3-D elastic modeling with surface topography by a Chebychev spectral method

Geophysics ◽  
1994 ◽  
Vol 59 (3) ◽  
pp. 464-473 ◽  
Author(s):  
E. Tessmer ◽  
D. Kosloff
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Surface Topography ◽  
Spectral Method ◽  
Seismic Wave ◽  
Numerical Scheme ◽  
Rough Surfaces ◽  
Fourier Method ◽  
Seismic Wave Propagation ◽  
Rectangular Grid

The 3-D numerical Chebychev modeling scheme accounts for surface topography. The method is based on spectral derivative operators. Spatial differencing in horizontal directions is performed by the Fourier method, whereas vertical derivatives are carried out by a Chebychev method that allows for the incorporation of boundary conditions into the numerical scheme. The method is based on the velocity‐stress formulation. The implementation of surface topography is done by mapping a rectangular grid onto a curved grid. Boundary conditions are applied by means of characteristic variables. The study of surface effects of seismic wave propagation in the presence of surface topography is important, since nonray effects such as diffractions and scattering at rough surfaces must be considered. Several examples show this. The 3-D modeling alogrithm can serve as a tool for understanding these phenomena since it computes the full wavefield.

A spectral method for seismic wave propagation in elastic media

Wave Motion ◽  
1990 ◽  
Vol 12 (5) ◽  
pp. 415-427 ◽  
Author(s):  
P. Berg ◽  
F. If ◽  
O. Skovgaard
Keyword(s):  
Wave Propagation ◽  
Spectral Method ◽  
Seismic Wave ◽  
Seismic Wave Propagation ◽  
Elastic Media

An efficient fully spectral method for constant-Q seismic-wave propagation

2018 ◽  
Author(s):  
Khemraj Shukla ◽  
José M. Carcione ◽  
Reynam C. Pestana ◽  
Juan Santos ◽  
Priyank Jaiswal
Keyword(s):  
Wave Propagation ◽  
Spectral Method ◽  
Seismic Wave ◽  
Seismic Wave Propagation

Finite‐difference modeling of SH‐wave propagation in nonwelded contact media

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1656-1663 ◽  
Author(s):  
Raphael A. Slawinski ◽  
Edward S. Krebes
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Seismic Wave ◽  
Seismic Wave Propagation ◽  
Displacement Discontinuity ◽  
Contact Boundary ◽  
Discrete Equation ◽  
Sh Wave ◽  
Grid Points

Many geological structures of interest are known to exhibit fracturing. Fracturing directly affects seismic wave propagation because, depending on its scale, fracturing may give rise to scattering and/or anisotropy. A fracture may be described mathematically as an interface in nonwelded contact (i.e., as a displacement discontinuity). This poses a difficulty for finite‐difference modeling of seismic wave propagation in fractured media, because the standard heterogeneous approach assumes welded contact. In the past, this difficulty has been circumvented by incorporating nonwelded contact into the medium parameters using equivalent medium theory. We present an alternate method based on the homogeneous approach to finite differencing, whereby nonwelded contact boundary conditions are imposed explicitly. For simplicity, we develop the method in the SH‐wave case. In the homogeneous approach, nonwelded contact boundary conditions are discretized by introducing auxiliary, so‐called fictitious, grid points. Wavefield values at fictitious grid points are then used in the discrete equation of motion, so that the time‐evolved wavefield satisfies the correct boundary conditions. Although not as general as the heterogeneous approach, the homogeneous approach has the advantage of being relatively simple and manifestly satisfying nonwelded contact boundary conditions. For fractures aligned with the numerical grid, the homogeneous and heterogeneous approaches yield identical results. In particular, in both approaches nonwelded contact results in a larger maximum stable time step size than in the welded contact case.

An improved immersed boundary finite-difference method for seismic wave propagation modeling with arbitrary surface topography

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. T311-T322 ◽  
Author(s):  
Wenyi Hu
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Surface Topography ◽  
Seismic Wave ◽  
Difference Method ◽  
Forward Modeling ◽  
Seismic Wave Propagation ◽  
Immersed Boundary

To accurately simulate seismic wave propagation for the purpose of developing modern land data processing tools, especially full-waveform inversion (FWI), we have developed an efficient high-order finite-difference forward-modeling algorithm with the capability of handling arbitrarily shaped free-surface topography. Unlike most existing forward-modeling algorithms using curvilinear grids to fit irregular surface topography, this finite-difference algorithm, based on an improved immersed boundary method, uses a regular Cartesian grid system without suffering from staircasing error, which is inevitable in a conventional finite-difference method. In this improved immersed boundary finite-difference (IBFD) algorithm, arbitrarily curved surface topography is accounted for by imposing the free-surface boundary conditions at exact boundary locations instead of using body-conforming grids or refined grids near the boundaries, thus greatly reducing the complexity of its preprocessing procedures and the computational cost. Furthermore, local continuity, large curvatures, and subgrid curvatures are represented precisely through the employment of the so-called dual-coordinate system — a local cylindrical and a global Cartesian coordinate. To properly describe the wave behaviors near complex free-surface boundaries (e.g., overhanging structures and thin plates, or other fine geometry features), the wavefields in a ghost zone required for the boundary condition enforcement are reconstructed accurately by introducing a special recursive interpolation technique into the algorithm, which substantially simplifies the boundary treatment procedures and further improves the numerical performance of the algorithm, as demonstrated by the numerical experiments. Numerical examples revealed the performance of the IBFD method in comparison with a conventional finite-difference method.

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