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Journal articles on the topic 'Staggered grids finite difference'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Bernth, Henrik, and Chris Chapman. "A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids." GEOPHYSICS 76, no. 3 (2011): WA43—WA50. http://dx.doi.org/10.1190/1.3555530.

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Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, the dispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-difference calculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and four in three dimensions, which decouple in isotropic media. Therefore the Lebedev scheme will have twice or four times the computational requirements, memory, and CPU as the Virieux grid but can be used with general anisotropy. In two dimensions, the rotated staggered grid is exactly equivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numerical dispersion in finite-difference grids depends on the direction of propagation and the grid type and parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computational requirements for the two grid types, the dispersion, averaged over propagation direction and medium velocity are calculated. Setting the parameters so the average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field data and at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

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2

Di Bartolo, Leandro, Leandro Lopes, and Luis Juracy Rangel Lemos. "High-order finite-difference approximations to solve pseudoacoustic equations in 3D VTI media." GEOPHYSICS 82, no. 5 (2017): T225—T235. http://dx.doi.org/10.1190/geo2016-0589.1.

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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.

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3

Igel, Heiner, Peter Mora, and Bruno Riollet. "Anisotropic wave propagation through finite‐difference grids." GEOPHYSICS 60, no. 4 (1995): 1203–16. http://dx.doi.org/10.1190/1.1443849.

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An algorithm is presented to solve the elastic‐wave equation by replacing the partial differentials with finite differences. It enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models. The space derivatives are calculated using discrete convolution sums, while the time derivatives are replaced by a truncated Taylor expansion. A centered finite difference scheme in Cartesian coordinates is used for the space derivatives leading to staggered grids. The use of finite difference approximations to the partial derivatives results in a frequency‐dependent error in the group and phase velocities of waves. For anisotropic media, the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the Hook sum. This interpolation induces an additional error in the wave properties. The overall error depends on the precision of the derivative and interpolation operators, the anisotropic symmetry system, its orientation and the degree of anisotropy. The dispersion relation for the homogeneous case was derived for the proposed scheme. Since we use a general description of convolution sums to describe the finite difference operators, the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model. In particular, phase and group velocities of the three wave types can be determined in any direction. We demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed.

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4

Huang, Huaxiong, and Ming Li. "Finite-difference approximation for the velocity-vorticity formulation on staggered and non-staggered grids." Computers & Fluids 26, no. 1 (1997): 59–82. http://dx.doi.org/10.1016/s0045-7930(96)00028-x.

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5

Armfield, S. W. "Finite difference solutions of the Navier-Stokes equations on staggered and non-staggered grids." Computers & Fluids 20, no. 1 (1991): 1–17. http://dx.doi.org/10.1016/0045-7930(91)90023-b.

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6

Zhang, Jie, Fan Shun Meng, and Yang Sen Li. "The Study of the Difference Methods with Variable Grids Seismic Wave Numerical Simulation in Multi-Scale Complex Media." Advanced Materials Research 1055 (November 2014): 254–58. http://dx.doi.org/10.4028/www.scientific.net/amr.1055.254.

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In the process of seismic wave field numerical simulation using finite difference method, the simulation accuracy and computational efficiency is one of the keys to the problem which is especially important to the numerical simulation of small scale geological body which velocity changes violently. In order to describe the local structure of medium subtly and guarantee the efficiency of the simulation, this article introduces the variable grid finite difference method to the staggered grid high-order finite difference numerical simulation on the basic of the traditional staggered grid finite difference algorithm to improve the staggered grid spatial algorithm and avoid the reduction of the simulation accuracy and computational efficiency caused by the interpolation factor. The results show that the variable staggered grid numerical simulation of finite difference algorithm can accurately depict the space variation of underground medium physical properties to further enhance the adaptability of numerical simulation of complex medium, it also can provide reliable basis for wave field imaging and the combined interpretation of p-wave and s-wave.

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7

Pitarka, Arben. "3D Elastic finite-difference modeling of seismic motion using staggered grids with nonuniform spacing." Bulletin of the Seismological Society of America 89, no. 1 (1999): 54–68. http://dx.doi.org/10.1785/bssa0890010054.

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Abstract This article provides a technique to model seismic motions in 3D elastic media using fourth-order staggered-grid finite-difference (FD) operators implemented on a mesh with nonuniform grid spacing. The accuracy of the proposed technique has been tested through comparisons with analytical solutions, conventional 3D staggered-grid FD with uniform grid spacing, and reflectivity methods for a variety of velocity models. Numerical tests with nonuniform grids suggest that the method allows sufficiently accurate modeling when the grid sampling rate is at least 6 grid points per shortest shear wavelength. The applicability for a finite fault with non-uniform distribution of point sources is also confirmed. The use of nonuniform spacing improves the efficiency of the FD methods when applied to large-scale structures by partially avoiding the spatial oversampling introduced by the uniform spacing in zones with high velocity. The significant reduction in computer memory that can be obtained by the new technique improves the efficiency of the 3D-FD method at handling shorter wavelengths, larger areas, or more realistic 3D velocity structures.

