Shear Wave Artifacts Research Articles

3D reverse-time migration for pure P-wave in orthorhombic media

Compared with the transverse isotropic (TI) medium, the orthorhombic anisotropic medium has both horizontal and vertical symmetry axes and it can be approximated as a set of vertical fissures developed in a group of horizontal strata. Although the full-elastic orthorhombic anisotropic wave equation can accurately simulate seismic wave propagation in the underground media, a huge computational cost is required in seismic modeling, migration, and inversion. The conventional coupled pseudo-acoustic wave equations based on acoustic approximation can be used to significantly reduce the cost of calculation. However, these equations usually suffer from unwanted shear wave artifacts during wave propagation, and the presence of these artifacts can significantly degrade the imaging quality. To solve these problems, we derived a new pure P-wave equation for orthorhombic media that eliminates shear wave artifacts while compromising computational efficiency and accuracy. In addition, the derived equation involves pseudo-differential operators and it must be solved by 3D FFT algorithms. In order to reduce the number of 3D FFT, we utilized the finite difference and pseudo-spectral methods to conduct 3D forward modeling. Furthermore, we simplified the equation by using elliptic approximation and implemented 3D reverse-time migration (RTM). Forward modeling tests on several homogeneous and heterogeneous models confirm that the accuracy of the new equation is better than that of conventional methods. 3D RTM imaging tests on three-layer and SEG/EAGE 3D salt models confirm that the ORT media have better imaging quality.

Least‐squares reverse time migration based on an efficient pure qP‐wave equation

AbstractSeismic waves propagating in anisotropic earth media suffer from waveform distortion, which leads to degraded resolution in seismic data imaging if this adverse effect is not corrected. Additionally, the wavefields simulated by the traditional coupled pseudo‐acoustic wave equations contain undesired shear wave artefacts and are unstable when Thomsen's anisotropy parameter ε is less than δ. To address these issues, many pure qP‐wave equations are proposed to describe wave propagation in anisotropic media. However, these equations are either low precision or high accuracy but computationally expensive. Considering the trade‐off between the computation efficiency and simulation accuracy, we derive a pure qP‐wave equation from the exact dispersion formula in tilted transverse isotropic media. The newly derived equation has only two higher order partial derivatives, which can better balance the accuracy and efficiency than the previous pure qP‐wave equations. The least‐squares reverse time migration method can balance amplitudes, suppress migration artefacts and improve imaging resolution. Then, based on the newly derived pure qP‐wave equation, we derive the Born modelling operator and adjoint migration operator and develop an anisotropic least‐squares reverse time migration method. To achieve the accelerated convergence, we introduce a generalized minimum residual algorithm to implement the anisotropic least‐squares reverse time migration method. Numerical simulation results show that the proposed pure qP‐wave equation outperforms the previous equations in balancing the trade‐off between accuracy and efficiency. In addition, synthetic examples demonstrate the effectiveness and robustness of our proposed anisotropic least‐squares reverse time migration method in correcting the deviations due to anisotropic effects.

Pseudo-acoustic anisotropic reverse-time migration of an ocean-bottom cable dataset acquired in the North Sea

ABSTRACT To improve the computational efficiency of reverse-time migration (RTM) for vertically transverse isotropic (VTI) media, various acoustic approximations of the elastic wave equations have been presented. Among these, the pseudo-acoustic wave equation, which combines differential and scalar operators, has the advantage that it does not produce shear wave artefacts. In this study, we investigate the feasibility of pseudo-acoustic anisotropic RTM (PA-RTM) for the analysis of synthetic and observed ocean-bottom cable (OBC) datasets of the Volve oil field in the North Sea. To analyse the influence of anisotropic parameters on RTM images and the sensitivity of data components to seismic anisotropy, we perform PA-RTM using synthetic data by incorporating various background models, as well as pressure wavefields and vertical particle acceleration. The synthetic experiments demonstrate that the anisotropic parameter ε plays an important role, whereas δ is negligible in PA-RTM for VTI media, and that vertical particle acceleration is less affected by seismic anisotropy than the pressure wavefields. Our experiments using observed data show that reflectors are better imaged by PA-RTM than by isotropic-RTM, particularly near the reservoir below the strongly anisotropic region, which is supported by angle-domain common image gathers. These results indicate that PA-RTM using only the vertical component is a suitable option for VTI media when insufficient seismic anisotropy data are available.

