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1.
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. 相似文献
2.
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. 相似文献
3.
This paper presents a Lebedev finite difference scheme on staggered grids for the numerical simulation of wave propagation in an arbitrary 3D anisotropic elastic media. The main concept of the scheme is the definition of all the components of each tensor (vector) appearing in the elastic wave equation at the corresponding grid points, i.e., all of the stresses are stored in one set of nodes while all of the velocity components are stored in another. Meanwhile, the derivatives with respect to the spatial directions are approximated to the second order on two‐point stencils. The second‐order scheme is presented for the sake of simplicity and it is easy to expand to a higher order. Another approach, widely‐known as the rotated staggered grid scheme, is based on the same concept; therefore, this paper contains a detailed comparative analysis of the two schemes. It is shown that the dispersion condition of the Lebedev scheme is less restrictive than that of the rotated staggered grid scheme, while the stability criteria lead to approximately equal time stepping for the two approaches. The main advantage of the proposed scheme is its reduced computational memory requirements. Due to a less restrictive dispersion condition and the way the media parameters are stored, the Lebedev scheme requires only one‐third to two‐thirds of the computer memory required by the rotated staggered grid scheme. At the same time, the number of floating point operations performed by the Lebedev scheme is higher than that for the rotated staggered grid scheme. 相似文献
4.
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
5.
Theory of equivalent staggered‐grid schemes: application to rotated and standard grids in anisotropic media 
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下载免费PDF全文 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. 相似文献
6.
Pure acoustic wave propagation in transversely isotropic media by the pseudospectral method 总被引：4，自引：0，他引：4
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. 相似文献
7.
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. 相似文献
8.
Anisotropic reverse-time migration for tilted TI media 总被引：1，自引：0，他引：1
Seismic anisotropy in dipping shales results in imaging and positioning problems for underlying structures. We develop an anisotropic reverse‐time depth migration approach for P‐wave and SV‐wave seismic data in transversely isotropic (TI) media with a tilted axis of symmetry normal to bedding. Based on an accurate phase velocity formula and dispersion relationships for weak anisotropy, we derive the wave equation for P‐wave and SV‐wave propagation in tilted transversely isotropic (TTI) media. The accuracy of the P‐wave equation and the SV‐wave equation is analyzed and compared with other acoustic wave equations for TTI media. Using this analysis and the pseudo‐spectral method, we apply reverse‐time migration to numerical and physical‐model data. According to the comparison between the isotropic and anisotropic migration results, the anisotropic reverse‐time depth migration offers significant improvements in positioning and reflector continuity over those obtained using isotropic algorithms. 相似文献
9.
Numerical simulation of 2D wave propagation on unstructured grids using explicit differential operators 总被引：2，自引：0，他引：2
We present a numerical method of simulating seismic wave propagation on unstructured 2D grids. The algorithm is based on the velocity–stress formulation of the elastic wave equation and therefore uses a staggered grid approach. Unlike finite-element or spectral-element methods, which can also handle flexible unstructured grids, we use explicit differential operators for the calculation of spatial derivatives in each time step. As shown in previous work, three types of these operators are used, and their particular performance is analysed and compared with standard explicit finite-difference operators on regular quadratic and hexagonal grids. Our investigations are especially focused on the influence of grid irregularity, sampling rate (i.e. gridpoints per wavelength) and numerical anisotropy on the accuracy of numerical seismograms. The results obtained from the various methods are therefore compared with analytical solutions. The algorithm is then applied to a number of models that are difficult to handle using (quasi-)regular grid methods. Such alternative techniques may be useful in modelling the full wavefield of bodies with complex geometries (e.g. cylindrical bore-hole samples, 2D earth models) and, because of their local character, they are well suited for parallelization. 相似文献
10.