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8

KÄSER, MARTIN, HEINER IGEL, MALCOLM SAMBRIDGE, and JEAN BRAUN. "A COMPARATIVE STUDY OF EXPLICIT DIFFERENTIAL OPERATORS ON ARBITRARY GRIDS." Journal of Computational Acoustics 09, no. 03 (2001): 1111–25. http://dx.doi.org/10.1142/s0218396x01000838.

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We compare explicit differential operators for unstructured grids and their accuracy with the aim of solving time-dependent partial differential equations in geophysical applications. As many problems suggest the use of staggered grids we investigate different schemes for the calculation of space derivatives on two separate grids. The differential operators are explicit and local in the sense that they use only information of the function in their nearest neighborhood, so that no matrix inversion is necessary. This makes this approach well-suited for parallelization. Differential weights are obtained either with the finite-volume method or using natural neighbor coordinates. Unstructured grids have advantages concerning the simulation of complex geometries and boundaries. Our results show that while in general triangular (hexagonal) grids perform worse than standard finite-difference approaches, the effects of grid irregularities on the accuracy of the space derivatives are comparably small for realistic grids. This suggests that such a finite-difference-like approach to unstructured grids may be an alternative to other irregular grid methods such as the finite-element technique.

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9

Bohlen, Thomas, and Erik H. Saenger. "Accuracy of heterogeneous staggered-grid finite-difference modeling of Rayleigh waves." GEOPHYSICS 71, no. 4 (2006): T109—T115. http://dx.doi.org/10.1190/1.2213051.

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Heterogeneous finite-difference (FD) modeling assumes that the boundary conditions of the elastic wavefield between material discontinuities are implicitly fulfilled by the distribution of the elastic parameters on the numerical grid. It is widely applied to weak elastic contrasts between geologic formations inside the earth. We test the accuracy at the free surface of the earth. The accuracy for modeling Rayleigh waves using the conventional standard staggered-grid (SSG) and the rotated staggered grid (RSG) is investigated. The accuracy tests reveal that one cannot rely on conventional numerical dispersion discretization criteria. A higher sampling is necessary to obtain acceptable accuracy. In the case of planar free surfaces aligned with the grid, 15 to 30 grid points per minimum wavelength of the Rayleigh wave are required. The widely used explicit boundary condition, the so-called image method, produces similar accuracy and requires approximately half the sampling of the wavefield compared to heterogeneous free-surface modeling. For a free-surface not aligned with the grid (surface topography), the error increases significantly and varies with the dip angle of the interface. For an irregular interface, the RSG scheme is more accurate than the SSG scheme. The RSG scheme, however, requires 60 grid points per minimum wavelength to achieve good accuracy for all dip angles. The high computation requirements for 3D simulations on such fine grids limit the application of heterogenous modeling in the presence of complex surface topography.

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10

Pérez Solano, C. A., D. Donno, and H. Chauris. "Finite-difference strategy for elastic wave modelling on curved staggered grids." Computational Geosciences 20, no. 1 (2016): 245–64. http://dx.doi.org/10.1007/s10596-016-9561-8.

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11

Butt, M. M., and Y. Yuan. "A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations." Numerical Mathematics: Theory, Methods and Applications 10, no. 3 (2017): 639–55. http://dx.doi.org/10.4208/nmtma.2017.m1637.

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AbstractA full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, form a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

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12

Rubio, Felix, Mauricio Hanzich, Albert Farrés, Josep de la Puente, and José María Cela. "Finite-difference staggered grids in GPUs for anisotropic elastic wave propagation simulation." Computers & Geosciences 70 (September 2014): 181–89. http://dx.doi.org/10.1016/j.cageo.2014.06.003.

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13

O'Reilly, Ossian, Tomas Lundquist, Eric M. Dunham, and Jan Nordström. "Energy stable and high-order-accurate finite difference methods on staggered grids." Journal of Computational Physics 346 (October 2017): 572–89. http://dx.doi.org/10.1016/j.jcp.2017.06.030.

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14

de la Puente, Josep, Miguel Ferrer, Mauricio Hanzich, José E. Castillo, and José M. Cela. "Mimetic seismic wave modeling including topography on deformed staggered grids." GEOPHYSICS 79, no. 3 (2014): T125—T141. http://dx.doi.org/10.1190/geo2013-0371.1.

