自适应网格细化算法模拟地震波传播
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摘要
数值方法求解波动方程一直都是地球物理学界和计算数学界重要的研究课题。数值方法模拟地震波传播现象,可以为天然地震预测和地震波勘探提供理论依据。现有的数值方法使用非常细化的网格,可以得到很好的地震波传播模拟效果,但是采用非常细化的网格计算时,必然会增加计算时间和计算存储量。研究高效的数值方法模拟地震波传播现象,成为求解地震波动方程的关键问题。
由于地下介质分布比较复杂,介质之间存在着波速变化,给地震波传播模拟造成了很大困难。高分辨率有限体积法可以得到波在不连续介质中传播的高分辨率的数值解,本文使用高分辨率有限体积法求解声波方程和弹性波方程,构造了波传播的算子,并对波传播算子进行修正限制;数值算例表明:该方法不但可以得到高分辨率的波场快照,而且有效地控制了数值振荡。通过数值解和精确解比较、频散分析,证明了高分辨率有限体积法模拟震波传播的有效性。
为了提高地震波传播模拟的计算效率,本文提出了使用自适应网格细化算法结合高分辨率有限体积法求解地震声波和弹性波方程。首先,使用高分辨率有限体积法得到基网格上的数值解,接着,对数值解进行截断误差估计,根据预先设定的阈值,确定需要进行网格细化的区域,将得到的细化网格嵌套在基网格上。自适应网格细化算法自动生成层层嵌套的细化网格,时间步长与空间步长同时细化,采用树状数据结构存储网格点。本文分别使用自适应网格和非自适应网格求解地震声波和弹性波方程,并对计算时间和计算存储量进行比较,结果表明自适应网格细化算法的高效性。
提升算法得到的第二代小波变换相对于小波变换运算速度更快,并且保持了多尺度分解的特性。本文利用第二代小波的快速变换和多尺度分解性质,构造了多尺度层层嵌套的自适应网格求解二维声波方程,网格点的空间导数使用有限差分法得到,并给出了声波数值传播的波场快照、计算精度的分析和计算效率的比较。数值实验结果,说明了基于第二代小波的自适应网格求解声波方程的可行性。
Numerical method solving wave equation has been always an important research topic in the fields of geophysics and computiaonal mathematics. Numerical modeling of seismic wave propagation phenomena can provide a theoretical basis for earthquake prediction and seismic exploration. Existing numerical methods modeling of wave propagtion can gain good numerical solution using the fine mesh, but the computional time and computional storage will be increased. It has been the key to solving seismic wave equation that using high efficient numerical method modeling of seismic wave propagation phenomena.
It is difficult for numerical modeling of seismic wave propagtion, because the subsurface media has complex distribution and wave volocity changing between discontinuous media. High resolution numerical solution can be obtained wave propagation in discontinuous media using high resolution finite volume method. In the thesis, high resolution finite volume method has been used to solve acoustic and elastic wave equations. The wave propagation operator is constructed, and it is implemented by wave correction limited. Numerical examples show that the numerical method can not only obtain high resolution snapshots of wave field, but also effectively control of numerical oscillations. The comparison of numerical solutions and exact solutions is given, and dispersion analysis is obtained. It is proved that the numerical method is effective for solving seismic wave equations.
In order to improve the computational efficiency for modeling of seismic wave propagation, a combination of adaptive mesh refinement and high resolution finite volume algorithm is presented for solving seismic acoustic and elastic wave equations. The numerical solutions are obtained using high resolution finite volume method on the coarse mesh, then the local truncation error is gained by the error estimation procedure. If the truncation error is greater than a given threshold value, the local rectangular region is refined. The fine mesh will be nested in the coarse mesh. Adaptive mesh refinement algorithm can automatic generated of nested-type of adaptive mesh, and the time step with space step is refined. The grid pionts are stored by a tree data structure. Using adaptive mesh and non-adaptive solve of seismic acoustic and elastic equations, the computational time and compuation storage will be compared, the results indicated that adaptive mesh refinement algorithm is high efficient.
Second generation wavelet transform is obtained by lifting scheme, its compational speed is faster than wavelet transform, and it maintains the characteristic of multi-level decomposition. In the thesis, second generation wavelet is used to construct nested-type adaptive mesh solving of two-dimensional acoustic wave equation, and the spatial derivatives of grid points are obtained by finite difference method. The snapshots of wave field are obtained by numerical modeling of acoustic wave propagation. Compuational accuracy analysis and the comparesion of computational efficiency are given. The applicability of adaptive mesh based on second generation wavelet solving wave equation is proved by numerical experiment results.
