A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids

Geophysics ◽  
2011 ◽  
Vol 76 (3) ◽  
pp. WA43-WA50 ◽  
Author(s):  
Henrik Bernth ◽  
Chris Chapman
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Isotropic Case ◽  
Anisotropic Elastic ◽  
Memory Accesses ◽  
Average Dispersion

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.

A simplified Lax‐Wendroff correction for staggered‐grid FDTD modeling of electromagnetic wave propagation in frequency‐dependent media

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1369-1377 ◽  
Author(s):  
Tim Bergmann ◽  
Joakim O. Blanch ◽  
Johan O. A. Robertsson ◽  
Klaus Holliger
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Dispersive Media ◽  
Frequency Dependent ◽  
Elegant Method ◽  
Constitutive Parameters

The Lax‐Wendroff correction is an elegant method for increasing the accuracy and computational efficiency of finite‐difference time‐domain (FDTD) solutions of hyperbolic problems. However, the conventional approach leads to implicit solutions for staggered‐grid FDTD approximations of Maxwell’s equations with frequency‐dependent constitutive parameters. To overcome this problem, we propose an approximation that only retains the purely acoustic, i.e., lossless, terms of the Lax‐Wendroff correction. This modified Lax‐Wendroff correction is applied to an O(2, 4) accurate staggered‐grid FDTD approximation of Maxwell’s equations in the radar frequency range (≈10 MHz–10 GHz). The resulting pseudo-O(4, 4) scheme is explicit and computationally efficient and exhibits all the major numerical characteristics of an O(4, 4) accurate FDTD scheme, even for strongly attenuating and dispersive media. The numerical properties of our approach are constrained by classical numerical dispersion and von Neumann‐Routh stability analyses, verified by comparisons with pertinent 1-D analytical solutions and illustrated through 2-D simulations in a variety of surficial materials. Compared to the O(2, 4) scheme, the pseudo-O(4, 4) scheme requires 64% fewer grid points in two dimensions and 78% in three dimensions to achieve the same level of numerical accuracy, which results in large savings in core memory.

A parameter-modified method for implementing surface topography in elastic-wave finite-difference modeling

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T313-T332 ◽  
Author(s):  
Jian Cao ◽  
Jing-Bo Chen
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Staggered Grid ◽  
Seismic Modeling ◽  
Time Step ◽  
Modified Method ◽  
Grid Scheme ◽  
Grid Points

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.

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.

Anisotropic finite-difference algorithm for modeling elastic wave propagation in fractured coalbeds

Geophysics ◽  
2012 ◽  
Vol 77 (1) ◽  
pp. C13-C26 ◽  
Author(s):  
Zhenglin Pei ◽  
Li-Yun Fu ◽  
Weijia Sun ◽  
Tao Jiang ◽  
Binzhong Zhou
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Taylor Series Expansion ◽  
Real Data ◽  
Elastic Wave Propagation ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
S Wave ◽  
Data Set

The simulation of wave propagations in coalbeds is challenged by two major issues: (1) strong anisotropy resulting from high-density cracks/fractures in coalbeds and (2) numerical dispersion resulting from high-frequency content (the dominant frequency can be higher than 100 Hz). We present a staggered-grid high-order finite-difference (FD) method with arbitrary even-order ([Formula: see text]) accuracy to overcome the two difficulties stated above. First, we derive the formulae based on the standard Taylor series expansion but given in a neat and explicit form. We also provide an alternative way to calculate the FD coefficients. The detailed implementations are shown and the stability condition for anisotropic FD modeling is examined by the eigenvalue analysis method. Then, we apply the staggered-grid FD method to 2D and 3D coalbed models with dry and water-saturated fractures to study the characteristics of the 2D/3C elastic wave propagation in anisotropic media. Several factors, like density and direction of vertical cracks, are investigated. Several phenomena, like S-wave splitting and waveguides, are observed and are consistent with those observed in a real data set. Numerical results show that our formulae can correlate the amplitude and traveltime anisotropies with the coal seam fractures.

Refined finite‐difference simulations using local integral forms: Application to telluric fields in two dimensions

Geophysics ◽  
1982 ◽  
Vol 47 (5) ◽  
pp. 825-831 ◽  
Author(s):  
John F. Hermance
Keyword(s):  
Finite Difference ◽  
Differential Forms ◽  
Integral Form ◽  
Closed Surface ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Difference Operators ◽  
Anomalous Field ◽  
Integral Forms ◽  
Point Operator

This paper describes a new finite‐difference form for simulating the behavior of telluric fields near electrical inhomogeneities. The technique involves a local integration of the electric current density crossing a closed surface surrounding a mesh node. To illustrate the concept, a two‐dimensional (2-D) model is considered, but it is readily possible to generalize to three dimensions. The resulting expressions, which are accurate to second degree everywhere, have the form of nine‐point finite‐difference operators, but they have a higher precision than those derived from the usual differential forms which result in five‐point operators. In particular, the new form accounts for cross‐derivative [Formula: see text] effects in the region about each node. Including this term can provide significant improvements in accuracy near sharp, localized discontinuities, where the anomalous field decays rapidly (as 1/r or [Formula: see text]) with distance. An analytical solution is compared to finite‐difference calculations using both the conventional five‐point differential form and the new nine‐point integral form developed here. The results suggest that, in some cases, one might expect at least a factor of three improvement when using the nine‐point operator instead of the five‐point operator. This is particularly true in the vicinity of localized structures where the curvilinear character of the distorted field is most pronounced and one would expect the cross‐derivative term to be large.

