成层半空间出平面自由波场的界面子波算法
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摘要
一般工程的地震反应分析中地震波假设为竖直向上入射的体波,场地自由波场可简化为成层半空间模型的一维波动问题,通过数值方法获得出平面自由场。在数值计算方法中以显式有限差分法为代表的时间域方法是当前地震波模拟的主流方法,其精度通常受到计算方法、模型尺寸、网格大小、边界设定等诸多条件的影响,数值算法稳定性还受到许多计算参数的限制。为了改进时域数值方法中存在的这些缺点和不足,本文结合分层均匀介质中的理论地震图合成方法提出了一种新的时域数值算法。其基本思想是将计算节点直接设置在波阻抗界面处,各个层内不再设置计算节点,然后根据地震波在波阻抗界面处的反射、透射关系及其在相邻界面之间的传播时间建立计算界面节点振动的数学表达式,据此计算得到各层界面节点的振动,并通过对自由面节点进行适当处理得到成层半空间出平面自由场。界面节点振动可看作为入射波通过波阻抗界面处产生的次一级子波源,本文称之为界面子波,因此把这一计算方法称为界面子波算法。数值算例表明该方法具有很高的精度和很快的计算效率。理论上还可将这一方法推广至二维波动情形,即地震波倾斜入射条件下成层半空间出平面自由场的计算。
The wave motion of a free field for general engineering can be simplified as a 1-D wave motion of an elastic layered half-space model,approximate solutions of which can be obtained by numerical methods.For such problems,the seismic waves are assumed to be vertical body waves propagating in the vertical direction,and site strata are regarded as nearly horizontal stratified structures.Even though there are many types of algorithms for seismic response analysis,all algorithms can be broadly classified into two main types.The first are numerical methods in the time domain,and the second are numerical methods in the frequency domain.However,numerical methods in the time domain,such as the finite difference method,finite element method,and boundary element method,are currently used.When these methods are used to calculate the free wave field in a layered half space,it is necessary to first discretize the computational region.The definite-solution problem of the continuous wave field is transformed to the problem of numerical computation of the discrete element nodes by methods of mathematical physics.The numerical formulas are usually expressed as a group of equations or explicit iteration schemes step-by-step in the time direction.However,the precision of approximate solutions computed by these numerical methods is affected by many factors,such as the mathematical algorithm,model range,mesh size,time step,and boundary condition.Inputting improper parameters will cause instability of the numerical algorithms,even causing no results to be obtained after a large amount of computation.Considering the generalized reflection transmission coefficient matrix method for synthetic seismograms,a new method is proposed,hich provides improvements for numerical methods in the time domain for solving problems of 1-D wave motion in an elastic layered half-space.When the method is used to compute a wave field in a layered half space,the element nodes are set at the wave impedance interfaces,which are called interfacial nodes.According to the wave motion principle of superposition,the wave field values between layers can be computed from the interface nodes,in which none of the nodes are set.Interfacial node values are made in accordance with the refracting and reflecting regulations at wave impedance interfaces and traveling time of waves between wave impedance interfaces,the expressions of which can be written as a group of time delay equations.Interfacial node values can be obtained after solving time delay equations.The wave field values in a layered half space can be obtained from one of the interface nodes at the free surface.Considering Huygens principle,the motion at interfacial nodes can be regarded as secondary sources or wavelet sources,when seismic waves pass through impedance interfaces.Therefore,the motion at interfacial nodes is called an interfacial wavelet,and the above method for determining the free wave field in a layered half space is called the interfacial wavelet method.The interfacial wavelet method is suitable for 1-D wave motion problems corresponding to the wave field of horizontal layered media caused by normal incidence.Two numerical results demonstrate that the proposed method has high accuracy and fast computing speed.In theory,the method can also be used to solve 2-D wave motion problems corresponding to the wave field of a horizontal layered media at oblique incidence.
The wave motion of a free field for general engineering can be simplified as a 1-D wave motion of an elastic layered half-space model,approximate solutions of which can be obtained by numerical methods.For such problems,the seismic waves are assumed to be vertical body waves propagating in the vertical direction,and site strata are regarded as nearly horizontal stratified structures.Even though there are many types of algorithms for seismic response analysis,all algorithms can be broadly classified into two main types.The first are numerical methods in the time domain,and the second are numerical methods in the frequency domain.However,numerical methods in the time domain,such as the finite difference method,finite element method,and boundary element method,are currently used.When these methods are used to calculate the free wave field in a layered half space,it is necessary to first discretize the computational region.The definite-solution problem of the continuous wave field is transformed to the problem of numerical computation of the discrete element nodes by methods of mathematical physics.The numerical formulas are usually expressed as a group of equations or explicit iteration schemes step-by-step in the time direction.However,the precision of approximate solutions computed by these numerical methods is affected by many factors,such as the mathematical algorithm,model range,mesh size,time step,and boundary condition.Inputting improper parameters will cause instability of the numerical algorithms,even causing no results to be obtained after a large amount of computation.Considering the generalized reflection transmission coefficient matrix method for synthetic seismograms,a new method is proposed,hich provides improvements for numerical methods in the time domain for solving problems of 1-D wave motion in an elastic layered half-space.When the method is used to compute a wave field in a layered half space,the element nodes are set at the wave impedance interfaces,which are called interfacial nodes.According to the wave motion principle of superposition,the wave field values between layers can be computed from the interface nodes,in which none of the nodes are set.Interfacial node values are made in accordance with the refracting and reflecting regulations at wave impedance interfaces and traveling time of waves between wave impedance interfaces,the expressions of which can be written as a group of time delay equations.Interfacial node values can be obtained after solving time delay equations.The wave field values in a layered half space can be obtained from one of the interface nodes at the free surface.Considering Huygens principle,the motion at interfacial nodes can be regarded as secondary sources or wavelet sources,when seismic waves pass through impedance interfaces.Therefore,the motion at interfacial nodes is called an interfacial wavelet,and the above method for determining the free wave field in a layered half space is called the interfacial wavelet method.The interfacial wavelet method is suitable for 1-D wave motion problems corresponding to the wave field of horizontal layered media caused by normal incidence.Two numerical results demonstrate that the proposed method has high accuracy and fast computing speed.In theory,the method can also be used to solve 2-D wave motion problems corresponding to the wave field of a horizontal layered media at oblique incidence.
