The Numerical Modeling of 3-D Elastic Wave Equation Using a High-Order,Staggered-Grid, Finite Difference Scheme

This article provides the application of the high-order, staggered-grid, finite-difference scheme to model elastic wave propagation in 3-D isotropic media. Here, we use second-order, temporal-and high-order spatial finite-difference formulations with a staggered grid for discretization of the 3-D elastic wave equations of motion. The set of absorbing boundary conditions based on paraxial approximations of 3-D elastic wave equations are applied to the numerical boundaries. The trial resuits for the salt model show that the numerical dispersion is decreased to a minimum extent, the accuracy high and diffracted waves abundant. It also shows that this method can be used for modeling wave propagation in complex media with the lateral variation of velocity.     

适用于更广渗透率范围含泥质砂岩的广义粘弹性BISQ模型

To simultaneously take into account the Biot-flow mechanism, the squirt-flow mechanism, and the frame-viscoelasticity mechanism, a generalized viscoelastic BISQ （Biot/squirt） model is developed for wave propagation in clay-bearing sandstones based on Dvorkin＇s elastic BISQ model. The present model is extended to a wide range of permeabilities （k 〉 0.05 mD） by introducing a dimensionless correction factor for viscoelastic parameters, defined as a function of the permeability and the clay content. We describe the frame＇s stress-strain relationship of the clay-bearing sandstones by the differential constitutive equations of generalized viscoelasticity and then derive the viscoelastic-wave dynamic equations. With the assumption of a plane-wave solution, we finally yield the phase velocities and the attenuation coefficients by solving the dynamic wave equations in the frequency and wave number domain. The comparison of numerical results and experimental data shows that the generalized viscoelastic BISQ model is applicable for modeling the wave propagation in most of the sandstones mainly bearing kaolinite clay.     

Dynamic responses of a sediment-filled valley with a fluid layer subject to incident waves

The diffraction of elastic waves by a sedimentary valley in a homogeneous elastic half-space is studied in this paper. The sediment-filled valley is composed of a fluid layer over a soft soil deposit whose characteristics may be significant and should be carefully considered when designing long span bridges with high piers. The method of analysis adopted in the paper is to decompose the problem into an interior region and an exterior region. In the exterior region, the scattered wave fields are constructed with the linear combinations of two independent sets of Lamb’s singular solutions, i.e., the integralsolutions for two concentrated surface loads in two directions; and their derivatives are used to represent the scattered wave fields. A technique is proposed to calculate the integrals in the wave-number domain based on the method of steepest descent. For the interior region, the wave fields for the fluid layer and soft soil deposit are expressed in terms of wave functions which satisfy the equation of motion. The continuity condition at the interface of the media is satisfied in the least square sense. The effects of geometric topography, soil amplification and fluid layer subject to different types of incident harmonic plane waves are analyzed and discussed..     

基于BISQ模型的三维双相各向异性介质数值模拟

Biot-flow and squirt-flow are the two most important fluid flow mechanisms in porous media containing fluids. Based on the BISQ （Biot-Squirt） model where the two mechanisms are treated simultaneously, the elastic wave-field simulation in the porous medium is limited to two-dimensions and two-components （2D2C） or two-dimensions and three-components （2D3C）. There is no previous report on wave simulation in three- dimensions and three-components. Only through three dimensional numerical simulations can we have an overall understanding of wave field coupling relations and the spatial distribution characteristics between the solid and fluid phases in the dual-phase anisotropic medium. In this paper, based on the BISQ equation, we present elastic wave propagation in a three dimensional dual-phase anisotropic medium simulated by the staggered-grid high-order finite-difference method. We analyze the resulting wave fields and show that the results are an improvement.     

