Application of perfectly matched layer for scalar arbitrarily wide-angle wave equations

Geophysics ◽  
2013 ◽  
Vol 78 (1) ◽  
pp. T29-T39 ◽  
Author(s):  
Hanming Chen ◽  
Hui Zhou ◽  
He Lin ◽  
Shangxu Wang
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Evanescent Waves ◽  
Absorbing Boundary Conditions ◽  
Wide Angle ◽  
Absorbing Boundary ◽  
Computation Cost ◽  
Propagating Waves ◽  
Derivation Procedure

Arbitrarily wide-angle wave equation (AWWE) is a space domain, high-order one-way wave equation (OWWE). Its accuracy can be arbitrarily increased, and it is amenable to easy numerical implementation. Those properties make it outstanding among the existing OWWEs and further enable it to be a desirable tool for migration. We extend the perfectly matched layer (PML) to 3D scalar AWWE to provide a good approach to suppress artifacts arising at truncation boundaries. We follow the concept of complex coordinate stretching, and the derivation procedure of PML for AWWE is straightforward. An existing finite-difference scheme is adopted to fit the split PML formulation and its stability is observed through numerical examples. The performance of the developed PML condition is compared with two different wave-equation based absorbing boundary conditions. Numerical results illustrate that the PML condition used in AWWE propagator can effectively absorb the propagating waves and evanescent waves at a price of limited additional computation cost.

Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T301-T311 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xueyuan Huang ◽  
Yanjie Zhou
Keyword(s):  
Boundary Condition ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
Complex Frequency ◽  
Absorbing Boundary ◽  
Time Layer ◽  
The Time Domain

The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.

Stability of wide‐angle absorbing boundary conditions for the wave equation

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1153-1163 ◽  
Author(s):  
R. A. Renaut ◽  
J. Petersen
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Least Squares ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Wide Angle ◽  
Courant Number ◽  
Absorbing Boundary ◽  
Wide Range ◽  
Artificial Boundaries

Numerical solution of the two‐dimensional wave equation requires mapping from a physical domain without boundaries to a computational domain with artificial boundaries. For realistic solutions, the artificial boundaries should cause waves to pass directly through and thus mimic total absorption of energy. An artificial boundary which propagates waves in one direction only is derived from approximations to the one‐way wave equation and is commonly called an absorbing boundary. Here we investigate order 2 absorbing boundary conditions which include the standard paraxial approximation. Absorption properties are compared analytically and numerically. Our numerical results confirm that the [Formula: see text] or Chebychev‐Padé approximations are best for wide‐angle absorption and that the Chebychev or least‐squares approximations are best for uniform absorption over a wide range of incident angles. Our results also demonstrate, however, that the boundary conditions are stable for varying ranges of Courant number (ratio of time step to grid size). We prove that there is a stability barrier on the Courant number specified by the coefficients of the boundary conditions. Thus, proving stability of the interior scheme is not sufficient. Furthermore, waves may radiate spontaneously from the boundary, causing instability, even if the stability bound on the Courant number is satisfied. Consequently, the Chebychev and least‐squares conditions may be preferred for wide‐angle absorption also.

scholarly journals ABSORBING BOUNDARY CONDITIONS FOR A WAVE EQUATION WITH A TEMPERATURE-DEPENDENT SPEED OF SOUND

2013 ◽  
Vol 21 (02) ◽  
pp. 1250028 ◽  
Author(s):  
IGOR SHEVCHENKO ◽  
MANFRED KALTENBACHER ◽  
BARBARA WOHLMUTH
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Speed Of Sound ◽  
Wave Equations ◽  
Focused Ultrasound ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
High Intensity Focused Ultrasound ◽  
Temperature Dependent ◽  
Absorbing Boundary

In this work, new absorbing boundary conditions (ABCs) for a wave equation with a temperature-dependent speed of sound are proposed. Based on the theory of pseudo-differential calculus, first- and second-order ABCs for the one- and two-dimensional wave equations are derived. Both boundary conditions are local in space and time. The well-posedness of the wave equation with the developed ABCs is shown through the reduction of the original problem to an equivalent one for which the uniqueness and existence of the solution has already been established. Although the second-order ABC is more accurate, the numerical realization is more challenging. Here we use a Lagrange multiplier approach which fits into the abstract framework of saddle point formulations and yields stable results. Numerical examples illustrating stability, accuracy and flexibility of the ABCs are given. As a test setting, we perform computations for a high-intensity focused ultrasound (HIFU) application, which is a typical thermo-acoustic multi-physics problem.

