A nonreflecting boundary condition for discrete acoustic and elastic wave equations

Geophysics ◽  
1985 ◽  
Vol 50 (4) ◽  
pp. 705-708 ◽  
Author(s):  
Charles Cerjan ◽  
Dan Kosloff ◽  
Ronnie Kosloff ◽  
Moshe Reshef
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Wave Equations ◽  
Fourier Method ◽  
Boundary Region ◽  
Solution Methods ◽  
Elastic Wave Equations ◽  
Discrete Methods ◽  
Grid Points ◽  
Nonreflecting Boundary Condition

One of the nagging problems which arises in application of discrete solution methods for wave‐propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh. These undesired events eventually override the actual seismic signals which propagate in the modeled region. The solution to avoiding boundary effects used to be to enlarge the numerical mesh, thus delaying the side reflections and wraparound longer than the range of times involved in the modeling. Obviously this solution considerably increases the expense of computation. More recently, nonreflecting boundary conditions were introduced for the finite‐difference method (Clayton and Enquist, 1977; Reynolds, 1978). These boundary conditions are based on replacing the wave equation in the boundary region by one‐way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh. This approach has been relatively successful, except that its effectiveness degrades for events which impinge on the boundaries at shallow angles. It is also not clear how to apply this type of boundary condition to global discrete methods such as the Fourier method for which all grid points are coupled.

scholarly journals Absorbing boundary conditions for acoustic and elastic wave equations

1977 ◽  
Vol 67 (6) ◽  
pp. 1529-1540 ◽  
Author(s):  
Robert Clayton ◽  
Björn Engquist
Keyword(s):  
Boundary Conditions ◽  
Elastic Wave ◽  
Wave Equations ◽  
Absorbing Boundary Conditions ◽  
Limited Area ◽  
Absorbing Boundary ◽  
Physical Behavior ◽  
Wide Range ◽  
Elastic Wave Equations ◽  
Numerical Wave

abstract Boundary conditions are derived for numerical wave simulation that minimize artificial reflections from the edges of the domain of computation. In this way acoustic and elastic wave propagation in a limited area can be efficiently used to describe physical behavior in an unbounded domain. The boundary conditions are based on paraxial approximations of the scalar and elastic wave equations. They are computationally inexpensive and simple to apply, and they reduce reflections over a wide range of incident angles.

A transparent boundary technique for numerical modeling of elastic waves

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 963-966 ◽  
Author(s):  
Jianlin Zhu
Keyword(s):  
Numerical Modeling ◽  
Boundary Conditions ◽  
Elastic Waves ◽  
Wave Equations ◽  
Normal Incidence ◽  
Absorbing Boundary Conditions ◽  
S Wave ◽  
Absorbing Boundary ◽  
S Waves ◽  
Elastic Wave Equations

In numerical modeling of wave motions, strong reflections from artificial model boundaries may contaminate or mask true reflections from the interior model interfaces. Hence, developing a kind of exterior model boundary transparent to the outgoing waves is of critical importance. Among proposed solutions, e.g., Smith (1974), Kausel and Tassoulas (1981), and Higdon (1991), the most widely used may be the Clayton and Engquist (1977) method of absorbing boundary conditions, based on paraxial approximations for acoustic and elastic‐wave equations. However, absorbing boundary conditions make the reflection coefficients zero only for normal incidence, and suppression of reflected S-waves (Clayton and Engquist, 1977) becomes poorer as the ratio of P- to S-wave velocity ([Formula: see text]) becomes larger.

The velocity-stress finite-difference method with a rotated staggered grid applied to seismic wave propagation in a fractured medium

Geophysics ◽  
2020 ◽  
Vol 85 (2) ◽  
pp. T89-T100
Author(s):  
Kang Wang ◽  
Suping Peng ◽  
Yongxu Lu ◽  
Xiaoqin Cui
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Wave Equations ◽  
Staggered Grid ◽  
Contact Boundary ◽  
Fractured Medium ◽  
Elastic Wave Equations ◽  
Fracture Interface

To enable a mathematical description, geologic fractures are considered as infinitely thin planes embedded in a homogeneous medium. These fracture structures satisfy linear slip boundary conditions, namely, a discontinuous displacement and continuous stress. The general finite-difference (FD) method described by the elastic wave equations has challenges when attempting to simulate the propagation of waves at the fracture interface. The FD method expressed by velocity-stress variables with the explicit application of boundary conditions at the fracture interface facilitates the simulation of wave propagation in fractured discontinuous media that are described by elastic wave equations and linear slip interface conditions. We have developed a new FD scheme for horizontal and vertical fracture media. In this scheme, a fictitious grid is introduced to describe the discontinuous velocity at the fracture interface and a rotated staggered grid is used to accurately indicate the location of the fracture. The new FD scheme satisfies nonwelded contact boundary conditions, unlike traditional approaches. Numerical simulations in different fracture media indicate that our scheme is accurate. The results demonstrate that the reflection coefficient of the fractured interface varies with the incident angle, wavelet frequency, and normal and tangential fracture compliances. Our scheme and conclusions from this study will be useful in assessing the properties of fractures, enabling the proper delineation of fractured reservoirs.

scholarly journals A note on resonant frequencies for a system of elastic wave equations

2004 ◽  
Vol 2004 (64) ◽  
pp. 3485-3498
Author(s):  
Luis A. Cortés-Vega
Keyword(s):  
Boundary Condition ◽  
Elastic Wave ◽  
Simple Proof ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Scattering Problem ◽  
Resonant Frequencies ◽  
Direct Scattering ◽  
Stationary Approach ◽  
Elastic Wave Equations

We present a rather simple proof of the existence of resonant frequencies for the direct scattering problem associated to a system of elastic wave equations with Dirichlet boundary condition. Our approach follows techniques similar to those in Cortés-Vega (2003). The proposed technique relies on a stationary approach of resonant frequencies, that is, the poles of the analytic continuation of the solution.

scholarly journals Multiblock SBP-SAT Methodology of Symmetric Matrix Form of Elastic Wave Equations on Curvilinear Grids

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Cheng Sun ◽  
Zai-Lin Yang ◽  
Guan-Xi-Xi Jiang ◽  
Yong Yang
Keyword(s):  
Numerical Simulation ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Wave Equations ◽  
Symmetric Matrix ◽  
Matrix Form ◽  
Curvilinear Grids ◽  
Elastic Wave Equations ◽  
Application Potential ◽  
Interface Boundary Conditions

A stable and accurate finite-difference discretization of first-order elastic wave equations is derived in this work. To simplify the origin and proof of the formulas, a symmetric matrix form (SMF) for elastic wave equations is presented. The curve domain is discretized using summation-by-parts (SBP) operators, and the boundary conditions are weakly enforced using the simultaneous-approximation-term (SAT) technique, which gave rise to a provably stable high-order SBP-SAT method via the energy method. In addition, SMF can be extended to wave equations of different types (SH wave and P-SV wave) and dimensions, which can simplify the boundary derivation process and improve its applicability. Application of this approximation can divide the domain into a multiblock context for calculation, and the interface boundary conditions of blocks can also be used to simulate cracks and other structures. Several numerical simulation examples, including actual elevation within the area of Lushan, China, are presented, which verifies the viability of the framework present in this paper. The applicability of simulating elastic wave propagation and the application potential in the seismic numerical simulation of this method are also revealed.

Sign in / Sign up

Export Citation Format

Share Document