由于地下介质分布比较复杂,介质之间存在着波速变化,给地震波传播模拟造成了很大困难。高分辨率有限体积法可以得到波在不连续介质中传播的高分辨率的数值解,本文使用高分辨率有限体积法求解声波方程和弹性波方程,构造了波传播的算子,并对波传播算子进行修正限制;数值算例表明:该方法不但可以得到高分辨率的波场快照,而且有效地控制了数值振荡。通过数值解和精确解比较、频散分析,证明了高分辨率有限体积法模拟震波传播的有效性。
为了提高地震波传播模拟的计算效率,本文提出了使用自适应网格细化算法结合高分辨率有限体积法求解地震声波和弹性波方程。首先,使用高分辨率有限体积法得到基网格上的数值解,接着,对数值解进行截断误差估计,根据预先设定的阈值,确定需要进行网格细化的区域,将得到的细化网格嵌套在基网格上。自适应网格细化算法自动生成层层嵌套的细化网格,时间步长与空间步长同时细化,采用树状数据结构存储网格点。本文分别使用自适应网格和非自适应网格求解地震声波和弹性波方程,并对计算时间和计算存储量进行比较,结果表明自适应网格细化算法的高效性。
提升算法得到的第二代小波变换相对于小波变换运算速度更快,并且保持了多尺度分解的特性。本文利用第二代小波的快速变换和多尺度分解性质,构造了多尺度层层嵌套的自适应网格求解二维声波方程,网格点的空间导数使用有限差分法得到,并给出了声波数值传播的波场快照、计算精度的分析和计算效率的比较。数值实验结果,说明了基于第二代小波的自适应网格求解声波方程的可行性。
Numerical method solving wave equation has been always an important research topic in the fields of geophysics and computiaonal mathematics. Numerical modeling of seismic wave propagation phenomena can provide a theoretical basis for earthquake prediction and seismic exploration. Existing numerical methods modeling of wave propagtion can gain good numerical solution using the fine mesh, but the computional time and computional storage will be increased. It has been the key to solving seismic wave equation that using high efficient numerical method modeling of seismic wave propagation phenomena.
It is difficult for numerical modeling of seismic wave propagtion, because the subsurface media has complex distribution and wave volocity changing between discontinuous media. High resolution numerical solution can be obtained wave propagation in discontinuous media using high resolution finite volume method. In the thesis, high resolution finite volume method has been used to solve acoustic and elastic wave equations. The wave propagation operator is constructed, and it is implemented by wave correction limited. Numerical examples show that the numerical method can not only obtain high resolution snapshots of wave field, but also effectively control of numerical oscillations. The comparison of numerical solutions and exact solutions is given, and dispersion analysis is obtained. It is proved that the numerical method is effective for solving seismic wave equations.
In order to improve the computational efficiency for modeling of seismic wave propagation, a combination of adaptive mesh refinement and high resolution finite volume algorithm is presented for solving seismic acoustic and elastic wave equations. The numerical solutions are obtained using high resolution finite volume method on the coarse mesh, then the local truncation error is gained by the error estimation procedure. If the truncation error is greater than a given threshold value, the local rectangular region is refined. The fine mesh will be nested in the coarse mesh. Adaptive mesh refinement algorithm can automatic generated of nested-type of adaptive mesh, and the time step with space step is refined. The grid pionts are stored by a tree data structure. Using adaptive mesh and non-adaptive solve of seismic acoustic and elastic equations, the computational time and compuation storage will be compared, the results indicated that adaptive mesh refinement algorithm is high efficient.
Second generation wavelet transform is obtained by lifting scheme, its compational speed is faster than wavelet transform, and it maintains the characteristic of multi-level decomposition. In the thesis, second generation wavelet is used to construct nested-type adaptive mesh solving of two-dimensional acoustic wave equation, and the spatial derivatives of grid points are obtained by finite difference method. The snapshots of wave field are obtained by numerical modeling of acoustic wave propagation. Compuational accuracy analysis and the comparesion of computational efficiency are given. The applicability of adaptive mesh based on second generation wavelet solving wave equation is proved by numerical experiment results.
引文
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