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

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.

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.

Time-space-domain temporal high-order staggered-grid finite-difference schemes by combining orthogonality and pyramid stencils for 3D elastic-wave propagation

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T259-T282 ◽  
Author(s):  
Shigang Xu ◽  
Yang Liu ◽  
Zhiming Ren ◽  
Hongyu Zhou
Keyword(s):  
Dispersion Relation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Large Time ◽  
High Order ◽  
Staggered Grid ◽  
Elastic Wave Equation ◽  
Space Domain ◽  
First Order ◽  
Time Space

The presently available staggered-grid finite-difference (SGFD) schemes for the 3D first-order elastic-wave equation can only achieve high-order spatial accuracy, but they exhibit second-order temporal accuracy. Therefore, the commonly used SGFD methods may suffer from visible temporal dispersion and even instability when relatively large time steps are involved. To increase the temporal accuracy and stability, we have developed a novel time-space-domain high-order SGFD stencil, characterized by ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracies ([Formula: see text]), to numerically solve the 3D first-order elastic-wave equation. The core idea of this new stencil is to use a double-pyramid stencil with an operator length parameter [Formula: see text] together with the conventional second-order SGFD to approximate the temporal derivatives. At the same time, the spatial derivatives are discretized by the orthogonality stencil with an operator length parameter [Formula: see text]. We derive the time-space-domain dispersion relation of this new stencil and determine finite-difference (FD) coefficients using the Taylor-series expansion. In addition, we further optimize the spatial FD coefficients by using a least-squares (LS) algorithm to minimize the time-space-domain dispersion relation. To create accurate and reasonable P-, S-, and converted wavefields, we introduce the 3D wavefield-separation technique into our temporal high-order SGFD schemes. The decoupled P- and S-wavefields are extrapolated by using the P- and S-wave dispersion-relation-based FD coefficients, respectively. Moreover, we design an adaptive variable-length operator scheme, including operators [Formula: see text] and [Formula: see text], to reduce the extra computational cost arising from adopting this new stencil. Dispersion and stability analyses indicate that our new methods have higher accuracy and better stability than the conventional ones. Using several 3D modeling examples, we demonstrate that our SGFD schemes can yield greater temporal accuracy on the premise of guaranteeing high-order spatial accuracy. Through effectively combining our new stencil, LS-based optimization, large time step, variable-length operator, and graphic processing unit, the computational efficiency can be significantly improved for the 3D case.

Sign in / Sign up

Export Citation Format

Share Document