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.

True-amplitude linearized waveform inversion with the quasi-elastic wave equation

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R827-R844 ◽  
Author(s):  
Zongcai Feng ◽  
Gerard Schuster
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Waveform Inversion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Elastic Wave Equation ◽  
P Waves ◽  
Time Migration

We present a quasi-elastic wave equation as a function of the pressure variable, which can accurately model PP reflections with elastic amplitude variation with offset effects under the first-order Born approximation. The kinematic part of the quasi-elastic wave equation accurately models the propagation of P waves, whereas the virtual-source part, which models the amplitudes of reflections, is a function of the perturbations of density and Lamé parameters [Formula: see text] and [Formula: see text]. The quasi-elastic wave equation generates a scattering radiation pattern that is exactly the same as that for the elastic wave equation, and only requires the solution of two acoustic wave equations for each shot gather. This means that the quasi-elastic wave equation can be used for true-amplitude linearized waveform inversion (also known as least-squares reverse time migration) of elastic PP reflections, in which the corresponding misfit gradients are with respect to the perturbations of density and the P- and S-wave impedances. The perturbations of elastic parameters are iteratively updated by minimizing the [Formula: see text]-norm of the difference between the recorded PP reflections and the predicted pressure data modeled from the quasi-elastic wave equation. Numerical tests on synthetic and field data indicate that true-amplitude linearized waveform inversion using the quasi-elastic wave equation can account for the elastic PP amplitudes and provide a robust estimate of the perturbations of P- and S-wave impedances and, in some cases, the density. In addition, true-amplitude linearized waveform inversion provides images with a wider bandwidth and fewer artifacts because the PP amplitudes are accurately explained. We also determine the 2D scalar quasi-elastic wave equation for P-SV reflections and the 3D vector equation for PS reflections.

Temporal high-order staggered-grid finite-difference schemes for elastic wave propagation

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T207-T224 ◽  
Author(s):  
Zhiming Ren ◽  
Zhen Chun Li
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
High Order ◽  
Elastic Wave Propagation ◽  
Staggered Grid ◽  
S Wave ◽  
Order Accuracy ◽  
Spatial Derivatives

The traditional high-order finite-difference (FD) methods approximate the spatial derivatives to arbitrary even-order accuracy, whereas the time discretization is still of second-order accuracy. Temporal high-order FD methods can improve the accuracy in time greatly. However, the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation. We have developed two temporal high-order staggered-grid FD (SFD) schemes for modeling elastic wave propagation. A new stencil containing the points on the axis and a few off-axial points is introduced to approximate the spatial derivatives. We derive the dispersion relations of the elastic wave equation based on the new stencil, and we estimate FD coefficients by the Taylor series expansion (TE). The TE-based scheme can achieve ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracy ([Formula: see text]). We further optimize the coefficients of FD operators using a combination of TE and least squares (LS). The FD coefficients at the off-axial and axial points are computed by TE and LS, respectively. To obtain accurate P-, S-, and converted waves, we extend the wavefield decomposition into the temporal high-order SFD schemes. In our modeling, P- and S-wave separation is implemented and P- and S-wavefields are propagated by P- and S-wave dispersion-relation-based FD operators, respectively. We compare our schemes with the conventional SFD method. Numerical examples demonstrate that our TE-based and TE + LS-based schemes have greater accuracy in time and better stability than the conventional method. Moreover, the TE + LS-based scheme is superior to the TE-based scheme in suppressing the spatial dispersion. Owing to the high accuracy in the time and space domains, our new SFD schemes allow for larger time steps and shorter operator lengths, which can improve the computational efficiency.

Elastic reverse time migration using acoustic propagators

Geophysics ◽  
2018 ◽  
Vol 83 (5) ◽  
pp. S399-S408 ◽  
Author(s):  
Yunyue Elita Li ◽  
Yue Du ◽  
Jizhong Yang ◽  
Arthur Cheng ◽  
Xinding Fang
Keyword(s):  
Wave Equations ◽  
Mode Conversion ◽  
Computational Cost ◽  
Body Waves ◽  
Constant Density ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Isotropic Constant ◽  
Time Migration

Elastic wave imaging has been a significant challenge in the exploration industry due to the complexities in wave physics and numerical implementation. We have separated the governing equations for P- and S-wave propagation without the assumptions of homogeneous Lamé parameters to capture the mode conversion between the two body waves in an isotropic, constant-density medium. The resulting set of two coupled second-order equations for P- and S-potentials clearly demonstrates that mode conversion only occurs at the discontinuities of the shear modulus. Applying the Born approximation to the new equations, we derive the PP, PS, SP, and SS imaging conditions from the first gradients of waveform matching objective functions. The resulting images are consistent with the physical perturbations of the elastic parameters, and, hence, they are automatically free of the polarity reversal artifacts in the converted images. When implementing elastic reverse time migration (RTM), we find that scalar wave equations can be used to back propagate the recorded P-potential, as well as individual components in the vector field of the S-potential. Compared with conventional elastic RTM, the proposed elastic RTM implementation using acoustic propagators not only simplifies the imaging condition, it but also reduces the computational cost and the artifacts in the images. We have determined the accuracy of our method using 2D and 3D numerical examples.

