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.

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.

scholarly journals On wave matrices, and some properties of the wave equation

1936 ◽  
Vol 157 (890) ◽  
pp. 1-27 ◽  
Keyword(s):  
Wave Equation ◽  
Wave Theory ◽  
Order Equation ◽  
Second Order ◽  
Order Operator ◽  
First Order ◽  
Second Order Wave Equation

1—Wave matrices became important in wave theory as the result of the use of them made by Dirac to express the operator of the second order wave equation as the square of a linear one, and hence obtain a first order equation. Thus, p 2 representing the second order operator, the equation p 2 Ψ = 0, may be factorized, and written (∑ E α p α ) (∑ E α p α ) Ψ = 0, (α = 1, 2, . . . , n ), giving the first order equation ∑ E α p α Ψ = 0, (1) if the p α commute with themselves and with the E α , and if the E α are matrix roots of +1 or of —1, which satisfy E α E β = — E β E a (β ≠ α). (2)

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