层状半无限空间中波动问题的数值模拟

本文对波动方程首先进行富里叶—贝赛尔积分变换,在波数k域内构成(z,t)的有限差分隐格式进行迭代,由此计算出纵向非均匀的层状模型的合成地震图。对含有低速层和高速薄层的几种模型做了对比计算,通过时间场与空间场的波动分析,揭示了几种主要震相的传播与形成过程。计算结果表明,无论高速层的厚薄如何。反射波始终很强烈。但初至首波在薄层构造中不清晰,一种属转换型的续至薄层首波震相值得注意;低速层的顶界面难以形成能量较强的上行波,因此在推断低速层埋深上存在不确定性。     

用于弹性波模拟的交错网格预处理和多次校正的Chebyshev谱元法

本文基于弹性波动方程,从其弱形式出发,利用Galerkin变分原理,通过对方程进行空间和时间上的离散,在空间域中引入预条件共轭梯度的逐元算法,在时间域中引入时间积分的交错网格预处理/多次校正算法,发展了弹性波模拟的Chebyshev谱元算法。针对均匀固体介质和具有倾斜分层的分区均匀固体介质模型,通过与有限差分算法结果相比较验证其精度的可信性,同时利用该算法模拟了弹性波在具有水平分层的任意起伏自由表面模型中的传播,并分析了其传播特点。研究表明,我们提出的交错网格预处理/多次校正算法的Chebyshev谱元算法,保留了有限元法的优势,并且采用了具有最优张量乘积技术的元到元的算法,能够处理带有起伏自由表面的复杂介质模型,它具有比有限元法收敛快,计算效率较高等优点,特别适合于复杂结构和复杂介质中的弹性波传播的数值模拟。     

基于Shannon奇异核理论的褶积微分算子在地震波场模拟中的应用

本文在前人工作的基础上,建立了一种基于Shannon奇异核的交错网格褶积微分算子方法.文中不仅详细讨论了影响算子精度的各种因素,同时也着重分析了其在弹性波模拟中的频散关系和稳定性条件.通过和交错网格有限差分算子比较,发现该算子即使在高波数域也具有较高的精度.均匀介质中的数值试验也表明,该方法9点格式就基本上达到了解析解精度.而分层均匀介质和复杂介质中的地震波数值模拟也同时证实了该方法精度高,稳定性好,是一种研究复杂介质中地震波传播的有效数值方法.     

基于高阶双渐近透射边界的重力坝-层状地基动力相互作用分析

时域高阶双渐近透射边界能够同时模拟层状介质中行波和快衰波的传播,具有很高的计算精度和计算效率.本文将高阶双渐近透射边界推广应用到多层层状地基系统弹性波传播问题的模拟,采用广义特征值分解分析该透射边界的数值稳定性,通过移谱法消除导致数值不稳定的虚假模态.将高阶双渐近透射边界以超单元的形式直接嵌入到近场有限元方程,建立了有限元-高阶双渐近透射边界时域耦合分析模型,并将其应用到重力坝-层状地基动力相互作用分析.数值算例分析结果表明,该时域耦合分析模型具有很高的精度和计算效率,适用于实际重力坝工程的地震响应分析.     

Caustics in a periodically layered transversely isotropic medium with vertical symmetry axis

Wave propagation in a finely layered medium is a very important topic in seismic modelling and inversion. Here we analyse non‐vertical wave propagation in a periodically layered transversely isotropic (VTI) medium and show that the evanescent (attenuation) zones in the frequency‐horizontal slowness domain result in caustics in the group velocity domain. These caustics, which may appear for both the quasi‐compressional (qP) and quasi‐shear (qSV) wave surfaces are frequency dependent but display weak dependence at low frequencies. The caustics computed for a specific frequency differ from those observed at the low‐ and high‐frequency limits. We illustrate these caustics with a few numerical examples and snapshots computed for both qP‐ and qSV‐wave types.     

