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.

Staggered-Grid High-Order Differencemethod for the First-Order Elastic Wave Equations

2000 ◽  
Vol 43 (3) ◽  
pp. 441-449 ◽  
Author(s):  
Liang-Guo DONG ◽  
Zai-Tian MA ◽  
Jing-Zhong CAO
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
High Order ◽  
Staggered Grid ◽  
First Order ◽  
Elastic Wave Equations

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.

High-order finite-difference approximations to solve pseudoacoustic equations in 3D VTI media

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T225-T235 ◽  
Author(s):  
Leandro Di Bartolo ◽  
Leandro Lopes ◽  
Luis Juracy Rangel Lemos
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
High Order ◽  
Staggered Grid ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Grid Scheme

Pseudoacoustic algorithms are very fast in comparison with full elastic ones for vertical transversely isotropic (VTI) modeling, so they are suitable for many applications, especially reverse time migration. Finite differences using simple grids are commonly used to solve pseudoacoustic equations. We have developed and implemented general high-order 3D pseudoacoustic transversely isotropic formulations. The focus is the development of staggered-grid finite-difference algorithms, known for their superior numerical properties. The staggered-grid schemes based on first-order velocity-stress wave equations are developed in detail as well as schemes based on direct application to second-order stress equations. This last case uses the recently presented equivalent staggered-grid theory, resulting in a staggered-grid scheme that overcomes the problem of large memory requirement. Two examples are presented: a 3D simulation and a prestack reverse time migration application, and we perform a numerical analysis regarding computational cost and precision. The errors of the new schemes are smaller than the existing nonstaggered-grid schemes. In comparison with existing staggered-grid schemes, they require 25% less memory and only have slightly greater computational cost.

Accuracy of finite‐difference and finite‐element modeling of the scalar and elastic wave equations

Geophysics ◽  
1984 ◽  
Vol 49 (5) ◽  
pp. 533-549 ◽  
Author(s):  
Kurt J. Marfurt
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Elastic Wave ◽  
Frequency Domain ◽  
Poisson’S Ratio ◽  
Wave Equations ◽  
Cost Effective ◽  
Poisson's Ratio ◽  
Solution Technique ◽  
Elastic Wave Equations

Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite‐difference and finite‐element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time‐domain and frequency‐domain techniques. In addition to versatility and ease of implementation, it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors and transmission coefficients. This paper shows that in homogeneous media the explicit finite‐element and finite‐difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson’s ratio less than 0.3. Finite‐elements are superior to finite‐differences when modeling elastic media with Poisson’s ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central‐differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency‐domain finite‐element solutions employing a weighted average of consistent and lumped masses yield the most accurate results, and they promise to be the most cost‐effective method for CDP, well log, and interactive modeling.

scholarly journals Elastic-wave reverse-time migration based on decoupled elastic-wave equations and inner-product imaging condition

2016 ◽  
Vol 13 (6) ◽  
pp. 953-963 ◽  
Author(s):  
Peng Yong ◽  
Jianping Huang ◽  
Zhenchun Li ◽  
Wenyuan Liao ◽  
Luping Qu ◽  
...  
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
Inner Product ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Imaging Condition ◽  
Time Migration ◽  
Elastic Wave Equations

The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. T89-T100
Author(s):  
Kang Wang ◽  
Suping Peng ◽  
Yongxu Lu ◽  
Xiaoqin Cui
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Wave Equations ◽  
Staggered Grid ◽  
Contact Boundary ◽  
Fractured Medium ◽  
Elastic Wave Equations ◽  
Fracture Interface

To enable a mathematical description, geologic fractures are considered as infinitely thin planes embedded in a homogeneous medium. These fracture structures satisfy linear slip boundary conditions, namely, a discontinuous displacement and continuous stress. The general finite-difference (FD) method described by the elastic wave equations has challenges when attempting to simulate the propagation of waves at the fracture interface. The FD method expressed by velocity-stress variables with the explicit application of boundary conditions at the fracture interface facilitates the simulation of wave propagation in fractured discontinuous media that are described by elastic wave equations and linear slip interface conditions. We have developed a new FD scheme for horizontal and vertical fracture media. In this scheme, a fictitious grid is introduced to describe the discontinuous velocity at the fracture interface and a rotated staggered grid is used to accurately indicate the location of the fracture. The new FD scheme satisfies nonwelded contact boundary conditions, unlike traditional approaches. Numerical simulations in different fracture media indicate that our scheme is accurate. The results demonstrate that the reflection coefficient of the fractured interface varies with the incident angle, wavelet frequency, and normal and tangential fracture compliances. Our scheme and conclusions from this study will be useful in assessing the properties of fractures, enabling the proper delineation of fractured reservoirs.

Sign in / Sign up

Export Citation Format

Share Document