A recipe for stability of finite‐difference wave‐equation computations

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 967-969 ◽  
Author(s):  
Larry R. Lines ◽  
Raphael Slawinski ◽  
R. Phillip Bording
Keyword(s):  
Inverse Problem ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Numerical Stability ◽  
Short Note ◽  
Seismic Wave Propagation ◽  
Difference Wave ◽  
Von Neumann ◽  
The Stability

Finite‐difference solutions to the wave equation are pervasive in the modeling of seismic wave propagation (Kelly and Marfurt, 1990) and in seismic imaging (Bording and Lines, 1997). That is, they are useful for the forward problem (modeling) and the inverse problem (migration). In computational solutions to the wave equation, it is necessary to be aware of conditions for numerical stability. In this short note, we examine a convenient recipe for insuring stability in our finite‐difference solutions to the wave equation. The stability analysis for finite‐difference solutions of partial differential equations is handled using a method originally developed by Von Neumann and described by Press et al. (1986, p. 827–830).

An improved optimal nearly analytical discretized method for 2D scalar wave equation in heterogeneous media based on the modified nearly analytical discrete operator

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T349-T362 ◽  
Author(s):  
Yushu Chen ◽  
Guojie Song ◽  
Zhihui Xue ◽  
Hao Jing ◽  
Haohuan Fu ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Fourth Order ◽  
Computational Efficiency ◽  
Numerical Stability ◽  
Seismic Wave Propagation ◽  
Numerical Dispersion ◽  
Energy Error ◽  
Long Time ◽  
Coarse Grids

Conventional finite-difference methods will encounter strong numerical dispersion on coarse grids or when higher frequencies are used. The nearly analytic discrete method (NADM) and its improved version, developed recently, can suppress numerical dispersion efficiently. However, the NADM is imperfect, especially in long-time seismic wave propagation simulation. To overcome this limitation, by minimizing the energy-error function, we have developed a modified optimal nearly analytical discretized (MNAD) method. The stencil of MNAD for the 2D wave equation is a novel diamond stencil. MNAD has two major advantages: The first one is that MNAD can suppress numerical dispersion effectively on coarse grids, which significantly improved computational efficiency and reduced memory demands. The second advantage is that the energy error of MNAD is much smaller after long-time simulation. The numerical dispersion analysis shows that the maximum phase velocity error was 5.92% even if only two sampling points were adopted in each minimum wavelength. To simulate a wavefield without visible numerical dispersion, the computational speed of MNAD, measured by CPU time, was approximately 4.32 times and 1.43 times comparing with the fourth-order Lax-Wendroff correction (LWC) method and the optimal nearly analytical discretized method (ONADM), respectively. MNAD also shows good numerical stability. Its CFL condition increased 24.3% comparing with that of ONADM, from 0.523 to 0.650. The total energy error was less than 1.5% after 300-s simulation whereas the error of other numerical schemes, such as the fourth-order and eighth-order LWC, etc., was up to more than 10% under the same computational parameters. Numerical results showed that our MNAD method performed well in computational efficiency, simulation accuracy, and numerical stability, as well as provided a useful tool for large-scale, long-time seismic wave propagation simulation.

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.

A Smoothing Scheme for Seismic Wave Propagation Simulation with a Mixed FEM of Diffusionized Wave Equation

2021 ◽  
pp. 2150061
Author(s):  
Ryuta Imai ◽  
Naoki Kasui ◽  
Masayuki Yamada ◽  
Koji Hada ◽  
Hiroyuki Fujiwara
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Seismic Wave ◽  
Time Integration ◽  
Coupled System ◽  
Seismic Wave Propagation ◽  
Implicit Time ◽  
Integration Scheme ◽  
Weak Formulation ◽  
Mixed Fem

In this paper, we propose a smoothing scheme for seismic wave propagation simulation. The proposed scheme is based on a diffusionized wave equation with the fourth-order spatial derivative term. So, the solution requires higher regularity in the usual weak formulation. Reducing the diffusionized wave equation to a coupled system of diffusion equations yields a mixed FEM to ease the regularity. We mathematically explain how our scheme works for smoothing. We construct a semi-implicit time integration scheme and apply it to the wave equation. This numerical experiment reveals that our scheme is effective for filtering short wavelength components in seismic wave propagation simulation.

Finite difference forward modelling across the Tyrrhenian basin

2021 ◽  
Author(s):  
Chiara Nardoni ◽  
Luca De Siena ◽  
Fabio Cammarano ◽  
Elisabetta Mattei ◽  
Fabrizio Magrini
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Wave Attenuation ◽  
Software Tool ◽  
Seismic Wave Propagation ◽  
Tyrrhenian Sea ◽  
Forward Modelling ◽  
Crustal Thinning ◽  
Energy Leakage ◽  
Radiative Transfer Equations

<p>Strong lateral variations in medium properties affect the response of seismic wavefields. The Tyrrhenian Sea is ideally suited to explore these effects in a mixed continental-oceanic crust that comprises magmatic systems. The study aims at investigating the effects of crustal thinning and sedimentary layers on wave propagation, especially the reverberating (e.g., Lg) phases, across the oceanic basin. We model regional seismograms (600-800 km) using the software tool OpenSWPC (Maeda et al., 2017, EPS) based on the finite difference simulation of the wave equation. The code simulates the seismic wave propagation in heterogeneous viscoelastic media including the statistical velocity fluctuations as well as heterogeneous topography, typical of mixed settings. This approach allows to evaluate the role of interfaces and layer thicknesses on phase arrivals and direct and coda attenuation measurements. The results are compared with previous simulations of the radiative-transfer equations. They provide an improved understanding of the complex wave attenuation and energy leakage in the mantle characterizing the southern part of the Tyrrhenian Sea and the Italian peninsula. The forward modelling is to be embedded in future applications of attenuation, absorption and scattering tomography performed with MuRAT (the Multi-Resolution Attenuation Tomography code – De Siena et al. 2014, JVGR) available at https://github.com/LucaDeSiena/MuRAT.</p>

Implicit interpolation in reverse‐time migration

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 906-917 ◽  
Author(s):  
Jinming Zhu ◽  
Larry R. Lines
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Real Data ◽  
Seismic Sources ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Difference Wave ◽  
Time Migration ◽  
Seismic Acquisition ◽  
Recorded Data

Reverse‐time migration applies finite‐difference wave equation solutions by using unaliased time‐reversed recorded traces as seismic sources. Recorded data can be sparsely or irregularly sampled relative to a finely spaced finite‐difference mesh because of the nature of seismic acquisition. Fortunately, reliable interpolation of missing traces is implicitly included in the reverse‐time wave equation computations. This implicit interpolation is essentially based on the ability of the wavefield to “heal itself” during propagation. Both synthetic and real data examples demonstrate that reverse‐time migration can often be performed effectively without the need for explicit interpolation of missing traces.

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