基于八阶NAD算子的保辛分部Runge-Kutta方法及其波场模拟(英文)

基于声波方程扩充的哈密尔顿系统,本文给出了空间精度为八阶的近似解析离散化(NAD)保辛分部Runge-Kutta方法,简称八阶NSPRK方法。该方法采用八阶精度的近似解析离散算子近似空间高阶偏微分算子,并使用二阶精度的辛分部Runge-Kutta方法进行时间离散。我们从理论和数值计算两个方面研究了八阶NSPRK方法的稳定性条件和数值频散关系,并同四阶NSPRK方法、八阶Lax-Wendroff(LWC)方法和八阶交错网格(SG)方法进行了比较。结果表明八阶NSPRK方法压制数值频散的能力显著优于传统数值计算方法。与四阶NSPRK方法和传统四阶辛格式(SPRK)方法相比,八阶NSPRK方法具有最小的数值误差和最高的计算效率:在达到同样消除数值频散的前提下,八阶NSPRK方法的计算速度约为四阶NSPRK方法的2.5倍、为四阶SPRK方法的3.4倍;八阶NSPRK方法的存储量仅为四阶NSPRK方法的47.17%、为四阶SPRK方法的49.41%。在双层介质、非均匀介质和Marmousi等复杂速度模型中,八阶NSPRK方法模拟得到的波场快照非常清晰,无可见数值频散。这些结果表明,八阶NSPRK方法在粗网格条件下能有效地压制数值频散,从而能够极大地节省计算内存,提高计算速度。总体而言,八阶NSPRK方法是一种在地震探测领域和地震学研究中有着巨大应用潜力的数值计算方法。     

基于保辛算法的声波叠前逆时偏移

叠前逆时偏移是目前成像精度最高的地震偏移方法之一,其实现过程中的一个重要步骤是数值求解全波方程,所以快速有效求解全波方程的数值算法对逆时偏移至关重要. 四阶近似解析辛可分Runge-Kutta (NSPRK) 方法是近年发展的一种具有高效率、高精度的数值求解波动方程的保辛差分方法, 能在粗网格条件下有效压制数值频散, 从而提高计算效率, 节省计算机内存需求量. 本文利用四阶NSPRK方法构造的基本思想,发展了具有六阶空间精度的NSPRK方法,并对新的六阶NSPRK方法进行了详细的稳定性和数值频散分析,以及计算效率比较和波场模拟. 同时将该方法用于声波叠前逆时偏移中, 得到一种时间上保辛、空间具有六阶精度、低数值频散、可应用大步长进行波场延拓并能长时计算的叠前逆时偏移方法,对Sigsbee2B模型进行了偏移成像, 并和四阶NSPRK方法、传统的六阶差分方法、四阶Lax-Wendroff correction (LWC) 方法进行了对比. 数值结果表明, 基于六阶NSPRK方法的叠前逆时偏移能得到更好的成像结果, 是一种优于四阶NSPRK方法、传统的六阶差分方法、四阶LWC叠前逆时偏移的方法, 尤其是在粗网格情况下具有更明显的优越性.     

地震波动方程的局部间断有限元方法数值模拟

近年来,随着地震波数值模拟对计算精度和效率的要求越来越高,间断有限元方法开始受到越来越多的关注.本文中,针对具有吸收边界条件的二维地震声波波动方程,作者提出了一种基于局部间断有限元方法的数值模拟算法.该算法在空间上使用局部间断有限元方法进行离散,在时间上采用了显式蛙跳格式.在这种时空离散的组合方式下,每个时间步上,此算法在空间剖分的每个单元上的求解计算是相互独立的,因而具有极高的并行性.通过数值算例,我们将该算法与连续有限元方法进行了比较.结果表明,本算法不仅具有对起伏构造的良好适应性,而且在计算效率和计算精度等方面,都具有优越性.     

