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.

An analytic signal based accurate time-domain visco-acoustic wave equation from the Constant-Q theory

Geophysics ◽  
2021 ◽  
pp. 1-58
Author(s):  
Hongwei Liu ◽  
Yi Luo
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Time Domain ◽  
Seismic Waves ◽  
Heterogeneous Media ◽  
Relaxation Property ◽  
Analytic Signal ◽  
Acoustic Wave Equation ◽  
Real Component ◽  
Frequency Components

We present a concise time-domain wave equation to accurately simulate wave propagation in visco-acoustic media. The central idea behind this work is to dismiss the negative frequency components from a time-domain signal by converting the signal to its analytic format. The negative frequency components of any analytic signal are always zero, meaning we can construct the visco-acoustic wave equation to honor the relaxation property of the media for positive frequencies only. The newly proposed complex-valued wave equation (CWE) represents the wavefield with its analytic signal, whose real part is the desired physical wavefield, while the imaginary part is the Hilbert transform of the real component. Specifically, this CWE is accurate for both weak and strong attenuating media in terms of both dissipation and dispersion and the attenuation is precisely linear with respect to the frequencies. Besides, the CWE is easy and flexible to model dispersion-only, dissipation-only or dispersion-plus-dissipation seismic waves. We have verified these CWEs by comparing the results with analytical solutions, and achieved nearly perfect matching. Except for the homogeneous Q media, we have also extended the CWEs to heterogeneous media. The results of the CWEs for heterogeneous Q media are consistent with those computed from the nonstationary operator based Fourier Integral method and from the Standard Linear Solid (SLS) equations.

scholarly journals Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160798 ◽  
Author(s):  
Wenjun Cai ◽  
Huai Zhang ◽  
Yushun Wang
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Acoustic Waves ◽  
Conservation Law ◽  
Heterogeneous Media ◽  
Numerical Dispersion ◽  
Discrete Version ◽  
Symplectic Method ◽  
Ode System ◽  
Strang Splitting

We propose a novel stable and efficient dissipation-preserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude. Through introducing the conformal multi-symplectic structure, the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation. The proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law. More specifically, two subsystems in conjunction with the original damped wave equation are derived. One is actually the conservative Hamiltonian wave equation and the other is a dissipative linear ordinary differential equation (ODE) system. Standard symplectic method is devoted to the conservative system, whereas the analytical solution is obtained for the ODE system. An explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the Strang splitting technique. Stability analysis and convergence tests are given thereafter. A benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation. Further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media.

Multiscale modeling of acoustic wave propagation in 2D media

Geophysics ◽  
2014 ◽  
Vol 79 (2) ◽  
pp. T61-T75 ◽  
Author(s):  
Richard L. Gibson ◽  
Kai Gao ◽  
Eric Chung ◽  
Yalchin Efendiev
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Heterogeneous Media ◽  
Coarse Grid ◽  
Basis Functions ◽  
Seismic Survey ◽  
Fine Scale ◽  
Acoustic Wave Propagation ◽  
Fine Grid

Conventional finite-difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications, but they face significant challenges when material properties vary significantly over distances less than the grid size. This challenge is likely to occur in reservoir characterization studies, because important reservoir heterogeneity can be present on scales of several meters to ten meters. Here, we describe a new multiscale finite-element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine- and coarse-scale grids. The wave equation is solved on a coarse grid, but it uses basis functions that are generated from the fine grid and allow the representation of the fine-scale variation of the wavefield on the coarser grid. Time stepping also takes place on the coarse grid, providing further speed gains. Another important property of the method is that the basis functions are only computed once, and time savings are even greater when simulations are repeated for many source locations. We first present validation results for simple test models to demonstrate and quantify potential sources of error. These tests show that the fine-scale solution can be accurately approximated when the coarse grid applies a discretization up to four times larger than the original fine model. We then apply the multiscale algorithm to simulate a complete 2D seismic survey for a model with strong, fine-scale scatterers and apply standard migration algorithms to the resulting synthetic seismograms. The results again show small errors. Comparisons to a model that is upscaled by averaging densities on the fine grid show that the multiscale results are more accurate.

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.

Interface conditions for acoustic and elastic wave propagation

Geophysics ◽  
1991 ◽  
Vol 56 (2) ◽  
pp. 168-181 ◽  
Author(s):  
J. S. Sochacki ◽  
J. H. George ◽  
R. E. Ewing ◽  
S. B. Smithson
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Heterogeneous Media ◽  
Elastic Wave Propagation ◽  
Integration Scheme ◽  
Acoustic Wave Equation ◽  
Sh Wave ◽  
Numerical Schemes ◽  
Tangential Stresses

The divergence theorem is used to handle the physics required at interfaces for acoustic and elastic wave propagation in heterogeneous media. The physics required at regular and irregular interfaces is incorporated into numerical schemes by integrating across the interface. The technique, which can be used with many numerical schemes, is applied to finite differences. A derivation of the acoustic wave equation, which is readily handled by this integration scheme, is outlined. Since this form of the equation is equivalent to the scalar SH wave equation, the scheme can be applied to this equation also. Each component of the elastic P‐SV equation is presented in divergence form to apply this integration scheme, naturally incorporating the continuity of the normal and tangential stresses required at regular and irregular interfaces.

Theory of a 2.5-D acoustic wave equation for constant density media

Geophysics ◽  
1991 ◽  
Vol 56 (12) ◽  
pp. 2114-2117 ◽  
Author(s):  
Christopher L. Liner
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Symmetry Axis ◽  
Constant Density ◽  
Efficient Computation ◽  
Acoustic Wave Equation ◽  
Out Of Plane ◽  
Acoustic Media ◽  
D Wave

The theory of 2.5-dimensional (2.5-D) wave propagation (Bleistein, 1986) allows efficient computation of 3-D wavefields in c(x, z) acoustic media when the source and receivers lie in a common y-plane (assumed to be y = 0 in this paper). It is really a method of efficiently computing an inplane 3-D wavefield in media with one symmetry axis. The idea is to raytrace the wavefield in the (x, z)-plane while allowing for out‐of‐plane spreading. In this way 3-D amplitude decay is honored without 3-D ray tracing. This theory has its conceptual origin in work by Ursin (1978) and Hubral (1978). Bleistein (1986) gives an excellent overview and detailed reference to earlier work.

Error Analysis of Chebyshev Spectral Element Methods for the Acoustic Wave Equation in Heterogeneous Media

2018 ◽  
Vol 26 (03) ◽  
pp. 1850035 ◽  
Author(s):  
Saulo Pomponet Oliveira
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Error Analysis ◽  
Heterogeneous Media ◽  
Spectral Element ◽  
Variable Density ◽  
Dirichlet Boundary Conditions ◽  
Acoustic Wave Equation ◽  
Polynomial Degree ◽  
Mesh Parameter

This work concerns the error analysis of the spectral element method with Gauss–Lobatto–Chebyshev collocation points with the implicit Newmark average acceleration scheme for the two-dimensional acoustic wave equation. The analysis is restricted to homogeneous Dirichlet boundary conditions, constant compressibility and variable density. The proposed error estimates are optimal with respect to the mesh parameter although suboptimal on the polynomial degree. Numerical examples illustrate the theoretical results.

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