Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

Staggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.     

Finite-difference modelling of S-wave splitting in anisotropic media

We have implemented a 3D finite‐difference scheme to simulate wave propagation in arbitrary anisotropic media. The anisotropic media up to orthorhombic symmetry were modelled using a standard staggered grid scheme and beyond (monoclinic and triclinic) using a rotated staggered grid scheme. The rationale of not using rotated staggered grid for all types of anisotropic media is that the rotated staggered grid schemes are more expensive than standard staggered grid schemes. For a 1D azimuthally anistropic medium, we show a comparison between the seismic data generated by our finite‐difference code and by the reflectivity algorithm; they are in excellent agreement. We conducted a study on zero‐offset shear‐wave splitting using the finite‐difference modelling algorithm using the rotated staggered grid scheme. Our S‐wave splitting study is mainly focused on fractured media. On the scale of seismic wavelenghts, small aligned fractures behave as an equivalent anisotropic medium. We computed the equivalent elastic properties of the fractures and the background in which the fractures were embedded, using low‐frequency equivalent media theories. Wave propagation was simulated for both rotationally invariant and corrugated fractures embedded in an isotropic background for one, or more than one, set of fluid‐filled and dry fractures. S‐wave splitting was studied for dipping fractures, two vertical non‐orthogonal fractures and corrugated fractures. Our modelling results confirm that S‐wave splitting can reveal the fracture infill in the case of dipping fractures. S‐wave splitting has the potential to reveal the angle between the two vertical fractures. We also notice that in the case of vertical corrugated fractures, S‐wave splitting is sensitive to the fracture infill.     

Theory of equivalent staggered‐grid schemes: application to rotated and standard grids in anisotropic media

The previous finite‐difference numerical schemes designed for direct application to second‐order elastic wave equations in terms of displacement components are strongly dependent on Poisson's ratio. This fact makes theses schemes useless for modelling in offshore regions or even in onshore regions where there is a high Poisson's ratio material. As is well known, the use of staggered‐grid formulations solves this drawback. The most common staggered‐grid algorithms apply central‐difference operators to the first‐order velocity–stress wave equations. They have been one of the most successfully applied numerical algorithms for seismic modelling, although these schemes require more computational memory than those mentioned based on second‐order wave equations. The goal of the present paper is to develop a general theory that enables one to formulate equivalent staggered‐grid schemes for direct application to hyperbolic second‐order wave equations. All the theory necessary to formulate these schemes is presented in detail, including issues regarding source application, providing a general method to construct staggered‐grid formulations to a wide range of cases. Afterwards, the equivalent staggered‐grid theory is applied to anisotropic elastic wave equations in terms of only velocity components (or similar displacements) for two important cases: general anisotropic media and vertical transverse isotropy media using, respectively, the rotated and the standard staggered‐grid configurations. For sake of simplicity, we present the schemes in terms of velocities in the second‐ and fourth‐order spatial approximations, with second‐order approximation in time for 2D media. However, the theory developed is general and can be applied to any set of second‐order equations (in terms of only displacement, velocity, or even stress components), using any staggered‐grid configuration with any spatial approximation order in 2D or 3D cases. Some of these equivalent staggered‐grid schemes require less computer memory than the corresponding standard staggered‐grid formulation, although the programming is more evolved. As will be shown in theory and practice, with numerical examples, the equivalent staggered‐grid schemes produce results equivalent to corresponding standard staggered‐grid schemes with computational advantages. Finally, it is important to emphasize that the equivalent staggered‐grid theory is general and can be applied to other modelling contexts, e.g., in electrodynamical and poroelastic wave propagation problems in a systematic and simple way.     