煤层中存在的裂隙会导致介质表现为各向异性,本文以HTI型煤层为例,结合各向异性介质弹性矩阵和各向异性裂隙理论,推导出不同充填物的垂直裂隙中各向异性参数表达式,将其应用于地震波响应分析;通过改进的交错网格差分法和各向异性Christoffel方程波场分解法,得到地震波合成记录和分解后的P波和SV波记录;将Thomsen群速度与相速度公式,经过坐标轴旋转变换,得到HTI型煤层中不同各向异性参数的地震波速度响应表达式;建立不同类型煤层地质模型,分析了裂隙密度、裂隙充填物以及煤层厚度等参数变化时的地震波响应特征.研究结果为分析垂向裂隙各向异性薄煤层地震波传播规律提供工具,为选用相应地震数据进行地震波各向异性参数反演提供依据. 相似文献
11.
各向异性介质纯P波方程完全不受横波的干扰,在一定程度上可以减缓由于介质各向异性引起的数值不稳定,本文推导了具有垂直对称轴的横向各向同性(VTI)介质纯P波一阶速度-应力方程.由于纯P波方程存在一个分数形式的伪微分算子,无法直接采用有限差分法求解.针对该问题,本文采用伪谱法和高阶有限差分法联合求解波动方程,重点分析了混合法求解纯P波一阶速度-应力方程的稳定性问题,并给出了混合法求解纯P波方程的稳定性条件.数值模拟结果表明纯P波方程伪谱法和高阶有限差分混合法能够进行复杂介质的正演模拟,在强变速度、变密度的地球介质中仍然具有较好的稳定性. 相似文献
12.
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. 相似文献
13.
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. 相似文献
14.
随着计算机硬件技术的发展以及高分辨率勘探需求的增加,我们希望能够更准确地模拟地下介质,得到更丰富的地层信息.然而,传统的声学假设并不能描述实际地层所存在各向异性和黏滞性,使得成像分辨率较低.为了实现深部储层的高精度成像,本文同时考虑了介质的各向异性和黏滞性,从TI介质弹性波的基本理论出发,结合各向异性GSLS理论,并通过声学近似方法导出基于GSLS模型的各向异性衰减拟声波方程.数值模拟表明该方程既能准确地描述各向异性介质下的准P波运动学规律,又能体现地层的吸收衰减效应;模型逆时偏移结果表明,在实现成像过程中考虑各向异性和黏滞性的影响,能对高陡构造清晰成像,且剖面振幅相对均衡,分辨率较高. 相似文献
15.
The airborne electromagnetic (AEM) method has a high sampling rate and survey flexibility. However, traditional numerical modeling approaches must use high-resolution physical grids to guarantee modeling accuracy, especially for complex geological structures such as anisotropic earth. This can lead to huge computational costs. To solve this problem, we propose a spectral-element (SE) method for 3D AEM anisotropic modeling, which combines the advantages of spectral and finite-element methods. Thus, the SE method has accuracy as high as that of the spectral method and the ability to model complex geology inherited from the finite-element method. The SE method can improve the modeling accuracy within discrete grids and reduce the dependence of modeling results on the grids. This helps achieve high-accuracy anisotropic AEM modeling. We first introduced a rotating tensor of anisotropic conductivity to Maxwell’s equations and described the electrical field via SE basis functions based on GLL interpolation polynomials. We used the Galerkin weighted residual method to establish the linear equation system for the SE method, and we took a vertical magnetic dipole as the transmission source for our AEM modeling. We then applied fourth-order SE calculations with coarse physical grids to check the accuracy of our modeling results against a 1D semi-analytical solution for an anisotropic half-space model and verified the high accuracy of the SE. Moreover, we conducted AEM modeling for different anisotropic 3D abnormal bodies using two physical grid scales and three orders of SE to obtain the convergence conditions for different anisotropic abnormal bodies. Finally, we studied the identification of anisotropy for single anisotropic abnormal bodies, anisotropic surrounding rock, and single anisotropic abnormal body embedded in an anisotropic surrounding rock. This approach will play a key role in the inversion and interpretation of AEM data collected in regions with anisotropic geology. 相似文献
16.
