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.

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.

A stable auxiliary differential equation perfectly matched layer condition combined with low-dispersive symplectic methods for solving second-order elastic wave equations

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T193-T206 ◽  
Author(s):  
Xiao Ma ◽  
Yangjia Li ◽  
Jiaxing Song
Keyword(s):  
Differential Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Elastic Model ◽  
Complex Frequency ◽  
Symplectic Method ◽  
Symplectic Methods ◽  
Compression Parameter

The stable implementation of the perfectly matched layer (PML), one of the most effective and popular artificial boundary conditions, has attracted much attention these years. As a type of low-dispersive and symplectic method for solving seismic wave equations, the nearly-analytic symplectic partitioned Runge-Kutta (NSPRK) method has been combined with split-field PML (SPML) and convolutional complex-frequency shifted PML (C-CFS-PML) previously to model acoustic and short-time elastic wave modelings, not yet successfully applied to long-time elastic wave propagation. In order to broaden the application of NSPRK and more general symplectic methods for second-order seismic models, we formulate an auxiliary differential equation (ADE)-CFS-PML with a stabilizing grid compression parameter. This includes deriving the ADE-CFS-PML equations and formulating an adequate time integrator to properly embed their numerical discretizations in the main symplectic numerical methods. The resulting (N)SPRK+ADE-CFS-PML algorithm can help break through the constraint of at most second-order temporal accuracy that used to be imposed on SPML and C-CFS-PML. Especially for NSPRK, we implement the strategy of neglecting the treatment of third-order spatial derivatives in the PML domain and obtain an efficient absorption effect. Related acoustic and elastic wave simulations illustrate the enhanced numerical accuracy of our ADE-CFS-PML compared with SPML and C-CFS-PML. The elastic wave simulation in a homogeneous isotropic medium shows that compared to NSPRK+C-CFS-PML, the NSPRK+ADE-CFS-PML is numerically stable throughout a simulation time of 2 s. The synthetic seismograms of the 2D acoustic SEG salt model and the two-layer elastic model demonstrate the effectiveness of NSPRK+ADE-CFS-PML for complex elastic models. The stabilization effect of the grid compression parameter is verified in the final homogeneous isotropic elastic model with free-surface boundary.

scholarly journals A Well-Posed and Discretely Stable Perfectly Matched Layer for Elastic Wave Equations in Second Order Formulation

2012 ◽  
Vol 11 (5) ◽  
pp. 1643-1672 ◽  
Author(s):  
Kenneth Duru ◽  
Gunilla Kreiss
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Order System ◽  
Stable Node ◽  
Complex Frequency ◽  
Elastic Wave Equations ◽  
Anisotropic Elastic ◽  
Well Posed

AbstractWe present a well-posed and discretely stable perfectly matched layer for the anisotropic (and isotropic) elastic wave equations without first re-writing the governing equations as a first order system. The new model is derived by the complex coordinate stretching technique. Using standard perturbation methods we show that complex frequency shift together with a chosen real scaling factor ensures the decay of eigen-modes for all relevant frequencies. To buttress the stability properties and the robustness of the proposed model, numerical experiments are presented for anisotropic elastic wave equations. The model is approximated with a stable node-centered finite difference scheme that is second order accurate both in time and space.

Analyses of Waves and the Wave Resistance due to Transom-Stern Ships

1969 ◽  
Vol 13 (02) ◽  
pp. 155-167
Author(s):  
Bohyun Yim
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Boundary ◽  
Exact Formulation ◽  
Wave Problem ◽  
Ship Wave ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

Waves and the wave resistance due to a ship with a transom stern are analyzed starting with an exact formulation of the general ship wave problem. Green's theorem is utilized together with the well-known Green's function which satisfies the linear free-surface boundary condition on the free surface. Here it is required to take at least the second-order terms in the formal development. It is found that the immersed transom stern acts to cancel stern waves. Using a limited form of Michell's wave-resistance formula, the wave resistance of the transom-stern ship is analyzed

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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