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1.
Application of a perfectly matched layer in seismic wavefield simulation with an irregular free surface 总被引：2，自引：0，他引：2 
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下载免费PDF全文 Haiqiang Lan Jingyi Chen Zhongjie Zhang Youshan Liu Jianguo Zhao Ruiqi Shi 《Geophysical Prospecting》2016,64(1):112-128
Recently, an effective and powerful approach for simulating seismic wave propagation in elastic media with an irregular free surface was proposed. However, in previous studies, researchers used the periodic condition and/or sponge boundary condition to attenuate artificial reflections at boundaries of a computational domain. As demonstrated in many literatures, either the periodic condition or sponge boundary condition is simple but much less effective than the well‐known perfectly matched layer boundary condition. In view of this, we intend to introduce a perfectly matched layer to simulate seismic wavefields in unbounded models with an irregular free surface. We first incorporate a perfectly matched layer into wave equations formulated in a frequency domain in Cartesian coordinates. We then transform them back into a time domain through inverse Fourier transformation. Afterwards, we use a boundary‐conforming grid and map a rectangular grid onto a curved one, which allows us to transform the equations and free surface boundary conditions from Cartesian coordinates to curvilinear coordinates. As numerical examples show, if free surface boundary conditions are imposed at the top border of a model, then it should also be incorporated into the perfectly matched layer imposed at the top‐left and top‐ right corners of a 2D model where the free surface boundary conditions and perfectly matched layer encounter; otherwise, reflections will occur at the intersections of the free surface and the perfectly matched layer, which is confirmed in this paper. So, by replacing normal second derivatives in wave equations in curvilinear coordinates with free surface boundary conditions, we successfully implement the free surface boundary conditions into the perfectly matched layer at the top‐left and top‐right corners of a 2D model at the surface. A number of numerical examples show that the perfectly matched layer constructed in this study is effective in simulating wave propagation in unbounded media and the algorithm for implementation of the perfectly matched layer and free surface boundary conditions is stable for long‐time wavefield simulation on models with an irregular free surface. 相似文献
2.
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. 相似文献
3.
Flow and deformation failure of sandy slopes 总被引：2，自引：0，他引：2
The effects of earthquake induced pore pressure on seismic and post seismic stability conditions of cohesionless slopes are investigated with reference to the infinite slope scheme. In cohesionless slopes the shear strength reduction caused by pore pressure build-up may lead the slope to a deformation failure or to a flow failure if liquefaction conditions are approached. Two critical values of the seismic induced pore pressure ratio are introduced to evaluate the effect of shear strength reduction on the slope failure mechanism. The results are given in the form of stability charts and a procedure for the evaluation of the seismic stability condition is described. The procedure gives useful information about the failure mechanism that slopes may exhibit and the displacement analysis which should be carried out. 相似文献
4.
This paper presents a simple, flexible way of introducing stress-free boundary conditions for including cracks and cavities
in 2D elastic media by a finite difference method (FDM). The surfaces of cracks and cavities are discretized in a staircase
on a rectangular grid scheme. When zero-stress is applied to free surfaces, the resulting finite difference schemes require
a set of adjacent fictitious points. These points are classified based on the geometry of the free surface and their displacement
is computed as a prior step to later calculation of motion on the crack surface. The use of this extra line of points does
not involve a significant drain on computational resources. However, it does provide explicit finite difference schemes and
the construction of displacement on the free surfaces by using the correct physical boundary conditions. An accuracy analysis
compares the results to an analytical solution. This quantitative analysis uses envelope and phase misfits. It estimates the
minimum number of points per wavelength necessary to achieve suitable results. Finally, the method is employed to compute
displacement in various models with cavities in the P-SV formulation. The results show suitable construction of the reflected
P and S waves from the free surface as well as diffraction produced by these cavities. 相似文献
5.
This article presents a new approach for the implementation of a planar-free surface boundary condition. It is based on a
vertical grid-size reduction above the free surface during the explicit computation of a free surface boundary condition.
