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1.
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
2.
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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We present a discrete modelling scheme which solves the elastic wave equation on a grid with vertically varying grid spacings. Spatial derivatives are computed by finite-difference operators on a staggered grid. The time integration is performed by the rapid expansion method. The use of variable grid spacings adds flexibility and improves the efficiency since different spatial sampling intervals can be used in regions with different material properties. In the case of large velocity contrasts, the use of a non-uniform grid avoids spatial oversampling in regions with high velocities. The modelling scheme allows accurate modelling up to a spatial sampling rate of approximately 2.5 gridpoints per shortest wavelength. However, due to the staggering of the material parameters, a smoothing of the material parameters has to be applied at internal interfaces aligned with the numerical grid to avoid amplitude errors and timing inaccuracies. The best results are obtained by smoothing based on slowness averaging. To reduce errors in the implementation of the free-surface boundary condition introduced by the staggering of the stress components, we reduce the grid spacing in the vertical direction in the vicinity of the free surface to approximately 10 gridpoints per shortest wavelength. Using this technique we obtain accurate results for surface waves in transversely isotropic media. 相似文献
4.
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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New alternative formulations of exact boundary conditions for arbitrary three-dimensional (3D) free-surface topographies on seismic media have been derived. They are shown to be equivalent to previously published formulations, thereby verifying the validity of each set of formulations. The top of a curved grid represents the free-surface topography while the interior of the grid represents the physical medium. We assume the velocity–stress version of the viscoelastic wave equations to be valid in this grid before transforming the equations to a rectangular grid. In order to perform the numerical discretization we apply the latter version of the equations for seismic wave propagation simulation in the medium. The numerical discretization of the free-surface topography boundary conditions by second-order finite differences (FDs) is shown, as well as the spatially unconditional stability of the resulting system of equations. The FD order is increased by two for each point away from the free surface up to eight, which is the order used in the interior. We use staggered grids in both space and time and the second-order leap-frog and Crank– Nicholson methods for wavefield time propagation. An application using parameters typical of teleseismic earthquakes and explosions is presented using a 200 × 100 km2 area of real topography from southwestern Norway over a homogeneous medium. A dipping plane wave simulates a teleseismic P-wave incident on the surface topography. Results show conversion from P- to Rg- (short period fundamental mode Rayleigh) waves in the steepest and/or roughest topography, as well as attenuated waves in valleys and fjords. The codes are parallelized for simulation on fast supercomputers and PC-clusters to model high frequencies and/or large areas. 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
8.
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
9.
Improving modelling and inversion in refraction seismics with a first-order Eikonal solver 总被引：1，自引：0，他引：1
Isabelle Lecomte Håvar Gjøystdal ers Dahle & Ole Christian Pedersen 《Geophysical Prospecting》2000,48(3):437-454
A first-order Eikonal solver is applied to modelling and inversion in refraction seismics. The method calculates the traveltime of the fastest wave at any point of a regular grid, including head waves as used in refraction. The efficiency, robustness and flexibility of the method give a very powerful modelling tool to find both traveltimes and raypaths. Comparisons with finite-difference data show the validity of the results. Any arbitrarily complex model can be studied, including the exact topography of the surface, thus avoiding static corrections. Later arrivals are also obtained by applying high-slowness masks over the high-velocity zones. Such an efficient modelling tool may be used interactively to invert for the model, but a better method is to apply the refractor-imaging principle of Hagedoorn to obtain the refractors from the picked traveltime curves. The application of this principle has already been tried successfully by previous authors, but they used a less well-adapted Eikonal solver. Some of their traveltimes were not correct in the presence of strong velocity variations, and the refractor-imaging principle was restricted to receiver lines along a plane surface. With the first-order Eikonal solver chosen, any topography of the receiving surface can be considered and there is no restriction on the velocity contrast. Based on synthetic examples, the Hagedoorn principle appears to be robust even in the case of first arrivals associated with waves diving under the refractor. The velocities below the refractor can also be easily estimated, parallel to the imaging process. In this way, the model can be built up successively layer by layer, the refractor-imaging and velocity-mapping processes being performed for each identified refractor at a time. The inverted model could then be used in tomographic inversions because the calculated traveltimes are very close to the observed traveltimes and the raypaths are available. 相似文献
10.