引文
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[6]赵密,杜修力,刘晶波,等.P-SV波斜入射时成层半空间自由场的时域算法[J].地震工程学报,2013,35(1);84-90.ZHAO Mi,DU XIU-li,LIU Jing-bo,et al.Time-domain Method for Free Field in Layered Half Space under P-SV Waves of Oblique Incidence[J].China Earthquake Engineering Journal,2013,35(1);84-90.(in Chinese)
[7]范留明.非均匀层状介质一维波动方程波精确解的有限差分算法[J].岩土力学,2013,34(9):2715-2720.Fan Liu-ming.A new Kind of Finite Difference Scheme to Obtain Exact Solutions of One-dimensional Wave Equation in Heterogeneous Layer Media[J].Rock and Soil Mechanics,2013,34(9);2715-2720.(in Chinese)
[8]袁修贵,李远禄.基于小波的一维线性波动方程的数值解法[J].数学理论与应用,2003,23(1):11-14.YUAN Xiu-gui,LI Yuan-lu.Wavelet-based Numerical Solution of 1-D Linear Wave Equation[J].Mathematical Theory and Applications,2003,23(1):11-14.(in Chinese)
[9]张钦礼,牛莉.一维波动方程的小波解法[J].华北航天工业学院学报,2001,11(1):45-48.ZHANG Qin-li,NIU Li.Wavelet Solution of One-dimensional Wave Equation[J].Journal of North China Institute of Astronautic Engineering,2001,11(1):45-48.(in Chinese)
[2]刘晶波,王艳.成层半空间出平面自由波场的一维化时域算法[J].力学学报,2006,38(2):219-225.LIU Jing-bo,WANG Yan.A 1-D Time-domain Method for 2-D Wave Motion in Elastic Layered Half-space by Antiplane Wave Oblique Incidence[J].Chinese Journal of Theoretical and Applied Mechanics,2006,38(2):219-225.(in Chinese)
[3]姚振兴,郑天愉.计算综合地震图的广义反射、透射系数矩阵和离散波数方法(二)——对不同深度点源的算法[J].地球物理学报,1984,27(4):338-348.YAO Zhen-xing,ZHENG Tian-yu.A Generalized Reflectiontransmission Coefficient Matrix and Discrete Wavenumber Method for Synthetic Seismograms(Ⅱ)——For Multiple Sources at Different Depths[J].Acta Geophsics of Sinca,1984,27(4):338-348.(in Chinese)
[4]李旭,陈运泰.合成地震图的广义反射透射系数矩阵方法[J].地展地磁观测与研究,1996,17(3):1-20.LI Xu,CHEN yun-tai.The Generalized Reflection-transmission Coefficient Matrix Method for Synthetic Seismograms[J].Seismological and Geomagnetic Observation and Research,1996,17(3):1-20.(in Chinese)
[5]廖振鹏,刘恒,谢志南.波动数值模拟的一种显式方法——一维波动[J].力学学报,2009,41(3):350-360.LIAO Zhen-peng,LIU Heng,XIE Zhi-nan.An Explicit Method for Numerical Simulation of Wave Motion——1-D Wave Motion[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(3):350-360.(in Chinese)
[6]赵密,杜修力,刘晶波,等.P-SV波斜入射时成层半空间自由场的时域算法[J].地震工程学报,2013,35(1);84-90.ZHAO Mi,DU XIU-li,LIU Jing-bo,et al.Time-domain Method for Free Field in Layered Half Space under P-SV Waves of Oblique Incidence[J].China Earthquake Engineering Journal,2013,35(1);84-90.(in Chinese)
[7]范留明.非均匀层状介质一维波动方程波精确解的有限差分算法[J].岩土力学,2013,34(9):2715-2720.Fan Liu-ming.A new Kind of Finite Difference Scheme to Obtain Exact Solutions of One-dimensional Wave Equation in Heterogeneous Layer Media[J].Rock and Soil Mechanics,2013,34(9);2715-2720.(in Chinese)
[8]袁修贵,李远禄.基于小波的一维线性波动方程的数值解法[J].数学理论与应用,2003,23(1):11-14.YUAN Xiu-gui,LI Yuan-lu.Wavelet-based Numerical Solution of 1-D Linear Wave Equation[J].Mathematical Theory and Applications,2003,23(1):11-14.(in Chinese)
[9]张钦礼,牛莉.一维波动方程的小波解法[J].华北航天工业学院学报,2001,11(1):45-48.ZHANG Qin-li,NIU Li.Wavelet Solution of One-dimensional Wave Equation[J].Journal of North China Institute of Astronautic Engineering,2001,11(1):45-48.(in Chinese)
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