3D acoustic wave modelling with time-space domain dispersion-relation-based finite-difference schemes and hybrid absorbing boundary conditions

2011 ◽  
Vol 42 (3) ◽  
pp. 176-189 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Dispersion Relation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Finite Difference Schemes ◽  
Absorbing Boundary ◽  
Space Domain ◽  
Wave Modelling ◽  
Time Space

Optimization based determination of highly absorbing boundary conditions for linear finite difference schemes

2016 ◽  
Vol 365 ◽  
pp. 45-69 ◽  
Author(s):  
A. Schirrer ◽  
E. Talic ◽  
G. Aschauer ◽  
M. Kozek ◽  
S. Jakubek
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Absorbing Boundary

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 mesh-free finite difference based on least squares for 2D acoustic-wave modeling

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. T143-T157 ◽  
Author(s):  
Bei Li ◽  
Yang Liu ◽  
Mrinal K. Sen ◽  
Zhiming Ren
Keyword(s):  
Dispersion Relation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Least Squares ◽  
Absolute Error ◽  
Step Size ◽  
Space Domain ◽  
Wave Modeling ◽  
Time Space ◽  
Mesh Free

Finite-difference (FD) methods approximate derivatives through a weighted summation of function values from neighboring nodes. Traditionally, these neighboring nodes are assumed to be distributed regularly, such as in square or rectangular lattices. To improve geometric flexibility, one option is to develop FD in a mesh-free discretization, in which scattered nodes can be placed suitably with respect to irregular boundaries or arbitrarily shaped anomalies without coordinate transformation or forming any triangles or tetrahedra, etc. These mesh-free FDs have had successful applications, especially in computational geoscience. However, they are all space-domain FD schemes, in which FD coefficients are derived by approximating spatial derivatives individually in the space domain. For acoustic-wave modeling, it has been proven that space-domain FD methods normally have higher dispersion error than time-space-domain FD methods, which determine FD coefficients by approximating the time-space-domain dispersion relation. Now, we have developed a time-space-domain mesh-free FD based on minimizing the absolute error of the dispersion relation by least-squares (LS) for 2D acoustic-wave modeling. The matrix used to solve for FD coefficients in our method is determined by the spatial distribution of the nodes in a local FD stencil, whereas the temporal step size and velocity information are considered in the right side of the linear system. This feature of considering both spatial and temporal effects allows our proposed mesh-free LS-based FD to obtain greater temporal accuracy adaptive to different Courant-Friedrichs-Lewy parameters than pure space-domain mesh-free FDs. Under several 2D acoustic scenarios, the advantage was proven by comparing our method with radial-basis-function-generated FD, which is one of the most popular mesh-free FDs and has been applied in elastic wave modeling.

Absorbing boundary conditions for 3-D acoustic and elastic finite-difference calculations

1989 ◽  
Vol 79 (1) ◽  
pp. 211-218
Author(s):  
Wen-Fong Chang ◽  
George A. McMechan
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Absorbing Boundary Conditions ◽  
Absorbing Boundary

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.

Acoustic and elastic modeling by optimal time-space-domain staggered-grid finite-difference schemes

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T17-T40 ◽  
Author(s):  
Zhiming Ren ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Computational Cost ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Equation Modeling ◽  
Optimal Time ◽  
Finite Difference Schemes ◽  
Space Domain ◽  
Time Space

Staggered-grid finite-difference (SFD) methods are widely used in modeling seismic-wave propagation, and the coefficients of finite-difference (FD) operators can be estimated by minimizing dispersion errors using Taylor-series expansion (TE) or optimization. We developed novel optimal time-space-domain SFD schemes for acoustic- and elastic-wave-equation modeling. In our schemes, a fourth-order multiextreme value objective function with respect to FD coefficients was involved. To yield the globally optimal solution with low computational cost, we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least-squares (LS) to derive the optimal SFD coefficients by minimizing the relative error of time-space-domain dispersion relations over a given frequency range. To ensure the robustness of our schemes, a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold. Moreover, the hybrid absorbing boundary condition was applied to remove artificial boundary reflections. We compared our optimal SFD with the conventional, TE-based time-space-domain, and LS-based SFD schemes. Dispersion analysis and numerical simulation results suggested that the new SFD schemes had a smaller numerical dispersion than the other three schemes when the same operator lengths were adopted. In addition, our LS-based time-space-domain SFD can obtain the same modeling accuracy with shorter spatial operator lengths. We also derived the stability condition of our schemes. The experiment results revealed that our new LS-based SFD schemes needed a slightly stricter stability condition.

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