M-ADAPTATION METHOD FOR ACOUSTIC WAVE EQUATION ON SQUARE MESHES

2012 ◽  
Vol 20 (04) ◽  
pp. 1250022 ◽  
Author(s):  
VITALIY GYRYA ◽  
KONSTANTIN LIPNIKOV
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Time Integration ◽  
Adaptive Strategy ◽  
Numerical Dispersion ◽  
Acoustic Wave Equation ◽  
Optimal Method ◽  
Long Time ◽  
The Stability ◽  
The Time Domain

A novel adaptive strategy, dubbed m-adaptation, is developed for solving the acoustic wave equation (in the time domain) on square meshes. The finite element, the finite difference and a few other more recent methods are shown to be particular members of the mimetic family. Analysis of the parametric family of mimetic discretization methods is performed to find the optimal member that eliminates the numerical dispersion at the fourth-order (as in Ref. 1) and the numerical anisotropy at the sixth-order (higher than in Ref. 1). The stability condition for the optimal method is derived that turns out to be comparable to the classical Courant condition. The numerical experiments show that the new approach is consistently better than the classical methods for reducing a long-time integration error.

An optimized wave equation for seismic modeling and reverse time migration

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA153-WCA158 ◽  
Author(s):  
Faqi Liu ◽  
Guanquan Zhang ◽  
Scott A. Morton ◽  
Jacques P. Leveille
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Computational Cost ◽  
Numerical Dispersion ◽  
New Wave ◽  
Acoustic Wave Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

The acoustic wave equation has been widely used for the modeling and reverse time migration of seismic data. Numerical implementation of this equation via finite-difference techniques has established itself as a valuable approach and has long been a favored choice in the industry. To ensure quality results, accurate approximations are required for spatial and time derivatives. Traditionally, they are achieved numerically by using either relatively very fine computation grids or very long finite-difference operators. Otherwise, the numerical error, known as numerical dispersion, is present in the data and contaminates the signals. However, either approach will result in a considerable increase in the computational cost. A simple and computationally low-cost modification to the standard acoustic wave equation is presented to suppress numerical dispersion. This dispersion attenuator is one analogy of the antialiasing operator widely applied in Kirchhoff migration. When the new wave equation is solved numerically using finite-difference schemes, numerical dispersion in the original wave equation is attenuated significantly, leading to a much more accurate finite-difference scheme with little additional computational cost. Numerical tests on both synthetic and field data sets in both two and three dimensions demonstrate that the optimized wave equation dramatically improves the image quality by successfully attenuating dispersive noise. The adaptive application of this new wave equation only increases the computational cost slightly.

scholarly journals An Unconditionally Stable Method for Solving the Acoustic Wave Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Zhi-Kai Fu ◽  
Li-Hua Shi ◽  
Zheng-Yu Huang ◽  
Shang-Chen Fu
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Testing Procedure ◽  
Basis Functions ◽  
Acoustic Wave Equation ◽  
Stable Method ◽  
Time Step ◽  
Orthogonal Functions ◽  
Unconditionally Stable ◽  
The Time Domain

An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL) condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.

Simulation of acoustic wavefields in heterogeneous media: A robust method for automatic suppression of numerical dispersion

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. T99-T110 ◽  
Author(s):  
Dinghui Yang ◽  
Guojie Song ◽  
Biaolong Hua ◽  
Henri Calandra
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Large Scale ◽  
Heterogeneous Media ◽  
Numerical Dispersion ◽  
Acoustic Wave Equation ◽  
Discrete Method ◽  
Computational Costs ◽  
Refined Method

Numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in large-scale computation. The nearly analytic discrete method (NADM) and its improved version for suppressing numerical dispersion were developed recently. This new method is a refinement of two previous methods and further increases the ability of suppressing numerical dispersion for modeling acoustic wave propagation in 2D heterogeneous media, which uses acoustic wave displacement, particle velocity, and their gradients to restructure the acoustic wavefield via the truncated Taylor expansion and the high-order interpolation approximate method. For the method proposed, we investigate its implementation and compare it with the higher-order Lax-Wendroff correction (LWC) scheme, the original nearly analytic discrete method (NADM) and its im-proved version with regard to numerical dispersion, computational costs, and computer storage requirements. The numerical dispersion relations provided by the refined algorithm for 1D and 2D cases are analyzed, as well as the numerical results obtained by this method against the exact solution for the 2D acoustic case. Numerical results show that the refined method gives no visible numerical dispersion for very large spatial grid increments. It can simulate high-frequency wave propagation for a given grid interval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized, when too few samples per wavelength are used, or when models have large velocity contrasts. Unlike the high-order LWC methods, our present method can save substantial computational costs and memory requirements because very large grid increments can be used. The refined method can be used for the simulation of large-scale wavefields.

scholarly journals Iterative finite-difference solution analysis of acoustic wave equation in the Laplace-Fourier domain

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. T29-T36 ◽  
Author(s):  
Evan Schankee Um ◽  
Michael Commer ◽  
Gregory A. Newman
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Iterative Methods ◽  
Pressure Field ◽  
Solution Space ◽  
Skin Depth ◽  
Numerical Dispersion ◽  
Acoustic Wave Equation ◽  
Grid Sampling

We have investigated numerical characteristics of iterative solutions to the acoustic wave equation in the Laplace-Fourier (LF) domain. We transformed the time-domain acoustic wave equation into the LF domain; the transformed equation was discretized with finite differences and was solved with iterative methods. Finite-difference modeling experiments demonstrate that iterative methods require an infinitesimal stopping tolerance to accurately compute the pressure field especially at long offsets. To understand the requirement for such infinitesimal tolerance values, we analyzed the evolution of intermediate solution vectors, residual vectors, and search direction vectors during the iteration. The analysis showed that the requirement arises from the fact that in the solution space, the amplitude of the pressure field varies more than sixty orders of magnitude on the common log scale. Accordingly, we propose a rule of thumb for choosing a proper stopping tolerance value. We also examined numerical dispersion errors in terms of the grid sampling resolutions per skin depth and wavelength. We found that despite the similarity of the form of the acoustic wave and electromagnetic diffusion equations, the former is different from the latter due to the fact that in the LF domain, the skin depth of the acoustic wave equation is decoupled from its wavelength. This aspect requires that in the LF domain, its grid size be determined by considering the minimum grid sampling resolutions based not only the wavelength but also the skin depth.

2-D acoustic wave equation inversion combining plane‐wave seismograms with well logs

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 735-741 ◽  
Author(s):  
Shoudong Wang ◽  
Jiaqi Liu ◽  
Ganquan Xie
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Plane Wave ◽  
Seismic Waves ◽  
Calculated Data ◽  
Well Logs ◽  
Acoustic Wave Equation ◽  
Inversion Procedure ◽  
The Difference ◽  
The Stability

Seismic waves can be described approximately by an acoustic wave equation. An inversion procedure for obtaining velocities using well logs and plane‐wave reflection seismograms is described. The forward problem is based on the 2-D acoustic wave equation. The inversion is performed by minimizing the difference between the calculated data and the observed data using the least‐squares regularization method. The numerical example shows that the stability and the robustness of the algorithm to noisy data is improved by introducing well logs into the inversion process.

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