正交曲线坐标系下的地震波数值模拟技术研究
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摘要
有限差分方法是一种高效、灵活且适应性很强的方法,被广泛应用于各种复杂介质的地震波数值模拟中,是目前勘探地震学和天然地震学中研究地震波传播规律不可或缺的重要工具。但是,当有不规则的起伏地表存在时,现有的各种有限差分方法不能很好的实施自由边界条件或者是在实施自由边界条件时需要进行复杂的坐标旋转及(或)插值运算,这不仅会影响到数值模拟结果的精度,同时也会增加求解过程的复杂性,降低模拟效率。
目前,用于模拟起伏地表条件下地震波传播的各种有限差分方法在实施自由边界条件时所采用的数值离散方法常受限于所选用的网格坐标系统,不能很好的对起伏地形条件下的地震波传播的影响进行模拟。本论文从以下几个方面对这种方法进行改进：(1)采用了流体力学中广泛使用的正交曲线网格,能够对起伏地表进行贴体的网格剖分,相比于常规的阶梯状网格近似方法,具有更好的近似效果,可以有效的避免阶梯状网格近似在数值模拟时产生的角点散射和绕射干扰；(2)选用张量形式的波动方程作为起伏地表地震波数值模拟的控制方程,这种形式的波动方程适用于任意曲线坐标系；(3)引入空气动力学中的选择性滤波同位网格有限差分算法,对张量形式的波动方程进行离散求解。这种差分算法能够适应于复杂非均匀介质,同时还避免了交错网格有限差分格式在曲线坐标系中数值精度不高,需要进行复杂插值运算等缺陷；(4)在正交曲线坐标系下,采用应力镜像法来实施边界条件,能使其简便易行,且与笛卡尔坐标系下的自由边界条件具有相似的形式；(5)将非分裂的ADE-PML吸收边界条件引入正交曲线坐标系下的地震波数值模拟中,用来消除人工边界反射的影响。
本文将正交曲线网格生成技术、选择性滤波同位网格有限差分算法、应用镜像法自由边界条件和ADE-PML吸收边界条件结合起来用于起伏地表条件下地震波传播规律的研究,并进行了对比验证。结果表明,这种方法具有较高的数值精度,能够模拟任意起伏地形下的地震波传播。
最后,采用本文算法对山峰,山谷以及正弦函数起伏模型下的地震波进行了数值模拟,研究了这些地形条件下的波场传播特征响应。
With its advantage of high-efficiency, flexibility and high-adaptation, finite differnce method is widely used to simulate seismic wave propagation in various complex medium. This method has played an important role in the study of properties of seismic wave propagation in both exploration seismology and earthquake seismology. However, when irregularly topographic free surface exists, most of the current finite-differnce approaches can not achieve good results in implementing the free-surface boundary conditions and always involve complicated coordinate rotations and interpolations, which may not only affect the numerical accuracy of simulations, but make the forward modeling more complicated and low-efficient.
At present, all finite difference methods used to synthetize seismic wave propagation in the presence of topography are limited to the coordinate system chosen. The discrete free surface conditions are not so well suited to model seismic waves when dealing with irregular surface. In this study, several efforts are made to improve the algorithm:(1) We introduced the orthogonally curvilinear grids to discretize the irregular physical domain. This coordinate system is body-conformal and much more accurate than staircased discretization to topography. Thus, corner wave scatterings and diffractions are avoided as a result;(2) Tensorial wave equations are introduced to simulate wave propagation as the control equation, which is well suited to arbitraryly curvilinear coordinate system;(3) We used selective filtering non-staggered finite difference method, which is commonly used in aeroacoustics, to solve tensorial wave equations. This algorithm is adaptive to complex heterogeneous medium and complicated interpolations are avoided which is the drawback of staggered-grid finite difference method;(4) In orthogonally curvilinear coordinate system, the free-surface boundary conditions are imposed by stress imaging method, which makes the boundary condtions simply to implement and has the similar form of that in Cartesian coordinate system;(5) An unsplit ADE-PML absorbing boundary condtion is utilized to attenuate reflected energy from artifical boundaries in curvilinear coordinate system.
In the present study, orthogonally curvilinear grids, selective filtering non-staggered finite-difference method, stress imaging free-surface boundary condition as well as ADE-PML absorbing boundary condition are integrated to simulate seismic wave propagation in the presence of topographic free surface. Validations are made and results showed that the proposed algorithm can be a good alternative in modeling seimic wave in the case of arbitrary topography.
