‐time migration uses the two‐way wave equation for the backward wave propagation of recorded seismic traces and ray tracing or the eikonal equation for the forward traveltime computation (the excitation‐time imaging principle). Consequently, reverse‐time migration differs little from most other one‐way wave equation or ray‐tracing migration methods which expect only primary reflection events. Because it is almost impossible to attenuate multiples without degrading primaries, there has been a compelling need to devise a tool to use multiples constructively in data processing rather than attempting to destroy them. Furthermore, multiples and other nonreflecting wave types can enhance boundary imaging and amplitude recovery if a full two‐way wave equation is used for migration. The new approach solves the two‐way wave equation for both forward and backward directions of wave propagation using a finite‐difference technique. Thus, it handles all types of acoustic waves such as reflection (primary and multiples), refraction, diffraction, transmission, and any combination of these waves. During the imaging process, all these different types of wavefields collapse at the boundaries where they are generated or altered. The process goes through four main steps. First, a source function (wavelet) marches forward using the full two‐way scalar wave equation from a source location toward all directions. Second, the recorded traces in a shot gather march backward using the full two‐way scalar wave equation from all receiver points in the gather toward all directions. Third, the two forward‐ and backward‐propagated wavefields are correlated and summed for all time indices. And fourth, a Laplacian image reconstruction operator is applied to the correlated image frame. This technique can be applied to all types of seismic data: surface seismic, vertical seismic profile (VSP), crosswell seismic, vertical cable seismic, ocean‐bottom cable (OBC) seismic, etc. Because it migrates all wave types, the input data require no or minimal preprocessing (demultiple should not be done, but near‐surface or acquisition‐related problems might need to be corrected). Hence, it is only a one‐step process from the raw field gathers to a final depth image. External noise in the raw data will not correlate with the forward wavefield except for some coincidental matching; therefore, it is usually unnecessary to do signal enhancement processing before the depth imaging with multiples. The input velocity model could be acquired from various methods such as iterative focusing analysis or tomography, as in other prestack depth migration methods. The new method has been applied to data sets from a simple multiple‐generating model, the Marmousi model, and a real offset VSP. The results show accurate imaging of primaries and multiples with overall significant improvements over conventionally imaged sections.

Approximation of pure acoustic seismic wave propagation in TTI media

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB97-WB107 ◽  
Author(s):  
Chunlei Chu ◽  
Brian K. Macy ◽  
Phil D. Anno
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Computational Cost ◽  
P Wave ◽  
Anisotropy Axis ◽  
S Wave ◽  
Time Migration ◽  
Tti Media

Pseudoacoustic anisotropic wave equations are simplified elastic wave equations obtained by setting the S-wave velocity to zero along the anisotropy axis of symmetry. These pseudoacoustic wave equations greatly reduce the computational cost of modeling and imaging compared to the full elastic wave equation while preserving P-wave kinematics very well. For this reason, they are widely used in reverse time migration (RTM) to account for anisotropic effects. One fundamental shortcoming of this pseudoacoustic approximation is that it only prevents S-wave propagation along the symmetry axis and not in other directions. This problem leads to the presence of unwanted S-waves in P-wave simulation results and brings artifacts into P-wave RTM images. More significantly, the pseudoacoustic wave equations become unstable for anisotropy parameters [Formula: see text] and for heterogeneous models with highly varying dip and azimuth angles in tilted transversely isotropic (TTI) media. Pure acoustic anisotropic wave equations completely decouple the P-wave response from the elastic wavefield and naturally solve all the above-mentioned problems of the pseudoacoustic wave equations without significantly increasing the computational cost. In this work, we propose new pure acoustic TTI wave equations and compare them with the conventional coupled pseudoacoustic wave equations. Our equations can be directly solved using either the finite-difference method or the pseudospectral method. We give two approaches to derive these equations. One employs Taylor series expansion to approximate the pseudodifferential operator in the decoupled P-wave equation, and the other uses isotropic and elliptically anisotropic dispersion relations to reduce the temporal frequency order of the P-SV dispersion equation. We use several numerical examples to demonstrate that the newly derived pure acoustic wave equations produce highly accurate P-wave results, very close to results produced by coupled pseudoacoustic wave equations, but completely free from S-wave artifacts and instabilities.

Stable P-wave modeling for reverse-time migration in tilted TI media

Geophysics ◽  
2011 ◽  
Vol 76 (2) ◽  
pp. S65-S75 ◽  
Author(s):  
Eric Duveneck ◽  
Peter M. Bakker
Keyword(s):  
Wave Equations ◽  
Symmetry Axis ◽  
Second Order ◽  
P Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
First Derivative ◽  
Time Migration ◽  
Mixed Derivatives ◽  
Spatial Derivatives

