To: “Elimination of numerical dispersion in finite‐difference modeling and migration by flux‐corrected transport,” by Tong Fei and Ken Larner (November‐December 1995 GEOPHYSICS, 60, p. 1830–1841.

Geophysics ◽  
1996 ◽  
Vol 61 (3) ◽  
pp. 922-922
Keyword(s):  
Finite Difference ◽  
Numerical Dispersion ◽  
Flux Corrected Transport ◽  
And Migration ◽  
Memorial University Of Newfoundland

Acknowledgments: The authors would like to thank Jinming Zhu from Memorial University of Newfoundland in St. John’s, Newfoundland, for identifying the errors in the figures.

scholarly journals Elimination of numerical dispersion in finite‐difference modeling and migration by flux‐corrected transport

Geophysics ◽  
1995 ◽  
Vol 60 (6) ◽  
pp. 1830-1842 ◽  
Author(s):  
Tong Fei ◽  
Ken Larner
Keyword(s):  
Finite Difference ◽  
Numerical Dispersion ◽  
Correction Procedure ◽  
Reverse Time ◽  
Flux Correction ◽  
Finite Difference Approximations ◽  
Difference Approximations ◽  
Flux Corrected Transport ◽  
Wavefield Continuation ◽  
And Migration

Finite‐difference acoustic‐wave modeling and reverse‐time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well. However, conventional finite‐difference methods for solving the acoustic‐ or elastic‐wave equation suffer from numerical dispersion when too few samples per wavelength are used. The flux‐corrected transport (FCT) algorithm, adapted from hydrodynamics, reduces the numerical dispersion in finite‐difference wavefield continuation. The flux‐correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion. Incorporating the flux‐correction procedure in conventional finite‐difference modeling or reverse‐time migration can provide finite‐difference solutions with no numerical dispersion even for impulsive sources. The FCT correction, which can be applied to finite‐difference approximations of any order in space and time, is an efficient alternative to use for finite‐difference approximations of increasing order. Through demonstrations of modeling and migration on both synthetic and field data, we show the benefits of the FCT algorithm, as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse.

Staggered-Grid Finite Difference Method for Numerical Simulation of the Formulated BISQ Model

2013 ◽  
Vol 756-759 ◽  
pp. 4742-4746
Author(s):  
Xin Ming Zhang ◽  
Chang Ming Fu ◽  
Ting Ting Guo
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Layer Method ◽  
Bisq Model ◽  
Flux Corrected Transport ◽  
Transport Method

In this paper, based on the formulated BISQ model including the Biot-flow and squirt-flow mechanism simultaneously, the elastic wave propagation in the isotropic porous medium filled with fluids is simulated by the staggered grid finite difference method. The perfectly matched layer method and the flux-corrected transport method are used to eliminate the effect of boundary reflection and numerical dispersion effect. The results of numerical simulation demonstrate that the method is effective.

Controlling Numerical Dispersion by Variably Timed Flux Updating in Two Dimensions

1982 ◽  
Vol 22 (03) ◽  
pp. 409-419 ◽  
Author(s):  
R.G. Larson
Keyword(s):  
Finite Difference ◽  
Pressure Gradient ◽  
Single Point ◽  
Companion Paper ◽  
Two Dimensions ◽  
Independent Component ◽  
Numerical Dispersion ◽  
Coarse Mesh ◽  
Fixed Concentration ◽  
Order Of Magnitude

