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.

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 Space-time spectral element method solution for the acoustic wave equation and its dispersion analysis

2017 ◽  
Vol 38 (6) ◽  
pp. 303-313 ◽  
Author(s):  
Yanhui Geng ◽  
Guoliang Qin ◽  
Jiazhong Zhang ◽  
Wenqiang He ◽  
Zhenzhong Bao ◽  
...  
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Spectral Element Method ◽  
Spectral Element ◽  
Dispersion Analysis ◽  
Space Time ◽  
Acoustic Wave Equation ◽  
Element Method

Finite Element θ-Schemes for the Acoustic Wave Equation

2011 ◽  
Vol 3 (1) ◽  
pp. 181-203 ◽  
Author(s):  
Samir Karaa
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Stability And Convergence ◽  
Optimal Error Estimates ◽  
Finite Difference Schemes ◽  
Time Interval ◽  
Acoustic Wave Equation ◽  
Time Step ◽  
Polynomial Degree

AbstractIn this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1 + ∆ts), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and ∆t the time step.

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 Extracting angle domain common image gather with variable density acoustic-wave equation

2021 ◽  
Vol 18 (2) ◽  
pp. 1-8
Author(s):  
Yuzhu Liu ◽  
Weigang Liu ◽  
Jizhong Yang ◽  
Liangguo Dong
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Velocity Model ◽  
Variable Density ◽  
Constant Density ◽  
Acoustic Wave Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Density Perturbations

Abstract Angle domain common image gathers (ADCIGs), commonly regarded as important prestacked gathers, provide the information required for velocity model construction and the phase and amplitude information needed for subsurface structures in oil/gas exploration. Based on the constant-density acoustic-wave equation assumption, the ADCIGs generated from reverse time migration ignore the fact that the subsurface density varies with location. Consequently, the amplitude versus angle (AVA) analysis extracted from these ADCIGs is not accurate. To partially solve this problem and to improve the accuracy of the AVA analysis, we developed amplitude-preserving ADCIGs suitable for density variations with the assumption of acoustic approximation. The Poynting vector approach, which is efficient and computationally inexpensive, was used to calculate the high-resolution wavefield propagation. The ADCIGs generated from the velocity and density perturbations match the theoretical AVA relationship better than ADCIGs with constant density. The extraction of the AVA analysis of the various combinations of the subsurface medium indicates that the density is non-negligible, especially when the density contrast is sharp. Numerical examples based on a layered model verify our conclusions.

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