时间-空间高阶精度矩形交错网格隐式有限差分声波正演模拟

有限差分方法因其操作简单、计算消耗低而成为地震勘探领域中最为常用的数值模拟方法之一,然而用离散的显式差分算子数值逼近地震波动方程中的连续导数容易导致数值频散,并且基于正方形网格离散形式的有限差分方法对不同地质模型的适应性较低.针对一阶变密度声波方程的数值模拟,本文发展了一种适用于矩形网格离散形式的时间高阶空间隐式有限差分格式,可以有效压制时间和空间频散,同时灵活的网格剖分增强了其应用的广泛性.基于本文矩形交错网格时间高阶空间隐式有限差分格式的时空域频散关系和变量替换的思想,首先采用泰勒级数展开方法求解不同方向的非轴上时间差分系数及轴上空间差分系数,使本文差分格式可以获得任意偶数阶时间和空间精度.为了进一步提高本文差分格式在更大波数区域的空间模拟精度,我们采用线性优化方法来求取新的轴上空间差分系数用于一阶变密度声波方程的波场迭代求解中.频散、稳定性分析及数值模拟算例表明：相比于传统十字形空间域隐式有限差分格式,本文矩形交错网格时间高阶空间隐式有限差分格式在精度、稳定性和效率方面均具有优势.     

二维时间-空间域声波全波形速度反演方法研究

我们采用区域分解(物理上分割模型,使用基于MPI的分布式存储架构的计算集群,从而节约单个CPU内核的内存使用量,加快正演数值模拟的计算速度)和炮并行(能够加快计算速度)的双并行算法,进行L-BFGS算法的二维时间-空间域声波全波形速度反演.我们采用多尺度的策略,只需要使用三个离散频率(5 Hz,8 Hz,12 Hz),从数据的低频成分开始反演,将低频的反演结果作为高频反演时的初始速度模型,依次反演数据的高频成分,进行Marmousi理论模型的全波形反演数值试验.数值试验反演所恢复得到的速度证实了:二维时间-空间域声波全波形速度反演方法计算灵活,可以适用于任何的采集观测系统,对地震数据可以方便地加时窗;使用多个计算节点同时计算多炮时,能够多倍提高数值计算效率.二维时间-空间域声波全波形速度反演所恢复得到的速度模型的分辨率较高.     

四阶龙格-库塔方法的一种改进算法及地震波场模拟

提出了求解波动方程的四阶龙格-库塔方法的一种改进算法.首先将原四阶龙格-库塔方法合并为两级格式, 然后在第一级中引入加权参数以获得加权算法. 针对这种改进方法,研究了它的稳定性条件; 对一维问题导出了频散关系, 给出了数值频散结果,并与四阶的 Lax-Wendroff (LWC) 方法和位移-应力交错网格方法进行了对比; 对二维问题, 使用我们的改进方法、四阶LWC和交错网格三种方法进行了声波波场模拟, 并进行了计算效率分析和不同方法计算结果的比较; 最后选取两个层状介质模型进行了声波和弹性波波场模拟. 数值结果表明,本文的改进方法具有非常弱的数值频散和高的计算效率, 是一种在地震勘探领域具有巨大应用潜力的数值方法.     

二维时间空间域和频率空间域声波全波形速度反演方法的对比研究

本文将二维时间空间域和频率空间域声波全波形速度反演方法分别应用到Marmousi模型,进行数值试验.两种方法均采用相同的观测系统和其他的参数,理论模型的数值试验结果证实了:使用较多的计算集群的CPU进行二维频率空间域直接法声波全波形反演时,其加速有限(正演数值模拟的计算量主要用于稀疏矩阵的LU分解,炮点计算波场时为线性关系).二维时间空间域声波全波形反演计算时更灵活,多炮同时计算时,可以多倍提高其计算效率;二维声波全波形速度反演时,直接法求解频率空间域的计算速度远快于时间空间域,所需要的计算机内存也比时间空间域少.二维声波全波形速度反演时,相比较于时间空间域的方法,频率空间域直接法声波全波形反演具有计算速度快和节省计算机内存需求的优势.     

二维地震波场的组合型紧致有限差分数值模拟

地震波场数值模拟在地球物理勘探和地震学中具有重要的支撑作用.本文将组合型紧致差分格式用于声波和弹性波方程的数值模拟中.根据泰勒级数展开和声波方程,建立了位移场时间四阶离散格式,并将组合型紧致差分格式用于位移场空间导数的求取,然后对该差分格式进行了精度分析、误差分析、频散分析和稳定性分析.理论研究结果表明：①该差分格式为时间四阶、空间六阶精度,与常规七点六阶中心差分和五点六阶紧致差分相比,具有更小的截断误差和更高的模拟精度;②每个波长仅需要5.6个采样点,且满足稳定性条件的库郎数为0.792,可以使用粗网格和较大时间步长进行计算.所以该方法具有占用内存少、计算效率高和低数值频散等优势.最后,本文进行了二维各向同性完全弹性介质的声波和弹性波方程的数值模拟,实验结果表明本文提出的方法具有更高的计算精度,能够大幅度的节约计算量和内存需求,对于三维大尺度模型问题具有更好的适应性.     

