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1.
New formulations of boundary conditions at an arbitrary two-dimensional (2D) free-surface topography are derived. The top of a curved grid represents the free-surface topography while the grid's interior represents the physical medium. The velocity–stress version of the viscoelastic wave equations is assumed to be valid in this grid. However, the rectangular grid version attained by grid transformation is used to model wave propagation in this work in order to achieve the numerical discretization. We show the detailed solution of the particle velocities at the free surface resulting from discretizing the boundary conditions by second-order finite-differences (FDs). The resulting system of equations is spatially unconditionally stable. The FD order is gradually increased with depth up to eighth order inside the medium. Staggered grids are used in both space and time, and the second-order leap-frog and Crank–Nicholson methods are used for time-stepping. We simulate point sources at the surface of a homogeneous medium with a plane free surface containing a hill and a trench. Applying parameters representing exploration surveys, we present examples with a randomly realized surface topography generated by a 1D von Kármán function of order 1. Viscoelastic simulations are presented using this surface with a homogeneous medium and with a layered, randomized medium realization, all generating significant scattering. 相似文献
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We have pursued two-dimensional (2D) finite-difference (FD) modelling of seismic scattering from free-surface topography. Exact free-surface boundary conditions for the particle velocities have been derived for arbitrary 2D topographies. The boundary conditions are combined with a velocity–stress formulation of the full viscoelastic wave equations. A curved grid represents the physical medium and its upper boundary represents the free-surface topography. The wave equations are numerically discretized by an eighth-order FD method on a staggered grid in space, and a leap-frog technique and the Crank–Nicholson method in time.
In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back-scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter-crust heterogeneities. 相似文献
In order to demonstrate the capabilities of the surface topography modelling technique, we simulate incident point sources with a sinusoidal topography in seismic media of increasing complexities. We present results using parameters typical of exploration surveys with topography and heterogeneous media. Topography on homogeneous media is shown to generate significant scattering. We show additional effects of layering in the medium, with and without randomization, using a von Kármán realization of apparent anisotropy. Synthetic snapshots and seismograms indicate that prominent surface topography can cause back-scattering, wave conversions and complex wave patterns which are usually discussed in terms of inter-crust heterogeneities. 相似文献
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本文以基于改进BISQ模型的二维双相各向同性介质一阶速度-应力方程为基础，推导出了曲线坐标系下对应的方程，然后采用低频散、低耗散的同位网格MacCormack有限差分法来离散方程，并采用紧致的单边MacCormack差分格式结合牵引力镜像法来施加自由地表边界条件，实现了地震波场数值模拟.曲线网格有限差分法采用贴体网格来描述自由表面，地表的网格线紧贴地形，避免了台阶近似造成的数值散射.数值模拟结果表明，在双相介质起伏自由地表和分界面处，各类波型复杂的反射透射规律可以清晰展现，曲线网格有限差分法可以精确地解决地震波在含起伏地表的双相各向同性介质中的传播问题. 相似文献
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In this paper, we deduced the corresponding first-order velocity–stress equation for curvilinear coordinates from the first-order velocity–stress equation based on the modified Biot/squirt model for a two-dimensional two-phase medium. The equations are then numerically solved by an optimized high-order non-staggered finite difference scheme, that is, the dispersion relation preserving/optimization MacCormack scheme. To implement undulating free-surface topography, we derive an analytical relationship between the derivatives of the particle velocity components and use the compact finite-difference scheme plus a traction-image method. In the undulating free surface and the undulating subsurface interface of two-phase medium, the complex reflected wave and transmitted wave can be clearly recognized in the numerical simulation results. The simulation results show that the curvilinear-grid finite-difference method, which uses a body-conforming grid to describe the undulating surface, can accurately reduce the numerical scattering effect of seismic wave propagation caused by the use of ladder-shaped grid to fit the surfaces when undulating topography is present in a two-phase isotropic medium. 相似文献
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Seismic wavefield modeling in media with fluid-filled fractures and surface topography 总被引：3，自引：0，他引：3
We present a finite difference (FD) method for the simulation of seismic wave fields in fractured medium with an irregular (non-flat) free surface which is beneficial for interpreting exploration data acquired in mountainous regions. Fractures are introduced through the Coates-Schoenberg approach into the FD scheme which leads to local anisotropic properties of the media where fractures are embedded. To implement surface topography, we take advantage of the boundary-conforming grid and map a rectangular grid onto a curved one. We use a stable and explicit second-order accurate finite difference scheme to discretize the elastic wave equations (in a curvilinear coordinate system) in a 2D heterogeneous transversely isotropic medium with a horizontal axis of symmetry (HTI). Efficiency tests performed by different numerical experiments clearly illustrate the influence of an irregular free surface on seismic wave propagation in fractured media which may be significant to mountain seismic exploration. The tests also illustrate that the scattered waves induced by the tips of the fracture are re-scattered by the features of the free surface topography. The scattered waves provoked by the topography are re-scattered by the fractures, especially Rayleigh wave scattering whose amplitudes are much larger than others and making it very difficult to identify effective information from the fractures. 相似文献
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The Numerical Modeling of 3-D Elastic Wave Equation Using a High-Order,Staggered-Grid, Finite Difference Scheme 总被引：1，自引：0，他引：1
XiaFan DongLiangguo MaZaitian 《应用地球物理》2004,1(1):38-41
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. 相似文献
9.