Elastic wave propagation in inhomogeneous anisotropic media

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Dynamic stiffness matrix of a poroelastic multi-layered site and its Green＇s functions

Few studies of wave propagation in layered saturated soils have been reported in the literature. In this paper, a general solution of the equation of wave motion in saturated soils, based on one kind of practical Biot‘s equation,was deduced by introducing wave potentials. Then exact dynamic-stiffness matrices for a poroelastic soil layer and halfspace were derived, which extended Wolf‘s theory for an elastic layered site to the case of poroelasticity, thus resolving a fundamental problem in the field of wave propagation and soil-structure interaction in a poroelastic layered soil site. By using the integral transform method, Green‘s functions of horizontal and vertical uniformly distributed loads in a poroelastic layered soil site were given. Finally, the theory was verified by numerical examples and dynamic responses by comparing three different soil sites. This study has the following advantages: all parameters in the dynamic-stiffness matrices have explicitly physical meanings and the thickness of the sub-layers does not affect the precision of the calculation which is very convenient for engineering applications. The present theory can degenerate into Wolf‘s theory and yields numerical results approaching those for an ideal elastic layered site when porosity tends to zero.     

Extended reflectivity method for modelling the propagation of diffusive–viscous wave in dip‐layered media

The reflectivity method plays an important role in seismic modelling. It has been used to model different types of waves propagating in elastic and anelastic media. The diffusive–viscous wave equation was proposed to investigate the relationship between frequency dependence of reflections and fluid saturation. It is also used to describe the attenuation property of seismic wave in a fluid‐saturated medium. The attenuation of diffusive–viscous wave is mainly characterised by the effective attenuation parameters in the equation. Thus, it is essential to obtain those parameters and further characterise the features of the diffusive–viscous wave. In this work, we use inversion method to obtain the effective attenuation parameters through quality factor to investigate the characteristics of diffusive–viscous wave by comparing with those of the viscoacoustic wave. Then, the reflection/transmission coefficients in a dip plane‐layered medium are studied through coordinate transform and plane‐wave theory. Consequently, the reflectivity method is extended to compute seismograms of diffusive–viscous wave in a dip plane multi‐layered medium. Finally, we present two models to simulate the propagation of diffusive–viscous wave in a dip plane multi‐layered medium by comparing the results with those in a viscoacoustic medium. The numerical results demonstrate the validity of our extension of reflectivity method to the diffusive–viscous medium. The numerical examples in both time domain and time–frequency domain show that the reflections from a dip plane interface have significant phase shift and amplitude change compared with the results of horizontal plane interface due to the differences in reflection/transmission coefficients. Moreover, the modelling results show strong attenuation and phase shift in the diffusive–viscous wave compared to those of the viscoacoustic wave.     

Modelling of viscoacoustic wave propagation in transversely isotropic media using decoupled fractional Laplacians

A new wave equation is derived for modelling viscoacoustic wave propagation in transversely isotropic media under acoustic transverse isotropy approximation. The formulas expressed by fractional Laplacian operators can well model the constant-Q (i.e. frequency-independent quality factor) attenuation, anisotropic attenuation, decoupled amplitude loss and velocity dispersion behaviours. The proposed viscoacoustic anisotropic equation can keep consistent velocity and attenuation anisotropy effects with that of qP-wave in the constant-Q viscoelastic anisotropic theory. For numerical simulations, the staggered-grid pseudo-spectral method is implemented to solve the velocity–stress formulation of wave equation in the time domain. The constant fractional-order Laplacian approximation method is used to cope with spatial variable-order fractional Laplacians for efficient modelling in heterogeneous velocity and Q media. Simulation results for a homogeneous model show the decoupling of velocity dispersion and amplitude loss effects of the constant-Q equation, and illustrate the influence of anisotropic attenuation on seismic wavefields. The modelling example of a layered model illustrates the accuracy of the constant fractional-order Laplacian approximation method. Finally, the Hess vertical transversely isotropic model is used to validate the applicability of the formulation and algorithm for heterogeneous media.     

双变参数标量纵波方程正演模拟方法

常见弹性波动理论的建立是基于介质均匀这一基本假设,实际介质的非均匀性非常普遍.为研究连续介质中波的传播特征,本文从弹性力学中建立弹性波动方程的三个基本方程出发,考虑连续介质弹性参数的空变特征,建立非均匀介质的弹性波动方程,利用Alkhalifah声学近似思想建立位移表征的纵波波动方程,利用本征值问题求解方法建立标量波频率-波数域传播算子,从而建立描述纵波传播的标量波方程,其中波函数为纵波位移的散度,不同于均匀介质标量波方程的波函数为位移势.随后推导含PML边界波动方程差分格式并建立不同模型数值模拟进行数值试算,与均匀假设标量波方程和变密度方程对比证明本方法的准确性和稳定性.     