Broadband finite-difference Q-compensated reverse time migration algorithm for tilted transverse isotropic media

[Formula: see text]-compensated reverse time migration (Q-RTM) is effective for improving the seismic imaging quality degraded by low-[Formula: see text] anomalies. However, it is difficult to apply existing pseudospectral-based Q-RTM methods to large-scale problems due to the obstacles to high-efficient parallelization posed by the global pseudospectral operators. On the other hand, finite-difference Q-RTM is intrinsically appropriate for domain decomposition and parallel computation, thus being suitable for industrial-sized problems but facing a twofold challenge: (1) to effectively compensate the phase in a broad-bandwidth sense during the wave back-propagation process and (2) to accurately handle the tilted transverse isotropic (TTI) medium with attenuation. We have developed a new framework of finite-difference Q-RTM algorithm by expanding the linear viscoacoustic constitutive relation to a series of integer-order differential terms and a unique integral term that can decouple the amplitude and the phase to allow accurate compensation in a broader frequency range. This framework has two typical implementations: (1) optimizing the frequency-dependent phase velocity (while fixing the negative constant [Formula: see text]) and (2) optimizing the [Formula: see text] value (while fixing the frequency-dependent phase velocity). We generalize this broadband finite-difference Q-RTM algorithm to TTI media, where an artificial [Formula: see text] is applied to suppress the shear-wave artifacts induced by the acoustic TTI approximation. Numerical examples demonstrate that this Q-RTM method accurately compensates the phase and amplitude in a broad frequency range of 5–70 Hz and produces high-quality images. Due to the local nature of finite-difference operators, this algorithm is expected to outperform the existing pseudospectral-based Q-RTM methods in terms of computational efficiency and implementation convenience for real-world Q-RTM projects.

Mono-component multiparameter acoustic full waveform inversion in vertically transverse isotropic media using converted vector wavefields

In multiparameter full-waveform inversion (FWI) with acoustic-approximation for vertically transverse isotropic (VTI) media, it is important to choose an appropriate modeling technique for computational efficiency and numerical stability. In addition, because the gradient is determined by the modeling algorithm used in the FWI process, we need to examine if the modeling algorithm yields a gradient similar to that obtained from the elastic scattering theory. The decomposed equation with mono-component has been proposed because it is computationally efficient and does not generate shear-wave artifacts. However, the decomposed equation has the limitation that its numerical scattering potentials for the perturbations of anisotropic parameters do not properly simulate the theoretical elastic scattering potentials. To overcome the limitation, we propose using the vector virtual sources to compute the gradient in the FWI algorithm by converting the mono-component (pressure) source and receiver wavefields into the two-component particle displacements using the equation of motion on the staggered grids. Unlike the conventional method, our approach properly simulates PP radiation patterns for anisotropic parameters obtained under the elastic assumption, while retaining computational efficiency achieved by using the mono-component. Numerical examples show that anisotropic parameters are updated in the correct gradient direction by our method in the FWI process.

A partial-low-rank method for solving acoustic wave equation

Numerical solutions of the acoustic wave equation, especially in anisotropic media, is crucial to seismic modeling, imaging and inversion as it provides efficient, practical, and stable approximate representation of the medium. However, a clean implementation (free of shear wave artifacts and dispersion) of wave propagation, especially in anisotropic media, requires an integral operator, the direct evaluation of which is extremely expensive. Recently, the low-rank method was proposed to provide a good approximation to the integral operator utilizing Fourier transforms. Thus, we propose to split the integral operator into two terms. The first term provides a differential operator that approximates that can be approximated with a standard finite-difference method. We, then, apply the low-rank approximation on the residual term of the finite-difference operator. We implement the two terms in two complementing steps, in which the spectral step corrects for any errors admitted by the finite difference step. Even though we utilize finite-difference approximations, the resulting algorithm admits spectral accuracy. Also, through the finite difference step, the method can deal approximately with the free surface and absorbing boundary conditions in a straight forward manner. Numerical examples show that the method is of high accuracy and efficiency.

Propagation Imaging in the Demonstration of Common Shear Wave Artifacts.

In this article, we describe our experience with shear wave propagation imaging (SWPI) as an adjunct to 2-dimensional (2D) shear wave elastography (SWE) in a cohort of patients being evaluated for diffuse liver disease. Two-dimensional SWE has been extensively studied in previous publications; however, 2D SWE using propagation images has not been widely described in the literature to date. We observed that when certain artifacts occurred on the color elastograms, highly characteristic changes to shear wave propagation contours were seen, which can help clarify the cause of the artifacts. To our knowledge, the use of SWPI to explain the etiology of artifacts has never been published before. The artifacts described in this article include the capsule reverberation artifact, penetration limitation or dropout artifact, artifact due to blood vessels, shadowing artifact, tissue motion artifact, and near-field distortion/precompression artifact. Hence, the purpose of this article is to show examples of common artifacts seen on 2D SWE as depicted on corresponding SWPI to demonstrate that both types of image displays are complementary to each other.