Highly accurate absorbing boundary conditions for wide-angle wave equations

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. S85-S97 ◽  
Author(s):  
A. Homayoun Heidari ◽  
Murthy N. Guddati
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Absorbing Boundary Conditions ◽  
Wide Angle ◽  
Absorbing Boundary ◽  
New Class ◽  
Continuation Problem ◽  
Explicit Finite Difference ◽  
Migration Modeling ◽  
The Waves

We develop a new class of absorbing boundary conditions (ABCs) to prevent unwanted artifacts and wraparounds associated with aperture truncation in migration/modeling using high-order, one-way wave equations. The fundamental approach behind the proposed development is the efficient discretization of the half-space, beyond the boundary of interest, using midpoint-integrated imaginary finite elements, an idea recently utilized in the development of effective one-way wave equations. The proposed absorbing boundary conditions essentially add absorbing layers at the aperture truncation points. We derive the absorbing boundary conditions, analyze their properties, and develop a stable explicit finite-difference scheme to solve the downward-continuation problem modified by these boundary conditions. With the help of numerical examples, we conclude that with as few as three absorbing layers, i.e., two additional gridpoints, the waves can be absorbed completely, thus preventing associated artifacts.

A Perfectly Matched Layer Technique Applied to Lattice Spring Model in Seismic Wavefield Forward Modeling for Poisson’s Solids

2021 ◽  
Author(s):  
Jinxuan Tang ◽  
Hui Zhou ◽  
Chuntao Jiang ◽  
Muming Xia ◽  
Hanming Chen ◽  
...  
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Forward Modeling ◽  
Seismic Wave Propagation ◽  
Perfectly Matched Layer ◽  
Spring Model ◽  
Absorbing Boundary ◽  
Lattice Spring Model ◽  
Elastic Wave Equations ◽  
Inner Region

ABSTRACT As a complementary way to traditional wave-equation-based forward modeling methods, lattice spring model (LSM) is introduced into seismology for wavefield modeling owing to its remarkable stability, high-calculation accuracy, and flexibility in choosing simulation meshes, and so forth. The LSM simulates seismic-wave propagation from a micromechanics perspective, thus enjoying comprehensive characterization of elastic dynamics in complex media. Incorporating an absorbing boundary condition (ABC) is necessary for wavefield modeling to avoid the artificial reflections caused by truncated boundaries. To the best of our knowledge, the perfectly matched layer (PML) method has been a routine ABC in the wave-equation-based numerical modeling of wave physics. However, it has not been used in the nonwave-equation-based LSM simulations. In this work, we want to apply PML to LSM to attenuate the boundary reflections. We divide the whole simulation region into PML region and inner region, PML region surrounds the inner region. To incorporate PML to LSM, we establish elastic-wave equations corresponding to LSM. The simulation in the PML region is conducted using the established wave equations and the simulation in the inner region is conducted using LSM. Three simulation examples show that the PML scheme is effective and outperforms Gaussian ABC.

Application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide-angle wave equation

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T313-T321 ◽  
Author(s):  
Hanming Chen ◽  
Hui Zhou ◽  
Yanqi Li
Keyword(s):  
Wave Equation ◽  
Computational Cost ◽  
Grazing Incidence ◽  
Equation System ◽  
Perfectly Matched Layer ◽  
Staggered Grid ◽  
Complex Frequency ◽  
Wide Angle ◽  
First Order ◽  
General Complex

A classical split perfectly matched layer (PML) method has recently been applied to the scalar arbitrarily wide-angle wave equation (AWWE) in terms of displacement. However, the classical split PML obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence. Our goal was to improve the computational efficiency of AWWE and to enhance the suppression of edge reflections by applying a convolutional PML (CPML). We reformulated the original AWWE as a first-order formulation and incorporated the CPML with a general complex frequency shifted stretching operator into the renewed formulation. A staggered-grid finite-difference (FD) method was adopted to discretize the first-order equation system. For wavefield depth continuation, the first-order AWWE with the CPML saved memory compared with the original second-order AWWE with the conventional split PML. With the help of numerical examples, we verified the correctness of the staggered-grid FD method and concluded that the CPML can efficiently absorb evanescent and propagating waves.

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