Multi‐block finite‐difference method for 3D elastodynamic simulations in anisotropic subhorizontally layered media 
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下载免费PDF全文 Prediction of elastic full wavefields is required for reverse time migration, full waveform inversion, borehole seismology, seismic modelling, etc. We propose a novel algorithm to solve the Navier wave equation, which is based on multi‐block methodology for high‐order finite‐difference schemes on curvilinear grids. In the current implementation, the blocks are subhorizontal layers. Smooth anisotropic heterogeneous media in each layer can have strong discontinuities at the interfaces. A curvilinear adaptive hexahedral grid in blocks is generated by mapping the original 3D physical domain onto a parametric cube with horizontal layers and interfaces. These interfaces correspond to the main curvilinear physical contrast interfaces of a subhorizontally layered formation. The top boundary of the parametric cube handles the land surface with smooth topography. Free‐surface and solid–solid transmission boundary conditions at interfaces are approximated with the second‐order accuracy. Smooth media in the layers are approximated up to sixth‐order spatial schemes. All expected properties of the developed algorithm are demonstrated in numerical tests using corresponding parallel message passing interface code. 相似文献
11.
本研究实现了一套基于有限差分（FD）方法的大地电磁测深数据带地形三维反演算法及代码.其中,在大地电磁场正演数值模拟方面,开发了起伏地形条件下基于交错网格剖分、有限差分方法的大地电磁测深三维正演代码;在满足平面波场假设的前提下,使用长方体网格剖分模拟三维起伏地形,实现了带地形三维正演计算;并设计理论模型进行试算,经试算结果与前人的有限元法计算结果对比,验证了所研发的带地形三维正演计算的正确性与可靠性.在反演方面,本研究基于非线性共轭梯度方法编写了大地电磁测深带地形三维反演代码,试验了不同的共轭梯度搜索因子β,避免了目标函数对海森矩阵（参数二次导数矩阵）的显式计算和存储,初步实现了大地电磁资料的带地形三维反演.最后,对一系列理论模型进行正演计算,利用其生成的合成数据模拟实测数据进行反演,并与现有的不带地形大地电磁测深三维反演结果比较,检验了所研发的带地形三维反演计算的可靠性与稳定性. 相似文献
12.
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
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《Advances in water resources》1998,22(3):239-245
In this paper, we examine the behavior of internal Kelvin waves on an f-plane in finite-difference models using the Arakawa C-grid. The dependence of Kelvin wave phase speed on offshore grid resolution and propagation direction relative to the numerical grid is illustrated by numerical experiments for three different geometries: (1) Kelvin wave propagating along a straight coastline; (2) Kelvin wave propagating at a 45° angle to the numerical grid along a stairstep coastline with stairstep size equal to the grid spacing; (3) Kelvin wave propagating at a 45° angle to the numerical grid along a coarse resolution stairstep coastline with stairstep size greater than the grid spacing. It can be shown theoretically that the phase speed of a Kelvin wave propagating along a straight coastline on an Arakawa C-grid is equal to the analytical inviscid wave speed and is not dependent on offshore grid resolution. However, we found that finite-difference models considerably underestimate the Kelvin wave phase speed when the wave is propagating at an angle to the grid and the grid spacing is comparable with the Rossby deformation radius. In this case, the phase speed converges toward the correct value only as grid spacing decreases well below the Rossby radius. A grid spacing of one-fifth the Rossby radius was required to produce results for the stairstep boundary case comparable with the straight coast case. This effect does not appear to depend on the resolution of the coastline, but rather on the direction of wave propagation relative to the grid. This behavior is important for modeling internal Kelvin waves in realistic geometries where the Rossby radius is often comparable with the grid spacing, and the waves propagate along irregular coastlines.©1998 Published by Elsevier Science Limited. All rights reserved 相似文献
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A three‐dimensional numerical modelling system is developed to study transformation processes of water resources in alluvial fan and river basin along the middle reaches of the Heihe River Basin, Northwest China, an arid and semi‐arid region. Integrating land utilization, remote sensing and geographic information systems, we have developed a numerical modelling system that can be used to quantify the effects of land use and anthropogenic activities on the groundwater system as well as to investigate the interaction between surface water and groundwater. Various hydraulic measurements are used to identify and calibrate the hydraulic boundary conditions and spatial distributions of hydraulic parameters. In the modelling study, various water exchanges and human effects on the watershed system are considered. These include water exchange between surface water and groundwater, groundwater pumping, lateral water recharges from mountain areas, land utilization, and infiltration and evaporation in the irrigation and non‐irrigation areas. The modelling system provides a quantitative method to describe spatial and temporal distributions and transformations between various water resources, and it has application to other watersheds in arid and semi‐arid areas. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
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.