Effective ellipsoidal models for wavefield extrapolation in tilted orthorhombic media

Wavefield computations using the ellipsoidally anisotropic extrapolation operator offer significant cost reduction compared to that for the orthorhombic case, especially when the symmetry planes are tilted and/or rotated. However, ellipsoidal anisotropy does not provide accurate wavefield representation or imaging for media of orthorhombic symmetry. Therefore, we propose the use of ‘effective ellipsoidally anisotropic’ models that correctly capture the kinematic behaviour of wavefields for tilted orthorhombic (TOR) media. We compute effective velocities for the ellipsoidally anisotropic medium using kinematic high-frequency representation of the TOR wavefield, obtained by solving the TOR eikonal equation. The effective model allows us to use the cheaper ellipsoidally anisotropic wave extrapolation operators. Although the effective models are obtained by kinematic matching using high-frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The proposed methodology offers a much better cost versus accuracy trade-off for wavefield computations in TOR media, particularly for media of low to moderate anisotropic strength. Furthermore, the computed wavefield solution is free from shear-wave artefacts as opposed to the conventional finite-difference based TOR wave extrapolation scheme. We demonstrate applicability and usefulness of our formulation through numerical tests on synthetic TOR models.

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.

Acoustic anisotropic wavefields through perturbation theory

Solving the anisotropic acoustic wave equation numerically using finite-difference methods introduces many problems and media restriction requirements, and it rarely contributes to the ability to resolve the anisotropy parameters. Among these restrictions are the inability to handle media with [Formula: see text] and the presence of shear-wave artifacts in the solution. Both limitations do not exist in the solution of the elliptical anisotropic acoustic wave equation. Using perturbation theory in developing the solution of the anisotropic acoustic wave equation allows direct access to the desired limitation-free solutions, that is, solutions perturbed from the elliptical anisotropic background medium. It also provides a platform for parameter estimation because of the ability to isolate the wavefield dependency on the perturbed anisotropy parameters. As a result, I derive partial differential equations that relate changes in the wavefield to perturbations in the anisotropy parameters. The solutions of the perturbation equations represented the coefficients of a Taylor-series-type expansion of the wavefield as a function of the perturbed parameter, which is in this case [Formula: see text] or the tilt of the symmetry axis. The expansion with respect to the symmetry axis allows use of an acoustic transversely isotropic media with a vertical symmetry axis (VTI) kernel to estimate the background wavefield and the corresponding perturbation coefficients. The VTI extrapolation kernel is about one-fourth the cost of the transversely isotropic model with a tilt in the symmetry axis kernel. Thus, for a small symmetry axis tilt, the cost of migration using a first-order expansion can be reduced. The effectiveness of the approach was demonstrated on the Marmousi model.

An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation

The pure P-wave equation for modelling and migration in tilted transversely isotropic (TTI) media has attracted more and more attention in imaging seismic data with anisotropy. The desirable feature is that it is absolutely free of shear-wave artefacts and the consequent alleviation of numerical instabilities generally suffered by some systems of coupled equations. However, due to several forward–backward Fourier transforms in wavefield updating at each time step, the computational cost is significant, and thereby hampers its prevalence. We propose to use a hybrid pseudospectral (PS) and finite-difference (FD) scheme to solve the pure P-wave equation. In the hybrid solution, most of the cost-consuming wavenumber terms in the equation are replaced by inexpensive FD operators, which in turn accelerates the computation and reduces the computational cost. To demonstrate the benefit in cost saving of the new scheme, 2D and 3D reverse-time migration (RTM) examples using the hybrid solution to the pure P-wave equation are carried out, and respective runtimes are listed and compared. Numerical results show that the hybrid strategy demands less computation time and is faster than using the PS method alone. Furthermore, this new TTI RTM algorithm with the hybrid method is computationally less expensive than that with the FD solution to conventional TTI coupled equations.

Decoupled equations for reverse time migration in tilted transversely isotropic media

Conventional modeling and migration for tilted transversely isotropic (TTI) media may suffer from numerical instabilities and shear wave artifacts due to the coupling of the P-wave and SV-wave modes in the TTI coupled equations. Starting with the separated P- and SV-phase velocity expressions for vertical transversely isotropic (VTI) media, we extend these decoupled equations for modeling and reverse time migration (RTM) in acoustic TTI media. Compared with the TTI coupled equations published in the geophysical literature, the new TTI decoupled equations provide a more stable solution due to the complete separation of the P-wave and SV-wave modes. The pseudospectral method is the most convenient method to implement these equations due to the form of wavenumber expressions and has the added benefit of being highly accurate and thus avoiding numerical dispersion. The rapid expansion method (REM) in time is employed to produce a broad band numerically stable time evolution of the wavefields. Synthetic results validate the proposed TTI decoupled equations and show that modeling and RTM in TTI media with the decoupled equations remain numerically stable even for models with strong anisotropy and sharp contrasts.