We present an approach for P-wave modeling in inhomogeneous transversely isotropic media with tilted symmetry axis (TTI media), suitable for anisotropic reverse-time migration. The proposed approach is based on wave equations derived from first principles — the equations of motion and Hooke’s law — under the acoustic TI approximation. Consequently, no assumptions are made about the spatial variation of medium parameters. A rotation of the stress and strain tensors to a local coordinate system, aligned with the TI-symmetry axis, makes it possible to benefit from the simple and sparse form of the TI-elastic tensor in that system. The resulting wave equations can be formulated either as a set of five first-order or as a set of two second-order partial differential equations. For the constant-density case, the second-order TTI wave equations involve mixed and nonmixed second-order spatial derivatives with respect to global, nonrotated coordinates. We propose a numerical implementation of these equations using high-order centered finite differences. To minimize modeling artifacts related to the use of centered first-derivative operators, we use discrete second-derivative operators for the nonmixed second-order spatial derivatives and repeated discrete first-derivative operators for the mixed derivatives. Such a combination of finite-difference operators leads to a stable wave propagator, provided that the operators are designed properly. In practice, stability is achieved by slightly weighting down terms that contain mixed derivatives. This has a minor, practically negligible, effect on the kinematics of wave propagation. The stability of the presented scheme in inhomogeneous TTI models with rapidly varying anisotropic symmetry axis direction is demonstrated with numerical examples.

An approach for attenuation-compensating multidimensional constant-Q viscoacoustic reverse time migration

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. S33-S46
Author(s):  
Ali Fathalian ◽  
Daniel O. Trad ◽  
Kristopher A. Innanen
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Velocity Model ◽  
Seismic Wave Propagation ◽  
Image Resolution ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Phase Dispersion ◽  
Time Migration ◽  
Amplitude Attenuation

Simulation of wave propagation in a constant-[Formula: see text] viscoacoustic medium is an important problem, for instance, within [Formula: see text]-compensated reverse time migration (RTM). Processes of attenuation and dispersion influence all aspects of seismic wave propagation, degrading the resolution of migrated images. To improve the image resolution, we have developed a new approach for the numerical solution of the viscoacoustic wave equation in the time domain and we developed an associated viscoacoustic RTM ([Formula: see text]-RTM) method. The main feature of the [Formula: see text]-RTM approach is compensation of attenuation effects in seismic images during migration by separation of amplitude attenuation and phase dispersion terms. Because of this separation, we are able to compensate the amplitude loss effect in isolation, the phase dispersion effect in isolation, or both effects concurrently. In the [Formula: see text]-RTM implementation, an attenuation-compensated operator is constructed by reversing the sign of the amplitude attenuation and a regularized viscoacoustic wave equation is invoked to eliminate high-frequency instabilities. The scheme is tested on a layered model and a modified acoustic Marmousi velocity model. We validate and examine the response of this approach by using it within an RTM scheme adjusted to compensate for attenuation. The amplitude loss in the wavefield at the source and receivers due to attenuation can be recovered by applying compensation operators on the measured receiver wavefield. Our 2D and 3D numerical tests focus on the amplitude recovery and resolution of the [Formula: see text]-RTM images as well as the interface locations. Improvements in all three of these features beneath highly attenuative layers are evident.

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

Geophysics ◽  
2014 ◽  
Vol 79 (1) ◽  
pp. C1-C18 ◽  
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.

Polarization-based wave-equation migration velocity analysis in acoustic media

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. U77-U88 ◽  
Author(s):  
Qunshan Zhang ◽  
George A. McMechan
Keyword(s):  
Wave Equation ◽  
Velocity Ratio ◽  
Empirical Relation ◽  
Synthetic Data ◽  
P Wave ◽  
Reverse Time ◽  
Seismic Reflection Data ◽  
Prestack Migration ◽  
Time Migration ◽  
Acoustic Media

The source extrapolation step in wave-equation prestack reverse-time migration gives wavefield polarization information, which can be used to generate angle-domain common-image gathers (ADCIGs) from seismic reflection data from acoustic media. Concatenation of P-wave polarization segments gives wavefield propagation paths (“wavepaths”), which are similar to the raypaths in ray-based velocity tomography. The ADCIGs provide residual depth moveout (RMO) information, from which a system of linear equations is constructed for tomography to solve for the velocity ratio used for velocity updating. An empirical relation between the RMO data and the velocity ratio updates reduces the amount of computation, and is stabilized by the feedback provided by the iterative loop through prestack migration, to RMO, to velocity update, to prestack migration. Correcting the RMOs to flatten the ADGIGs is the convergence condition. Synthetic data for a layered model with a fault successfully illustrates the method.

Time and depth remigration in elliptically anisotropic media using image-wave propagation

Geophysics ◽  
2007 ◽  
Vol 72 (1) ◽  
pp. S1-S9 ◽  
Author(s):  
Jörg Schleicher ◽  
Rafael Aleixo
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Differential Equations ◽  
Vertical Velocity ◽  
Wave Equations ◽  
Anisotropic Media ◽  
Velocity Analysis ◽  
Acoustic Wave Equation ◽  
Migration Velocity Analysis ◽  
Kinematic Properties

The image-wave equations for the problems of depth and time remigration in elliptically anisotropic media are second-order partial differential equations similar to the acoustic-wave equation. The propagation variable is the vertical velocity or the medium ellipticity rather than time. These differential equations are derived from the kinematic properties of anisotropic remigration. The objective is to construct subsurface images that correspond to different vertical velocity and/or different degrees of medium anisotropy directly from a single migrated image. In this way, anisotropy panels can be obtained in a way completely analogous to velocity panels for migration velocity analysis. A simple numerical example demonstrates the validity of the theory.

Sign in / Sign up

Export Citation Format

Share Document