Paleoreconstructions suggest that during the Last Glacial Maximum (LGM) the North Atlantic circulation was noticeably different from its present state. However, the glacial salt conveyor belt is believed to be similar to the present-day’s conveyor, albeit weaker and shallower because of an increased freshwater flux in high-latitudes. We present here the investigation of the conveyor operation based on ocean circulation modelling using two numerical models in parallel. The GFDL primitive equation model and a planetary geostrophic model are employed to address the problem of the paleocirculation modelling in cases of uncertain and sparse data comprising the glacial surface boundary conditions. The role of different simplifications that may be used in the ocean climate studies, including the role of grid resolution, bottom topography, coast-line, etc., versus glacial-interglacial changes of the ocean surface climatology is considered. The LGM reverse conveyor gyre appeared to be the most noticeable feature of the glacial-to-interglacial alteration of the ocean circulation. The reversed upper-ocean conveyor, weaker and subducting ‘normal’ conveyor in the intermediate depths, and the change of the deep-ocean return flow route are robust signatures of the glacial North Atlantic climate. The results are found to be ‘model-independent’ and fairly insensitive to all factors other than the onset of the glacial surface conditions. 相似文献
19.
The staggered grid finite-difference method is a powerful tool in seismology and is commonly used to study earthquake source
dynamics. In the staggered grid finite-difference method stress and particle velocity components are calculated at different
grid points, and a faulting problem is a mixed boundary problem, therefore different implementations of fault boundary conditions
have been proposed. Viriuex and Madariaga (1982) chose the shear stress grid as the fault surface, however, this method has
several problems: (1) Fault slip leakage outside the fault, and (2) the stress bump beyond the crack tip caused by S waves
is not well resolved. Madariaga et al. (1998) solved the latter problem via thick fault implementation, but the former problem remains and causes a new issue;
displacement discontinuity across the slip is not well modeled because of the artificial thickness of the fault. In the present
study we improve the implementation of the fault boundary conditions in the staggered grid finite-difference method by using
a fictitious surface to satisfy the fault boundary conditions. In our implementation, velocity (or displacement) grids are
set on the fault plane, stress grids are shifted half grid spacing from the fault and stress on the fictitious surface in
the rupture zone is given such that the interpolated stress on the fault is equal to the frictional stress. Within the area
which does not rupture, stress on the fictitious surface is given a condition of no discontinuity of the velocity (or displacement).
Fault normal displacement (or velocity) is given such that the normal stress on the fault is continuous across the fault.
Artificial viscous damping is introduced on the fault to avoid vibration caused by onset of the slip. Our implementation has
five advantages over previous versions: (1) No leakage of the slip prior to rupture and (2) a zero thickness fault, (3) stress
on the fault is reliably calculated, (4) our implementation is suitable for the study of fault constitutive laws, as slip
is defined as the difference between displacement on the plane of z = + 0 and that of z = − 0, and (5) cessation of slip is achieved correctly. 相似文献