三维弹性波正演模拟中的混合吸收边界条件（英文）

地震波场数值模拟中不可避免地会出现边界反射,一般采用吸收边界条件以压制人工边界反射。目前常用的分裂式完全匹配层(PML)边界条件需要在边界处进行特殊处理,尤其是在三维情况下需要将变量分裂为三个分量,增加了数值模拟的计算时间和内存占用量。与分裂式PML吸收边界条件相比,混合吸收边界条件(HABC)具有易于实现、计算量小和吸收效果好等优点,可以提高三维波动方程数值模拟的计算效率。本文将基于一阶Higdon单程波方程的混合吸收边界条件从二维计算域发展到三维,提出了适用于三维弹性波数值模拟的混合吸收边界条件。均匀模型以及复杂模型的三维数值模拟结果表明,混合吸收边界条件与传统的完全匹配层边界条件相比,具有效率高、吸收效果好的优势。     

适于弹性波方程无网格有限差分数值解的PML与CFS-PML吸收边界条件（英文）

无网格有限差分法能有效提高数值模拟的几何灵活性,且无需网格映射或复杂的网格生成过程。RBF-FD (基于径向基函数的有限差分)是最常用的无网格有限差分法之一,可以准确模拟地震波在非矩形计算域中的传播。本文提出适于弹性波方程无网格有限差分数值解的PML (完全匹配层)吸收边界条件,可以应用于非矩形速度模型的边界。但是PML吸收边界对近掠射波、低频波的吸收效果不好。为此,我们继续提出适于弹性波方程无网格有限差分数值解的CFS-PML(复频移完全匹配层)吸收边界条件。本文所提两种边界条件均是通过构造辅助微分方程,得到不分裂时域表达式,具有存储量小、便于编程实现的特点。模拟结果表明,两种PML吸收边界条件都能有效地消除无网格有限差分数值模拟的人工边界反射。此外,本文所提CFS-PML相较PML对近掠射波和损耗波的吸收效果更好。     

三维弹性波数值模拟中改进的NPML研究

在地震波传播数值模拟的过程中,需要使用吸收边界条件达到衰减人为边界反射的目的,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.但在三维地震波传播数值模拟中直接应用常规的PML吸收边界条件存在编程实现复杂、计算资源需求高等不足之处.本文基于传统的NPML吸收边界条件,提出了一种改进的方法.该方法通过引入伸展函数和辅助变量,把PML介质中的三维弹性波动方程分解成正常计算项和衰减项,其中衰减项的计算只在PML区域内部进行,其值只与坐标轴的方向相关,可通过离散数值迭代求解.分析和试验结果表明,该方法避免了卷积运算,减少了辅助变量的个数,具有理论方程简洁、编程实现简单、计算资源需求少等优点,能更有效地解决三维地震波传播数值模拟中的吸收边界问题.     

不分裂卷积完全匹配层与旋转交错网格有限差分在孔隙弹性介质模拟中的应用

完全匹配层吸收边界在地震波模拟中已广泛使用,但常用的场分裂格式完全匹配层吸收边界(SPML)和传统的不分裂完全匹配层吸收边界(NPML)对极低频入射波或大角度入射波的边界吸收效果不好.一种无需分裂和显式卷积计算的完全匹配层吸收边界(CPML)不仅能够解决常规PML吸收边界的不足,而且具有存储量小、计算效率高、易于编程实现的特点.本文将这种完全匹配层(CPML)吸收边界引入到孔隙弹性介质速度-应力格式的旋转交错网格有限差分算法中,对完全匹配层吸收边界参数进行数值分析,得到一组优化的参数.孔隙弹性介质数值模拟结果表明这种不分裂卷积完全匹配层的吸收效果优于常规完全匹配层.     

CPML条件下黏弹介质地震波传播的有限差分模拟与特征分析

在波动方程有限差分波场数值模拟中,为了使计算得到的边界波场值更接近于真实的边界波场值,需要使用边界条件以减少来自计算区域边界的人为反射能量。传统的分裂式完全匹配层(SPML)吸收边界条件不能有效吸收掠射波,后来发展的不分裂卷积完全匹配层(CPML)能够较好地处理这个问题,并且CPML在处理边界问题时无需对波场进行非物理分裂。本文针对Kelvin-Voigt模型的黏弹性介质,采用高阶交错网格有限差分方法进行数值模拟,同时利用CPML吸收边界条件进行边界处理。数值模拟结果表明:与弹性波相比,大角度入射产生的低频掠射波对黏弹性波的影响更大;CPML吸收边界条件对低频掠射波的吸收效果比传统的SPML吸收边界条件更好。     

