Optimized staggered-grid finite-difference operators using window functions

We describe the application of the rotated staggered‐grid (RSG) finite‐difference technique to the wave equations for anisotropic and viscoelastic media. The RSG uses rotated finite‐difference operators, leading to a distribution of modeling parameters in an elementary cell where all components of one physical property are located only at one single position. This can be advantageous for modeling wave propagation in anisotropic media or complex media, including high‐contrast discontinuities, because no averaging of elastic moduli is needed. The RSG can be applied both to displacement‐stress and to velocity‐stress finite‐difference (FD) schemes, whereby the latter are commonly used to model viscoelastic wave propagation. With a von Neumann‐style anlysis, we estimate the dispersion error of the RSG scheme in general anisotropic media. In three different simulation examples, all based on previously published problems, we demonstrate the application and the accuracy of the proposed numerical approach.

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Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

Journal of Computational Methods in Physics ◽

10.1155/2014/108713 ◽

2014 ◽

Vol 2014 ◽

pp. 1-14 ◽

Cited By ~ 5

Author(s):

Ke-Yang Chen

Keyword(s):

Wave Equation ◽

Finite Difference ◽

Elastic Wave ◽

Computing Time ◽

Staggered Grid ◽

Difference Operators ◽

Elastic Wave Equation ◽

Absorbing Boundary ◽

Wave Separation ◽

Time Requirements

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.

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Optimised staggered grid finite difference operators based on the combined window function

Exploration Geophysics ◽

10.1080/08123985.2020.1737519 ◽

2020 ◽

Vol 51 (5) ◽

pp. 523-534

Author(s):

Ruiqian Cai ◽

Chengyu Sun ◽

Dunshi Wu ◽

Zhihao Qiao

Keyword(s):

Finite Difference ◽

Window Function ◽

Staggered Grid ◽

Difference Operators

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Optimized staggered-grid finite-difference operators based on combined window

10.1190/nsapc2015-021 ◽

2015 ◽

Author(s):

Hong Liu* ◽

Zhiyang Wang

Keyword(s):

Finite Difference ◽

Staggered Grid ◽

Difference Operators

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Staggered-Grid Finite Difference Method with Variable-Order Accuracy for Porous Media

Mathematical Problems in Engineering ◽

10.1155/2013/157071 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 6

Author(s):

Jinghuai Gao ◽

Yijie Zhang

Keyword(s):

Porous Media ◽

Finite Difference ◽

Finite Difference Methods ◽

Computation Time ◽

High Order ◽

Staggered Grid ◽

Difference Operators ◽

Variable Order ◽

Low Velocity ◽

Order Accuracy

The numerical modeling of wave field in porous media generally requires more computation time than that of acoustic or elastic media. Usually used finite difference methods adopt finite difference operators with fixed-order accuracy to calculate space derivatives for a heterogeneous medium. A finite difference scheme with variable-order accuracy for acoustic wave equation has been proposed to reduce the computation time. In this paper, we develop this scheme for wave equations in porous media based on dispersion relation with high-order staggered-grid finite difference (SFD) method. High-order finite difference operators are adopted for low-velocity regions, and low-order finite difference operators are adopted for high-velocity regions. Dispersion analysis and modeling results demonstrate that the proposed SFD method can decrease computational costs without reducing accuracy.

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Seismic modeling by optimizing regularized staggered-grid finite-difference operators using a time-space-domain dispersion-relationship-preserving method

Geophysics ◽

10.1190/geo2014-0078.1 ◽

2014 ◽

Vol 79 (5) ◽

pp. T277-T285 ◽

Cited By ~ 50

Author(s):

Yanfei Wang ◽

Wenquan Liang ◽

Zuhair Nashed ◽

Xiao Li ◽

Guanghe Liang ◽

...

Keyword(s):

Finite Difference ◽

Staggered Grid ◽

Difference Operators ◽

Seismic Modeling ◽

Space Domain ◽

Dispersion Relationship ◽

Time Space

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Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach

Geophysics ◽

10.1190/1.2753831 ◽

2007 ◽

Vol 72 (5) ◽

pp. SM147-SM153 ◽

Cited By ~ 62

Author(s):

Yixian Xu ◽

Jianghai Xia ◽

Richard D. Miller

Keyword(s):

Wave Propagation ◽

Finite Difference ◽

Rayleigh Waves ◽

Second Order ◽

Body Waves ◽

Staggered Grid ◽

Difference Operators ◽

Image Method ◽

Elastic Boundary ◽

Second Order In Time

The need for incorporating the traction-free condition at the air-earth boundary for finite-difference modeling of seismic wave propagation has been discussed widely. A new implementation has been developed for simulating elastic wave propagation in which the free-surface condition is replaced by an explicit acoustic-elastic boundary. Detailed comparisons of seismograms with different implementations for the air-earth boundary were undertaken using the (2,2) (the finite-difference operators are second order in time and space) and the (2,6) (second order in time and sixth order in space) standard staggered-grid (SSG) schemes. Methods used in these comparisons to define the air-earth boundary included the stress image method (SIM), the heterogeneous approach, the scheme of modifying material properties based on transversely isotropic medium approach, the acoustic-elastic boundary approach, and an analytical approach. The method proposed achieves the same or higher accuracy of modeled body waves relative to the SIM. Rayleigh waves calculated using the explicit acoustic-elastic boundary approach differ slightly from those calculated using the SIM. Numerical results indicate that when using the (2,2) SSG scheme for SIM and our new method, a spatial step of 16 points per minimum wavelength is sufficient to achieve 90% accuracy; 32 points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. When using the (2,6) SSG scheme for the two methods, a spatial step of eight points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. Our proposed method is physically reasonable and, based on dispersive analysis of simulated seismographs from a layered half-space model, is highly accurate. As a bonus, our proposed method is easy to program and slightly faster than the SIM.

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Finite‐difference modeling in transversely isotropic media

Geophysics ◽

10.1190/1.1443590 ◽

1994 ◽

Vol 59 (2) ◽

pp. 282-289 ◽

Cited By ~ 34

Author(s):

Eduardo L. Faria ◽

Paul L. Stoffa

Keyword(s):

Finite Difference ◽

Time Derivative ◽

Computational Cost ◽

Vertical Component ◽

Transversely Isotropic ◽

Staggered Grid ◽

Difference Operators ◽

Transversely Isotropic Media ◽

Isotropic Media ◽

Modeling Algorithm

We developed a modeling algorithm for transversely isotropic media that uses finite‐difference operators in a staggered grid. Staggered grid schemes are more stable than the conventional finite‐difference methods because the differences are actually based on half the grid spacing. This modeling algorithm uses the full elastic wave equation that makes possible the modeling of all kinds of waves propagating in transversely isotropic media. The spatial derivatives are represented by fourth‐order, finite‐difference operators while the time derivative is represented by a secondorder, finite‐difference operator. The algorithm has no limitation on the acquisition geometry or on the heterogeneity of the media. The program is currently formulated to work in a 2-D transversely isotropic medium but can readily be extended to 3-D. Snapshots can be obtained at any time with no additional computational cost. A four‐layer model is used to show the usefulness of the method. Horizontal and vertical component seismograms are modeled in transversely isotropic media and compared with seismograms modeled in the corresponding isotropic media.

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