Finite‐difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid

Geophysics ◽  
2004 ◽  
Vol 69 (2) ◽  
pp. 583-591 ◽  
Author(s):  
Erik H. Saenger ◽  
Thomas Bohlen
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Wave Equations ◽  
Physical Property ◽  
Anisotropic Media ◽  
Numerical Approach ◽  
Elementary Cell ◽  
Staggered Grid ◽  
Difference Operators ◽  
Von Neumann

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.

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.

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

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM147-SM153 ◽  
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.

Optimized staggered-grid finite-difference operators using window functions

2018 ◽  
Vol 15 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Ying-Jun Ren ◽  
Jian-Ping Huang ◽  
Peng Yong ◽  
Meng-Li Liu ◽  
Chao Cui ◽  
...  
Keyword(s):  
Finite Difference ◽  
Staggered Grid ◽  
Difference Operators ◽  
Window Functions

Generalization of von Neumann analysis for a model of two discrete half-spaces: The acoustic case

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM35-SM46 ◽  
Author(s):  
Matthew M. Haney
Keyword(s):  
Wave Propagation ◽  
Numerical Model ◽  
Finite Difference ◽  
Stability Criterion ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
Von Neumann ◽  
Transmission Coefficients ◽  
Strong Material ◽  
Von Neumann Analysis

Evaluating the performance of finite-difference algorithms typically uses a technique known as von Neumann analysis. For a given algorithm, application of the technique yields both a dispersion relation valid for the discrete time-space grid and a mathematical condition for stability. In practice, a major shortcoming of conventional von Neumann analysis is that it can be applied only to an idealized numerical model — that of an infinite, homogeneous whole space. Experience has shown that numerical instabilities often arise in finite-difference simulations of wave propagation at interfaces with strong material contrasts. These interface instabilities occur even though the conventional von Neumann stability criterion may be satisfied at each point of the numerical model. To address this issue, I generalize von Neumann analysis for a model of two half-spaces. I perform the analysis for the case of acoustic wave propagation using a standard staggered-grid finite-difference numerical scheme. By deriving expressions for the discrete reflection and transmission coefficients, I study under what conditions the discrete reflection and transmission coefficients become unbounded. I find that instabilities encountered in numerical modeling near interfaces with strong material contrasts are linked to these cases and develop a modified stability criterion that takes into account the resulting instabilities. I test and verify the stability criterion by executing a finite-difference algorithm under conditions predicted to be stable and unstable.

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