Love waves in an inhomogeneous fluid saturated porous layered half-space with linearly varying properties

The paper is concerned with the propagation of the Love waves in an inhomogeneous transversely isotropic fluid saturated porous layered half-space with linearly varying properties. The analysis is based on Biot's theory. Firstly, the dispersion equation in the complex form for the Love waves in an inhomogeneous porous layer is derived. Then the equation is solved by an iterative method. Detailed numerical calculation is presented for an inhomogeneous fluid saturated porous layer overlying a purely elastic half-space. The dispersion and attenuation of the Love waves are discussed. In addition, the upper and lower bounds of the Love wave speed are explored.     

A semi-analytical artificial boundary for time-dependent elastic wave propagation in two-dimensional homogeneous half space

This paper presents a time-dependent semi-analytical artificial boundary for numerically simulating elastic wave propagation problems in a two-dimensional homogeneous half space. A polygonal boundary is considered in the half space to truncate the semi-infinite domain, with an appropriate boundary condition imposed. Using the concept of the scaled boundary finite element method, the wave equation of the truncated semi-infinite domain is represented by the partial differential equation of non-constant coefficients. The resulting partial differential equation has only one spatial coordinate variable and time variable. Through introducing a few auxiliary functions at the truncated boundary, the resulting partial differential equations are further transformed into linear time-dependent equations. This allows an artificial boundary to be derived from the time-dependent equations. The proposed artificial boundary is local in time, global at the truncated boundary and semi-analytical in the finite element sense. Compared with the scaled boundary finite element method, the main advantage in using the proposed artificial boundary is that the requirement for solving a matrix form of Lyapunov equation to obtain the unit-impulse response matrix is avoided, so that computer efforts are significantly reduced. The related numerical results from some typical examples have demonstrated that the proposed artificial boundary is of high accuracy in dealing with time-dependent elastic wave propagation in two-dimensional homogeneous semi-infinite domains.     

Low‐frequency wave propagation in periodically layered media

In this paper, we consider wave propagation in a layered medium. Using the Baker‐Campbell‐Hausdorff series, we expand the logarithm of a propagator matrix in series of frequency. The series coefficients allow us to extend the effective Backus medium for low frequencies. The proposed technique is applied to vertical propagation in a periodically layered and binary medium as well as for a gradient medium. The velocity dispersion equations are derived for these media. We also consider the layered medium with monoclinic anisotropy. We illustrate the accuracy of the proposed method on synthetic and well‐log data.     

Seismic ground motion in a layered half-space due to a Haskell-type source. I: Theory

In this paper, closed-form analytic expressions for the frequency-wave number domain Fourier amplitudes of the displacement field at the free surface of a layered, anelastic half-space are established. The displacement field is caused by a seismic source described by a shear dislocation propagating with constant velocity over a rectangular fault (Haskell's model). Three-dimensional plane wave propagation is considered in the layered half-space using a propagator-based formalism. The wave radiation from the source is decoupled into P-SV and SH motions and the two problems are treated separately. First, analytic expressions are calculated for the displacement field at the free surface due to unidirectional unit impulses. Then, these expressions are used to compute solutions for the displacement field due to effective point sources associated with a pure strike slip and a pure dip slip. Finally, these solutions are combined and integrated over the rectangular fault area to establish closed-form analytic expressions of the total displacement field at the free surface.     

Hamilton体系及弹性波在层状介质中的传播问题

利用结构力学与最优控制的模拟理论,研究弹性波在层状介质中传播的数值计算方法. 将弹性波传播问题导向哈密顿(Hamilton)体系,在哈密顿体系中,推导出一种新的半解析单元,称之为动力-部分杂交元,由此导出一套哈密顿体系下的半解析数值计算方法. 本文给出了该方法在层状正交各向异性材料介质的弹性波传播问题的数值算例,分析了一定频率的弹性波在层状介质中传播时的位移、应力的模式. 计算结果展现了Hamilton体系和辛几何在弹性波传播问题研究的应用前景.     

