弥散黏滞性波动方程的吸收边界算法
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摘要
针对地震波数值模拟中因计算区域有界而产生的人工边界反射问题,提出了一种弥散黏滞性波动方程的吸收边界算法.在所研究的有界计算区域周围加入合适的吸收层,使得吸收层内的地震波随传播距离指数衰减而达到吸收边界反射的目的.利用傅里叶变换求得无界空间该波动方程的解;引入衰减函数构建有界空间相应的辅助方程,合理地选择衰减函数使得计算区域内辅助方程的解为原方程的解,而边界吸收层内的解呈指数衰减;运用有限差分法在均匀介质和均匀层状介质中求解弥散黏滞性波动方程,采用该方法对边界进行处理并与未加边界处理的波场快照和地震记录进行对比.实验结果表明,所提方法处理后的波场快照几乎无边界反射波存在,在空间位置为x=147m、z=747m的地震记录中,0.5s处边界反射波的振幅趋于0.所提方法也适用于声波方程和Stokes方程.
A new absorbing boundary method for diffusive-viscous wave equation(DVW) is proposed to deal with the artificial reflections from boundaries caused by a truncated computational domain in seismic wave numerical modeling.The main idea is to make the seismic waves exponentially attenuate with propagation distance by adding a proper absorbing layer around the boundary of computational domain.The solution of the wave equation in infinite homogenous space is obtained by using Fourier transform.Then,auxiliary equations are constructed by introducing decay functions such that the solution of auxiliary equations is the solution of the DVW equation in computational domain and decays exponentially at boundaries.The wave equation is finally solved by using finite-difference method in both homogeneous media and homogeneous layered media.Numerical results are given through comparison of the results from the proposed method and those of non-absorbing boundary condition,and show that boundary reflections are absorbed with the proposed method.The proposed method is also applicable to acoustic equation and Stokes equation.
A new absorbing boundary method for diffusive-viscous wave equation(DVW) is proposed to deal with the artificial reflections from boundaries caused by a truncated computational domain in seismic wave numerical modeling.The main idea is to make the seismic waves exponentially attenuate with propagation distance by adding a proper absorbing layer around the boundary of computational domain.The solution of the wave equation in infinite homogenous space is obtained by using Fourier transform.Then,auxiliary equations are constructed by introducing decay functions such that the solution of auxiliary equations is the solution of the DVW equation in computational domain and decays exponentially at boundaries.The wave equation is finally solved by using finite-difference method in both homogeneous media and homogeneous layered media.Numerical results are given through comparison of the results from the proposed method and those of non-absorbing boundary condition,and show that boundary reflections are absorbed with the proposed method.The proposed method is also applicable to acoustic equation and Stokes equation.
引文
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[8]KORNEEV V A.Slow waves in fractures filled with viscous fluid[J].Geophysics,2008,73(1):1-7.
[9]HE Zhenhua,XIONG Xiaojun,BIAN Lien.Numeri-cal simulation of seismic low-frequency shadows and its applications[J].Applied Geophysics,2008,5(4):301-306.
[10]边立恩,贺振华,黄德济.储层的地震低频响应及识别[J].石油物探,2008,47(6):573-576.BIAN Lien,HE Zhenhua,HUANG Deji.Seismic low-frequency response and identification of reservoir[J].Geophysics Prospecting for Petroleum,2008,47(6):573-576.
[11]王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34.WANG Shoudong.A perfect matched layer absorbing boundary condition for acoustic wave equation[J].Ge-ophysics Prospecting for Petroleum,2003,38(1):31-34.
[12]瑞克N H.黏弹性介质中的地震波[M].许云,译.北京:地质出版社,1981.
[2]ENGQUIST B,MAJDA A.Absorbing boundary con-ditions for the numerical simulation of waves[J].Ap-plied Mathematical Sciences,1977,74(5):1765-1766.
[3]CERJAN C,KOSLOFF D,KOSLOFF R,et al.A nonreflecting boundary condition for discrete acoustic and elastic wave equations[J].Geophysics,1985,50(4):705-708.
[4]BERENGER J P.A perfectly matched layer for the absorption of electromagnetic waves[J].Journal of Computational Physics,1994,114(2):185-200.
[5]HESTHAYEN J S.On the analysis and construction of perfectly matched layers for the linearized Euler equations[J].Journal of computational Physics,1998,142(1):129-147.
[6]KOMATITSCH D,TROMP J.A perfectly matched layer absorbing boundary condition for the second-or-der seismic wave equation[J].Geophysical Journal In-ternational,2003,154(1):146-153.
[7]KORNEEV V A,GOLOSHUBIN G M,DALEY TM,et al.Seismic low-frequency effects in monitoring fluid-saturated reservoir[J].Geophysics,2004,69(2):522-532.
[8]KORNEEV V A.Slow waves in fractures filled with viscous fluid[J].Geophysics,2008,73(1):1-7.
[9]HE Zhenhua,XIONG Xiaojun,BIAN Lien.Numeri-cal simulation of seismic low-frequency shadows and its applications[J].Applied Geophysics,2008,5(4):301-306.
[10]边立恩,贺振华,黄德济.储层的地震低频响应及识别[J].石油物探,2008,47(6):573-576.BIAN Lien,HE Zhenhua,HUANG Deji.Seismic low-frequency response and identification of reservoir[J].Geophysics Prospecting for Petroleum,2008,47(6):573-576.
[11]王守东.声波方程完全匹配层吸收边界[J].石油地球物理勘探,2003,38(1):31-34.WANG Shoudong.A perfect matched layer absorbing boundary condition for acoustic wave equation[J].Ge-ophysics Prospecting for Petroleum,2003,38(1):31-34.
[12]瑞克N H.黏弹性介质中的地震波[M].许云,译.北京:地质出版社,1981.
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