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1.
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
2.
Subsalt imaging is strongly dependent on the quality of the velocity model. However, rugose salt bodies complicate wavefield propagation and lead to subsalt multipathing, illumination gaps and shadow zones, which cannot be handled correctly by conventional traveltime‐based migration velocity analysis (MVA). We overcome these limitations by the wave‐equation MVA technique, introduced in a companion paper, and demonstrate the methodology on a realistic synthetic data set simulating a salt‐dome environment and a Gulf of Mexico data set. We model subsalt propagation using wave paths created by one‐way wavefield extrapolation. Those wave paths are much more accurate and robust than broadband rays, since they inherit the frequency dependence and multipathing of the underlying wavefield. We formulate an objective function for optimization in the image space by relating an image perturbation to a perturbation of the velocity model. The image perturbations are defined using linearized prestack residual migration, thus ensuring stability, relative to the first‐order Born approximation assumptions. Synthetic and real data examples demonstrate that wave‐equation MVA is an effective tool for subsalt velocity analysis, even when shadows and illumination gaps are present. 相似文献
3.
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
4.
Tariq Alkhalifah 《Geophysical Prospecting》2015,63(5):1126-1141
Wavefield extrapolation operators for elliptically anisotropic media offer significant cost reduction compared with that for the transversely isotropic case, particularly when the axis of symmetry exhibits tilt (from the vertical). However, elliptical anisotropy does not provide accurate wavefield representation or imaging for transversely isotropic media. Therefore, we propose effective elliptically anisotropic models that correctly capture the kinematic behaviour of wavefields for transversely isotropic media. Specifically, we compute source‐dependent effective velocities for the elliptic medium using kinematic high‐frequency representation of the transversely isotropic wavefield. The effective model allows us to use cheaper elliptic wave extrapolation operators. Despite the fact that the effective models are obtained by matching kinematics using high‐frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The methodology developed here offers a much better cost versus accuracy trade‐off for wavefield computations in transversely isotropic media, particularly for media of low to moderate complexity. In addition, the wavefield solution is free from shear‐wave artefacts as opposed to the conventional finite‐difference‐based transversely isotropic wave extrapolation scheme. We demonstrate these assertions through numerical tests on synthetic tilted transversely isotropic models. 相似文献
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本文理论分析了电子束沿地磁场穿越均匀、磁化等离子体密度跃变区域时,在弱磁场近似、哨声波激发、低频近似等几种典型情况下电子束流空间电荷波（Space charge wave）向电磁波的转换.先运用小信号假设求得电子束入射进均匀各向异性冷等离子体之后的色散关系和空间电荷波波数,然后借助于电磁波分量和电子束速度的边界条件,求解电子束在等离子浓度发生变化区域激发的波振幅,在几种典型情形下推导出空间电荷波转换为电磁波之间转换系数的近似解,给出了相应波辐射的坡印亭(Poynting)矢量表达式.结果表明,在渡越辐射（Transition radiation）情形下电子束可以在空间等离子体中激发出阿尔芬波（Alfven wave）和哨声波（Whistler wave）.所得结论可用于对主动空间试验结果的分析. 相似文献
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近年来,基于Lowrank近似的qP波正演模拟受到广泛关注。通常的Lowrank近似两步法波场外推虽然也能消除伪横波干扰,但是两步法依赖一个实值的相位函数,且在大时间步长不适用。基于Lowrank近似的qP波一步法波场外推方法具有处理复值相位函数的能力,对相位函数进行改进,加入速度梯度项,并成功应用于TTI介质qP波正演模拟。通过试验可知:基于Lowrank近似的一步法波场延拓具有传统两步法的优点,无伪横波干扰,在大时间步长波传播时比两步法更加稳定;各向异性情况下,一步法波场外推适用于任意倾角情况下,波场模拟结果清晰准确,没有发生不稳定。 相似文献
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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
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Riemannian wavefield extrapolation is a technique for one‐way extrapolation of acoustic waves. Riemannian wavefield extrapolation generalizes wavefield extrapolation by downward continuation by considering coordinate systems different from conventional Cartesian ones. Coordinate systems can conform with the extrapolated wavefield, with the velocity model or with the acquisition geometry. When coordinate systems conform with the propagated wavefield, extrapolation can be done accurately using low‐order kernels. However, in complex media or in cases where the coordinate systems do not conform with the propagating wavefields, low order kernels are not accurate enough and need to be replaced by more accurate, higher‐order kernels. Since Riemannian wavefield extrapolation is based on factorization of an acoustic wave‐equation, higher‐order kernels can be constructed using methods analogous to the one employed for factorization of the acoustic wave‐equation in Cartesian coordinates. Thus, we can construct space‐domain finite‐differences as well as mixed‐domain techniques for extrapolation. High‐order Riemannian wavefield extrapolation kernels improve the accuracy of extrapolation, particularly when the Riemannian coordinate systems does not closely match the general direction of wave propagation. 相似文献
