Second-Order Implicit Finite-Difference Schemes for Acoustic Wave Equation in Time-Space Domain

Geophysics ◽  
2021 ◽  
pp. 1-83
Author(s):  
Navid Amini ◽  
Changsoo Shin ◽  
Jaejoon Lee
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Second Order ◽  
Numerical Dispersion ◽  
Acoustic Wave Equation ◽  
Space Domain ◽  
Time Space ◽  
Implicit Finite Difference ◽  
2D And 3D

We propose compact implicit finite-difference (FD) schemes in time-space domain based on second-order FD approximation for accurate solution of the acoustic wave equation in 1D, 2D, and 3D. Our method is based on weighted linear combination of the second-order FD operators with different spatial orientations to mitigate numerical error anisotropy and weighted averaging of the mass acceleration term over the grid-points of the second-order FD stencil to reduce the overall numerical dispersion error. We present derivation of the schemes for 1D, 2D, and 3D cases and obtain their corresponding dispersion equations, then we find optimum weights by optimization of the time-space domain dispersion function and finally tabulate the optimized weights for each case. We analyze the numerical dispersion, stability and convergence rates of the proposed schemes and compare their numerical dispersion characteristics with the standard high-order ones. We also discuss efficient solution of the system of equations associated with the proposed implicit schemes using conjugate gradient method. The comparison of dispersion curves and the numerical solutions with the analytical and the pseudo-spectral solutions reveals that the proposed schemes have better performance than the standard spatial high-order schemes and remain stable for relatively large time-steps.

Acoustic Wave Equation Modeling with New Time-Space Domain Finite Difference Operators

2013 ◽  
Vol 56 (6) ◽  
pp. 840-850 ◽  
Author(s):  
LIANG Wen-Quan ◽  
YANG Chang-Chun ◽  
WANG Yan-Fei ◽  
LIU Hong-Wei
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Equation Modeling ◽  
Difference Operators ◽  
Acoustic Wave Equation ◽  
Space Domain ◽  
Time Space ◽  
New Time

Time-domain explicit finite-difference method based on the mixed-domain function approximation for acoustic wave equation

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T237-T248 ◽  
Author(s):  
Zhikai Wang ◽  
Jingye Li ◽  
Benfeng Wang ◽  
Yiran Xu ◽  
Xiaohong Chen
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Function Approximation ◽  
High Accuracy ◽  
Acoustic Wave Equation ◽  
Space Domain ◽  
Time Space ◽  
Explicit Finite Difference

Explicit finite-difference (FD) methods with high accuracy and efficiency are preferred in full-waveform inversion and reverse time migration. The Taylor-series expansion (TE)-based FD methods can only obtain high accuracy on a small wavenumber zone. We have developed a new explicit FD method with spatial arbitrary even-order accuracy based on the mixed [Formula: see text] (wavenumber)-space domain function approximation for the acoustic wave equation, and we derived the FD coefficients by minimizing the approximation error in a least-squares (LS) sense. The weighted pseudoinverse of mixed [Formula: see text]-space matrix is introduced into the LS optimization problem to improve the accuracy. The new method has an exact temporal derivatives discretization in homogeneous media and also has higher temporal and spatial accuracy in heterogeneous media. Approximation errors and numerical dispersion analysis demonstrate that the new FD method has a higher numerical accuracy than conventional TE-based FD and TE-based time-space domain dispersion-relation FD methods. Stability analysis reveals that our proposed method requires a slightly stricter stability condition than the TE-based FD and TE-based time-space domain dispersion-relation FD methods. Numerical tests in the homogeneous model, horizontally layered model, and 2D modified Sigsbee2 model demonstrate the accuracy, efficiency, and flexibility of the proposed new FD method.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

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.

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