Unsplit complex frequency shifted perfectly matched layer for second-order wave equation using auxiliary differential equations

2015 ◽  
Vol 138 (6) ◽  
pp. EL551-EL557 ◽  
Author(s):  
Yingjie Gao ◽  
Jinhai Zhang ◽  
Zhenxing Yao
Keyword(s):  
Wave Equation ◽  
Differential Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Complex Frequency ◽  
Second Order Wave Equation

scholarly journals Second-order scalar wave field modeling with a first-order perfectly matched layer

Solid Earth ◽  
2018 ◽  
Vol 9 (6) ◽  
pp. 1277-1298
Author(s):  
Xiaoyu Zhang ◽  
Dong Zhang ◽  
Qiong Chen ◽  
Yan Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Forward Modeling ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Staggered Grid ◽  
Scalar Wave ◽  
First Order ◽  
Second Order Wave Equation ◽  
Boundary Absorption

Abstract. The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary-matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary-matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

scholarly journals Second-order Scalar Wave Field Modeling with First-order Perfectly Matched Layer

2018 ◽  
Author(s):  
Xiaoyu Zhang ◽  
Dong Zhang ◽  
Qiong Chen ◽  
Yan Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Forward Modeling ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Staggered Grid ◽  
Scalar Wave ◽  
First Order ◽  
Second Order Wave Equation ◽  
Boundary Absorption

Abstract. The forward modeling of a scalar wave equation plays an important role in the numerical geophysical computations. The finite-difference algorithm in the form of a second-order wave equation is one of the commonly used forward numerical algorithms. This algorithm is simple and is easy to implement based on the conventional-grid. In order to ensure the accuracy of the calculation, absorption layers should be introduced around the computational area to suppress the wave reflection caused by the artificial boundary. For boundary absorption conditions, a perfectly matched layer is one of the most effective algorithms. However, the traditional perfectly matched layer algorithm is calculated using a staggered-grid based on the first-order wave equation, which is difficult to directly integrate into a conventional-grid finite-difference algorithm based on the second-order wave equation. Although a perfectly matched layer algorithm based on the second-order equation can be derived, the formula is rather complex and intermediate variables need to be introduced, which makes it hard to implement. In this paper, we present a simple and efficient algorithm to match the variables at the boundaries between the computational area and the absorbing boundary area. This new boundary matched method can integrate the traditional staggered-grid perfectly matched layer algorithm and the conventional-grid finite-difference algorithm without formula transformations, and it can ensure the accuracy of finite-difference forward modeling in the computational area. In order to verify the validity of our method, we used several models to carry out numerical simulation experiments. The comparison between the simulation results of our new boundary matched algorithm and other boundary absorption algorithms shows that our proposed method suppresses the reflection of the artificial boundaries better and has a higher computational efficiency.

Nonsplit complex-frequency-shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 2: Wavefield simulations

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T167-T179 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xijun He ◽  
Xueyuan Huang ◽  
Jiaxing Song
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Free Surface ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Surface Boundary ◽  
Complex Frequency ◽  
Symplectic Methods

The perfectly matched layer (PML) is an efficient artificial boundary condition that has been routinely implemented in seismic wave modeling. However, the effective combination of PML with symplectic numerical schemes for solving seismic wave equations has rarely been studied. In a companion paper, we have developed a complex-frequency-shifted convolutional PML (CPML) with a nonconstant compression grid parameter for solving the time-domain second-order seismic wave equation. Subsequently, we combine this CPML with two classes of symplectic methods to formulate symplectic partitioned Runge-Kutta (SPRK) + CPML and nearly analytic SPRK (NSPRK) + CPML, both of which are properly synchronized. To further investigate their validity, the two algorithms are then applied to acoustic and elastic wave simulations in typical geologic models, including a heterogeneous acoustic model, several isotropic and orthotropic elastic models, and an isotropic elastic model with a free-surface boundary. Relevant numerical results demonstrate the effectiveness of our CPML and combination algorithms. Specifically, the numerical accuracy and stability of the CPML that we develop are greatly improved compared with the classic split-field PML. Moreover, the final model with the free-surface boundary condition indicates that the nonconstant grid-compression parameter can eliminate the unstable modes at the free surface in the PML domain. The (N)SPRK + CPML that we propose is prospective for future application in other complex models and wave-equation-based migration and inversion.

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.

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.

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