Global SH-wave propagation in a 2D whole Moon model using the parallel hybrid PSM/FDM method

We present numerical modeling of SH-wave propagation for the recently proposed whole Moon model and try to improve our understanding of lunar seismic wave propagation. We use a hybrid PSM/FDM method on staggered grids to solve the wave equations and implement the calculation on a parallel PC cluster to improve the computing efficiency. Features of global SH-wave propagation are firstly discussed for a 100-km shallow and900-km deep moonquakes, respectively. Effects of frequency range and lateral variation of crust thickness are then investigated with various models. Our synthetic waveforms are finally compared with observed Apollo data to show the features of wave propagation that were produced by our model and those not reproduced by our models. Our numerical modeling show that the low-velocity upper crust plays significant role in the development of reverberating wave trains. Increasing frequency enhances the strength and duration of the reverberations.Surface multiples dominate wavefields for shallow event.Core–mantle reflections can be clearly identified for deep event at low frequency. The layered whole Moon model and the low-velocity upper crust produce the reverberating wave trains following each phases consistent with observation. However, more realistic Moon model should be considered in order to explain the strong and slow decay scattering between various phases shown on observation data.     

A whole-space transform formula of cylindrical wave functions for scattering problems

The theory of elastic wave scattering is a fundamental concept in the study of elastic dynamics and wave motion,and the wave function expansion technique has been widely used in many subjects.To supply the essential tools for solving wave scattering problems induced by an eccentric source or multi-sources as well as multi-scatters,a whole-space transform formula of cylindrical wave functions is presented and its applicability to some simple cases is demonstrated in this study.The transforms of wave functions in cylindrical coordinates can be classifi ed into two basic types: interior transform and exterior transform,and the existing Graf’s addition theorem is only suitable for the former.By performing a new replacement between the two coordinates,the exterior transform formula is fi rst deduced.It is then combined with Graf’s addition theorem to establish a whole-space transform formula.By using the whole-space transform formula,the scattering solutions by the sources outside and inside a cylindrical cavity are constructed as examples of its application.The effectiveness and advantages of the whole-space transform formula is illustrated by comparison with the approximate model based on a large cycle method.The whole-space transform formula presented herein can be used to perform the transform between two different cylindrical coordinates in the whole space.In addition,its concept and principle are universal and can be further extended to establish the coordinate transform formula of wave functions in other coordinate systems.     

In-plane dynamic Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic(TI)half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf(1985),and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.     

Stability of finite difference numerical simulations of acoustic logging-while-drilling with different perfectly matched layer schemes

In acoustic logging-while-drilling （ALWD） finite difference in time domain （FDTD） simulations, large drill collar occupies, most of the fluid-filled borehole and divides the borehole fluid into two thin fluid columns （radius -27 mm）. Fine grids and large computational models are required to model the thin fluid region between the tool and the formation. As a result, small time step and more iterations are needed, which increases the cumulative numerical error. Furthermore, due to high impedance contrast between the drill collar and fluid in the borehole （the difference is 〉30 times）, the stability and efficiency of the perfectly matched layer （PML） scheme is critical to simulate complicated wave modes accurately. In this paper, we compared four different PML implementations in a staggered grid finite difference in time domain （FDTD） in the ALWD simulation, including field-splitting PML （SPML）, multiaxial PML（M- PML）, non-splitting PML （NPML）, and complex frequency-shifted PML （CFS-PML）. The comparison indicated that NPML and CFS-PML can absorb the guided wave reflection from the computational boundaries more efficiently than SPML and M-PML. For large simulation time, SPML, M-PML, and NPML are numerically unstable. However, the stability of M-PML can be improved further to some extent. Based on the analysis, we proposed that the CFS-PML method is used in FDTD to eliminate the numerical instability and to improve the efficiency of absorption in the PML layers for LWD modeling. The optimal values of CFS-PML parameters in the LWD simulation were investigated based on thousands of 3D simulations. For typical LWD cases, the best maximum value of the quadratic damping profile was obtained using one do. The optimal parameter space for the maximum value of the linear frequency-shifted factor （a0） and the scaling factor （β0） depended on the thickness of the PML layer. For typical formations, if the PML thickness is 10 grid points, the global error can be reduced to 〈1% using the optimal PML parameters, and the error will decrease as the PML thickness increases.     