This technique is very much similar to the well-known stress imaging technique. VGR-stress imaging technique name is proposed
for this new free surface boundary condition (VGR stands for ‘vertical grid-size reduction’). To study the performance of
the proposed VGR-stress imaging technique, it was implemented in a newly developed second order accurate in time and fourth-order
accurate in space (2, 4) staggered grid SH-wave finite-difference (FD) algorithm with variable grid size. It was confirmed
that the effective thickness (ETH) of first soil layer becomes less by one-half of vertical grid size than the assigned thickness
(ATH), if stress imaging technique is used as a free surface boundary condition. The qualitative and quantitative results
of various numerical experiments revealed that the proposed VGR-stress imaging technique is better than the stress imaging
technique since it is free from the thickness discrepancy arising due to the use of images of stress components across the
free surface. On the basis of iterative numerical experiments, it was confirmed that the stability condition for this FD scheme
with variable grid size is
It was also inferred that at least five to six grid points per shortest wavelength are required to avoid the grid dispersion.
The maximum grid-spacing ratio up to 12.5 or even more did not affect the accuracy of (2,4) SH-wave algorithm. The obtained
reduction of 10.46 and 5.38 folds in the requirement of computational memory and time for a particular basin-edge model, as
compared with the homogeneous grid size, reflects the efficacy of the new FD algorithm. 相似文献
6.
We present an original implementation of the free-surface boundary condition in a mesh-free finite-difference method for simulating elastic wave propagation in the frequency domain. For elastic wave modelling in the frequency domain, the treatment of free surfaces is a key issue which requires special consideration. In the present study, the free-surface boundary condition is directly implemented at node positions located on the free-surface. Flexible nature of the mesh-free method for nodal distribution enables us to introduce topography into numerical models in an efficient manner. We investigate the accuracy of the proposed implementation by comparing numerical results with an analytical solution. The results show that the proposed method can calculate surface wave propagation even for an inclined free surface with substantial accuracy. Next, we calculate surface wave propagation in a model with a topographic surface using our method, and compare the numerical result with that using the finite-element method. The comparison shows the excellent agreement with each other. Finally, we apply our method to the SEG foothill model to investigate the effectiveness of the proposed method. Since the mesh-free method has high flexibility of nodal distribution, the proposed implementation would deal with models of topographic surface with sufficient accuracy and efficiency. 相似文献
7.
The complexity of near surface intensifies the diversity of seismic wave fields, which makes study on near surface wavefields important in many aspects. The strong absorption of low velocity layer can affect the resolution of seismic data, and free boundary can cause surface wave. Considering the above problems, we focus on the Rayleigh wavefields simulation using finite-difference wave equation of higher-order staggered grids and PML boundary conditions. Free boundary, buried source and overlying low velocity layer are taken into consideration and point explosion source is adopted. Through some numerical simulation with different parameters, we quantitatively analyze relationship between wave intensity and source depth, as well as the energy variation with propagation and obtain some practical knowledge and conclusions. 相似文献
8.
J. Wolf 《Continental Shelf Research》1982,2(4):215-229
As the capacity of computers increases, the size and resolution of numerical models can be increased. In tidal models, however, using an explicit scheme together with high spatial resolution results in an unreasonably small time-step demanded by the CFL condition for stability. This condition is usually the most restrictive and applies to the propagation of the gravity wave.A fully nonlinear three-dimensional model has been developed, using the Galerkin method in the vertical dimension, in which the gravity wave terms are treated by an alternating-direction implicit scheme, while the friction, viscous and advective terms are treated explicitly. This permits a stable solution with a longer time-step than that required in an equivalent explicit scheme, while not needing as much computational effort as a fully implicit scheme. This semi-implicit model is compared with an explicit model in terms of efficiency, accuracy, and stability. Tidal and wind-driven flows and free oscillations in a rectangular sea model are examined, using various boundary conditions. The semi-implicit scheme takes approximately 1.1 times as long to run (per time-step) on the CRAY-1 computer as the explicit scheme. 相似文献
9.