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The behaviour of the actual polarization of an electromagnetic wave or elastic S–wave is described by the coupling ray theory, which represents the generalization of both the zero–order isotropic and anisotropic ray theories and provides continuous transition between them. The coupling ray theory is usually applied to anisotropic common reference rays, but it is more accurate if it is applied to reference rays which are closer to the actual wave paths. In a generally anisotropic or bianisotropic medium, the actual wave paths may be approximated by the anisotropic–ray–theory rays if these rays behave reasonably. In an approximately uniaxial (approximately transversely isotropic) anisotropic medium, we can define and trace the SH (ordinary) and SV (extraordinary) reference rays, and use them as reference rays for the prevailing–frequency approximation of the coupling ray theory. In both cases, i.e. for the anisotropic–ray–theory rays or the SH and SV reference rays, we have two sets of reference rays. We thus obtain two arrivals along each reference ray of the first set and have to select the correct one. Analogously, we obtain two arrivals along each reference ray of the second set and have to select the correct one. In this paper, we suggest the way of selecting the correct arrivals. We then demonstrate the accuracy of the resulting prevailing–frequency approximation of the coupling ray theory using elastic S waves along the SH and SV reference rays in four different approximately uniaxial (approximately transversely isotropic) velocity models. 相似文献
10.
It is important to include the viscous effect in seismic numerical modelling and seismic migration due to the ubiquitous viscosity in an actual subsurface medium. Prestack reverse‐time migration (RTM) is currently one of the most accurate methods for seismic imaging. One of the key steps of RTM is wavefield forward and backward extrapolation and how to solve the wave equation fast and accurately is the essence of this process. In this paper, we apply the time‐space domain dispersion‐relation‐based finite‐difference (FD) method for visco‐acoustic wave numerical modelling. Dispersion analysis and numerical modelling results demonstrate that the time‐space domain FD method has great accuracy and can effectively suppress numerical dispersion. Also, we use the time‐space domain FD method to solve the visco‐acoustic wave equation in wavefield extrapolation of RTM and apply the source‐normalized cross‐correlation imaging condition in migration. Improved imaging has been obtained in both synthetic and real data tests. The migration result of the visco‐acoustic wave RTM is clearer and more accurate than that of acoustic wave RTM. In addition, in the process of wavefield forward and backward extrapolation, we adopt adaptive variable‐length spatial operators to compute spatial derivatives to significantly decrease computing costs without reducing the accuracy of the numerical solution. 相似文献
11.
Wavefield computations using the ellipsoidally anisotropic extrapolation operator offer significant cost reduction compared to that for the orthorhombic case, especially when the symmetry planes are tilted and/or rotated. However, ellipsoidal anisotropy does not provide accurate wavefield representation or imaging for media of orthorhombic symmetry. Therefore, we propose the use of ‘effective ellipsoidally anisotropic’ models that correctly capture the kinematic behaviour of wavefields for tilted orthorhombic (TOR) media. We compute effective velocities for the ellipsoidally anisotropic medium using kinematic high-frequency representation of the TOR wavefield, obtained by solving the TOR eikonal equation. The effective model allows us to use the cheaper ellipsoidally anisotropic wave extrapolation operators. Although the effective models are obtained by kinematic matching using high-frequency asymptotic, the resulting wavefield contains most of the critical wavefield components, including frequency dependency and caustics, if present, with reasonable accuracy. The proposed methodology offers a much better cost versus accuracy trade-off for wavefield computations in TOR media, particularly for media of low to moderate anisotropic strength. Furthermore, the computed wavefield solution is free from shear-wave artefacts as opposed to the conventional finite-difference based TOR wave extrapolation scheme. We demonstrate applicability and usefulness of our formulation through numerical tests on synthetic TOR models. 相似文献
12.