非均匀介质中地震波应力场的WNAD方法及其数值模拟

通过对近似解析离散化(NAD)方法的分析,给出了一种求解声波和弹性波方程的带权重的近似解析离散化(WNAD)方法,并用WNAD方法、Lax-Wendroff 修正格式(LWC)和二阶中心差分方法计算了二维波动方程初值问题的应力场数值误差.结果表明WNAD方法具有更高的数值精度.用WNAD方法、LWC和四阶交错网格法对二维非均匀介质中弹性波传播的应力场进行了数值模拟.应力场快照和地表地震记录表明,即使是在粗网格条件下WNAD方法的模拟结果仍无可见的数值频散和源噪声.另一方面,由于WNAD方法同时计算了地震位移和梯度场,使得应力的计算更为便捷和精确,而且WNAD方法中波位移梯度局部连接关系的使用使得应力在间断处能够自动近似地满足应力连续性.     

2.5维地震波数值模拟评述：声波模型

本文的目的是对基于声波模型的2.5维地震波数值模拟工作进行评述,以便能够找出其存在的问题和解决这些问题的可能途径,根据定义,2.5维问题是三维问题中的一种特殊情况,其特点是:(1)介质参数沿走向保持为常数;(2)场源具有球对称性;(3)场源和接收点均位于垂直于走向的直线上,与三维数值模拟问题不同,2.5维数值模拟问题分为两部分:(1)在垂直于走向的平面内用数值方法解相应的微分方程,这在实质上是二维问题;(2)采用积分变换或其他方法处理来自于计算平面外的影响,这实际上是将一个特殊的三维问题转化成为了无限多个(在离散情况下是有限多个)二维问题的叠加,与二维模型相比,2.5维模型能得到计算平面内的精确地震波振幅信息,鉴于声波模型是反射地震偏移成像理论和应用研究中的基本数据模型,所以对2.5维声波数值模拟的研究具有重要的意义,根据对计算平面外传播效应的处理方式可以将到目前为止提出的2.5维声波数值模拟方法分为四类:(1)几何射线法;(2)滤波校正法;(3)Fourier变换法;(4)近似波动方程法.其中,几何射线法具有直观、快速的特点,但是在焦散区内失效,滤波校正法只在均匀介质条件下严格成立,在一般条件下只是一种精度难以估计的近似,Fourier变换法是一种经典方法,其研究程度已经相当深入,该方法的基本思想是通过沿走向的Fourier变换将2.5维问题转化为有限多个二维问题,从而,对反变换的数值实现直接影响到该方法的精度和效率,近似波动方程法的宗旨是针对2.5维波动问题建立专门的波动方程,与Fourier变换法相比,近似波动方程法等同于一个二维数值模拟,因此可以大大地降低计算量,但是,根据笔者所掌握的资料,到目前为止提出的几个近似波动方程不是具有很大的振幅误差,就是难以进行数值计算,因此,有必要对近似波动方程的形式进行进一步的研究.     

Symplectic partitioned Runge-Kutta method based on the eighth-order nearly analytic discrete operator and its wavefield simulations

We propose a symplectic partitioned Runge-Kutta （SPRK） method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete （NAD） operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction （LWC） method and the eighth-order staggered-grid （SG）method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.     

A nearly analytic symplectic partitioned Runge–Kutta method based on a locally one-dimensional technique for solving two-dimensional acoustic wave equations

In this paper, we develop a new nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique for numerically solving two-dimensional acoustic wave equations. We first split two-dimensional acoustic wave equation into the local one-dimensional equations and transform each of the split equations into a Hamiltonian system. Then, we use both a nearly analytic discrete operator and a central difference operator to approximate the high-order spatial differential operators, which implies the symmetry of the discretized spatial differential operators, and we employ the partitioned second-order symplectic Runge–Kutta method to numerically solve the resulted semi-discrete Hamiltonian ordinary differential equations, which results in fully discretized scheme is symplectic unlike conventional nearly analytic symplectic partitioned Runge–Kutta methods. Theoretical analyses show that the nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique exhibits great higher stability limits and less numerical dispersion than the nearly analytic symplectic partitioned Runge–Kutta method. Numerical experiments are conducted to verify advantages of the nearly analytic symplectic partitioned Runge–Kutta method based on locally one-dimensional technique, such as their computational efficiency, stability, numerical dispersion and long-term calculation capability.     