A hybrid absorbing boundary condition for elastic staggered‐grid modelling

We recently proposed an efficient hybrid scheme to absorb boundary reflections for acoustic wave modelling that could attain nearly perfect absorptions. This scheme uses weighted averaging of wavefields in a transition area, between the inner area and the model boundaries. In this paper we report on the extension of this scheme to 2D elastic wave modelling with displacement‐stress formulations on staggered grids using explicit finite‐difference, pseudo‐implicit finite‐difference and pseudo‐spectral methods. Numerical modelling results of elastic wave equations with hybrid absorbing boundary conditions show great improvement for modelling stability and significant absorption for boundary reflections, compared with the conventional Higdon absorbing boundary conditions, demonstrating the effectiveness of this scheme for elastic wave modelling. The modelling results also show that the hybrid scheme works well in 2D rotated staggered‐grid modelling for isotropic medium, 2D staggered‐grid modelling for vertically transversely isotropic medium and 2D rotated staggered‐grid modelling for tilted transversely isotropic medium.     

一种基于全局优化的交错网格有限差分法

在地震波场数值模拟中, 交错网格有限差分技术得到了广泛的应用, 但是在弹性模量变化较大时, 通常会因插值而导致模拟误差增大. 旋转交错网格可以很好地克服这个缺点, 因而适合于各向异性介质正演模拟. 但是对于同样大小的网格单元, 旋转交错网格需要的步长比常规交错网格要大, 这会使梯度和散度算子的误差增大因而更易产生空间数值频散. 针对这些问题, 本文提出了旋转交错网格与紧致有限差分相结合的方法, 并基于模拟退火算法进行全局优化, 压制数值频散, 拓宽波数范围. 数值模拟结果表明, 此方法可以有效地压制数值频散, 且具有较高的模拟精度.      

Effective velocities in fractured media: a numerical study using the rotated staggered finite‐difference grid

The modelling of elastic waves in fractured media with an explicit finite‐difference scheme causes instability problems on a staggered grid when the medium possesses high‐contrast discontinuities (strong heterogeneities). For the present study we apply the rotated staggered grid. Using this modified grid it is possible to simulate the propagation of elastic waves in a 2D or 3D medium containing cracks, pores or free surfaces without hard‐coded boundary conditions. Therefore it allows an efficient and precise numerical study of effective velocities in fractured structures. We model the propagation of plane waves through a set of different, randomly cracked media. In these numerical experiments we vary the wavelength of the plane waves, the crack porosity and the crack density. The synthetic results are compared with several static theories that predict the effective P‐ and S‐wave velocities in fractured materials in the long wavelength limit. For randomly distributed and randomly orientated, rectilinear, non‐intersecting, thin, dry cracks, the numerical simulations of velocities of P‐, SV‐ and SH‐waves are in excellent agreement with the results of the modified (or differential) self‐consistent theory. On the other hand for intersecting cracks, the critical crack‐density (porosity) concept must be taken into account. To describe the wave velocities in media with intersecting cracks, we propose introducing the critical crack‐density concept into the modified self‐consistent theory. Numerical simulations show that this new formulation predicts effective elastic properties accurately for such a case.     

一种模拟非均匀介质中弹性波传播新的k-space方法

传统的伪谱(PS)方法,采用傅里叶变换(FT)计算空间导数具有很高的精度,每个波长仅需要两个采样点,而时间导数采用有限差分(FD)近似因而精度较低.当采用大时间步长时,由于时空精度不平衡,PS法存在不稳定性问题.原始的k-space方法可以有效地克服这些问题但是却无法适用于非均匀介质.为了提高原始k-space方法模拟非均匀介质波动方程的精度,我们提出了一种新的k-space算子族.它是用非均匀介质的变速度代替原k-space算子中的常数补偿速度构造得到,引入低秩近似可以高效求解.我们将构造的新的k-space算子应用于耦合的二阶位移波动方程,而不是交错网格一阶速度应力波动方程,使模拟弹性波的计算存储量减少.我们从数学上证明了基于二阶波动方程的k-space方法与基于一阶波动方程的k-space方法是等价的.数值模拟实验表明,与传统的PS、交错网格PS和原始的k-space方法相比,我们的新方法可以在时间和空间步长较大的均匀和非均匀介质中,为弹性波的传播提供更精确的数值解.在保持稳定性和精度的同时,采用较大的时空采样间隔,可以大大降低数值模拟的计算成本.     