C.J. Thomson 《Geophysical Prospecting》2012,60(1):64-92
The reflection operator for a simple flat‐lying interface can be thought of as the set of all its plane‐wave reflection coefficients or as the set of virtual surveys with sources and receivers along the interface. When there is dip, however, it is necessary to include the varying effects of propagation between the virtual‐survey level and the interface. Hence, step one in this paper is to derive the reflection operator for a dipping plane interface as observed at a datum level some distance away. The key assumption is that the aperture at the datum level is sufficient to characterize the reflector properties around a particular point. This translates into an assumption that the dip is moderate, though no explicit small‐angle approximation is required. The second step is to find the apparent reflection operator that would relate data that have been extrapolated from the datum towards and possibly beyond the reflector using an assumed migration velocity. This apparent reflection operator is closely related to extended common‐image gathers. The apparent reflection operator may be analysed asymptotically in terms of rays and other signals, shedding light on the structure of extended image gathers. In keeping with the virtual‐survey idea, the results are considered in a subsurface space‐time or slowness‐time domain at various extrapolation levels around the interface. An important distinction is drawn between using subsurface midpoint‐offset coordinates and the wavefield coordinates of the incident and reflected waves. The latter reveal more clearly the effects of dip, because they lead to a more asymmetric apparent reflection operator. Properties such as an up‐dip shift of a traveltime minimum and its associated curvature theoretically provide information about the reflector location and dip and the migration‐velocity error. The space‐time form of the reflection operator can be highly intricate around the offset‐time origin and it was described for a simple flat interface in a background paper. To avoid a layer of mathematics, the reflection‐operator formulas presented here are in the intermediate space‐frequency domain. They are analysed by considering their stationary‐phase and branch‐point high‐frequency contributions. There is no Born‐like assumption of weak reflector contrast and so wide‐angle, total reflection and head‐wave effects are included. Snell’s law is an explicit part of the theory. It is hoped that the work will therefore be a step towards the goal of unifying amplitude‐versus‐offset, imaging and waveform inversion. 相似文献
13.
Anisotropy is often observed due to the thin layering or aligned micro‐structures, like small fractures. At the scale of cross‐well tomography, the anisotropic effects cannot be neglected. In this paper, we propose a method of full‐wave inversion for transversely isotropic media and we test its robustness against structured noisy data. Optimization inversion techniques based on a least‐square formalism are used. In this framework, analytical expressions of the misfit function gradient, based on the adjoint technique in the time domain, allow one to solve the inverse problem with a high number of parameters and for a completely heterogeneous medium. The wave propagation equation for transversely isotropic media with vertical symmetry axis is solved using the finite difference method on the cylindrical system of coordinates. This system allows one to model the 3D propagation in a 2D medium with a revolution symmetry. In case of approximately horizontal layering, this approximation is sufficient. The full‐wave inversion method is applied to a crosswell synthetic 2‐component (radial and vertical) dataset generated using a 2D model with three different anisotropic regions. Complex noise has been added to these synthetic observed data. This noise is Gaussian and has the same amplitude f?k spectrum as the data. Part of the noise is localized as a coda of arrivals, the other part is not localized. Five parameter fields are estimated, (vertical) P‐wave velocity, (vertical) S‐wave velocity, volumetric mass and the Thomsen anisotropic parameters epsilon and delta. Horizontal exponential correlations have been used. The results show that the full‐wave inversion of cross‐well data is relatively robust for high‐level noise even for second‐order parameters such as Thomsen epsilon and delta anisotropic parameters. 相似文献
14.
T. Alkhalifah 《Geophysical Prospecting》2015,63(1):2-10
The double‐square‐root equation is commonly used to image data by downward continuation using one‐way depth extrapolation methods. A two‐way time extrapolation of the double‐square‐root‐derived phase operator allows for up and downgoing wavefields but suffers from an essential singularity for horizontally travelling waves. This singularity is also associated with an anisotropic version of the double‐square‐root extrapolator. Perturbation theory allows us to separate the isotropic contribution, as well as the singularity, from the anisotropic contribution to the operator. As a result, the anisotropic residual operator is free from such singularities and can be applied as a stand alone operator to correct for anisotropy. We can apply the residual anisotropy operator even if the original prestack wavefield was obtained using, for example, reverse‐time migration. The residual correction is also useful for anisotropic parameter estimation. Applications to synthetic data demonstrate the accuracy of the new prestack modelling and migration approach. It also proves useful in approximately imaging the Vertical Transverse Isotropic Marmousi model. 相似文献
15.