New approach to analyzing soil-building systems

A new method of analyzing seismic response of soil-building systems is introduced. The method is based on the discrete-time formulation of wave propagation in layered media for vertically propagating plane shear waves. Buildings are modeled as an extension of the layered soil media by assuming that each story in the building is another layer. The seismic response is expressed in terms of wave travel times between the layers, and the wave reflection and transmission coefficients at layer interfaces. The calculation of the response is reduced to a pair of simple finite-difference equations for each layer, which are solved recursively starting from the bedrock. Compared with commonly used vibration formulation, the wave propagation formulation provides several advantages, including the ability to incorporate soil layers, simplicity of the calculations, improved accuracy in modeling the mass and damping, and better tools for system identification and damage detection.     

基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移

时间域常Q黏声波方程,由于含分数阶时间导数项,数值求解需要大量内存,计算效率低,不利于地震偏移的实施.通过一系列近似,可将该方程简化为介质频散效应和衰减效应解耦的分数阶拉普拉斯算子黏声波方程,数值求解内存需求少,计算效率高.本文采用交错网格有限差分逼近时间导数,改进的伪谱法计算空间导数,PML吸收边界去除边界反射,对该方程进行数值离散和地震正演模拟,开展地震数据的黏声介质逆时偏移,实现波场逆时延拓过程中同时完成频散校正和衰减补偿.改善深层构造的成像精度,数值结果表明,基于分数阶拉普拉斯算子解耦的黏声介质地震正演模拟与逆时偏移可大幅度提高地震模拟计算效率,偏移剖面明显优于常规声波偏移剖面,极大改善深层构造的成像品质.     

流体饱和多孔隙介质弹性波方程边界元解法研究

基于流体饱和多孔隙各向同性介质模型,本文首先推导了流体饱和多孔隙介质中弹性波传播的频率域系统动力方程及边界积分方程,然后给出了流体饱和多孔隙介质弹性波方程的基本解,最后,利用本文给出的边界元方法对流体饱和多孔隙各向同性介质中的弹性波传播进行了数值模拟.结果表明：不论是从固相位移,还是液相位移的地震合成记录都能看到明显的慢速P波,本文提出的流体饱和多孔隙介质弹性波边界元法是有效可行的.     

土中Love波弥散特性

利用有限单元法及解析法建立和求解了土中Love波特征方程以及位移计算公式．计算结果表明,这一计算方法比纯解析法优越,可以用来分析均质和非均质上中Love波弥散性．本文利用这一方法详细讨论了Love波在上软下硬地基及软夹层地基中的传播特性和弥散特性．上软下硬地基Love波具有弥散性,土层的剪切波及厚度对Love波弥散曲线影响较大,而质量密度的相对变化对Love彼弥散曲线影响较小．软夹层地基中低频时Love波以第一模态波为主,现场所测为第一模态波波速;高频时存在多个高模态波,土中传播的波为这几个高模态波的叠加波,现场所测波速随两传感器的位置不同而有波动．     

地震波传播的哈密顿表述及辛几何算法

地震波传播过程本质上是能量在传播过程中逐步损耗直至殆尽的过程，而在实际应用中，常在无能量损耗假设下，用弹性波动方程或标量波动方程描述它.在哈密顿(Hamilton)体系表述下，地震波传播过程即为一个无限维的哈密顿系统随时间的演化过程.若不计能量损耗，波场演化过程实质上为一个单参数连续的辛变换，因而对应的数值算法应为辛几何算法.本文首先从地震波标量方程出发，给出哈密顿体系下地震波传播的表述，即任意两个时刻的波场是通过辛变换联系起来的.随后，把波场在时间和相空间离散化后，给出了用于波场计算的一些辛格式，如显式辛格式、隐式辛格式和蛙跳辛格式.并进一步讨论了有限差分格式和辛格式的异同.然后，应用显式辛格式和同阶的有限差分方法给出了同一理论速度模型下的波场和Marmousi速度模型下的单炮记录.数值结果表明，辛算法是一类可行的波场模拟的数值算法.在时间步长较小时，有限差分方法是辛算法的一个很好近似.文中的理论和方法，为地震波传播理论及实际应用研究提供了新的途径.