黏弹性与弹性介质中Rayleigh面波特性对比研究

Rayleigh面波的频散特性可以用来研究地表浅层结构. 本文使用时域有限差分法来模拟复杂黏弹性介质中的Rayleigh面波,研究了Q值对面波频散特性的影响.文中采用旋转交错网格有限差分,以非分裂卷积形式的完全匹配层为吸收边界,推出了求解二阶位移-应力各向同性黏弹性波动方程的数值方法.为了检验数值解的精度,首先将简单模型的正演结果与解析解对比,验证了方法的正确性;然后模拟了横向缓变层状介质和含有洞穴的介质中的面波,对弹性和黏弹性介质中的面波的频散特性进行对比分析.模拟结果表明浅层Q值对面波的频散特性有显著的影响;强吸收情况下,高阶面波的能量相对低阶面波能量显著增强.     

黏弹TTI介质中旋转交错网格高阶有限差分数值模拟

以Carcione黏弹各向异性理论为基础,给出了适用于黏弹性具有任意倾斜对称轴横向各向同性介质(黏弹TTI介质)的二维三分量一阶速度-应力方程,采用旋转交错网格任意偶数阶精度有限差分格式求解该方程,并推导出了二维黏弹TTI介质完全匹配层(PML)吸收边界条件公式和相应的旋转交错网格任意偶数阶精度有限差分格式,实现了该类介质的地震波场数值模拟.数值模拟结果表明:该方法模拟精度高,边界吸收效果好,可以得到高精度的波场快照和合成记录;并且波场快照和合成记录能较好地反映地下介质的各向异性特征和黏弹性特征.     

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.     

波场模拟中的数值频散分析与校正策略

波动方程有限差分法正演模拟,对认识地震波传播规律、进行地震属性研究、地震资料地质解释、储层评价等,均具有重要的理论和实际意义.但有限差分法本身固有存在着数值频散问题,数值频散在正演模拟中是一种严重的干扰,会降低波场模拟的精度与分辨率.针对TI介质波场模拟的交错网格有限差分方法,本文从空间网格离散、时间网格离散和算子近似等三个方面对其产生的数值频散进行了分析,并结合其他学者的研究成果给出了TI介质波场模拟中压制数值频散的方法与策略：在已知介质频散关系时,对差分算子可实施算子校正;通过提高差分方程的阶数来提高波场模拟精度;采用流体力学中守恒式方程的通量校正传输方法来压制波场模拟中的数值频散;在实际正演模拟时,采用交错网格高阶有限差分方程,不仅在空间上采用高阶差分,而且在时间上也要采用高阶差分,否则只在单一方向上（空间或时间）提高方程的阶数对压制数值频散也不会取得理想的效果.     

交错网格高阶差分法三维弹性波数值模拟

本文应用交错网格高阶有限差分方法模拟弹性波在三维各向同性介质中的传播。采用时间上二阶、空间上高阶近似的交错网格高阶差分公式求解三维弹性波位移-应力方程,并在计算边界处应用基于傍轴近似法得到的三维弹性波方程吸收边界条件。在此基础上进行了三维盐丘地质模型的地震波传播数值模拟试算。试算结果表明该方法模拟精度高,在很大程度上减小了数值频散,绕射波更加丰富,而且适用于介质速度具有纵向变化和横向变化的情况。     

Optimal implicit staggered‐grid finite‐difference schemes based on the sampling approximation method for seismic modelling