Post‐stack diffraction imaging in vertical transverse isotropy media using non‐hyperbolic moveout approximations 
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下载免费PDF全文 Diffractions carry valuable information about local discontinuities and small‐scale objects in the subsurface. They are still not commonly used in the process of geological interpretation. Many diffraction imaging techniques have been developed and applied for isotropic media, whereas relatively few techniques have been developed for anisotropic media. Ignoring anisotropy can result in low‐resolution images with wrongly positioned or spurious diffractors. In this article, we suggest taking anisotropy into account in two‐dimensional post‐stack domain by considering P‐wave non‐hyperbolic diffraction traveltime approximations for vertical transverse isotropy media, previously developed for reflection seismology. The accuracy of the final images is directly connected to the accuracy of the diffraction traveltime approximations. We quantified the accuracy of six different approximations, including hyperbolic moveout approximation, by the application of a post‐stack diffraction imaging technique on two‐dimensional synthetic data examples. 相似文献
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Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference (symplectic FFD) method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps. 相似文献
18.
Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media 下载免费PDF全文
下载免费PDF全文 Existing and commonly used in industry nowadays, closed‐form approximations for a P‐wave reflection coefficient in transversely isotropic media are restricted to cases of a vertical and a horizontal transverse isotropy. However, field observations confirm the widespread presence of rock beds and fracture sets tilted with respect to a reflection boundary. These situations can be described by means of the transverse isotropy with an arbitrary orientation of the symmetry axis, known as tilted transversely isotropic media. In order to study the influence of the anisotropy parameters and the orientation of the symmetry axis on P‐wave reflection amplitudes, a linearised 3D P‐wave reflection coefficient at a planar weak‐contrast interface separating two weakly anisotropic tilted tranversely isotropic half‐spaces is derived. The approximation is a function of the incidence phase angle, the anisotropy parameters, and symmetry axes tilt and azimuth angles in both media above and below the interface. The expression takes the form of the well‐known amplitude‐versus‐offset “Shuey‐type” equation and confirms that the influence of the tilt and the azimuth of the symmetry axis on the P‐wave reflection coefficient even for a weakly anisotropic medium is strong and cannot be neglected. There are no assumptions made on the symmetry‐axis orientation angles in both half‐spaces above and below the interface. The proposed approximation can be used for inversion for the model parameters, including the orientation of the symmetry axes. Obtained amplitude‐versus‐offset attributes converge to well‐known approximations for vertical and horizontal transverse isotropic media derived by Rüger in corresponding limits. Comparison with numerical solution demonstrates good accuracy. 相似文献
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相速度和偏振方向是研究地震波传播规律和描述介质特性的重要参数,在理论研究和实际应用中有重要作用.本文假定倾斜横向各向同性(TTI)介质对称轴位于观测坐标系XOZ面内,在此观测坐标系下直接推导了TTI介质弹性波相速度和偏振方向的解析表达式,再进一步利用Thomsen弱各向异性理论,推导了弱各向异性近似条件下弹性波相速度以及qP波和qSV波偏振方向表达式.理论分析和数值试例表明,在相速度方面,随着各向异性介质参数改变,qP波和qSH波速度变化较为平缓,qSV波速度变化较为剧烈.弹性波相速度近似式误差均较小,能较好地近似精确相速度.在偏振方向方面,SH波偏振方向只是传播方向和对称轴倾角的函数,而与各向异性参数无关,SH波偏振方向既垂直于传播方向,又垂直于TTI介质对称轴方向.除特定方向外,qP波和qSV波的偏振方向与传播方向均成一定角度,并且随TTI介质对称轴倾角的改变而改变;在精确和近似情况下,qP波和qSV波的偏振方向始终垂直;在精度允许范围内,偏振方向的弱各向异性近似式与理论解析式吻合较好. 相似文献