基于递归积分的复频移PML弹性波方程交错网格高阶差分法

常规完全匹配吸收边界(PML)对以近掠射角入射到界面上的波以及低频波、损耗波都会产生虚假边界反射.基于递归积分的不分裂复频移PML算法,利用复频移拉伸函数,极大地改善了PML边界条件的性能,我们进一步推导出基于递归积分的不分裂复频移PML弹性波方程交错网格高阶差分法,对长条形介质模型进行数值模拟,与常规PML算法进行比较说明该算法对以掠射角入射到PML界面的波以及PML层内损耗波的吸收效果.     

一阶弹性波方程数值模拟中的混合吸收边界条件

Liu和Sen（2010和2012）在地震波场数值模拟中提出一种混合吸收边界条件. 该方法具有计算量小、容易实现及吸收效果好等优点. 但现有的混合吸收边界条件是针对二阶位移-应力方程设计的,存在稳定性问题. 本文首先推导了两种速度-应力单程波方程：二阶Higdon单程波方程和一阶Higdon单程波方程. 进而提出基于一阶弹性波方程的混合吸收边界条件方法. 在内部区域和边界之间引入一个过渡区域,通过单程波与双程波方程平滑过渡来消除人工边界反射. 为了改善混合吸收边界条件的吸收效果和稳定性,我们采用了能同时吸收纵、横波反射的一阶单程波方程和与变量位置有关的加权系数. 为了验证混合吸收边界条件的有效性,将其与常规分裂完全匹配层（PML）方法进行了比较. 数值模拟结果表明,与PML边界条件相比,混合吸收边界条件在耗用更小计算时间和存储量的前提下,可以获得更好的吸收效果. 另外,本文提出的两种混合吸收边界条件中,混合一阶Higdon吸收边界条件具有更好的稳定性.     

基于PML边界条件的二阶电磁波动方程GPR时域有限元模拟

完全匹配层(PML)作为一种稳定高效的吸收边界条件,广泛应用于基于一阶电磁波动方程的探地雷达(GPR)数值模拟中.为解决基于二阶电磁波动方程的GPR数值模拟的吸收边界问题,本文借鉴二阶弹性波动方程的PML边界条件构建思想,提出了一种适合二阶电磁波动方程GPR时域有限元模拟的PML边界条件.从二阶电磁波动方程出发,基于复拉伸坐标变换,推导了PML算法的频域表达式;通过合理构造辅助微分方程,得到了PML算法的时域表达式,并以变分形式(弱形式)加载到GPR时域有限元方程中,实现了PML边界条件在二阶电磁波动方程GPR时域有限元模拟中的应用.在此基础上,对比了无边界条件、Sarma边界条件和PML边界条件下均匀模型的波场快照、单道波形、时域反射误差和能量衰减曲线,结果表明:PML边界条件的吸收效果要远优于Sarma边界条件,具有近似零反射系数.一个复杂介质模型的正演模拟验证了PML边界条件在非均匀地电结构中电磁波传播模拟的良好吸收效果.     

稀疏存储的显式有限元三角网格地震波数值模拟及其PML吸收边界条件

有限元法是复杂介质地震模拟的有力工具,它能比较客观地反映地震波的传播,比较细致地再现地震图像.但是,为了获得较精确的结果,有限元法模拟地震波的传播需要的网格点数多,具有计算量大和消耗内存多的缺点.针对上述缺点,本文对刚度矩阵采用压缩存储行(CSR)格式,以减少计算量并节省内存;采用集中质量矩阵得到对角的质量矩阵以提高有限元法(显式有限元)的计算效率;时间离散采用保能量的Newmark算法以提高有限元法的计算精度;采用变分形式(弱形式)的PML吸收边界条件对人工截断边界进行处理.通过与高精度的数值方法--谱元法的数值试验的对比表明,上述方法的引入可使有限元法在计算精度和计算效率方面均可取得比较显著的改进.为了获得相当的计算精度,相比于7阶谱元法,显式有限元法需要更精细的网格.然而,显式有限元法的计算速度比前者快近2倍,而内存需求仅为谱元法的1/4~1/6.     