A central difference method with low numerical dispersion for solving the scalar wave equation 总被引：4，自引：0，他引：4
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. 相似文献
17.
地下介质中普遍存在着各向异性,当前基于各向异性的地震波射线追踪多是在弱各向异性介质中进行且采用群速度近似表示方法,这些近似方法在强各项异性介质中会导致很大误差而无法真正模拟地震波的传播规律。根据地下普遍存在各向异性的事实和地震波基本传播规律,提出利用牛顿迭代法高效求解群速度,基于Paraview平台自动化构建三维地质模型,采用最短路径法进行地震波射线追踪模拟及可视化,实现对复杂三维地质的速度不均匀性和各向异性的表达,为三维地质模型的构建和地震波射线追踪模拟及可视化提供一种新思路,并以华北克拉通山西断陷带北部局部区域为例进行研究。结果表明,该方法能够减少由各向异性对地震波传播模拟造成的影响,清晰表达了研究区地质结构和各向异性特点,在对复杂三维地质结构的解读中能够较好应用。 相似文献
18.
Tariq Alkhalifah 《Geophysical Prospecting》2014,62(5):1089-1099
Spectral methods provide artefact‐free and generally dispersion‐free wavefield extrapolation in anisotropic media. Their apparent weakness is in accessing the medium‐inhomogeneity information in an efficient manner. This is usually handled through a velocity‐weighted summation (interpolation) of representative constant‐velocity extrapolated wavefields, with the number of these extrapolations controlled by the effective rank of the original mixed‐domain operator or, more specifically, by the complexity of the velocity model. Conversely, with pseudo‐spectral methods, because only the space derivatives are handled in the wavenumber domain, we obtain relatively efficient access to the inhomogeneity in isotropic media, but we often resort to weak approximations to handle the anisotropy efficiently. Utilizing perturbation theory, I isolate the contribution of anisotropy to the wavefield extrapolation process. This allows us to factorize as much of the inhomogeneity in the anisotropic parameters as possible out of the spectral implementation, yielding effectively a pseudo‐spectral formulation. This is particularly true if the inhomogeneity of the dimensionless anisotropic parameters are mild compared with the velocity (i.e., factorized anisotropic media). I improve on the accuracy by using the Shanks transformation to incorporate a denominator in the expansion that predicts the higher‐order omitted terms; thus, we deal with fewer terms for a high level of accuracy. In fact, when we use this new separation‐based implementation, the anisotropy correction to the extrapolation can be applied separately as a residual operation, which provides a tool for anisotropic parameter sensitivity analysis. The accuracy of the approximation is high, as demonstrated in a complex tilted transversely isotropic model. 相似文献
19.
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. 相似文献
20.
时间域的波场延拓方法在本质上都可以归结为对一个空间-波数域算子的近似.本文基于一阶波数-空间混合域象征,提出一种新的方法求解解耦的二阶位移弹性波方程.该方法采用交错网格,连续使用两次一阶前向和后向拟微分算子,推导得到了解耦的二阶位移弹性波方程的波场延拓算子.由于该混合域象征在伪谱算子的基础上增加了一个依赖于速度模型的补偿项,可以补偿由于采用二阶中心差分计算时间微分项带来的误差,有效地减少模拟结果的数值频散,提高模拟精度.然而,在非均匀介质中,直接计算该二阶的波场延拓算子,每一个时间步上需要做N次快速傅里叶逆变换,其中N是总的网格点数.为了减少计算量,提出了交错网格低秩分解方法;针对常规有限差分数值频散问题,本文将交错网格低秩方法与有限差分法结合,提出了交错网格低秩有限差分法.数值结果表明,交错网格低秩方法和交错网格低秩有限差分法具有较高的精度,对于复杂介质的地震波数值模拟和偏移成像具有重要的价值. 相似文献