Numerical study of the interface errors of finite-difference simulations of seismic waves

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. T219-T232 ◽  
Author(s):  
Dmitry Vishnevsky ◽  
Vadim Lisitsa ◽  
Vladimir Tcheverda ◽  
Galina Reshetova
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Order Of Convergence ◽  
Numerical Study ◽  
Second Order ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Solid Interface ◽  
Grid Scheme ◽  
Homogeneous Media

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.

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.

3D finite-difference modeling of elastic wave propagation in the Laplace-Fourier domain

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T137-T155 ◽  
Author(s):  
Petr V. Petrov ◽  
Gregory A. Newman
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Fourth Order ◽  
Order Approximation ◽  
Staggered Grid ◽  
Order System ◽  
Finite Difference Schemes ◽  
Fourier Domain ◽  
Krylov Methods ◽  
Complex Objects

With the recent interest in the Laplace-Fourier domain full waveform inversion, we have developed new heterogeneous 3D fourth- and second-order staggered-grid finite-difference schemes for modeling seismic wave propagation in the Laplace-Fourier domain. Our approach is based on the integro-interpolation technique for the velocity-stress formulation in the Cartesian coordinate system. Five averaging elastic coefficients and three averaging densities are necessary to describe the heterogeneous medium, with harmonic averaging of the bulk and shear moduli, and arithmetic averaging of density. In the fourth-order approximation, we improved the accuracy of the scheme using a combination of integral identities for two elementary volumes — “small” and “large” around spatial gridpoints where the wave variables are defined. Two solution approaches are provided, both of which are solved with iterative Krylov methods. In the first approach, the stress variables are eliminated and a linear system for the velocity components is solved. In the second approach, we worked directly with the first-order system of velocity and stress variables. This reduced the computer memory required to store the complex matrix, along with reducing (by 30%) the number of arithmetic operations needed for the solution compared to the fourth-order scheme for velocity only. Numerical examples show that our finite-difference formulations for elastic wavefield simulations can achieve more accurate solutions with fewer grid points than those from previously published second and fourth-order frequency-domain schemes. We applied our simulator to the investigation of wavefields from the SEG/EAGE model in the Laplace-Fourier domain. The calculation is sensitive to the heterogeneity of the medium and capable of describing the structures of complex objects. Our technique can also be extended to 3D elastic modeling within the time domain.

Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM147-SM153 ◽  
Author(s):  
Yixian Xu ◽  
Jianghai Xia ◽  
Richard D. Miller
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Rayleigh Waves ◽  
Second Order ◽  
Body Waves ◽  
Staggered Grid ◽  
Difference Operators ◽  
Image Method ◽  
Elastic Boundary ◽  
Second Order In Time

The need for incorporating the traction-free condition at the air-earth boundary for finite-difference modeling of seismic wave propagation has been discussed widely. A new implementation has been developed for simulating elastic wave propagation in which the free-surface condition is replaced by an explicit acoustic-elastic boundary. Detailed comparisons of seismograms with different implementations for the air-earth boundary were undertaken using the (2,2) (the finite-difference operators are second order in time and space) and the (2,6) (second order in time and sixth order in space) standard staggered-grid (SSG) schemes. Methods used in these comparisons to define the air-earth boundary included the stress image method (SIM), the heterogeneous approach, the scheme of modifying material properties based on transversely isotropic medium approach, the acoustic-elastic boundary approach, and an analytical approach. The method proposed achieves the same or higher accuracy of modeled body waves relative to the SIM. Rayleigh waves calculated using the explicit acoustic-elastic boundary approach differ slightly from those calculated using the SIM. Numerical results indicate that when using the (2,2) SSG scheme for SIM and our new method, a spatial step of 16 points per minimum wavelength is sufficient to achieve 90% accuracy; 32 points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. When using the (2,6) SSG scheme for the two methods, a spatial step of eight points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. Our proposed method is physically reasonable and, based on dispersive analysis of simulated seismographs from a layered half-space model, is highly accurate. As a bonus, our proposed method is easy to program and slightly faster than the SIM.

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