Lebedev网格改进差分系数TTI介质正演模拟方法研究

本文采用一种新的交错网格-Lebedev网格(LG)进行TTI介质的正演模拟研究,避免了Virieux标准交错网格(SSG)算法在处理TTI、单斜等各向异性介质时波场插值引入的数值误差,提高了模拟精度.在方法实现过程中,本文针对有限差分正演模拟面临的网格频散与边界反射两个关键性问题分别做了优化,并通过模型试算验证了它们的有效性与可行性:(1)结合最小二乘思想推导出新的频散改进差分系数(DIC),该系数比Taylor系数更能有效地压制粗网格引起的数值频散,可以节约内存,提高计算效率;(2)将分裂的多轴完全匹配层(M-PML)吸收边界条件引入到LG算法中,解决了传统PML边界条件在某些各向异性介质中的不稳定现象并且具有较好的边界吸收效果.     

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.     

三维逆时偏移GPU/CPU机群实现方案研究

叠前逆时偏移是当前最为准确的地震成像方法,由于计算量大、存储量大等原因需要合适的实现策略和高效的计算平台.本文以高阶有限差分逆时偏移为基础,重点讨论了在GPU上实现需要解决的显存不足问题和人工边界问题.利用区域分解技术可以在当前GPU上高效地实现任意生产规模的三维逆时偏移成像,不会受到GPU显存规模的制约.常规最佳匹配层边界条件边界区域控制方程与内部区域差异较大,不适于GPU高速运算.本文在GPU上实现近似最佳匹配层(NPML)边界条件,使得高阶有限差分计算不需要分支判断,边界区域辅助波场的存储量也较低,保证了在GPU上进行波场传播的高效性.三维理论数据和实际资料成像结果表明了本文方法的正确性.     

Hybrid absorbing boundary condition for three-dimensional elastic wave modeling

Edge reflections are inevitable in numerical modeling of seismic wavefields, and they are usually attenuated by absorbing boundary conditions. However, the commonly used perfectly matched layer (PML) boundary condition requires special treatment for the absorbing zone, and in three-dimensional (3D) modeling, it has to split each variable into three corresponding variables, which increases the computing time and memory storage. In contrast, the hybrid absorbing boundary condition (HABC) has the advantages such as ease of implementation, less computation time, and near-perfect absorption; it is thus able to enhance the computational efficiency of 3D elastic wave modeling. In this study, a HABC is developed from two-dimensional (2D) modeling into 3D modeling based on the 1st Higdon one way wave equations, and a HABC is proposed that is suitable for a 3D elastic wave numerical simulation. Numerical simulation results for a homogenous model and a complex model indicate that the proposed HABC method is more effective and has better absorption than the traditional PML method.     

Complex frequency-shifted multi-axial perfectly matched layer for frequency-domain seismic wavefield simulation in anisotropic media

Seismic anisotropy has an important influence on seismic data processing and interpretation. Although the frequency-domain seismic wavefield simulation has a problem of solving the large scale linear sparse matrix due to the computational limitations, it has some advantages over the time-domain seismic wavefield simulation including efficient inversion using only a limited number of frequency components and easy implementation of multiple sources. To accurately simulate seismic wave propagation in the frequency domain, we also need to choose the absorbing boundary conditions to absorb artificial reflections from edges of the model as we do in the time domain. Compared with the classical boundary conditions including the perfectly matched layer and complex frequency-shifted perfectly matched layer, the complex frequency-shifted multi-axial perfectly matched layer has been proven to effectively suppress the unwanted reflections at grazing incidence and solve the instability problem in the time-domain seismic numerical modelling in anisotropic elastic media. In this paper, we propose to extend the complex frequency-shifted multi-axial perfectly matched layer absorbing boundary condition to the frequency-domain seismic wavefield simulation in anisotropic elastic media. To test the validity of our proposed algorithm, we compare the results (snapshots and seismograms) of the frequency-domain seismic wavefield simulation with those of the time-domain modelling. The model studies indicate that the complex frequency-shifted multi-axial perfectly matched layer absorbing boundary condition is stable in the frequency-domain seismic wavefield simulation in anisotropic media, and provides better absorbing performance than the complex frequency-shifted perfectly matched layer boundary condition.     

完全匹配层在数值模拟孔隙介质声波传播中的应用

The nonsplitting perfectly matched layer （NPML） absorbing boundary condition （ABC） was first provided by Wang and Tang （2003） for the finite-difference simulation of elastic wave propagation in solids. In this paper, the method is developed to extend the NPML to simulating elastic wave propagation in poroelastic media. Biot＇s equations are discretized and approximated to a staggered-grid by applying a fourth-order accurate central difference in space and a second-order accurate central difference in time. A cylindrical twolayer seismic model and a borehole model are chosen to validate the effectiveness of the NPML. The results show that the numerical solutions agree well with the solutions of the discrete wavenumber （DW） method.     

