SH-wave propagation in heterogeneous media: Velocity‐stress finite‐difference method

Geophysics ◽  
1984 ◽  
Vol 49 (11) ◽  
pp. 1933-1942 ◽  
Author(s):  
Jean Virieux
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Layered Medium ◽  
Heterogeneous Media ◽  
Heterogeneous Medium ◽  
Head Wave ◽  
Sh Wave ◽  
Quarter Plane ◽  
Discrete Grid ◽  
Lateral Propagation

A new finite‐difference (FD) method is presented for modeling SH-wave propagation in a generally heterogeneous medium. This method uses both velocity and stress in a discrete grid. Density and shear modulus are similarly discretized, avoiding any spatial smoothing. Therefore, boundaries will be correctly modeled under an implicit formulation. Standard problems (quarter‐plane propagation, sedimentary basin propagation) are studied to compare this method with other methods. Finally a more complex example (a salt dome inside a two‐layered medium) shows the effect of lateral propagation on seismograms recorded at the surface. A corner wave, always in‐phase with the incident wave, and a head wave will appear, which will pose severe problems of interpretation with the usual vertical migration methods.

Interface conditions for acoustic and elastic wave propagation

Geophysics ◽  
1991 ◽  
Vol 56 (2) ◽  
pp. 168-181 ◽  
Author(s):  
J. S. Sochacki ◽  
J. H. George ◽  
R. E. Ewing ◽  
S. B. Smithson
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Heterogeneous Media ◽  
Elastic Wave Propagation ◽  
Integration Scheme ◽  
Acoustic Wave Equation ◽  
Sh Wave ◽  
Numerical Schemes ◽  
Tangential Stresses

The divergence theorem is used to handle the physics required at interfaces for acoustic and elastic wave propagation in heterogeneous media. The physics required at regular and irregular interfaces is incorporated into numerical schemes by integrating across the interface. The technique, which can be used with many numerical schemes, is applied to finite differences. A derivation of the acoustic wave equation, which is readily handled by this integration scheme, is outlined. Since this form of the equation is equivalent to the scalar SH wave equation, the scheme can be applied to this equation also. Each component of the elastic P‐SV equation is presented in divergence form to apply this integration scheme, naturally incorporating the continuity of the normal and tangential stresses required at regular and irregular interfaces.

Seismic waves in a quarter plane

1969 ◽  
Vol 59 (1) ◽  
pp. 347-368
Author(s):  
Z. S. Alterman ◽  
A. Rotenberg
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Finite Difference ◽  
Elastic Wave ◽  
Surface Waves ◽  
Seismic Waves ◽  
Initial Pulse ◽  
Surface Conditions ◽  
Quarter Plane ◽  
Boundary Lines

Abstract The equations for elastic wave propagation are solved by a finite difference scheme for the case of an elastic quarter plane. A point-source emitting a compressional pulse is located along the diagonal inside the quarter plane. Free-surface conditions are assumed on the boundary lines, so that the problem is nonseparable. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The effect of different finite difference formulations for the boundary conditions and the effect of different mesh sizes are studied. Various reflected volume and surface waves are identified, corner-generated surface waves are clearly seen in the seismograms and their particle motion is studied. The amplitude of the pulse observed at the corner is three times the amplitude of the initial pulse.

scholarly journals Fundamental Problems of the Electrodynamics of Heterogeneous Media

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
N. N. Grinchik ◽  
Yu. N. Grinchik
Keyword(s):  
Mathematical Model ◽  
Electric Double Layer ◽  
Layered Medium ◽  
Heterogeneous Media ◽  
Heterogeneous Medium ◽  
Surface Current ◽  
Charge Carriers ◽  
Thermal Fields ◽  
Dirichlet Theorem ◽  
Generalized Wave Equation

The consistent physic-mathematical model of propagation of an electromagnetic wave in a heterogeneous medium is constructed using the generalized wave equation and the Dirichlet theorem. Twelve conditions at the interfaces of adjacent media are obtained and justified without using a surface charge and surface current in explicit form. The conditions are fulfilled automatically in each section of counting schemes for calculations. A consistent physicomathematical model of interaction of nonstationaly electric and thermal fields in a layered medium with allowance or mass transfer is constructed. The model is based on the methods of thermodynamics and on the equations of an electromagnetic field and is formulated without explicit separation of the charge carriers and the charge of an electric double layer.

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.

Sign in / Sign up

Export Citation Format

Share Document