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Finite-difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography, due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography. As a consequence, alternatives to finite-difference methods have been proposed to circumvent these issues. We present a new numerical scheme for 3D elastic wave propagation in the presence of strong topography. This finite-difference scheme is based upon a staggered grid of the Lebedev type, or fully staggered grid (FSG). It uses a grid deformation strategy to make a regular Cartesian grid conform to a topographic surface. In addition, the scheme uses a mimetic approach to accurately solve the free-surface condition and hence allows for a less restrictive grid spacing criterion in the computations. The scheme can use high-order operators for the spatial derivatives and obtain low-dispersion results with as few as six points per minimum wavelength. A series of tests in 2D and 3D scenarios, in which our results are compared to analytical and numerical solutions obtained with other numerical approaches, validate the accuracy of our scheme. The resulting FSG mimetic scheme allows for accurate and efficient seismic wave modelling in the presence of very rough topographies with the advantage of using a structured staggered grid.

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15

Mittet, Rune. "Small-scale medium variations with high-order finite-difference and pseudospectral schemes." GEOPHYSICS 86, no. 5 (2021): T387—T399. http://dx.doi.org/10.1190/geo2020-0210.1.

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The accuracy of implementing interfaces with coarse-grid methods such as the pseudospectral method and high-order finite differences has been considered to be low. Our focus is on variations in interface locations and on inclusions that are significantly smaller than the grid step sizes. Classic implementations of these staggered-grid high-order methods are used. Band-limited versions of the Heaviside step function are used to manipulate the material-parameter grids. Interfaces can be implemented with accuracy that is one order of magnitude smaller than the step size. Small medium inclusions, diffractors, can, be three orders of magnitude smaller in area than the typical cell size and still be modeled with good accuracy. If the method used to implement an interface or to implement a small-scale inclusion is viewed as a filter, then this filter must be accurate up to the spatial Nyquist wavenumber of the simulation grid.

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16

van ’t Hof, Bas, and Mathea J. Vuik. "Symmetry-preserving finite-difference discretizations of arbitrary order on structured curvilinear staggered grids." Journal of Computational Science 36 (September 2019): 101008. http://dx.doi.org/10.1016/j.jocs.2019.06.005.

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17

Rui, Hongxing, and Ming Sun. "A locking-free finite difference method on staggered grids for linear elasticity problems." Computers & Mathematics with Applications 76, no. 6 (2018): 1301–20. http://dx.doi.org/10.1016/j.camwa.2018.06.023.

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18

Hošek, Radim, and Bangwei She. "Stability and consistency of a finite difference scheme for compressible viscous isentropic flow in multi-dimension." Journal of Numerical Mathematics 26, no. 3 (2018): 111–40. http://dx.doi.org/10.1515/jnma-2017-0010.

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Abstract Motivated by the work of Karper [29], we propose a numerical scheme to compressible Navier-Stokes system in spatial multi-dimension based on finite differences. The backward Euler method is applied for the time discretization, while a staggered grid, with continuity and momentum equations on different grids, is used in space. The existence of a solution to the implicit nonlinear scheme, strictly positivity of the numerical density, stability and consistency of the method for the whole range of physically relevant adiabatic exponents are proved. The theoretical part is complemented by computational results that are performed in two spatial dimensions.

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19

Vasmel, Marlies, and Johan O. A. Robertsson. "Exact wavefield reconstruction on finite-difference grids with minimal memory requirements." GEOPHYSICS 81, no. 6 (2016): T303—T309. http://dx.doi.org/10.1190/geo2016-0060.1.

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Wavefield injection in finite-difference (FD) grids can be described by the method of multiple point sources. The method teaches how synthetically generated wavefields and wavefield constituents can be reconstructed from surface recordings using a combination of monopole and dipole sources on an injection surface surrounding the model. We show how to properly record surface wavefields and inject point sources in staggered FD grids, in a way that is consistent with the order of spatial accuracy of the FD scheme. The description is general and can be used for schemes of any order. Only one or two surface wavefields are required to reconstruct the original wavefields or wavefield constituents to numerical precision, independent of the order of spatial accuracy of the FD stencil. We have applied the method for the separation of up- and downgoing wavefields and for source wavefield reconstruction for reverse time migration. Our implementation enables accurate source wavefield reconstruction with optimally minimal memory requirements.

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20

Huang, Jianping, Wenyuan Liao, and Zhenchun Li. "A multi-block finite difference method for seismic wave equation in auxiliary coordinate system with irregular fluid–solid interface." Engineering Computations 35, no. 1 (2018): 334–62. http://dx.doi.org/10.1108/ec-12-2016-0438.