Coupled and splitting bedload sediment transport models based on a modified flux-wave approach

Numerical modeling of free-surface flow over a mobile bed with predominantly bedload sediment transport can be done by solving the shallow water and Exner equations using coupled and splitting approaches.The coupled method uses a coupling of the governing equations at the same time step leading to a non-conservative solution.The splitting method solves the Exner and the shallow water equations in a separate manner,and is only capable of modeling weak free-surface and bedload interactions.In the current study,an extended version of a Godunov-type wave propagation algorithm is presented for modeling of morphodynamic systems using both coupled and splitting approaches.In the introduced coupled method the entire morphodynamic system is solved in the form of a conservation law.For the splitting technique,a new wave Riemann decomposition is defined which enables the scheme to be utilized for mild and strong interactions.To consider the bedload sediment discharge within the Exner equation,the Smart and Meyer-Peter&Müller formulae are used.It was found that the coupled solution gives accurate predictions for all investigated flow regimes including propagation over a dry-state using a Courant-Friedrichs-Lewy(CFL)number equal to 0.6.Furthermore,the splitting method was able to model all flow regimes with a lower CFL number of 0.3.     

Global SH-wavefield calculation for a two-dimensional whole- Earth model with the parallel hybrid PSM/FDM algorithm

We present a parallel hybrid algorithm based on pseudospectral method （PSM） and finite difference method （FDM） for two-dimensional （2-D） global SH- wavefield simulation. The whole-Earth model is taken as a cross section of spherical Earth, and corresponding wave equations are defined in 2-D cylindrical coordinates. Spatial derivatives in the wave equations are approximated with efficient and high accuracy PSM in the lateral and high-order FDM in the radial direction on staggered grids. This algorithm allows us to divide the whole-Earth into sub-domains in radial direction and implement efficient parallel computing on PC cluster, while retains high accuracy and efficiency of PSM in lateral direction. A transformation of moment tensor between 3-D spherical Earth and our 2-D model was proposed to give corre- sponding moment tensor components used in 2-D modeling. Comparison of modeling results with those obtained by direct solution method shows very good accuracy of our algorithm. We also demonstrate its feasibility with a lateral heterogeneous whole-Earth model with localized velocity perturbation.     

起伏地表弹性波传播的间断Galerkin有限元数值模拟方法

间断Galerkin有限元法（DG-FEM）作为一种有效的高阶有限元法受到了国内外学者的广泛关注.本文基于任意高阶间断Galerkin有限元法对弹性波方程进行空间离散,并将离散后所得的非齐次线性常微分方程系统齐次化,最后结合针对齐次问题的强稳定性保持龙格库塔（SSP Runge-Kutta）算法,将DG-FEM推广至时间任意高阶精度.另外,借鉴近最佳匹配层（NPML）的思想,基于复频移（CFS）拉伸坐标变换推导了一种新的PML吸收边界条件（简称为CFS-NPML）,该CFS-NPML能够与DG-FEM算法很好地结合,形成有效的起伏地表地震波传播数值模拟技术.数值试验结果表明,DG-FEM具有高阶精度,可以适应任意复杂起伏地表和复杂构造情况下的弹性波传播数值模拟.同时,CFS-NPML对包括面波等震相的人为边界反射都具有良好的吸收效果.     

