Discrete phase space I. Finite difference operators and lattice Schrödinger wave equation

1994 ◽  
Vol 7 (1) ◽  
pp. 21-38 ◽  
Author(s):  
A. Das ◽  
P. Smoczynski
Keyword(s):  
Wave Equation ◽  
Phase Space ◽  
Finite Difference ◽  
Difference Operators ◽  
Discrete Phase

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

Numerical analysis of a narrow-angle, one-way, elastic-wave equation and extension to curvilinear coordinates

Geophysics ◽  
2006 ◽  
Vol 71 (5) ◽  
pp. T137-T146 ◽  
Author(s):  
D. A. Angus ◽  
C. J. Thomson
Keyword(s):  
Wave Equation ◽  
Numerical Analysis ◽  
Finite Difference ◽  
Velocity Structure ◽  
Dispersion Analysis ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
Propagation Path ◽  
Difference Operators ◽  
Narrow Angle

In this paper, we review the finite-difference implementation of a narrow-angle, one-way vector wave equation for elastic, 3D media. Extrapolation is performed in the frequency domain, where the second-order-accurate lateral spatial-difference operators are sufficiently accurate for narrow-angle propagation. We perform a numerical analysis of the finite-difference scheme to highlight the stability and dispersion characteristics. The von Neumann stability criterion indicates that extracting a reference phase during the extrapolation step noticeably improves the forward marching scheme, and dispersion analysis shows that numerical grid anisotropy is minimal for the propagation path lengths, source pulse spectral content, and angular range of forward propagation of interest. Although the algorithm is reasonable, its computational efficiency is limited by the second-order-accurate extrapolation step; therefore, the extrapolation scheme can be improved. We extend the Cartesian narrow-angle formulation to curvilinear coordinates, where the computational grid tracks the true wavefront in a reference medium and the wavefield derivative normal to the reference wavefront is evaluated locally using the Cartesian propagator. An example of curvilinear extrapolation for a simple model consisting of a high-velocity sphere within a homogeneous background velocity structure shows that the narrow-angle propagator is capable of modeling frequency-dependent geometric spreading and diffraction effects in curvilinear coordinates.

scholarly journals Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ke-Yang Chen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Computing Time ◽  
Staggered Grid ◽  
Difference Operators ◽  
Elastic Wave Equation ◽  
Absorbing Boundary ◽  
Wave Separation ◽  
Time Requirements

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.

Implicit finite-difference simulations of seismic wave propagation

Geophysics ◽  
2012 ◽  
Vol 77 (2) ◽  
pp. T57-T67 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Integration Method ◽  
Time Integration ◽  
Difference Operators ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Time Integration Method ◽  
Implicit Finite Difference ◽  
Splitting Time

We propose a new finite-difference modeling method, implicit both in space and in time, for the scalar wave equation. We use a three-level implicit splitting time integration method for the temporal derivative and implicit finite-difference operators of arbitrary order for the spatial derivatives. Both the implicit splitting time integration method and the implicit spatial finite-difference operators require solving systems of linear equations. We show that it is possible to merge these two sets of linear systems, one from implicit temporal discretizations and the other from implicit spatial discretizations, to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm. We give the complete derivations of the implicit splitting time integration method and the implicit spatial finite-difference operators, and present the resulting discretized formulas for the scalar wave equation. We conduct a thorough numerical analysis on grid dispersions of this new implicit modeling method. We show that implicit spatial finite-difference operators greatly improve the accuracy of the implicit splitting time integration simulation results with only a slight increase in computational time, compared with explicit spatial finite-difference operators. We further verify this conclusion by both 2D and 3D numerical examples.

A new family of finite-difference schemes to solve the heterogeneous acoustic wave equation

Geophysics ◽  
2012 ◽  
Vol 77 (5) ◽  
pp. T187-T199 ◽  
Author(s):  
Leandro Di Bartolo ◽  
Cleberson Dors ◽  
Webe J. Mansur
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Staggered Grid ◽  
Order System ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Memory Usage ◽  
Acoustic Wave Equation ◽  
New Family

Equivalent staggered grid scheme (ESG) is a new family of schemes based on the finite-difference method (FDM). The method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic FDM approaches. The main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations (SSG), but with the same memory requirements of simple grid schemes. Hence, it results in a reduction of memory usage by 33% in 2D and 50% in 3D problems, compared to the memory usage of SSG. The first-order system of equations in terms of pressure and velocity is not used here. Instead, the formulation is based on applying new central difference operators to the second-order acoustic wave equation in terms of pressure, obtaining the same level of accuracy and stability as the SSG schemes. The equivalence between the ESG and SSG is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

On invariant subspaces for nonlinear finite-difference operators

1998 ◽  
Vol 128 (6) ◽  
pp. 1293-1308 ◽  
Author(s):  
Victor A. Galaktionov
Keyword(s):  
Finite Difference ◽  
Differential Operators ◽  
Invariant Subspaces ◽  
Reaction Diffusion ◽  
Difference Operators ◽  
Diffusion Type ◽  
Nonlinear Differential Operators ◽  
Discrete Operators ◽  
Spatial Discretisation ◽  
Lower Dimensional

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.

A method for correcting acoustic finite-difference amplitudes for elastic effects

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. T243-T255 ◽  
Author(s):  
James W. D. Hobro ◽  
Chris H. Chapman ◽  
Johan O. A. Robertsson
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Effective Means ◽  
Cost Effective ◽  
P Wave ◽  
S Waves ◽  
Velocity Models ◽  
Acoustic Simulation ◽  
Wide Range ◽  
Elastic Simulation

We present a new method for correcting the amplitudes of arrivals in an acoustic finite-difference simulation for elastic effects. In this method, we selectively compute an estimate of the error incurred when the acoustic wave equation is used to approximate the behavior of the elastic wave equation. This error estimate is used to generate an effective source field in a second acoustic simulation. The result of this second simulation is then applied as a correction to the original acoustic simulation. The overall cost is approximately twice that of an acoustic simulation but substantially less than the cost of an elastic simulation. Because both simulations are acoustic, no S-waves are generated, so dispersed converted waves are avoided. We tested the characteristics of the method on a simple synthetic model designed to simulate propagation through a strong acoustic impedance contrast representative of sedimentary geology. It corrected amplitudes to high accuracy for reflected arrivals over a wide range of incidence angles. We also evaluated results from simulations on more complex models that demonstrated that the method was applicable in realistic sedimentary models containing a wide range of seismic contrasts. However, its accuracy was reduced for wide-angle reflections from very high impedance contrasts such as a shallow top-salt interface. We examined the influence of modeling at coarse grid resolutions, in which converted S-waves in the equivalent elastic simulation are dispersed. These results provide some validation for the accuracy of the method when applied using finite-difference grids designed for acoustic modeling. The method appears to offer a cost-effective means of modeling elastic amplitudes for P-wave arrivals in a useful range of velocity models. It has several potential applications in imaging and inversion.

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