Finite‐difference modeling of wave propagation in a fluid–solid configuration

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 618-624 ◽  
Author(s):  
Robbert van Vossen ◽  
Johan O. A. Robertsson ◽  
Chris H. Chapman
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Fourth Order ◽  
Spatial Sampling ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Imaging Method ◽  
Complex Structures ◽  
Scholte Waves ◽  
Scholte Wave

Finite‐difference (FD) techniques are widely used to model wave propagation through complex structures. Two main sources of error can be identified: (1) from numerical dispersion and numerical anisotropy and (2) by modeling the response of internal grid boundaries. Conventional discretization criteria to reduce the effects of numerical dispersion and numerical anisotropy have long been established (5‐8 gridpoints per wavelength for a fourth‐order accurate FD scheme). We analyze the second source of errors, comparing different staggered‐grid FD solutions to the Cagniard‐de Hoop solution in models with fluid–solid contacts. Our results confirm that it is sufficient to rely on conventional discretization criteria if the fluid–solid interface is aligned with the grid. If accurate modeling of the Scholte wave is required, then a new imaging method we propose should be used to allow for conventional sampling of the wavefield to minimize numerical dispersion. However, for an interface not aligned with the grid (irregular interfaces), a spatial sampling of at least 15 gridpoints per minimum wavelength is required to obtain acceptable results, particularly in seismic seabed applications where Scholte waves may need to be modeled more accurately.

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.

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.

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.

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