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Purpose The purpose of this paper is to develop a new finite difference method for solving the seismic wave propagation in fluid-solid media, which can be described by the acoustic and viscoelastic wave equations for the fluid and solid parts, respectively. Design/methodology/approach In this paper, the authors introduced a coordinate transformation method for seismic wave simulation method. In the new method, the irregular fluid–solid interface is transformed into a horizontal interface. Then, a multi-block coordinate transformation method is proposed to mesh every layer to curved grids and transforms every interface to horizontal interface. Meanwhile, a variable grid size is used in different regions according to the shape and the velocity within each region. Finally, a Lebedev-standard staggered coupled grid scheme for curved grids is applied in the multi-block coordinate transformation method to reduce the computational cost. Findings The instability in the auxiliary coordinate system caused by the standard staggered grid scheme is resolved using a curved grid viscoelastic wave field separation strategy. Several numerical examples are solved using this new method. It has been shown that the new method is stable, efficient and highly accurate in solving the seismic wave equation defined on domain with irregular fluid–solid interface. Originality/value First, the irregular fluid–solid interface is transformed into a horizontal interface by using the coordinate transformation method. The conversion between pressures and stresses is easy to implement and adaptive to different irregular fluid–solid interface models, because the normal stress and shear stress vanish when the normal angle is 90° in the interface. Moreover, in the new method, the strong false artificial boundary reflection and instability caused by ladder-shaped grid discretion are resolved as well.

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21

Barton, I. E., and R. Kirby. "Finite difference scheme for the solution of fluid flow problems on non-staggered grids." International Journal for Numerical Methods in Fluids 33, no. 7 (2000): 939–59. http://dx.doi.org/10.1002/1097-0363(20000815)33:7<939::aid-fld38>3.0.co;2-#.

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22

Fornberg, Bengt. "High-Order Finite Differences and the Pseudospectral Method on Staggered Grids." SIAM Journal on Numerical Analysis 27, no. 4 (1990): 904–18. http://dx.doi.org/10.1137/0727052.

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23

Li, Xiaoli, and Hongxing Rui. "Superconvergence of a fully conservative finite difference method on non-uniform staggered grids for simulating wormhole propagation with the Darcy–Brinkman–Forchheimer framework." Journal of Fluid Mechanics 872 (June 10, 2019): 438–71. http://dx.doi.org/10.1017/jfm.2019.399.

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In this paper, a finite difference scheme on non-uniform staggered grids is proposed for wormhole propagation with the Darcy–Brinkman–Forchheimer framework in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration and auxiliary flux with second-order superconvergence in different discrete norms are established rigorously and carefully on non-uniform grids. We also obtain second-order superconvergence for some terms of the $H^{1}$ norm of the velocity on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the proposed scheme.

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24

Faria, Eduardo L., and Paul L. Stoffa. "Finite‐difference modeling in transversely isotropic media." GEOPHYSICS 59, no. 2 (1994): 282–89. http://dx.doi.org/10.1190/1.1443590.

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We developed a modeling algorithm for transversely isotropic media that uses finite‐difference operators in a staggered grid. Staggered grid schemes are more stable than the conventional finite‐difference methods because the differences are actually based on half the grid spacing. This modeling algorithm uses the full elastic wave equation that makes possible the modeling of all kinds of waves propagating in transversely isotropic media. The spatial derivatives are represented by fourth‐order, finite‐difference operators while the time derivative is represented by a secondorder, finite‐difference operator. The algorithm has no limitation on the acquisition geometry or on the heterogeneity of the media. The program is currently formulated to work in a 2-D transversely isotropic medium but can readily be extended to 3-D. Snapshots can be obtained at any time with no additional computational cost. A four‐layer model is used to show the usefulness of the method. Horizontal and vertical component seismograms are modeled in transversely isotropic media and compared with seismograms modeled in the corresponding isotropic media.

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Ren, Ying-Jun, Jian-Ping Huang, Peng Yong, Meng-Li Liu, Chao Cui, and Ming-Wei Yang. "Optimized staggered-grid finite-difference operators using window functions." Applied Geophysics 15, no. 2 (2018): 253–60. http://dx.doi.org/10.1007/s11770-018-0668-7.

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26

Jiang, Luqian, and Wei Zhang. "TTI equivalent medium parametrization method for the seismic waveform modelling of heterogeneous media with coarse grids." Geophysical Journal International 227, no. 3 (2021): 2016–43. http://dx.doi.org/10.1093/gji/ggab310.