弹性波正演模拟中改进的非分裂式PML实现方法(英文)

在弹性波有限差分正演模拟中,吸收边界条件常用来吸收截断边界处引入的不期望边界反射,其中完全匹配层(PML)吸收边界条件被认为是目前最理想的吸收边界条件。但是PML吸收边界条件的传统实现却存在着很大不足:全局分裂式PML吸收边界条件实现简单但是需要占用太多内存;局部分裂式PML吸收边界条件需要考虑多个边界和角点区域,编程实现非常复杂;非分裂式PML吸收边界条件由于涉及卷积运算,计算量很大。本文基于非分裂式PML吸收边界条件,结合复频移伸展函数,提出了一种新的数值实现方法,其计算方程简单、占用内存小、编程实现容易,是对PML介质理论数值实现的改进和完善。     

求解二阶解耦弹性波方程的低秩分解法和低秩有限差分法

时间域的波场延拓方法在本质上都可以归结为对一个空间-波数域算子的近似.本文基于一阶波数-空间混合域象征,提出一种新的方法求解解耦的二阶位移弹性波方程.该方法采用交错网格,连续使用两次一阶前向和后向拟微分算子,推导得到了解耦的二阶位移弹性波方程的波场延拓算子.由于该混合域象征在伪谱算子的基础上增加了一个依赖于速度模型的补偿项,可以补偿由于采用二阶中心差分计算时间微分项带来的误差,有效地减少模拟结果的数值频散,提高模拟精度.然而,在非均匀介质中,直接计算该二阶的波场延拓算子,每一个时间步上需要做N次快速傅里叶逆变换,其中N是总的网格点数.为了减少计算量,提出了交错网格低秩分解方法;针对常规有限差分数值频散问题,本文将交错网格低秩方法与有限差分法结合,提出了交错网格低秩有限差分法.数值结果表明,交错网格低秩方法和交错网格低秩有限差分法具有较高的精度,对于复杂介质的地震波数值模拟和偏移成像具有重要的价值.     

求解3D弹性波方程的并行WNAD方法及其TI介质中的波场模拟

准确模拟TTI介质中弹性波的传播是研究地震各向异性、AVO反演的基础. 在二维加权近似解析离散化(WNAD)算法的基础上, 本文发展的并行WNAD算法是一种研究三维横向各向同性(TI)介质中弹性波传播的、快速高效的数值模拟方法. 我们首先介绍三维WNAD方法的构造过程, 然后与经典的差分格式--交错网格(SG)算法进行了比较. 理论分析和数值算例表明, WNAD算法比交错网格算法更适合在高性能计算机上进行大规模弹性波场模拟. 同时, 本文利用并行的WNAD方法研究了弹性波在TTI介质中的传播规律, 观测了TI介质中弹性波传播的重要特征:横波分离、体波耦合和速度各向异性等. 在TTI介质分界面处, 弹性波产生更加复杂的折射、反射和波型转化, 使得波场非常复杂, 研究和辨别不同类型的波能够加深我们对由裂隙诱导的各向异性介质的认识.     

A nearly analytic exponential time difference method for solving 2D seismic wave equations

In this paper, we propose a nearly analytic exponential time difference (NETD) method for solving the 2D acoustic and elastic wave equations. In this method, we use the nearly analytic discrete operator to approximate the high-order spatial differential operators and transform the seismic wave equations into semi-discrete ordinary differential equations (ODEs). Then, the converted ODE system is solved by the exponential time difference (ETD) method. We investigate the properties of NETD in detail, including the stability condition for 1-D and 2-D cases, the theoretical and relative errors, the numerical dispersion relation for the 2-D acoustic case, and the computational efficiency. In order to further validate the method, we apply it to simulating acoustic/elastic wave propagation in multilayer models which have strong contrasts and complex heterogeneous media, e.g., the SEG model and the Marmousi model. From our theoretical analyses and numerical results, the NETD can suppress numerical dispersion effectively by using the displacement and gradient to approximate the high-order spatial derivatives. In addition, because NETD is based on the structure of the Lie group method which preserves the quantitative properties of differential equations, it can achieve more accurate results than the classical methods.     