We propose new implicit staggered‐grid finite‐difference schemes with optimal coefficients based on the sampling approximation method to improve the numerical solution accuracy for seismic modelling. We first derive the optimized implicit staggered‐grid finite‐difference coefficients of arbitrary even‐order accuracy for the first‐order spatial derivatives using the plane‐wave theory and the direct sampling approximation method. Then, the implicit staggered‐grid finite‐difference coefficients based on sampling approximation, which can widen the range of wavenumber with great accuracy, are used to solve the first‐order spatial derivatives. By comparing the numerical dispersion of the implicit staggered‐grid finite‐difference schemes based on sampling approximation, Taylor series expansion, and least squares, we find that the optimal implicit staggered‐grid finite‐difference scheme based on sampling approximation achieves greater precision than that based on Taylor series expansion over a wider range of wavenumbers, although it has similar accuracy to that based on least squares. Finally, we apply the implicit staggered‐grid finite difference based on sampling approximation to numerical modelling. The modelling results demonstrate that the new optimal method can efficiently suppress numerical dispersion and lead to greater accuracy compared with the implicit staggered‐grid finite difference based on Taylor series expansion. In addition, the results also indicate the computational cost of the implicit staggered‐grid finite difference based on sampling approximation is almost the same as the implicit staggered‐grid finite difference based on Taylor series expansion.     

A central difference method with low numerical dispersion for solving the scalar wave equation

In this paper, we propose a nearly‐analytic central difference method, which is an improved version of the central difference method. The new method is fourth‐order accurate with respect to both space and time but uses only three grid points in spatial directions. The stability criteria and numerical dispersion for the new scheme are analysed in detail. We also apply the nearly‐analytic central difference method to 1D and 2D cases to compute synthetic seismograms. For comparison, the fourth‐order Lax‐Wendroff correction scheme and the fourth‐order staggered‐grid finite‐difference method are used to model acoustic wavefields. Numerical results indicate that the nearly‐analytic central difference method can be used to solve large‐scale problems because it effectively suppresses numerical dispersion caused by discretizing the scalar wave equation when too coarse grids are used. Meanwhile, numerical results show that the minimum sampling rate of the nearly‐analytic central difference method is about 2.5 points per minimal wavelength for eliminating numerical dispersion, resulting that the nearly‐analytic central difference method can save greatly both computational costs and storage space as contrasted to other high‐order finite‐difference methods such as the fourth‐order Lax‐Wendroff correction scheme and the fourth‐order staggered‐grid finite‐difference method.     

等价各向异性介质一阶拟声波 方程正演模拟及其效率分析

为克服各向异性弹性波动方程正演模拟的局限,本文研究了各向异性介质拟声波方程的交错网格有限差分数值解法.首先,从VTI介质胡克定律和qP-qSV波频散关系两种思路出发,通过声假设近似,给出了两种不同形式的VTI介质一阶拟声波方程,并通过引入波场的伪速度分量,推导了一种新的VTI介质一阶应力-速度方程,并通过旋转坐标系将其推广到TTI介质中;其次,构造了一阶拟声波方程的交错网格高阶有限差分格式,并推导了相应的PML边界条件;最后,对本文方法中固有的qSV人为干扰波的产生机制和压制方法进行了简单讨论.数值结果表明:3种一阶拟声波方程在运动学和动力学上是等价的,相对于各向异性弹性波正演模拟,其节省了内存,提高了计算效率;各向异性因素会影响反射波旅行时和振幅等波场特征,在后续的处理、反演和解释中不可忽略;VTI介质HESS模型的逆时偏移结果也验证了本文方法的合理性.      

Modified image algorithm to simulate seismic channel waves in 3D tunnel model with rugged free surfaces

The existence of rugged free‐surface three‐dimensional tunnel conditions in the coal seams, caused either by geological or mining processes, will inevitably influence wave propagation characteristics when the seismic waves go through the coal mines. Thus, a modified image algorithm has been developed to account for seismic channel waves propagating through this complicated topography with irregular free surfaces. Moreover, the seismic channel waves commonly exhibit damped and dispersive signatures, which is not only because of their own unique sandwich geometry of rock–coal–rock but also because of the viscoelastic behavior of coal. Considering the complexity of programming in three‐dimensional tunnel models with rugged free surfaces, an optimized vacuum grid search algorithm, enabling to model highly irregular topography and to compute efficiently, is also proposed when using high‐order staggered finite‐difference scheme to simulate seismic channel wave propagations in viscoelastic media. The numerical simulations are implemented to investigate the accuracy and stability of the method and the impact of coal's viscoelastic behavior on seismic channel wave propagation characteristics. The results indicate that the automatic vacuum grid search algorithm can be easily merged into high‐order staggered finite‐difference scheme, which can efficiently be applied to calculate three‐dimensional tunnel models with rugged free surfaces in the viscoelastic media. The simulation also suggests that the occurrence of a three‐dimensional tunnel with free surfaces has a remarkable influence on the seismic channel wave propagation characteristics and elastic energy distribution.     