Controlling Numerical Dispersion by Variably Timed Flux Updating in One Dimension

1982 ◽  
Vol 22 (03) ◽  
pp. 399-408 ◽  
Author(s):  
R.G. Larson
Keyword(s):  
Finite Difference ◽  
Truncation Error ◽  
Hyperbolic Equations ◽  
Numerical Technique ◽  
Material Balance ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Point Tracking ◽  
Multidimensional Problems ◽  
Grid Orientation

Abstract The one-dimensional (1D) material balance equations for multiphase multicomponent transport in porous media can be cast into forms, analogous to characteristic equations, that express explicitly the velocities at which fixed values of concentration are propagated. Use of these concentration-velocity equations to control the frequency with which component fluxes from finite-difference gridblocks ate updated leads to greatly reduced numerical dispersion, as demonstrated in miscible flooding, waterflooding, surfactant flooding, and other example problems. Introduction Accurate numerical simulation of enhanced oil-recovery processes, such as CO2, surfactant, thermal, or caustic flooding can involve calculations of phase behavior, interfacial tension, relative permeabilities, viscosities, heat and mass transfer, and even chemical reactions, thereby requiring considerable computational effort for each meshpoint or gridblock at each timestep. It is therefore impractical to resolve the steep concentration or thermal gradients often present in these processes by resorting to ultrafine meshes. Because the mathematical description of such processes is often unavoidably complex, it is important that the numerical technique be simple and ruggedly insensitive to the details of the process description, if one is to avoid becoming ensnarled in cumbersome and tedious programming and debugging.Although the finite-difference method's simplicity is its great advantage, its accuracy is seriously deficient, at least when one is using the simplest and most obvious discretizations. Central-difference discretization leads to artificial oscillations and overshoot, and upstream differencing leads to artificial smearing of sharp fronts-i.e., numerical dispersion or truncation error. Upstream difference solutions in two or three dimensions often show a significant dependence on grid orientation. Suggested improvements in the finite-difference technique, such as "transfer of overshoot," "truncation error analysis," or "two-point upstream weighting," still have significant numerical dispersion, grid orientation or oscillation errors.The method of characteristics, or point tracking, incurs no numerical dispersion or overshoot errors, but for general multicomponent, multidimensional problems, computer programs based on these techniques can become labyrinthine in their complexity.The finite-element, or variational, methods hold the potential of significantly reducing overshoot and/or numerical dispersion below that produced by finite difference, but implementation is considerably more complicated and time-consuming.The method of random choice, a technique developed for solving sets of multidimensional hyperbolic equations that appear in gas dynamics, recently has been employed in reservoir simulation. This method is somewhat akin to point tracking, propagating discontinuous fronts without smearing or overshoot errors.A new numerical technique is presented here that has the form and simplicity of finite difference, but utilizes variably timed flux updating (VTU) to gain a considerable improvement in accuracy. The technique is potentially applicable to general multicomponent, multidimensional problems. In this and a companion paper (see Pages 409-419), however, the technique is restricted to problems governed by the following equations. SPEJ P. 399^

Accurate and efficient seismic modeling in random media

Geophysics ◽  
1993 ◽  
Vol 58 (4) ◽  
pp. 576-588 ◽  
Author(s):  
Guido Kneib ◽  
Claudia Kerner
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Random Media ◽  
Polynomial Interpolation ◽  
Random Medium ◽  
High Order ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Seismic Modeling

The optimum method for seismic modeling in random media must (1) be highly accurate to be sensitive to subtle effects of wave propagation, (2) allow coarse sampling to model media that are large compared to the scale lengths and wave propagation distances which are long compared to the wavelengths. This is necessary to obtain statistically meaningful overall attributes of wavefields. High order staggered grid finite‐difference algorithms and the pseudospectral method combine high accuracy in time and space with coarse sampling. Investigations for random media reveal that both methods lead to nearly identical wavefields. The small differences can be attributed mainly to differences in the numerical dispersion. This result is important because it shows that errors of the numerical differentiation which are caused by poor polynomial interpolation near discontinuities do not accumulate but cancel in a random medium where discontinuities are numerous. The differentiator can be longer than the medium scale length. High order staggered grid finite‐difference schemes are more efficient than pseudospectral methods in two‐dimensional (2-D) elastic random media.

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.

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