Constraints on the anisotropic parameters for pseudoelastic vertical transverse isotropy wave equations and the applications on imaging carbonate reservoirs

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. C85-C94 ◽  
Author(s):  
Houzhu (James) Zhang ◽  
Hongwei Liu ◽  
Yang Zhao
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Velocity ◽  
Wave Equations ◽  
Transverse Isotropy ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Computation Efficiency ◽  
The Stability ◽  
Vertical Transverse Isotropy

Seismic anisotropy is an intrinsic elastic property. Appropriate accounting of anisotropy is critical for correct and accurate positioning seismic events in reverse time migration. Although the full elastic wave equation may serve as the ultimate solution for modeling and imaging, pseudoelastic and pseudoacoustic wave equations are more preferable due to their computation efficiency and simplicity in practice. The anisotropic parameters and their relations are not arbitrary because they are constrained by the energy principle. Based on the investigation of the stability condition of the pseudoelastic wave equations, we have developed a set of explicit formulations for determining the S-wave velocity from given Thomsen’s parameters [Formula: see text] and [Formula: see text] for vertical transverse isotropy and tilted transverse isotropy media. The estimated S-wave velocity ensures that the wave equations are stable and well-posed in the cases of [Formula: see text] and [Formula: see text]. In the case of [Formula: see text], a common situation in carbonate, a positive value of S-wave velocity is needed to avoid the wavefield instability. Comparing the stability constraints of the pseudoelastic- with the full-elastic wave equation, we conclude that the feasible range of [Formula: see text] and [Formula: see text] was slightly larger for the pseudoelastic assumption. The success of achieving high-accuracy images and high-quality angle gathers using the proposed constraints is demonstrated in a synthetic example and a field example from Saudi Arabia.

Pseudo-analytical finite-difference elastic-wave extrapolation based on the k-space method

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. T1-T14 ◽  
Author(s):  
Xufei Gong ◽  
Qizhen Du ◽  
Qiang Zhao ◽  
Chengfeng Guo ◽  
Pengyuan Sun ◽  
...  
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Wave Equations ◽  
Computational Cost ◽  
Superior Performance ◽  
First Order ◽  
Time Migration ◽  
Elastic Wave Equations ◽  
Temporal Dispersion ◽  
Space Method

Cost-effective elastic-wave modeling is the key to practical elastic reverse time migration and full-waveform inversion implementations. We have developed an efficient elastic pseudo-analytical finite-difference (PAFD) scheme for elastic-wave extrapolation. The elastic PAFD scheme is based on a modified pseudo-spectral method, k-space method, in which a pseudo-analytical operator is used to ensure the high accuracy of elastic-wave extrapolation. However, the k-space method is motivated for a pure wave mode, and thus its application in coupled first-order elastic-wave equations may cause the elastic pseudo-analytical operators to suffer from crosstalk between the P- and S-wavefields. The approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal. This is done, first, by performing a predictor-corrector strategy in first-order elastic-wave equations to eliminate those errors during wave extrapolation. Considering the massive computational cost in the spectral domain, we have developed an efficient elastic PAFD implementation, in which an innovative model-adaptive finite-difference coefficient-predicted scheme is provided to reduce the computational cost of elastic pseudo-analytical operator differencing. Dispersion analysis demonstrates the flexibility with varying velocity and superior performance of our PAFD scheme for spatial and temporal dispersion suppression than the existing Taylor-expansion-based scheme. Under the same simulation parameters, several numerical examples prove that the elastic PAFD scheme can provide more accurate simulation results, whereas the conventional scheme suffers from spatial or temporal dispersion errors, even in complex heterogeneous media.

Experimental study on the effects of fractures on elastic wave propagation in synthetic layered rocks

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. D441-D451 ◽  
Author(s):  
Tianyang Li ◽  
Ruihe Wang ◽  
Zizhen Wang ◽  
Yuzhong Wang
Keyword(s):  
Wave Propagation ◽  
Fractal Dimension ◽  
Power Spectrum ◽  
Elastic Wave ◽  
Power Law ◽  
Layered Media ◽  
Physical Models ◽  
P Wave ◽  
Fracture Density ◽  
S Wave

