【作者】 李宾；

【导师】 杜启振；

【作者基本信息】 中国石油大学 ， 地球探测与信息技术， 2009， 硕士

【摘要】 地震波数值模拟一直是人们理解和认识波在各向异性介质中传播特性的重要手段。在数值模拟的方法方面,有限差分方法以其计算速度快、边界处理简单等优势受到人们的青睐,但是有限差分数值模拟方法有两个关键问题需要考虑,即数值频散和人工边界条件。本文重点研究了高阶交错网格有限差分数值模拟中的数值频散和人工边界条件,给出了相应的改进策略,然后基于均匀横向各向同性(Vertical Transverse Isotropy,简称VTI)介质P波、P-SV转换波和SH波一阶速度-应力波动方程应用高阶交错网格有限差分法进行了波场正演模拟,并实现了弹性波波场分离。为了消弱数值频散,本文将紧致差分格式和交错网格技术相结合,推导了横向各向同性介质一阶速度-应力波动方程的紧致交错网格差分格式,对比分析了紧致交错网格差分格式、交错网格差分格式以及紧致差分格式的截断误差主项,并利用Fourier误差分析方法分析了上述三种差分格式的近似精度。基于最优化的理论,分别引入强约束条件和弱约束条件,构造了Lagrange函数;然后通过求取条件极值得到了一阶导数的交错网格优化差分算子,进一步提高了差分格式的计算精度。针对人工边界造成的截断误差问题,本文总结了几种常用的人工边界条件,指出了它们的优缺点;详细地分析了衰减边界条件吸收效果的影响因素,提出了改进的衰减函数及其应用原则。基于完全匹配层(Perfectly Matched Layer,简称PML)边界条件,本文实现了二维VTI介质P-SV波和SH波一阶速度-应力波动方程的裂化PML边界条件,并将提出的衰减函数应用原则应用到PML边界条件中。分别应用改进的衰减边界条件和PML边界条件进行数值模拟,均获得了较好的效果。为方便分析波场特征,有必要对正演模拟得到的矢量波场进行分离。基于纵波为无旋场、横波为无散场,本文实现了纵波和转换横波波场分离,数值算例证实了方法的有效性。更多还原

【Abstract】 Seismic numerical modeling is an improtant tool to understand the wave propagation characteristics in anstropic medium. In the numerical simulation field, by virtue of its high efficiency and easy implementation of boundary conditions, the finite difference method is attrative. However, it has two key questions, which are called numerical dispersion and aritifical boundary conditions. Based on the theory of staggered-grid finite difference, firstly, we focus on the numerical dispersion as well as the artifical boundary conditons, and put forward the corresponding improvement strategies; Secondly, we simulate seismic wave propagation in 2D vertical transversely isotropic media, and analyze the seismic numerical modeling results; Finally, we successfully separate the elastic wavefield.To deal with the numerical dispersion, firstly, by combining the staggered-grid technology with the compact finite difference scheme, we derive a compact staggered-grid finite difference scheme in vertical transverse isotropic medium from the first-order velocity-stress wave equations. With the comparison of the principal truncation error terms of the compact staggered-grid finite difference scheme, the staggered-grid finite difference scheme and the compact finite difference scheme, the thesis analyzes the approximation accuracy of the above three schemes through Fourier analysis. Secondly, based on the optimization theory, we construct two forms of Lagrange function by introducing strong and weak constraint, respectively, and further obtain the optimization operator for staggred-grid finite difference via the solution of conditional extremum value, which is also compared with the normal Taylor expansion operator.In order to weaken the artifical boundary reflection, we summarize several kinds of typical artifical boundary conditions, and point out their shortcomings, respectively. After analyzing the influence factors of the sponge methods in detail, we present the improved damping functions and their application principles. The perfectly matched layer (PML) absorbing boundary condition (ABC) has been widely used for seismic numerical modeling. In the paper, we implement the split PMLs for the velocity-stress formulation of elastodynamics, and adapt the proposed application principles to the PMLs. The simulation results have shown the vadity of the improved sponge methods as well as the PMLs.In order to analyze the wave propagation characteristics easily, it’s necessary to carry out the separation of the wavefields. Taking the divergence and the curl of the vectorial wavefield during the finite difference modeling, we separate P- and converted SV-waves in 2D elastic seismograms and get clear wavefields by introducing optimal difference operator. The rusults has also proved the validity of the improved strategies.更多还原