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SUMMARY In recent years, many higher-order and optimized schemes have been developed to reduce the dispersion error with the use of large grid spacing in finite-difference seismic waveform simulations. However, there are two problems in the application of coarse grids for heterogeneous media: the generation of artefact diffraction from the stair-step representation of non-planar interfaces and the inaccuracy of the calculated waveforms due to the interface error. Several equivalent medium parametrization approaches have been proposed to reduce the interface error of the finite-difference method in heterogeneous media. However, these methods are specifically designed for the standard (2,4) staggered-grid scheme and may not be accurate enough for coarse grids when higher-order and optimized schemes are used. In this paper, we develop a tilted transversely isotropic equivalent medium parametrization method to suppress the interface error and the artefact diffraction caused by the staircase approximation under the application of coarse grids. We use four models to demonstrate the effectiveness of the proposed method, and analyse the accuracy of each seismic phase related to the interface. The results show that our method can be used with higher-order and optimized schemes at 3 points per wavelength and produce satisfactory results.

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27

Gaspar, Francisco J., Francisco J. Lisbona, and Petr N. Vabishchevich. "A Numerical Model for the Radial Flow Through Porous and Deformable Shells." Computational Methods in Applied Mathematics 4, no. 1 (2004): 34–47. http://dx.doi.org/10.2478/cmam-2004-0003.

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AbstractEnergy estimates and convergence analysis of finite difference methods for Biot's consolidation model are presented for several types of radial ow. The model is written by a system of partial differential equations which depend on an integer parameter (n = 0; 1; 2) corresponding to the one-dimensional ow through a deformable slab and the radial ow through an elastic cylindrical or spherical shell respectively. The finite difference discretization is performed on staggered grids using separated points for the approximation of pressure and displacements. Numerical results are given to illustrate the obtained theoretical results.

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Gao, Longfei, David C. Del Rey Fernández, Mark Carpenter, and David Keyes. "SBP–SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids." Journal of Computational and Applied Mathematics 348 (March 2019): 421–44. http://dx.doi.org/10.1016/j.cam.2018.08.040.

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HUANG, Chao, and Liang-Guo DONG. "Staggered-Grid High-Order Finite-Difference Method in Elastic Wave Simulation with Variable Grids and Local Time-Steps." Chinese Journal of Geophysics 52, no. 6 (2009): 1324–33. http://dx.doi.org/10.1002/cjg2.1457.

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Levander, Alan R. "Fourth‐order finite‐difference P-SV seismograms." GEOPHYSICS 53, no. 11 (1988): 1425–36. http://dx.doi.org/10.1190/1.1442422.

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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.

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Hou, Junsheng, Robert K. Mallan, and Carlos Torres-Verdín. "Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials." GEOPHYSICS 71, no. 5 (2006): G225—G233. http://dx.doi.org/10.1190/1.2245467.

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This paper describes the implementation and successful validation of a new staggered-grid, finite-difference algorithm for the numerical simulation of frequency-domain electromagnetic borehole measurements. The algorithm is based on a coupled scalar-vector potential formulation for arbitrary 3D inhomogeneous electrically anisotropic media. We approximate the second-order partial differential equations for the coupled scalar-vector potentials with central finite differences on both Yee’s staggered and standard grids. The discretization of the partial differential equations and the enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. The accuracy and efficiency of the algorithm is assessed with examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic, 2D isotropic, and 3D anisotropic rock formations. The simulation examples consider vertical and deviated wells with and without borehole and mud-filtrate invasion regions. Simulation results obtained with the scalar-vector coupled potential formulation favorably compare in accuracy with results obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electrical conductivity. Our numerical exercises indicate that the coupled scalar-vector potential equations provide a general and consistent algorithmic formulation to simulate borehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex models of electrical conductivity.

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Liu, Yang, and Mrinal K. Sen. "An implicit staggered-grid finite-difference method for seismic modelling." Geophysical Journal International 179, no. 1 (2009): 459–74. http://dx.doi.org/10.1111/j.1365-246x.2009.04305.x.

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Yang, Lei, Hongyong Yan, and Hong Liu. "Least squares staggered-grid finite-difference for elastic wave modelling." Exploration Geophysics 45, no. 4 (2014): 255–60. http://dx.doi.org/10.1071/eg13087.

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Robertsson, Johan O. A., Joakim O. Blanch, and William W. Symes. "Viscoelastic finite‐difference modeling." GEOPHYSICS 59, no. 9 (1994): 1444–56. http://dx.doi.org/10.1190/1.1443701.