The perfectly matched layer is an efficient tool to simulate nonreflecting boundary condition at boundaries of a grid in the
finite-difference modeling of seismic wave propagation. We show relations between different formulations of the perfectly
matched layer with respect to their three key aspects — split/unsplit, classical/convolutional, with the general/special form
of the stretching factor. First we derive two variants of the split formulations for the general form of the stretching factor.
Both variants naturally lead to the convolutional formulations in case of the general form of the stretching factor. One of
them, L-split, reduces to the well-known classical split formulation in case of the special form of the stretching factor.
The other, R-split, remains convolutional even for the special form of the stretching factor. The R-split formulation eventually
leads to the equations identical with those obtained straightforwardly in the unsplit formulation.
We also present an alternative time discretization of the unsplit formulation that is slightly algorithmically simpler than
the discretization presented recently. We implement the discretization in the 3D velocity-stress staggered-grid finite-difference
scheme — 4th-order in the interior grid, 2nd-order in the perfectly matched layer. 相似文献
10.
基于应力、速度混合变量弹性波方程及任意四边形网格差分算子,给出了交错计算应力及速度的非规则网格弹性波应力一速度差分法该方法融合了有限元法能适应复杂形状边界及差分法无需计算刚度阵的特点,具有较高的计算精度,所需计算机存储空间较少,计算效率也很高．基于积分平衡方程引入了任意形状自由表面的边界条件,且通过局部滤波改善了自由表面边界条件的稳定性,使得该方法可应用于考虑地表形状影响的地震波数值模拟 相似文献
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During seismic wave propagation on a free surface, a strong material contrast boundary develops in response to interference by P- and S- waves to create a surfacewave phenomenon. To accurately determine the effects of this interface on surface-wave propagation, the boundary conditions must be accurately modeled. In this paper, we present a numerical approach based on the dynamic poroelasticity for a space–time-domain staggeredgrid finite-difference simulation in porous media that contain a free-surface boundary. We propose a generalized stess mirror formulation of the free-surface boundary for solids and fluids in porous media for the grid mesh on which lays the free-surface plane. Its analog is that used for elastic media, which is suitable for precise and stable Rayleigh-type surface-wave modeling. The results of our analysis of first kind of Rayleigh (R1) waves obtained by this model demonstrate that the discretization of the mesh in a similar way to that for elastic media can realize stable numerical solutions with acceptable precision. We present numerical examples demonstrating the efficiency and accuracy of our proposed method. 相似文献
12.
Christopher Juhlin 《Geophysical Prospecting》1995,43(6):843-858
The velocity-stress formulation for propagation of elastic seismic waves through 2D heterogeneous transversely isotropic media of arbitrary orientation is presented. The equations are recast into a finite-difference scheme and solved numerically using fourth-order spatial operators and a second-order temporal operator on a staggered grid. Absorbing, free-surface and symmetry boundary conditions have been implemented. Test cases compare well with other published solutions. Synthetic seismograms are calculated over two idealized models: (i) vertical fractures in granite with a dolerite sill reflector and (ii) a dipping anisotropic shale. Comparisons with the isotropic counterparts show significant differences which may have to be accounted for in seismic processing in the future. 相似文献
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A 2D numerical finite-difference algorithm accounting for surface topography is presented. Higher-order, dispersion-bounded, cost-optimized finite-difference operators are used in the interior of the numerical grid, while non-reflecting absorbing boundary conditions are used along the edges. Transformation from a curved to a rectangular grid achieves the modelling of the surface topography. We use free-surface boundary conditions along the surface. In order to obtain complete modelling of the effects of wave propagation, it is important to account for the surface topography, otherwise near-surface effects, such as scattering, are not modelled adequately. Even if other properties of the medium, for instance randomization, can improve numerical simulations, inclusion of the surface topography makes them more realistic. 相似文献
14.