Theoretical and quantitative evaluation of hybrid PML-ABCs for seismic wave simulation

A good artificial boundary treatment in a seismic wave grid-based numerical simulation can reduce the size of the computational region and increase the computational efficiency, which is becoming increasingly important for seismic migration and waveform inversion tasks requiring hundreds or thousands of simulations. Two artificial boundary techniques are commonly used: perfectly matched layers (PMLs), which exhibit the excellent absorption performance but impose a greater computational burden by using finite layers to gradually reduce wave amplitudes; and absorbing boundary conditions (ABCs), which have the high computational efficiency but are less effective in absorption because they employ the one-way wave equation at the exterior boundary. Naturally, PMLs have been combined with ABCs to reduce the number of PMLs, thus improving the computational efficiency; many studies have proposed such hybrid PMLs. Depending on the equations from which the ABCs are derived, there are two hybrid PML variants: the PML+unstretched ABC (UABC), in which the ABC is derived from a physical equation; or the PML+stretched ABC (SABC), in which the ABC is derived from the PML equation. Even though all the previous studies concluded that hybrid PMLs can improve the absorption performance, none of them quantified how many PMLs can be removed by combining the PML with the ABC compared with the pure PML. In this paper, we systematically study the absorption performance of the two hybrid PML variants. We develop a method to distinguish the artificial reflections from the PML-interior interface and those caused by the PML exterior boundary to accurately approximate the additional absorption achieved by using the UABC and the SABC. The reflection coefficients based on a theoretical derivation and numerical tests both show that the UABC amplifies most reflections and is not recommended in any situation; conversely, the SABC can always diminish reflections, but the additional absorption achieved by the SABC is relatively poor and cannot effectively reduce the number of PMLs. In contrast, we find that simply increasing the damping parameter improves absorption better than the PML+SABC. Our results show that the improvement in absorption achieved by combining the PML with either the SABC or the UABC is not better than that obtained by simply adjusting the damping profile of the PML; thus, combining the PML with the ABC is not recommended in practice.     

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.     

Perfectly matched layer boundary conditions for frequency-domain acoustic wave simulation in the mesh-free discretization

Mesh-free discretization, flexibly distributing nodes without computationally expensive meshing process, is able to deal with staircase problem, oversampling and undersampling problems and saves plenty of nodes through distributing nodes suitably with respect to irregular boundaries and model parameters. However, the time-domain mesh-free discretization usually exhibits poorer stability than that in regular grid discretization. In order to reach unconditional stability and easy implementation in parallel computing, we develop the frequency-domain finite-difference method in a mesh-free discretization, incorporated with two perfectly matched layer boundary conditions. Furthermore, to maintain the flexibility of mesh-free discretization, the nodes are still irregularly distributed in the absorbing zone, which complicates the situation of artificial boundary reflections. In this paper, we implement frequency-domain acoustic wave modelling in a mesh-free system. First, we present the perfectly matched layer boundary condition to suppress spurious reflections. Moreover, we develop the complex frequency shifted–perfectly matched layer boundary condition to improve the attenuation of grazing waves. In addition, we employ the radial-basis-function-generated finite difference method in the mesh-free discretization to calculate spatial derivatives. The numerical experiment on a rectangle homogeneous model shows the effectiveness of the perfectly matched layer boundary condition and the complex frequency shifted–perfectly matched layer boundary condition, and the latter one is better than the former one when absorbing large angle incident waves. The experiment on the Marmousi model suggests that the complex frequency shifted–perfectly matched layer boundary condition works well for complicated models.     

几种自由边界实施方法在完全 匹配层条件下的对比研究

传统的完全匹配层技术是一种能够较为有效地消除边界反射的边界条件,但是当表层为泊松比较高的自由表面时,该技术可能会产生不稳定的现象.针对传统的完全匹配层技术固有的不稳定和掠射情况下吸收效果不佳等缺陷,发展了多轴完全匹配层、卷积完全匹配层以及将两者结合的多轴卷积完全匹配层等3种边界条件.本文介绍了水平自由表面的不同处理方法以及传统、多轴、卷积和多轴卷积等4种完全匹配层条件的原理,通过二维半无限空间模型的交错网格有限差分正演模拟对比,分析了几种自由边界实施方法在这几种完全匹配层条件下的稳定性,并通过提取单道波形与解析解进行对比,定性分析了水平自由表面几种不同处理方法的准确性以及各自的适用条件. 结果表明,泊松比和水平自由表面实施方法对波场模拟效果及其稳定性有重要影响.