完全匹配层吸收边界在孔隙介质弹性波模拟中的应用

模拟弹性波在孔隙介质中传播，需要稳定有效的吸收边界来消除或尽可能的减小由人工边界引起的虚假反射. 本文在前人工作基础上，首次建立了弹性孔隙介质情况下完全匹配层吸收边界的高阶速度-应力交错网格有限差分算法，并详细讨论了完全匹配层的构建及其有限差分算法实现. 首先，本文通过均匀孔隙模型的数值解与解析解的对比，验证所提出的数值方法的正确性；然后，本文考察了完全匹配层对不同入射角度入射波和自由表面上的瑞利波的吸收性能，将完全匹配层与廖氏和阻尼吸收边界进行了对比，研究了这三种吸收边界在不同吸收厚度情况下对弹性波吸收能力. 数值结果表明，在孔隙介质中，完全匹配层作为吸收边界能十分有效地吸收衰减外行波，无论对体波还是面波，是一种高效边界吸收算法.     

基于精细积分法的三维弹性波数值模拟(英文)

波动方程有限差分法是地震数值模拟中的一种重要的方法,对理解和分析地震传播规律、分析地震属性和解释地震资料有着非常重要的意义。但是有限差分法由于其离散化的思想,产生了不稳定性。精细积分法在有限差分法的基础上,在时间域采用解析解的表达形式,在空间域保留任意差分格式,发展成为半解析的数值方法。本文结合并发展了以往学者的成果,推导了任意精细积分法的三维弹性波正演模拟计算公式,并对其稳定性进行了数值分析。在计算实例中,实现了精细积分法二维和三维弹性波模型的地震正演模拟,对计算结果的分析表明,精细积分法反射信号走时准确,稳定性好,弹性波场相较于声波波场,弹性波波场成分更为丰富,包含了更多波型成分(PP-和PS-反射波、透射波和绕射波),这对实际地震资料的解释和储层分析有重要的意义。实践证明,该方法可直接应用到弹性波的地质模型的数值模拟中。     

三角网格有限元法声波与弹性波模拟频散分析

本文对声波与弹性波方程进行有限元法离散,构造有限元法频散关系的一般特征值问题,分析了时间离散格式为中心差分的三角网格有限元法声波与弹性波模拟的频散特性. 比较了三种质量矩阵即分布式质量矩阵、集中质量矩阵和混合质量矩阵对有限元法频散的影响;选取四种典型三角网格,分析了混合质量矩阵有限元（MFEM）频散的方向各向异性;数值频散、方向各向异性随插值阶数的增加逐渐减弱,当空间为三阶插值时,频散主要表现为随采样率的变化而几乎无明显方向各向异性, 其频散幅值也较小. 控制其他影响因素不变的情况下,研究了不同波速比介质中弹性波的数值频散. 最后给出了三角网格MFEM的数值耗散性.     

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.     

A discontinuous Galerkin method for seismic wave propagation in coupled elastic and poroelastic media

Numerical simulation in coupled elastic and poroelastic media is important in oil and gas exploration. However, the interface between elastic and poroelastic media is a challenge to handle. In order to deal with the coupled model, the first-order velocity–stress wave equations are used to unify the elastic and poroelastic wave equations. In addition, an arbitrary high-order discontinuous Galerkin method is used to simulate the wave propagation in coupled elastic–poroelastic media, which achieves same order accuracy in time and space domain simultaneously. The interfaces between the two media are explicitly tackled by the Godunov numerical flux. The proposed forms of numerical flux can be used efficiently and conveniently to simulate the wave propagation at the interfaces of the coupled model and handle the absorbing boundary conditions properly. Numerical results on coupled elastic–poroelastic media with straight and curved interfaces are compared with those from a software that is based on finite element method and the interfaces are handled by boundary conditions, demonstrating the feasibility of the proposed scheme in dealing with coupled elastic–poroelastic media. In addition, the proposed method is used to simulate a more complex coupled model. The numerical results show that the proposed method is feasible to simulate the wave propagation in such a media and is easy to implement.