Modified image algorithm to simulate seismic channel waves in 3D tunnel model with rugged free surfaces 总被引：1，自引：0，他引：1 
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下载免费PDF全文 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. 相似文献
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Modelling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the grid‐characteristic method 下载免费PDF全文
下载免费PDF全文 Alena V. Favorskaya Michael S. Zhdanov Nikolay I. Khokhlov Igor B. Petrov 《Geophysical Prospecting》2018,66(8):1485-1502
This paper introduces a novel method of modelling acoustic and elastic wave propagation in inhomogeneous media with sharp variations of physical properties based on the recently developed grid‐characteristic method which considers different types of waves generated in inhomogeneous linear‐elastic media (e.g., longitudinal, transverse, Stoneley, Rayleigh, scattered PP‐, SS‐waves, and converted PS‐ and SP‐waves). In the framework of this method, the problem of solving acoustic or elastic wave equations is reduced to the interpolation of the solutions, determined at earlier time, thus avoiding a direct solution of the large systems of linear equations required by the FD or FE methods. We apply the grid‐characteristic method to compare wave phenomena computed using the acoustic and elastic wave equations in geological medium containing a hydrocarbon reservoir or a fracture zone. The results of this study demonstrate that the developed algorithm can be used as an effective technique for modelling wave phenomena in the models containing hydrocarbon reservoir and/or the fracture zones, which are important targets of seismic exploration. 相似文献
16.
The 2008 Wenchuan earthquake occurred in an active earthquake zone, i.e., Longmenshan tectonic zone. Seismic waves triggered
by this earthquake can be used to explore the characteristics of the fault rupture process and the hierarchical structure
of the Earth’s interior. We employ spectral element method incorporated with large-scale parallel computing technology, to
investigate the characteristics of seismic wave propagation excited by Wenchuan earthquake. We calculate synthetic seismograms
with one-point source model and three-point source model respectively. The AK135 model is employed as a prototype of our numerical
global Earth model. The Earth’s ellipticity, Earth’s medium attenuation, and topography data are taken into consideration.
These wave propagation processes are simulated by solving three-dimensional elastic wave governing equations. Three-dimensional
visualization of our numerical results displays the profile of the seismic wave propagation. The three-point source, which
is proposed from the latest investigations through field observation and reverse estimation, can better demonstrate the spatial
and temporal characteristics of the source rupture process than the one-point source. We take comparison of synthetic seismograms
with observational data recorded at 16 observatory stations. Primary results show that the synthetic seismograms calculated
from three-point source agree well with the observations. This can further reveal that the source rupture process of Wenchuan
earthquake is a multi-rupture process, which is composed by at least three or more stages of rupture processes.
Supported by National Basic Research Program of China (Grant No. 2004CB418406), National Natural Science Foundation of China
(Grant Nos. 40774049 and 40474038), and Computer Network Information Center, Chinese Academy of Sciences (Grant No. INF105-SCE-02-12) 相似文献
17.
Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations 总被引：2，自引：0，他引：2
The perfectly matched layer (PML) is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second-order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite-element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media. 相似文献
18.
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. 相似文献
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本文基于Kjartansson常Q模型理论，推导了常Q衰减介质中黏声波和黏弹性波的速度-应力方程，并采用基于二项式窗函数的优化交错网格有限差分方法进行了数值模拟，同时引入不分裂的复频移卷积完全匹配层(CPML)吸收边界条件，以消除边界反射.使用基于自适应时间步长记忆方法的中心差分近似时间分数阶导数，与常用的短时记忆方法相比，提高了波动方程的离散化精度和计算效率.通过对比均匀模型下声波的数值解与解析解，验证了算法的精确性，并进一步分析了不同品质因子下地震波的频散及衰减特征.对BP盐丘模型的数值模拟结果可以较好地反映本文数值方法对复杂介质的适应性及频散压制效果. 相似文献
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伴随层析成像（Adjoint Tomography）通过求解全波方程来准确模拟地震波在复杂介质中的传播,并利用波形信息来反演地下结构,是新一代的高分辨率成像方法.其中3-D伴随层析成像需要庞大的计算资源,而2-D反演相对更具计算效率.面波和远震体波是研究地壳上地幔速度结构的重要方法,它们对S波速度及Moho面的敏感度不同,通过联合反演,可以得到更为准确的S波速度结构及Moho面.通过两种数据的高度互补性,本文提出基于伴随方法的线性台阵背景噪声面波和远震体波联合成像方法,同时约束台阵下方S波速度结构及Moho面形态.我们将该方法应用到符合华北克拉通岩石圈典型结构特征的理论模型上,测试结果表明联合反演方法优势明显,相比于面波伴随层析成像,能获得更高分辨率的S波速度结构,同时能精准约束Moho面形态.相比于体波伴随层析成像,联合反演能有效压制高频假象,降低波形反演过程中的非线性化程度.本研究有望提供一种更为高效精准的线性台阵成像方法,搭建联合伴随层析成像理论框架,提升岩石圈成像分辨率,并为后续其他类型波形数据的引入提供思路和方法. 相似文献