I.D. James 《Continental Shelf Research》1988,8(12)
A three-dimensional numerical model is used to simulate the development of disturbances on shelf-sea coastal currents and fronts. The model, which has a free surface, uses a finite difference grid ☐ scheme based on sigma coordinates. It has a semi-implicit scheme for the barotropic flow and a hydrid advection scheme to retain sharp fronts. The results demonstrate that (i) eddy formation follows changes at the inflow of a coastal current, (ii) a simple radiation boundary condition at the outflow produces nearly identical results for different outflow boundary positions, (iii) eddy growth, with matching behaviour of surface and bottom fronts, follows a small displacement on a tidal mixing front and (iv) effects of friction and mixing can significantly alter the behaviour of the front and the relative strength of the cyclonic and anticyclonic eddies formed. 相似文献
10.
Jiří Nedoma 《Pure and Applied Geophysics》1973,110(1):1899-1910
Summary In the present paper the gravity field of the earth in the neighbourhood of the local disturbing masses is studied. The object of the method presented consists of the approximation of the disturbing potentialT
h
, which fulfils Laplace's equation outside disturbing masses, on the earth's surface the fundamental boundary value condition of gravity and in infinity it is to be regular by the approximation of the disturbing potential (or by the discrete disturbing potential)T
h
, which fulfils the respective finite difference approximation of Laplace's equation and the boundary value conditions in infinity and on the earth's surface. It is also shown that the approximation of the disturbing potentialT
h
has the same properties as the disturbing potentialT. The method under consideration will be derived quite generally without any hypothesis about the distribution of the mass between the earth's surface and the geoid. It commences from the gravity data related to the earth's surface only-from the given geodetic measurements. 相似文献
11.
瑞雷面波经常被用来反演地表浅层横波速度,受到越来越广泛的关注。对瑞雷波的研究一般都基于完全弹性介质,而实际地层更接近黏弹性介质,对黏弹性介质中的瑞雷面波进行模拟更具实际意义。本文采用广义标准线性体模型来描述黏弹性介质,并采用交错网格有限差分法对考虑水平自由表面的黏弹介质进行正演模拟,再与弹性介质中的结果进行对比分析。首先采用非线性最优化算法根据期望常数品质因子直接求取松弛时间来拟合常Q模型,并给出广义标准线性固体的具体算例,实施自由表面条件时采用声学-弹性边界近似法,通过剪切模量不变来考虑自由表面上、下横向应力保持连续的条件。对于非自由表面,采用非分裂的多轴卷积完全匹配层来吸收波场。然后对几种典型的数值模型进行正演模拟计算,数值解与解析解的对比验证了本文方法的准确性与有效性,正演结果的对比表明波场尤其是面波频散会受黏弹性影响,因此有必要在面波勘探中考虑黏弹性因素。 相似文献
12.
就大型近场波动的高效数值模拟而言,稳定实现高阶人工边界是一个尚未圆满解决的问题.本文针对使用多次透射公式的SH波动集中质量有限元模拟,依据GKS定理的群速度解释,进一步阐明了人工边界与内域离散格式耦合所导致高频失稳的机理,即两者支持群速度指向内域的外行高频平面谐波,波动能量自发地从人工边界进入內域,从而导致失稳,而这类谐波是由集中质量有限元离散引入的.本文提出了消除此种耦合失稳的一种方法:通过修改有限元刚度阵来改变内域离散格式,并保证修改格式的精度不低于原有格式的精度.理论分析和数值实验表明此法能稳定实现透射边界.本文研究结果具有推广应用前景. 相似文献
13.
Georgios Panou Nikos Yannakakis Demitris Delikaraoglou 《Studia Geophysica et Geodaetica》2013,57(2):203-216
In this paper the linear fixed altimetry-gravimetry boundary-value problem is analyzed with respect to the existence and uniqueness of the solution. Nowadays, it is possible to determine very precisely points on the physical surface of the Earth by 3D satellite positioning and the problem is to determine the disturbing potential in an unbounded domain representing the exterior of the Earth. In order to establish realistic boundary conditions, a Dirichlet condition is imposed at seas and an oblique derivative condition on land. Then, mathematical methods are used within the frame of functional analysis for attacking the problem under consideration. Specifically, the Stampacchia theorem is used to decide upon the existence and uniqueness of the weak solution of the problem in a weighted Sobolev space. Finally, we confirm that the condition of validity for such a theorem has a geometrical interpretation. 相似文献
14.
