Acoustic VTI modeling with a time-space domain dispersion-relation-based finite-difference scheme

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. A11-A17 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Numerical Modeling ◽  
Dispersion Relation ◽  
Finite Difference ◽  
Wave Equations ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
Dispersion Analysis ◽  
Space Domain ◽  
Time Space ◽  
Grid Points

The finite-difference (FD) method is widely used in numerical modeling of wave equations. Conventional FD stencils for space derivatives are usually designed in the space domain. When they are used to solve wave equations, it is difficult to satisfy the dispersion relations exactly. We have designed a spatial FD stencil based on a time-space domain dispersion relation to simulate wave propagation in an acoustic vertically transversely isotropic (VTI) medium. Two-dimensional dispersion analysis and numerical modeling demonstrate that this stencil has greater precision than one used in a conventional FD, when the same number of grid points is used in the calculation of the spatial derivatives. Thus, the spatial FD stencil based on time-space domain dispersion relation can be used to replace a conventional one such that we can achieve greater accuracy with almost no increase in computational cost.

Acoustic VTI Modeling Using an Optimal Time-Space Domain Finite-Difference Scheme

2016 ◽  
Vol 24 (04) ◽  
pp. 1650016 ◽  
Author(s):  
Hongyong Yan ◽  
Lei Yang ◽  
Xiang-Yang Li ◽  
Hong Liu
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Taylor Series Expansion ◽  
Transversely Isotropic ◽  
High Accuracy ◽  
Seismic Exploration ◽  
Numerical Dispersion ◽  
Optimal Time ◽  
Space Domain ◽  
Time Space

Finite-difference (FD) schemes have been used widely for solving wave equations in seismic exploration. However, the conventional FD schemes hardly guarantee high accuracy at both small and large wavenumbers. In this paper, we propose an optimal time-space domain FD scheme for acoustic vertical transversely isotropic (VTI) wave modeling. The optimal FD coefficients for the second-order spatial derivatives are derived by approaching the time-space domain dispersion relation of acoustic VTI wave equations with the combination of the Taylor-series expansion and the sampling interpolation. We perform numerical dispersion analyses and acoustic VTI modeling using the optimal time-space domain FD scheme. The numerical dispersion analyses show that the optimal FD scheme has smaller dispersion than the conventional FD scheme at large wavenumbers, and also preserves high accuracy at small wavenumbers. The acoustic VTI modeling examples also demonstrate that the optimal time-space domain FD scheme has greater accuracy compared with the conventional time-space domain FD scheme for the same modeling parameters.

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.

Time-space domain dispersion-relation-based mixed staggered-grid finite-difference methods

2017 ◽  
Author(s):  
Hu Ziduo* ◽  
Liu Wei ◽  
Wang Yanxiang ◽  
Yang Zhe ◽  
Wang Shujiang
Keyword(s):  
Dispersion Relation ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Staggered Grid ◽  
Space Domain ◽  
Time Space ◽  
Difference Methods

High-order finite-difference approximations to solve pseudoacoustic equations in 3D VTI media

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T225-T235 ◽  
Author(s):  
Leandro Di Bartolo ◽  
Leandro Lopes ◽  
Luis Juracy Rangel Lemos
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
High Order ◽  
Staggered Grid ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Grid Scheme

Pseudoacoustic algorithms are very fast in comparison with full elastic ones for vertical transversely isotropic (VTI) modeling, so they are suitable for many applications, especially reverse time migration. Finite differences using simple grids are commonly used to solve pseudoacoustic equations. We have developed and implemented general high-order 3D pseudoacoustic transversely isotropic formulations. The focus is the development of staggered-grid finite-difference algorithms, known for their superior numerical properties. The staggered-grid schemes based on first-order velocity-stress wave equations are developed in detail as well as schemes based on direct application to second-order stress equations. This last case uses the recently presented equivalent staggered-grid theory, resulting in a staggered-grid scheme that overcomes the problem of large memory requirement. Two examples are presented: a 3D simulation and a prestack reverse time migration application, and we perform a numerical analysis regarding computational cost and precision. The errors of the new schemes are smaller than the existing nonstaggered-grid schemes. In comparison with existing staggered-grid schemes, they require 25% less memory and only have slightly greater computational cost.

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.

Globally optimal finite-difference schemes based on least squares

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. T113-T132 ◽  
Author(s):  
Yang Liu
Keyword(s):  
Dispersion Relation ◽  
Finite Difference ◽  
Phase Velocity ◽  
Least Squares ◽  
Relative Error ◽  
Optimization Methods ◽  
Second Order ◽  
Space Domain ◽  
Time Space ◽  
The Given

Spatial finite-difference (FD) coefficients are usually determined by the Taylor-series expansion (TE) or optimization methods. The former can provide high accuracy on a smaller wavenumber or frequency zone, and the latter can give moderate accuracy on a larger zone. Present optimization methods applied to calculate FD coefficients are generally gradient-like or global optimization-like algorithms, and thus iterations are involved. They are more computationally expensive, and sometimes the global solution may not be found. I examined second-order spatial derivatives and computed the optimized spatial FD coefficients over the given wavenumber range using the least-squares (LS) method. The results indicated that the FD accuracy increased with increasing operator length and decreasing wavenumber range. Therefore, for the given error and operator length, globally optimal spatial FD coefficients can be easily obtained. Some optimal FD coefficients were given. I developed schemes to obtain optimized LS-based spatial FD coefficients by minimizing the relative error of space-domain dispersion relation for second-order derivatives and time-space-domain dispersion relation for the acoustic wave equation. I discovered that minimizing the relative error of the space-domain dispersion relation provides less phase velocity error for small wavenumbers, compared to minimizing the absolute error. I also found that minimizing the relative error of the time-space-domain dispersion relation can reduce relative errors of phase velocity. Accuracy analysis demonstrated the correctness and advantage of schemes. I gave three examples of 2D acoustic FD modeling for a homogeneous, a large velocity-contrast, and a heterogeneous model, respectively. LS-based spatial FD operators have variable lengths for different velocities. Modeling examples demonstrated that the proposed LS-based FD scheme can maintain the same modeling precision while using a shorter spatial FD operator length, thus reducing the computation cost relative to conventional TE-based FD schemes, particularly for the higher order.

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