Abstract The variably-timed flux updating (VTU) finite difference technique is extended to two dimensions. VTU simulations of miscible floods on a repeated five-spot pattern are compared with exact solutions and with solutions obtained by front tracking. It is found that for neutral and favorable mobility ratios. VTU gives accurate results even on a coarse mesh and reduces numerical dispersion by a factor of 10 or more over the level generated by conventional single-point (SP) upstream weighting. For highly unfavorable mobility ratios, VTU reduces numerical dispersion. but on a coarse mesh the simulation is nevertheless inaccurate because of the inherent inadequacy of the finite-difference estimation of the flow field. Introduction A companion paper (see Pages 399-408) introduced the one-dimensional version of VTU for controlling numerical dispersion in finite-difference simulation of displacements in porous media. For linear and nonlinear, one- and two-independent-component problems, VTU resulted in more than an order-of-magnitude reduction in numerical dispersion over conventional explicit. SP upstream-weighted simulations with the same number of gridblocks. In this paper, the technique is extended to two dimensional (2D) problems, which require solution of a set of coupled partial differential equations that express conservation of material components-i.e., (1) and (2) Fi, the fractional flux of component i, is a function of the set of s - 1 independent-component fractional concentrations {Ci}, which prevail at the given position and time., the dispersion flux, is given by an expression that is linear in the specie concentration gradients. The velocity, is proportional to the pressure gradient,. (3) where lambda, in general, can be a function of composition and of the magnitude of the pressure gradient. The premises on which Eqs. 1 through 3 rest are stated in the companion paper. VTU in Two Dimensions The basic idea of variably-timed flux updating is to use finite-difference discretization of time and space, but to update the flux of a component not every timestep, but with a frequency determined by the corresponding concentration velocity -i.e., the velocity of propagation of fixed concentration of that component. The concentration velocity is a function of time and position. In the formulation described here, the convected flux is upstream-weighted, and all variables except pressure are evaluated explicitly. As described in the companion paper (SPE 8027), the crux of the method is the estimation of the number of timesteps required for a fixed concentration to traverse from an inflow to an outflow face of a gridblock. This task is simpler in one dimension, where there is only one inflow and one outflow face per gridblock, than it is in two dimensions, where each gridblock has in general multiple inflow and outflow faces. SPEJ P. 409^

Fourth‐order finite‐difference P-SV seismograms

Geophysics ◽  
1988 ◽  
Vol 53 (11) ◽  
pp. 1425-1436 ◽  
Author(s):  
Alan R. Levander
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Fourth Order ◽  
Equations Of Motion ◽  
Dispersion Analysis ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Order System ◽  
Surface Boundary ◽  
Lamb’S Problem

I describe the properties of a fourth‐order accurate space, second‐order accurate time, two‐dimensional P-SV finite‐difference scheme based on the Madariaga‐Virieux staggered‐grid formulation. The numerical scheme is developed from the first‐order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga‐Virieux staggered‐grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic‐elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free‐surface or within a layer and to satisfy free‐surface boundary conditions. Benchmark comparisons of finite‐difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite‐difference and reflectivity solutions for elastic‐elastic and acoustic‐elastic layered models.

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.

A simple and exact acoustic wavefield modeling code for data processing, imaging, and interferometry applications

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. F17-F27 ◽  
Author(s):  
Erica Galetti ◽  
David Halliday ◽  
Andrew Curtis
Keyword(s):  
Finite Difference ◽  
Synthetic Data ◽  
Acoustic Modeling ◽  
Numerical Dispersion ◽  
Seismic Interferometry ◽  
Numerical Errors ◽  
Matlab Code ◽  
Modeling Code ◽  
Seismic Images ◽  
Methods Numerical

Improvements in industrial seismic, seismological, acoustic, or interferometric theory and applications often result in quite subtle changes in sound quality, seismic images, or information which are nevertheless crucial for improved interpretation or experience. When evaluating new theories and algorithms using synthetic data, an important aspect of related research is therefore that numerical errors due to wavefield modeling are reduced to a minimum. We present a new MATLAB code based on the Foldy method that models theoretically exact, direct, and scattered parts of a wavefield. Its main advantage lies in the fact that while all multiple scattering interactions are taken into account, unlike finite-difference or finite-element methods, numerical dispersion errors are avoided. The method is therefore ideal for testing new theory in industrial seismics, seismology, acoustics, and in wavefield interferometry in particular because the latter is particularly sensitive to the dynamics of scattering interactions. We present the theory behind the Foldy acoustic modeling method and provide examples of its implementation. We also benchmark the code against a good finite-difference code. Because our Foldy code was written and optimized to test new theory in seismic interferometry, examples of its application to seismic interferometry are also presented, showing its validity and importance when exact modeling results are needed.

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