A Hamiltonian Particle Method with a Staggered Particle Technique for Simulating Seismic Wave Propagation

We present a Hamiltonian particle method (HPM) with a staggered particle technique for simulating seismic wave propagation. In the conventional HPM, physical variables, such as particle displacement and stress, are defined at the center, i.e., at the same position, of each particle. As most seismic simulations using finite difference methods (FDM) are practiced with staggered grid techniques, we know the staggered alignment of space variables could improve the numerical accuracy. In the present study, we hypothesized that staggered technique could improve the numerical accuracy also in the HPM and tested the hypothesis. First, we conducted a plane wave analysis for the HPM with the staggered particles in order to verify the validity of our strategy. The comparison of grid dispersion in our strategy with that in the conventional one suggests that the accuracy would be improved dramatically by use of the staggered technique. It is also observed that the dispersion of waves is dependent on the propagation direction due to the difference in the average spacing of the neighboring two particles for the same parameters, as is usually observed in FDM with a rotated staggered grid. Next, we compared the results from the conventional Lamb’s problem using our HPM with those from an analytical approach in order to demonstrate the effectiveness of the staggered particle technique. Our results showed better agreement with the analytical solutions than those from HPM without the staggered particles. We conclude that the staggered particle technique would be a method to improve the calculation accuracy in the simulation of seismic wave propagation.     

六分量弹性波场数值模拟与分析

地震波传播过程中,质点的振动不仅包括三个独立的平移部分,还包括三个独立的旋转部分.本文基于一阶速度-应力弹性波方程,采用分裂完全匹配层(SPML)的吸收边界条件,推导了时间导数二阶精度和空间导数高阶精度的交错网格有限差分格式的弹性波速度与应力各分量计算公式,模拟了各向同性介质中均匀模型和层状模型下的六分量波场,并对二维各向同性层状模型下的三个分量地震记录做高分辨率线性拉东变换得到各自的频散能谱.数值模拟分析结果表明:(1)旋转分量的能量要比平动分量弱的多;(2)在平动分量上,面波能量强,频率低,反射P波能量较强,反射S波能量稍弱;在旋转分量上,反射P波能量很弱,S波能量强;(3)与平动分量相比,旋转分量的频散能谱效果更好,能看到基阶和完整的高阶面波,即旋转分量能反映更多的地下介质信息.     

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.     

TTI介质的交错网格伪P波正演方法

研究了三维弱各向异性近似下,利用伪P波(伪纵波)模拟弹性波场P分量在倾斜对称轴的横向各向同性(TTI)介质中的传播过程,并对比了分别基于弹性Hooke定律、弹性波投影和运动学色散方程所建立的三种二阶差分伪P波方程的正演特点.目前这些伪P波方程数值计算主要采用规则网格差分,但是规则网格在TTI模拟中有低效率、低精度以及不稳定的缺点.为了提高计算的精度,本文构建出相应方程的交错网格有限差分格式.通过对比伪P波方程在三维TTI介质中不同的数值模拟的表达形式,本文认为基于色散方程所建立的伪P波方程在模拟弹性波中P波传播的过程中具有最小的噪声.本文分析不同的各向同性对称轴空间角度的频散特征,并引入适当的横波速度维持计算的稳定.二维模型算例表明,本文提出的交错网格正演算法可以得到稳定光滑的伪P波正演波场.使用本文交错网格算法对二维BP TTI模型的逆时偏移也具有较稳定的偏移结果.