Fractures greatly increase the difficulty of oil and gas exploration and development in reservoirs consisting of interlayered carbonates and shales and increase the uncertainty of highly efficient development. The presence of fractures or layered media is also widely known to affect the elastic properties of rocks. The combined effects of fractures and layered media are still unknown. We have investigated the effects of fracture structure on wave propagation in interlayered carbonate and shale rocks using physical models based on wave theory and the similarity principle. We have designed and built two sets of layered physical models with randomly embedded predesigned vertically aligned fractures according to the control variate principle. We have measured the P- and S-wave velocities and attenuation and analyzed the effects of fracture porosity and aspect ratio (AR) on velocity, attenuation, and power spectral dimension of the P- and S-waves. The experimental results indicated that under conditions of low porosity ([Formula: see text]), Han’s empirical velocity-porosity relations and Wang’s attenuation-porosity relation combined with Wyllie’s time-average model are a good prediction for layered physical models with randomly embedded fractures. When the porosity is constant, the effect of different ARs on elastic wave properties can be described by a power law function. We have calculated the power spectrum fractal dimension [Formula: see text] of the transmitted signal in the frequency domain, which can supplement the S-wave splitting method for estimating the degree of anisotropy. The simple power law relation between the power spectrum fractal dimension of the P-waveform and fracture density suggests the possible use of P-waves for discriminating fracture density. The high precision and low error of this processing method give new ideas for rock anisotropy evaluation and fracture density prediction when only P-wave data are available.

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.

scholarly journals Anisotropic viscoacoustic wave modelling in VTI media using frequency-dependent complex velocity

2020 ◽  
Author(s):  
Yabing Zhang ◽  
Yang Liu ◽  
Shigang Xu
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Wave Equations ◽  
Transversely Isotropic ◽  
P Wave ◽  
Low Rank ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Rank Decomposition ◽  
Low Rank Decomposition

Abstract Under the conditions of acoustic approximation and isotropic attenuation, we derive the pseudo- and pure-viscoacoustic wave equations from the complex constitutive equation and the decoupled P-wave dispersion relation, respectively. Based on the equations, we investigate the viscoacoustic wave propagation in vertical transversely isotropic media. The favourable advantage of these formulas is that the phase dispersion and the amplitude dissipation terms are inherently separated. As a result, we can conveniently perform the decoupled viscoacoustic wavefield simulations by choosing different coefficients. In the computational process, a generalised pseudo-spectral method and a low-rank decomposition scheme are adopted to calculate the wavenumber-domain and mixed-domain propagators, respectively. Because low-rank decomposition plays an important role in the simulated procedure, we evaluate the approximation accuracy for different operators using a linear velocity model. To demonstrate the effectiveness and the accuracy of our method, several numerical examples are carried out based on the new pseudo- and pure-viscoacoustic wave equations. Both equations can effectively describe the viscoacoustic wave propagation characteristics in vertical transversely isotropic media. Unlike the pseudo-viscoacoustic wave equation, the pure-viscoacoustic wave equation can produce stable viscoacoustic wavefields without any SV-wave artefacts.

Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansion

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T299-T311
Author(s):  
Edvaldo S. Araujo ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Computational Cost ◽  
Rapid Expansion ◽  
Numerical Dispersion ◽  
Seismic Modeling ◽  
Time Step ◽  
Velocity Models ◽  
Time Migration

We have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method.

scholarly journals Reflection Full Waveform Inversion with Decoupled Elastic-wave Equations in Inhomogeneous Medium

2021 ◽  
Vol 64 (1) ◽  
Author(s):  
Zhanyuan Liang ◽  
Guochen Wu ◽  
Xiaoyu Zhang ◽  
Qingyang Li
Keyword(s):  
Elastic Wave ◽  
Inhomogeneous Medium ◽  
Wave Velocity ◽  
Wave Equations ◽  
Waveform Inversion ◽  
P Wave ◽  
Full Waveform Inversion ◽  
S Wave ◽  
Full Waveform ◽  
Elastic Wave Equations

Reflection full-waveform inversion (RFWI) can reduce the nonlinearity of inversion providing an accurate initial velocity model for full-waveform inversion (FWI) through the tomographic components (low-wavenumber). However, elastic-wave reflection full-waveform inversion (ERFWI) is more vulnerable to the problem of local minimum due to the complicated multi-component wavefield. Our algorithm first divides kernels of ERFWI into four subkernels based on the theory of decoupled elastic-wave equations. Then we try to construct the tomographic components of ERFWI with only single-component wavefields, similarly to acoustic inversions. However, the S-wave velocity is still vulnerable to the coupling effects because of P-wave components contained in the S-wavefield in an inhomogeneous medium. Therefore we develop a method for decoupling elastic- wave equations in an inhomogeneous medium, which is applied to the decomposition of kernels in ERFWI. The new decoupled system can improve the accuracy of S-wavefield and hence further reduces the high-wavenumber crosstalk in the subkernel of S-wave velocity after kernels are decomposed. The numerical examples of Sigsbee2A model demonstrate that our ERFWI method with decoupled elastic-wave equations can efficiently recover the low-wavenumber components of the model and improve the precision of the S-wave velocity.

Sign in / Sign up

Export Citation Format

Share Document