Seismic wavefield modeling based on time-domain symplectic and Fourier finite-difference method

Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference (symplectic FFD) method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps.     

最优化辛格式广义离散奇异核褶积微分算子地震波场模拟

将波动方程变换至Hamilton体系,构造了一种新的保结构算法,即最优化辛格式广义褶积微分算子(OSGCD). 在时间离散上,首先引入了Lie算子设计二级二阶辛格式,基于最小误差原理得到了优化的辛格式. 在空间离散上,引入广义离散奇异核褶积微分算子计算空间微分,提出了一种有效方法优化GCD并得到了稳定的算子系数. 针对本文发展的新方法,给出了OSGCD稳定性条件. 在数值实验中,将OSGCD与多种方法比较,从精度和计算效率两方面分析了OSGCD的计算优势,计算结果也表明OSGCD长时程以及非均匀介质中地震波模拟亦具有较强能力.     

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 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.     

A new kind of optimal second-order symplectic scheme for seismic wave simulations

Here we introduce generalized momentum and coordinate to transform seismic wave displacement equations into Hamiltonian system. We define the Lie operators associated with kinetic and potential energy, and construct a new kind of second order symplectic scheme, which is extremely suitable for high efficient and long-term seismic wave simulations. Three sets of optimal coefficients are obtained based on the principle of minimum truncation error. We investigate the stability conditions for elastic wave simulation in homogeneous media. These newly developed symplectic schemes are compared with common symplectic schemes to verify the high precision and efficiency in theory and numerical experiments. One of the schemes presented here is compared with the classical Newmark algorithm and third order symplectic scheme to test the long-term computational ability. The scheme gets the same synthetic surface seismic records and single channel record as third order symplectic scheme in the seismic modeling in the heterogeneous model.     

求解声波方程的辛RKN格式

将声波方程变换至Hamiltion体系,构造了适用于高效声波模拟的二阶显式辛Runge-Kutta-Nyström（RKN）格式,运用根数理论得到此格式的阶条件方程组. 针对两个自由度的辛条件方程组,根据三次项截断误差最小原理得到一种误差最小辛格式;通过分析声波的时间演进方程的稳定性,选择不同的辛系数使演进方程更稳定,并得到了另一种更为稳定辛格式;在频散关系分析中,选择使数值频散最小的辛系数,得到第三种最小频散辛格式. 在理论分析中,这组辛RKN格式相比常见格式在精度控制、数值频散压制以及稳定性提升等方面均具有明显优势;在数值实验中,通过具体算例验证了理论分析的正确性.     

Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method

Seismic wave propagation in transversely isotropic (TI) media is commonly described by a set of coupled partial differential equations, derived from the acoustic approximation. These equations produce pure P‐wave responses in elliptically anisotropic media but generate undesired shear‐wave components for more general TI anisotropy. Furthermore, these equations suffer from instabilities when the anisotropy parameter ε is less than δ. One solution to both problems is to use pure acoustic anisotropic wave equations, which can produce pure P‐waves without any shear‐wave contaminations in both elliptical and anelliptical TI media. In this paper, we propose a new pure acoustic transversely isotropic wave equation, which can be conveniently solved using the pseudospectral method. Like most other pure acoustic anisotropic wave equations, our equation involves complicated pseudo‐differential operators in space which are difficult to handle using the finite difference method. The advantage of our equation is that all of its model parameters are separable from the spatial differential and pseudo‐differential operators; therefore, the pseudospectral method can be directly applied. We use phase velocity analysis to show that our equation, expressed in a summation form, can be properly truncated to achieve the desired accuracy according to anisotropy strength. This flexibility allows us to save computational time by choosing the right number of summation terms for a given model. We use numerical examples to demonstrate that this new pure acoustic wave equation can produce highly accurate results, completely free from shear‐wave artefacts. This equation can be straightforwardly generalized to tilted TI media.