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Real earth media disperse and attenuate propagating mechanical waves. This anelastic behavior can be described well by a viscoelastic model. We have developed a finite‐difference simulator to model wave propagation in viscoelastic media. The finite‐difference method was chosen in favor of other methods for several reasons. Finite‐difference codes are more portable than, for example, pseudospectral codes. Moreover, finite‐difference schemes provide a convenient environment in which to define complicated boundaries. A staggered scheme of second‐order accuracy in time and fourth‐order accuracy in space appears to be optimally efficient. Because of intrinsic dispersion, no fixed grid points per wavelength rule can be given; instead, we present tables, which enable a choice of grid parameters for a given level of accuracy. Since the scheme models energy absorption, natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary. The viscoelastic scheme is only marginally more expensive than analogous elastic schemes. The efficient implementation of absorbing boundaries may therefore be a good reason for also using the viscoelastic scheme in purely elastic simulations. We illustrate our method and the importance of accurately modeling anelastic media through 2-D and 3-D examples from shallow marine environments.

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Wang, Jian, Xiaohong Meng, Hong Liu, Wanqiu Zheng, and Zhiwei Liu. "GPU Elastic Modeling Using an Optimal Staggered-Grid Finite-Difference Operator." Journal of Theoretical and Computational Acoustics 26, no. 02 (2018): 1850005. http://dx.doi.org/10.1142/s2591728518500056.

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Staggered-grid finite-difference forward modeling in the time domain has been widely used in reverse time migration and full waveform inversion because of its low memory cost and ease to implementation on GPU, however, high dominant frequency of wavelet and big grid interval could result in significant numerical dispersion. To suppress numerical dispersion, in this paper, we first derive a new weighted binomial window function (WBWF) for staggered-grid finite-difference, and two new parameters are included in this new window function. Then we analyze different characteristics of the main and side lobes of the amplitude response under different parameters and accuracy of the numerical solution between the WBWF method and some other optimum methods which denotes our new method can drive a better finite difference operator. Finally, we perform elastic wave numerical forward modeling which denotes that our method is more efficient than other optimum methods without extra computing costs.

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Gao, Jinghuai, and Yijie Zhang. "Staggered-Grid Finite Difference Method with Variable-Order Accuracy for Porous Media." Mathematical Problems in Engineering 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/157071.

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The numerical modeling of wave field in porous media generally requires more computation time than that of acoustic or elastic media. Usually used finite difference methods adopt finite difference operators with fixed-order accuracy to calculate space derivatives for a heterogeneous medium. A finite difference scheme with variable-order accuracy for acoustic wave equation has been proposed to reduce the computation time. In this paper, we develop this scheme for wave equations in porous media based on dispersion relation with high-order staggered-grid finite difference (SFD) method. High-order finite difference operators are adopted for low-velocity regions, and low-order finite difference operators are adopted for high-velocity regions. Dispersion analysis and modeling results demonstrate that the proposed SFD method can decrease computational costs without reducing accuracy.

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Fang, Gang, Sergey Fomel, Qizhen Du, and Jingwei Hu. "Lowrank seismic-wave extrapolation on a staggered grid." GEOPHYSICS 79, no. 3 (2014): T157—T168. http://dx.doi.org/10.1190/geo2013-0290.1.

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We evaluated a new spectral method and a new finite-difference (FD) method for seismic-wave extrapolation in time. Using staggered temporal and spatial grids, we derived a wave-extrapolation operator using a lowrank decomposition for a first-order system of wave equations and designed the corresponding FD scheme. The proposed methods extend previously proposed lowrank and lowrank FD wave extrapolation methods from the cases of constant density to those of variable density. Dispersion analysis demonstrated that the proposed methods have high accuracy for a wide wavenumber range and significantly reduce the numerical dispersion. The method of manufactured solutions coupled with mesh refinement was used to verify each method and to compare numerical errors. Tests on 2D synthetic examples demonstrated that the proposed method is highly accurate and stable. The proposed methods can be used for seismic modeling or reverse-time migration.

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Moczo, P. "3D Fourth-Order Staggered-Grid Finite-Difference Schemes: Stability and Grid Dispersion." Bulletin of the Seismological Society of America 90, no. 3 (2000): 587–603. http://dx.doi.org/10.1785/0119990119.

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Zhang, Qiang, Qi Zhen Du, and Xu Fei Gong. "Finite Difference Modeling of Elastic Wave Propagation." Advanced Materials Research 433-440 (January 2012): 4656–61. http://dx.doi.org/10.4028/www.scientific.net/amr.433-440.4656.

