Comparative study among implementations of several free-surface boundaries with perfectly matched layer conditions
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摘要: 传统的完全匹配层技术是一种能够较为有效地消除边界反射的边界条件，但是当表层为泊松比较高的自由表面时，该技术可能会产生不稳定的现象.针对传统的完全匹配层技术固有的不稳定和掠射情况下吸收效果不佳等缺陷，发展了多轴完全匹配层、卷积完全匹配层以及将两者结合的多轴卷积完全匹配层等3种边界条件.本文介绍了水平自由表面的不同处理方法以及传统、多轴、卷积和多轴卷积等4种完全匹配层条件的原理，通过二维半无限空间模型的交错网格有限差分正演模拟对比，分析了几种自由边界实施方法在这几种完全匹配层条件下的稳定性，并通过提取单道波形与解析解进行对比，定性分析了水平自由表面几种不同处理方法的准确性以及各自的适用条件. 结果表明，泊松比和水平自由表面实施方法对波场模拟效果及其稳定性有重要影响.Abstract: Classical perfectly matched layer (PML) absorbing boundary is certified an efficient method to suppress spurious edge reflections in seismic modeling. However, when modeling Rayleigh waves with the existence of free surface, the classical PML algorithm may become unstable in the case that Poisson’s ratio of the medium is high. On the basis of defects of classical PML, such as its inherent instability and the effects of grazing incidence, multiaxial perfectly matched layer, convolutional perfectly matched layer and multiaxial convolutional perfectly matched layer are developed. Six implementations of planar free-surface boundary conditions and principles of four perfectly matched layer methods are introduced, and forward modeling of 2D half-infinite model using staggered grid finite-difference method are given to study the stability of different methods. Through the comparison of waveform between numerical solution and analytical solution, we can illustrate the accuracy and applicability of these methods in this paper. The results show that Poisson’s ratio and implementations of planar free-surface have an important influence on the effects and stability of wavefield simulation.
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Keywords:
- PML /
- planar free-surface /
- Poisson’s ratio /
- stability /
- accuracy
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图 1 二维交错网格速度-应力空间分布图(引自Levander，1988)
Figure 1. Spatial stencils of two-dimensional staggered grid for the velocity and stress update Dot and open circle represent horizontal and verical components of velocity，respectively; solid square and hollow square separately represent normal stress and tangential stress
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图 2 传统PML条件下采用应力镜像-紧致差分法t=115，132，139和149ms时刻水平分量(a)与垂直分量(b)的波场快照(泊松比为0.25时)
Figure 2. Wavefield smapshots of horizontal(a)vertical components(b)computed by stress image and compact difference methed when the Poisson's ratio is 0.25 with the classical PML at time instant t=115，132，139 and 149 ms
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图 3 地震记录 (a)直接法；(b)应力镜像-2阶格式法；(c)应力镜像-降阶法；(d)应力镜像-2阶速度展开法；(e)应力镜像-虚拟层赋零法；(f)应力镜像-紧致差分法
Figure 3. Seismic records (a)Direct method；(b)Stress image and second-order scheme method；(c)Stress image and reduced-order method；(d)Stress image and second-order velocity evolving method；(e)Stress image and fictitious layer assigns to zero method；(f)Stress image and compact difference method
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图 4 传统PML(a)、M-PML(b)、C-PML(c)和MC-PML(d)条件下采用应力镜像-紧致差分法得到的t=139 ms时刻的波场快照(泊松比为0.48). 图中画圈处为不稳定机理
Figure 4. Wavefield snapshots computed by stress image and compact difference method when the Poisson’s ratio is 0.48 with PML(a)，M-PML(b)，C-PML(c) and MC-PML(d)at time instant t=139 ms. The red circle represents the instability
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图 5 传统PML边界条件下泊松比为0.48时t=115，132，139和149 ms时刻的波场快照 (a)应力镜像-紧致差分法；(b)应力镜像-降阶法；(c)应力镜像-2阶速度展开法
Figure 5. Wavefield snapshots when the Poisson’s ratio is 0.48 with the classical PML at time instant t=115，132，139 and 149 ms (a)Stress image and compact difference method；(b)Stress image and reduced-order method；(c)Stress image and second-order velocity evolving method
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图 6 单道波形(虚线)与解析解(实线)的对比 (a)直接法；(b)应力镜像-2阶格式法；(c)应力镜像-降阶法；(d)应力镜像-虚拟层赋零法；(e)应力镜像-2阶速度展开法；(f)应力镜像-紧致差分法
Figure 6. Comparison of waveforms between numerical(dashed line) and analytical solutions(solid line) (a)Direct method；(b)Stress image and second-order scheme method；(c)Stress image and reduced-order method；(d)Stress image and fictitious layer assigns to zero method；(e)Stress image and second-order velocity evolving method；(f)Stress image and compact difference method
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图 7 单道波形(虚线)与解析解(实线)对比 (a)直接法；(b)应力镜像-虚拟层赋零法；(c)应力镜像-2阶格式法
Figure 7. Comparison of waveforms between numerical(dashed line) and analytical solutions(solid line) (a)Direct method;(b)Stress image and fictitious layer assigns to zero method;(c)Stress image and second-order scheme method
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图 8 泊松比为0.25时6种自由表面处理方法在传统PML条件下的单道波形对比 ① 直达P波; ② 面波; ③ 反射P波; ④ 反射S波
Figure 8. Comparison of waveforms with six free-surface methods under the classical PML when the Possion’s ratio is 0.25 ① Direct P-wave; ② Surface wave; ③ Reflected P-wave; ④ Reflected S-wave
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图 9 泊松比为0.48时3种稳定自由表面处理方法在传统PML条件下的单道波形对比 ① 直达P波; ② 面波; ③ 反射P波; ④ 反射S波
Figure 9. Comparison of waveforms of three stable free-surface methods under the classical PML when the Possion’s ratio is 0.48 ① Direct P-wave; ② Surface wave; ③ Reflected P-wave; ④ Reflected S-wave
表 1 传统PML在不同水平自由表面边界条件下的对比总结
Table 1 Comparison among implementations of planar free-surface boundary conditions with the classical PML
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