Finally, simulation results are provided for hill and canyon models as well as for sine curve model to study the properties of wave propagation in topographic surface.
目前,用于模拟起伏地表条件下地震波传播的各种有限差分方法在实施自由边界条件时所采用的数值离散方法常受限于所选用的网格坐标系统,不能很好的对起伏地形条件下的地震波传播的影响进行模拟。本论文从以下几个方面对这种方法进行改进：(1)采用了流体力学中广泛使用的正交曲线网格,能够对起伏地表进行贴体的网格剖分,相比于常规的阶梯状网格近似方法,具有更好的近似效果,可以有效的避免阶梯状网格近似在数值模拟时产生的角点散射和绕射干扰；(2)选用张量形式的波动方程作为起伏地表地震波数值模拟的控制方程,这种形式的波动方程适用于任意曲线坐标系；(3)引入空气动力学中的选择性滤波同位网格有限差分算法,对张量形式的波动方程进行离散求解。这种差分算法能够适应于复杂非均匀介质,同时还避免了交错网格有限差分格式在曲线坐标系中数值精度不高,需要进行复杂插值运算等缺陷；(4)在正交曲线坐标系下,采用应力镜像法来实施边界条件,能使其简便易行,且与笛卡尔坐标系下的自由边界条件具有相似的形式；(5)将非分裂的ADE-PML吸收边界条件引入正交曲线坐标系下的地震波数值模拟中,用来消除人工边界反射的影响。
本文将正交曲线网格生成技术、选择性滤波同位网格有限差分算法、应用镜像法自由边界条件和ADE-PML吸收边界条件结合起来用于起伏地表条件下地震波传播规律的研究,并进行了对比验证。结果表明,这种方法具有较高的数值精度,能够模拟任意起伏地形下的地震波传播。
最后,采用本文算法对山峰,山谷以及正弦函数起伏模型下的地震波进行了数值模拟,研究了这些地形条件下的波场传播特征响应。
With its advantage of high-efficiency, flexibility and high-adaptation, finite differnce method is widely used to simulate seismic wave propagation in various complex medium. This method has played an important role in the study of properties of seismic wave propagation in both exploration seismology and earthquake seismology. However, when irregularly topographic free surface exists, most of the current finite-differnce approaches can not achieve good results in implementing the free-surface boundary conditions and always involve complicated coordinate rotations and interpolations, which may not only affect the numerical accuracy of simulations, but make the forward modeling more complicated and low-efficient.
At present, all finite difference methods used to synthetize seismic wave propagation in the presence of topography are limited to the coordinate system chosen. The discrete free surface conditions are not so well suited to model seismic waves when dealing with irregular surface. In this study, several efforts are made to improve the algorithm:(1) We introduced the orthogonally curvilinear grids to discretize the irregular physical domain. This coordinate system is body-conformal and much more accurate than staircased discretization to topography. Thus, corner wave scatterings and diffractions are avoided as a result;(2) Tensorial wave equations are introduced to simulate wave propagation as the control equation, which is well suited to arbitraryly curvilinear coordinate system;(3) We used selective filtering non-staggered finite difference method, which is commonly used in aeroacoustics, to solve tensorial wave equations. This algorithm is adaptive to complex heterogeneous medium and complicated interpolations are avoided which is the drawback of staggered-grid finite difference method;(4) In orthogonally curvilinear coordinate system, the free-surface boundary conditions are imposed by stress imaging method, which makes the boundary condtions simply to implement and has the similar form of that in Cartesian coordinate system;(5) An unsplit ADE-PML absorbing boundary condtion is utilized to attenuate reflected energy from artifical boundaries in curvilinear coordinate system.
In the present study, orthogonally curvilinear grids, selective filtering non-staggered finite-difference method, stress imaging free-surface boundary condition as well as ADE-PML absorbing boundary condition are integrated to simulate seismic wave propagation in the presence of topographic free surface. Validations are made and results showed that the proposed algorithm can be a good alternative in modeling seimic wave in the case of arbitrary topography.
Finally, simulation results are provided for hill and canyon models as well as for sine curve model to study the properties of wave propagation in topographic surface.
引文
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