The method of conformal mapping is applied to the analysis of transient flow toward parallel periodic drains in a semi-infinite aquifer taking into consideration the non-linear boundary conditions on the free surface. The mapping function is expressed as a power series in time and the seepage domain is mapped onto a domain of an auxiliary complex variable. Mapping is performed in such a manner that the free surface will always remain the real axis. Calculations are carried out for different ratios of drain depth to drain spacing using various drain diameter to depth ratios. 相似文献
15.
The three-dimensional scattering by a hemi-spherical canyon in an elastic half-space subjected to seismic plane and spherical waves has long been a challenging boundary-value problem. It has been studied by earthquake engineers and strong-motion seismologists to understand the amplification effects caused by surface topography. The scattered and diffracted waves will, in all cases, consist of both longitudinal (P-) and shear (S-) shear waves. Together, at the half-space surface, these waves are not orthogonal over the infinite plane boundary of the half-space. Thus, to simultaneously satisfy both zero normal and shear stresses on the plane boundary numerical approximation of the geometry and/or wave functions were required, or in some cases, relaxed (disregarded). This paper re-examines this boundary-value problem from the applied mathematics point of view, and aims to redefine the proper form of the orthogonal spherical-wave functions for both the longitudinal and shear waves, so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite canyon surface will be easy to satisfy. 相似文献
16.
17.
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
18.
从Maxwell方程组出发,推导了探地雷达(GPR)有限元波动方程.阐述了透射边界条件和Sarma边界条件的原理,推导了这两种边界条件的理论公式;通过在衰减层内加入过渡带优化了Sarma边界条件的加载方法,压制了介质区和衰减层交界面处的人为反射.考虑到透射边界条件与Sarma边界条件不同的理论机制,提出了一种结合透射边界条件和Sarma边界条件的混合边界条件,它利用Sarma边界条件对到达边界区域的GPR波能量衰减功能和透射边界对GPR波能量的透射功能,使GPR波经过Sarma边界条件的衰减吸收后,再通过透射边界条件将剩余能量透射出去,集成了二者的优势.并以二维均匀模型中的中心脉冲激励源方式为例,通过Matlab程序实现,以GPR的全波场快照的直观方式,对比了有、无边界条件及不同边界条件对人工截断边界的处理效果,说明了该混合边界条件对到达截断边界处的GPR波的处理优于单一边界条件.最后,以基于混合边界条件的有限单元法对两个典型的GPR地电模型进行了正演模拟,指导了GPR数据处理与工程实践. 相似文献
19.
20.
Zdeněk Martinec 《Studia Geophysica et Geodaetica》1990,34(4):313-326
Summary A new method for computing the potential coefficients of the Earth's external gravity field is presented. The gravimetric
boundary-value problem with a free boundary is reduced to the problem with a fixed known telluroid. The main idea of the derivation
consists in a continuation of the quantities from the physical surface to the telluroid by means of Taylor's series expansion
in such a way that the terms whose magnitudes are comparable with the accuracy of today's gravity measurements are retained.
Thus not only linear, but also non-linear terms are taken into account. Explicitly, the terms up to the order of the third
power of the Earth's flattening are retained. The non-linear boundary-value problem on the telluroid is solved by an iteration
procedure with successive approximations. In each iteration step the solution of the non-linear problem is estimated by the
solutions of two linear problems utilizing the fact that the non-linear boundary condition may be split into two parts; the
linear spherical approximation of the gravity anomaly whose magnitude is significantly greater than the others and the non-linear
ellipsoidal corrections. Finally, in order to solve the problem in terms of spherical harmonics, the transform method composed
of the fast Fourier transform and Gauss Legendre quadrature is theoretically outlined. Immediate data processing of gravity
data measured on the physical Earth's surface without any continuation of gravity measurements to a reference level surface
belongs to the main advantage of the presented method. This implies that no preliminary data handling is needed and that the
error data propagation is, consequently, maximally suppressed. 相似文献