Immersive boundary conditions: Theory, implementation, and examples

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. T97-T110 ◽  
Author(s):  
Filippo Broggini ◽  
Marlies Vasmel ◽  
Johan O. A. Robertsson ◽  
Dirk-Jan van Manen
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Seismic Wave Propagation ◽  
Local Domain ◽  
Time Step ◽  
Local Modeling ◽  
Numerical Precision ◽  
Computational Geophysics ◽  
Wavefield Extrapolation ◽  
Full Domain

Many applications in computational geophysics involve the modeling of seismic wave propagation on a set of closely related subsurface models. In such scenarios, it is of interest to recompute the seismic wavefields locally (only in the regions of change), instead of in the full subsurface model. We have developed a method for local acoustic wavefield recomputation that makes it possible to fully immerse a local modeling domain within a larger domain of arbitrary extent and complexity, such that the wave propagation in the full domain is completely accounted for. The method enables wavefield modeling on much smaller local domains, while relying on the up-front generation of a large number of Green’s functions and a wavefield extrapolation step at each time step of the simulation. A Kirchhoff-Helmholtz extrapolation integral is used to predict the interaction of the wavefield leaving the local domain with the exterior domain. The outward propagating wavefield and the wavefield reentering the local domain are applied as a boundary condition along the edges. Thanks to these dynamically calculated boundary conditions, all higher order long-range interactions between the two domains are fully accounted for. We have implemented the method in a conventional finite-difference time-domain scheme and determined that the locally calculated wavefields are equal to wavefields generated on the full domain to within numerical precision. The efficiency of the local modeling algorithm will greatly depend on the nature and size of the problem.

2.5-D modeling of seismic wave propagation: Boundary condition, stability criterion, and efficiency

Geophysics ◽  
1998 ◽  
Vol 63 (6) ◽  
pp. 2082-2090 ◽  
Author(s):  
Shunhua Cao ◽  
Stewart Greenhalgh
Keyword(s):  
Boundary Condition ◽  
Wave Propagation ◽  
Boundary Conditions ◽  
Traveling Waves ◽  
Time Domain ◽  
Stability Criterion ◽  
Spatial Location ◽  
Seismic Wave Propagation ◽  
Sampling Scheme ◽  
Maximum Value

The modeling of 3-D wave propagation in media having only 2-D variation in the elastic properties—so‐called 2.5-D modeling—is achieved using the wavenumber transform, in which multiple 2-D problems are solved, each one associated with a different strike‐direction wavenumber [Formula: see text] We derived a 2.5-D transmitting boundary condition in the frequency domain, which has no simple representation in the time domain. It yields significantly improved results over existing boundary conditions. For time‐domain methods, attenuating boundary conditions must be applied. The 2.5-D stability criterion changes from the 2-D to the 3-D criterion as the wavenumber increases from zero to the maximum value for traveling waves, respectively. In the frequency‐wavenumber ([Formula: see text]) domain at a given spatial location, the wavefield for a fixed frequency (ω) oscillates at progressively higher rates as wavenumber ([Formula: see text]) increases from zero to the maximum value for traveling waves. A nonuniform sampling scheme in wavenumber space, to exploit the oscillatory nature of the wavefield, yields significant efficiency improvement over the normal uniform sampling scheme.

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.

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.

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