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We present a staggered-grid finite difference scheme for velocity-stress equations to simulate the elastic wave propagating in transversely isotropic media. Instead of the widely used temporally second-order difference scheme, a temporally fourth-order scheme is obtained in this paper. We approximate the third-order spatial derivatives with 2N-order difference rather than second-order or other fixed order difference as before. Thus, it could be possible to make a balanced accuracy of O (Δt4+Δx2N) with arbitrary N. Related issues such as stability criterion, numerical dispersion, source loading and boundary condition are also discussed in this paper. The numerical modeling result indicates that the scheme is reliable.

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Janjić, Zaviša I., and Fedor Mesinger. "Response to small-scale forcing on two staggered grids used in finite-difference models of the atmosphere." Quarterly Journal of the Royal Meteorological Society 115, no. 489 (1989): 1167–76. http://dx.doi.org/10.1002/qj.49711548909.

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Gilles, L., S. C. Hagness, and L. Vázquez. "Comparison between Staggered and Unstaggered Finite-Difference Time-Domain Grids for Few-Cycle Temporal Optical Soliton Propagation." Journal of Computational Physics 161, no. 2 (2000): 379–400. http://dx.doi.org/10.1006/jcph.2000.6460.

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Wang, Jian, Xiao-Hong Meng, Hong Liu, Wan-Qiu Zheng, and Sheng Gui. "Cosine-modulated window function-based staggered-grid finite-difference forward modeling." Applied Geophysics 14, no. 1 (2017): 115–24. http://dx.doi.org/10.1007/s11770-017-0596-y.

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TAN, Handong, Qinfan YU, Booker JOHN, and Wenbo WEI. "Three-Dimensional Magnetotelluric Modeling using the Staggered-Grid Finite Difference Method." Chinese Journal of Geophysics 46, no. 5 (2003): 1011–20. http://dx.doi.org/10.1002/cjg2.420.

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Zhang, Yijie, and Jinghuai Gao. "A 3D staggered-grid finite difference scheme for poroelastic wave equation." Journal of Applied Geophysics 109 (October 2014): 281–91. http://dx.doi.org/10.1016/j.jappgeo.2014.08.007.

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de Groot-Hedlin, Catherine. "Finite-difference modeling of magnetotelluric fields: Error estimates for uniform and nonuniform grids." GEOPHYSICS 71, no. 3 (2006): G97—G106. http://dx.doi.org/10.1190/1.2195991.

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In the finite-difference (FD) method, one solves a set of discrete approximations to continuous differential equations; thus, the solutions only approximate the true values. For the magnetotelluric (MT) method, errors in the electric and magnetic fields computed by the staggered FD method are precisely quantifiable for a model with uniform conductivity. In this case, the errors in the electric and magnetic fields are equal in magnitude but increase with rising node separation. In this paper, I show that errors in MT responses, which rely on ratios of the field values, depend strongly on the method used to interpolate electric field values to the surface where the magnetic field is sampled. Analytic expressions for the FD estimates of the MT responses for a half-space are derived and compared for three different methods of electric field interpolation. The best results are achieved when the electric field values just above and below the surface are interpolated exponentially. For a half-space, the FD estimates of the MT responses are independent of node separation and are precisely equal to the analytic values when the electric field is interpolated exponentially. For models with sharp conductivity contrasts, the errors in the responses derived using this interpolation method increase with rising node spacing but still perform better than other examined interpolation methods. Varying the vertical node separation within a half-space model degrades the solution accuracy. The magnitude of the error depends primarily on the magnitude of the change in vertical node spacing. Lateral variations in the grid spacing do not necessarily yield errors in the FD solutions to the MT equations.

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Vishnevsky, Dmitry, Vadim Lisitsa, Vladimir Tcheverda, and Galina Reshetova. "Numerical study of the interface errors of finite-difference simulations of seismic waves." GEOPHYSICS 79, no. 4 (2014): T219—T232. http://dx.doi.org/10.1190/geo2013-0299.1.

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Numerical simulations of wave propagation produce different errors and the most well known is numerical dispersion, which is only valid for homogeneous media. However, there is a lack of error studies for heterogeneous media or even for the canonical case of media that have two constant velocity layers. The error associated with media that have two layers is called an interface error, and it typically converges to zero with a lower order of convergence compared to the theoretical convergence rate of the finite-difference schemes (FDS) for homogeneous media. We evaluated a detailed numerical study of the interface error for three staggered-grid FDS that are commonly used in the simulation of seismic-wave propagation. We determined that a standard staggered-grid scheme (SSGS) (also known as the Virieux scheme), a rotated staggered-grid scheme (RSGS), and a Lebedev scheme (LS) preserve the second order of convergence at horizontal/vertical solid-solid interfaces when the medium parameters have been properly modified, such as by harmonic averaging of finely layered media for the stiffness tensor and arithmetic mean for the density. However, for a fluid-solid interface aligned with the grid line, a second-order convergence can only be achieved by an SSGS. In addition, the presence of a fluid-solid interface reduces the order of convergence for the LS and the RSGS to a first order of convergence. The presence of inclined interfaces makes high-order (second and more) convergence impossible.

