地震研究

在利用谱元法求解地震波运动方程时需要对方程进行离散,然而由于震源项常包含δ函数,该函数是奇异函数,通过数值近似具有一定难度,因此需要采用特殊方式处理震源项。引入δ函数的2类离散形式,以谱元法求解地震波运动方程为例,详细给出了震源项离散形式。为了验证这种震源处理方法的有效性,开展了多组数值模拟实验。首先在规则网格下,模拟了δ函数离散形式处理单极震源和矩张量震源的情况; 其次在非规则网格下,模拟了δ函数离散形式处理地震点源情况。数值实验表明,在谱元法模拟地震波传播时,利用δ函数离散形式近似地震点源项,可得到高精度数值模拟结果。

When solving the seismic wave equation with the spectral element method,it is necessary to discretize the equation.However,due to the presence of the δ function in the source term,which is a singular function,achieving accurate numerical approximation poses certain challenges.Therefore,special techniques are required to handle the source term.Two types of discrete forms of delta function are introduced in this paper.Using the spectral element method for numerical solution of seismic wave propagation,the discretized form of the source term is presented in detail.To validate the effectiveness of this source treatment approach,several numerical experiments were conducted.Firstly,the discrete form of the δ function was simulated for handling single-force and moment tensor sources on regular grid models.Secondly, the discrete form of the δ function was simulated for handling seismic point sources on an irregular grid model. The numerical experiments demonstrate that when simulating seismic wave propagation with the spectral element method,approximating the point source term with the discrete form of delta function can help to obtain high-precision results from numerical simulation.

引言
1 谱元法求解波动方程基本原理
2 震源处理
3 数值算例与分析
4 结论

图1 第一类δ函数离散形式(δ～1)<br/>Fig.1 The first type of discrete form of δ function(δ～1)

图1 第一类δ函数离散形式(δ～1)
Fig.1 The first type of discrete form of δ function(δ～1)

表1 第二类δ函数离散数值(δ～2)<br/>Tab.1 The second type of discrete values of δ function(δ～2)

表1 第二类δ函数离散数值(δ～2)
Tab.1 The second type of discrete values of δ function(δ～2)

图2 波数谱函数H(k)(a)以及第二类δ函数离散形式(δ～2)(b)<br/>Fig.2 The wave number spectrum function H(k)(a) and the second type of discrete form of δ function(δ～2)(b)

图2 波数谱函数H(k)(a)以及第二类δ函数离散形式(δ～2)(b)
Fig.2 The wave number spectrum function H(k)(a) and the second type of discrete form of δ function(δ～2)(b)

图3 介质模型示意图<br/>Fig.3 The sketch of the medium model

图3 介质模型示意图
Fig.3 The sketch of the medium model

图4 接收器R1、R2记录的SEM合成波形与解析波形对比(a)以及两者的相对误差(b)<br/>Fig.4 The SEM synthetic waveforms and analytical waveforms of Receiver 1 and Receiver 2(a),and their relative errors(b)

图4 接收器R1、R2记录的SEM合成波形与解析波形对比(a)以及两者的相对误差(b)
Fig.4 The SEM synthetic waveforms and analytical waveforms of Receiver 1 and Receiver 2(a),and their relative errors(b)

图5 t=0.4 s时,使用M2、M3方法计算得到震源S1激发的地震波场的水平(a)和垂直(b)分量波场快照<br/>Fig.5 The snapshots of the horizontal(a)and vertical(b)component of the excitation wavefield by Source 1 calculated respectively by M2 and M3 when t=0.4 s[JZ)]

图5 t=0.4 s时,使用M2、M3方法计算得到震源S1激发的地震波场的水平(a)和垂直(b)分量波场快照
Fig.5 The snapshots of the horizontal(a)and vertical(b)component of the excitation wavefield by Source 1 calculated respectively by M2 and M3 when t=0.4 s[JZ)]

图6 震源为S1时,接收器R1、R2的SEM合成波形与参考波形(a)以及两者的相对误差(b)<br/>Fig.6 The SEM synthetic and analytical waveforms of the Source 1-excited waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)

图6 震源为S1时,接收器R1、R2的SEM合成波形与参考波形(a)以及两者的相对误差(b)
Fig.6 The SEM synthetic and analytical waveforms of the Source 1-excited waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)

图7 t=0.4 s时,使用M2、M3方法计算得到震源S2激发的地震波场的水平(a)和垂直(b)分量波场快照<br/>Fig.7 The snapshots of the horizontal(a)and vertical(b)components of the excitation wavefield by Source 2 calculated respectively by M2 and M3 when t=0.4 s[JZ)]

图7 t=0.4 s时,使用M2、M3方法计算得到震源S2激发的地震波场的水平(a)和垂直(b)分量波场快照
Fig.7 The snapshots of the horizontal(a)and vertical(b)components of the excitation wavefield by Source 2 calculated respectively by M2 and M3 when t=0.4 s[JZ)]

图8 震源为S2时,接收器R1、R2记录的SEM合成波形与参考波形(a)以及两者的相对误差(b)<br/>Fig.8 The SEM synthetic and analytical waveforms of the Source 2-excited waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)

图8 震源为S2时,接收器R1、R2记录的SEM合成波形与参考波形(a)以及两者的相对误差(b)
Fig.8 The SEM synthetic and analytical waveforms of the Source 2-excited waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)

图9 非规则网格模型示意图<br/>Fig.9 Schematic diagram of the irregular grid model

图9 非规则网格模型示意图
Fig.9 Schematic diagram of the irregular grid model

图10 非均匀网格模型中接收器R1、R2的SEM合成波形与参考波形(a)以及两者的相对误差(b)<br/>Fig.10 The SEM synthetic and analytical waveforms of the original waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)in the irregular grid model

图10 非均匀网格模型中接收器R1、R2的SEM合成波形与参考波形(a)以及两者的相对误差(b)
Fig.10 The SEM synthetic and analytical waveforms of the original waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)in the irregular grid model

图11 接收器R1、R2的FDM合成波形与解析波形对比(a)以及两者的相对误差(b)<br/>Fig.11 The FDM synthetic and analytical waveforms of the original waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)

图11 接收器R1、R2的FDM合成波形与解析波形对比(a)以及两者的相对误差(b)
Fig.11 The FDM synthetic and analytical waveforms of the original waveforms received by Receiver 1 and Receiver 2(a)and their relative errors(b)

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备注

引言

1 谱元法求解波动方程基本原理

2 震源处理

3 数值算例与分析

4 结论

期刊信息