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Shragge, Jeffrey, and Benjamin Tapley. "Solving the tensorial 3D acoustic wave equation: A mimetic finite-difference time-domain approach." GEOPHYSICS 82, no. 4 (2017): T183—T196. http://dx.doi.org/10.1190/geo2016-0691.1.

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Generating accurate numerical solutions of the acoustic wave equation (AWE) is a key computational kernel for many seismic imaging and inversion problems. Although finite-difference time-domain (FDTD) approaches for generating full-wavefield solutions are well-developed for Cartesian computational domains, several challenges remain when applying FDTD approaches to scenarios arguably best described by more generalized geometry. In particular, how best to generate accurate and stable FDTD solutions for scenarios involving grids conforming to complex topography or internal surfaces. We address these issues by developing a mimetic FDTD (MFDTD) approach that combines four key components: a tensorial 3D AWE, mimetic finite-difference (MFD) operators, fully staggered grids (FSGs), and MFD Robin boundary conditions (RBC). The tensorial formulation of the 3D AWE permits wave propagation to be specified on (semi-) analytically defined coordinate meshes designed to conform to complex domain boundaries. MFD operators allow for higher order FD stencils to be applied throughout the model domain, including the boundary region where implementing centered FD stencils can be problematic. The FSG approach combines wavefield information propagated on four complementary subgrids to ensure the existence of all wavefield gradients required for computing the tensorial Laplacian operator, and thereby avoids interpolation approximations. The RBCs are implemented with a flux-preserving mimetic boundary operator that forestalls introduction of nonphysical energy into the grid by enforcing underlying flux-conservation laws. After validating the 3D MFDTD scheme on a sheared Cartesian mesh, we generate 3D wavefield simulation examples for internal boundary (IB) and topographic coordinate systems. The numerical examples demonstrate that the MFDTD scheme is capable of providing accurate and low-dispersion impulse responses for scenarios involving distorted IB meshes conforming to water-bottom surfaces and topographic coordinate systems exhibiting 2.5 km of topographic relief and including steep (65°) slope angles.

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Cao, Jian, and Jing-Bo Chen. "A parameter-modified method for implementing surface topography in elastic-wave finite-difference modeling." GEOPHYSICS 83, no. 6 (2018): T313—T332. http://dx.doi.org/10.1190/geo2018-0098.1.

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Accurate seismic modeling with a realistic topography plays an essential role in onshore seismic migration and inversion. The finite-difference (FD) method is one of the most popular numerical tools for seismic modeling. But implementing the free surface on topography using the FD method is nontrivial. We have developed a stable and efficient parameter-modified (PM) method for modeling elastic-wave propagation in the presence of complex topography. This method is based on a standard staggered-grid scheme, and the stress-free condition is implemented on the rugged surface by modifying the redefined medium parameters at the discrete topography boundary points. This numerical treatment for topography needs to be performed only once before the wave simulation. In this way, we avoid the tedious handling of wavefield variables in every time step, and this boundary treatment can be integrated easily into existing staggered-grid FD modeling codes. A series of numerical tests in two dimensions and three dimensions indicate that with a spatial sampling of 15 grid points per minimum wavelength, our method is good enough to eliminate staircase diffractions and produces more accurate results than those obtained by some other staggered-grid-based numerical approaches. Numerical experiments on some more complex models also demonstrate the feasibility of our method in handling topography with strong variation and Poisson’s ratio discontinuity. In addition, this PM method can be used in a discontinuous-grid scheme in which only the regions near the irregular topography need to be oversampled, which is very important for improving its efficiency in real applications.

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Babaee, Hessam, and Sumanta Acharya. "A Hybrid Staggered/Semistaggered Finite-Difference Algorithm for Solving Time-Dependent Incompressible Navier-Stokes Equations on Curvilinear Grids." Numerical Heat Transfer, Part B: Fundamentals 65, no. 1 (2013): 1–26. http://dx.doi.org/10.1080/10407790.2013.827012.

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Gao, Longfei, David Ketcheson, and David Keyes. "On long-time instabilities in staggered finite difference simulations of the seismic acoustic wave equations on discontinuous grids." Geophysical Journal International 212, no. 2 (2017): 1098–110. http://dx.doi.org/10.1093/gji/ggx470.

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