An efficient wave extrapolation method for anisotropic media with tilt

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.     

VTI介质纯P波混合法正演模拟及稳定性分析

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.     

正交各向异性介质中qP波入射的二阶近似反射系数与透射系数

地震波在各向异性介质中以一个准P波(qP)和两个准S波(qS1和qS2)的形式传播.研究三种波的相速度、群速度以及偏振方向等传播性质能够为各向异性介质中的正反演问题提供有效支撑.具有比横向各向同性(TI)介质更一般对称性的正交各向异性介质通常需要9个独立参数对其进行描述,这使得对传播特征的计算更为复杂.当两个准S波速度相近时具有耦合性,从而令慢度的计算产生奇异性.因此,奇异点(慢度面的鞍点和交叉点)附近的反射与透射(R/T)系数的求解不稳定,会导致波场振幅不准确.本文首次通过结合耦合S波射线理论和基于迭代的各向异性相速度与偏振矢量的高阶近似解,得到了适用于正交各向异性介质以qP波入射所产生的二阶R/T系数的计算方法.与基于一阶近似的结果相比,基于二阶近似的方法提高了qP波R/T系数的精度,能得到一阶耦合近似无法表达的准确的qP-qS转换波的R/T系数解,且方法适用于较强的各向异性介质.     

Reflected waves in finely layered tilted orthorhombic media

Upscaling in seismics is a homogenization of finely layered media in the zero-frequency limit. An upscaling technique for arbitrary anisotropic layers has been developed by Schoenberg and Muir. Applying this technique to a stack of layers of orthorhombic (ORT) symmetry whose vertical symmetry planes are aligned, results in an effective homogeneous layer with orthorhombic symmetry. If the symmetry planes in a horizontal orthorhombic layer are rotated with respect to vertical, the medium is referred to as tilted orthorhombic (TOR) medium, and the stack composed of TOR layers in zero-frequency limit will produce an effective medium of a lower symmetry than orthorhombic. We consider a P-wave that propagates through a stack of thin TOR layers, then it is reflected (preserving the mode) at some interface below the stack, and then propagates back through the same stack. We propose to use a special modified medium for the upscaling in case of this sequential down- and up-propagation: each TOR layer in the stack is replaced by two identical TOR layers whose tilt angles have the opposite algebraic sign. In this modified medium, one-way propagation of a seismic wave (any wave mode) is equivalent to propagation of a pure-mode reflection in the original medium. We apply this idea to study the contribution from an individual layer from the stack and show how the approach can be applied to a stack of TOR layers. To demonstrate the applicability of the model, we use well log data for the upscaling. The model we propose for the upscaling can be used in well-seismic ties to correct the effective parameters obtained from well log data for the presence of tilt, if latter is confirmed by additional measurements (for example, borehole imaging).     

Effective wavefield extrapolation in anisotropic media: accounting for resolvable anisotropy

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.     

Generalized migration in frequency-wavenumber domain (MGF-K) in anisotropic media

In this paper, the background of MGF-K migration in dual domain (wavenumber-frequency K-F and space-time) in anisotropic media is presented. Algorithms for poststack (zero-offset) and prestack migration are based on downward extrapolation of acoustic wavefield by shift-phase with correction filter for lateral variability of medium’s parameters. In anisotropic media, the vertical wavenumber was determined from full elastic wavefield equations for two dimensional (2D) tilted transverse isotropy (TTI) model. The method was tested on a synthetic wavefield for TTI anticlinal model (zero-offset section) and on strongly inhomogeneous vertical transverse isotropy (VTI) Marmousi model. In both cases, the proper imaging of assumed media was obtained.     

Elastic wave‐vector decomposition in heterogeneous anisotropic media

The goal of wave‐mode separation and wave‐vector decomposition is to separate a full elastic wavefield into three wavefields with each corresponding to a different wave mode. This allows elastic reverse‐time migration to handle each wave mode independently. Several of the previously proposed methods to accomplish this task require the knowledge of the polarisation vectors of all three wave modes in a given anisotropic medium. We propose a wave‐vector decomposition method where the wavefield is decomposed in the wavenumber domain via the analytical decomposition operator with improved computational efficiency using low‐rank approximations. The method is applicable for general heterogeneous anisotropic media. To apply the proposed method in low‐symmetry anisotropic media such as orthorhombic, monoclinic, and triclinic, we define the two S modes by sorting them based on their phase velocities (S1 and S2), which are defined everywhere except at the singularities. The singularities can be located using an analytical condition derived from the exact phase‐velocity expressions for S waves. This condition defines a weight function, which can be applied to attenuate the planar artefacts caused by the local discontinuity of polarisation vectors at the singularities. The amplitude information lost because of weighting can be recovered using the technique of local signal–noise orthogonalisation. Numerical examples show that the proposed approach provides an effective decomposition method for all wave modes in heterogeneous, strongly anisotropic media.     

轴对称各向异性介质波动方程深度偏移的对称非平稳相移方法

由所建立的三维qP波相速度表示式出发,导出并解析求解各向异性介质中的频散方程,得到三维各向异性介质中的相移算子,进而将以相移算子为基础的对称非平稳相移方法推广到各向异性介质,发展了一个三维各向异性介质的深度偏移方法. 文中使用的各向异性介质的速度模型与现行的各向异性构造的速度估计方法一致,将各向同性、弱各向异性及强各向异性统一在一个模型中. 所建立的各向异性介质对称非平稳相移波场延拓算子可以同时适应速度及各向异性参数横向变化;文中给出的算例虽然是针对二维VTI介质的,但所提出的算法同样适用于三维TI介质.     

Anisotropic reverse-time migration for tilted TI media

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.     

In-Plane Ray Tracing with Calculation of Out-of-Plane Geometrical Spreading in Anisotropic Media

The objective is to provide, in one single paper, a complete collection of equations governing kinematic and dynamic ray tracing related to a symmetry plane of an anisotropic medium. Well known systems for kinematic ray tracing and in-plane dynamic ray tracing are reformulated for the purpose of clarity, by taking advantage of a vector representation of the Christoffel matrix elements and related quantities. A generalized formula is derived for the integrand in out-of-plane dynamic ray tracing, pertaining to a monoclinic medium. Integrands corresponding to non-tilted orthorhombic and transversely isotropic media are obtained as special cases.     

High-order kernels for Riemannian wavefield extrapolation

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.     

Elastic wave mode separation for tilted transverse isotropy media

Seismic waves propagate through the earth as a superposition of different wave modes. Seismic imaging in areas characterized by complex geology requires techniques based on accurate reconstruction of the seismic wavefields. A crucial component of the methods in this category, collectively known as wave‐equation migration, is the imaging condition that extracts information about the discontinuities of physical properties from the reconstructed wavefields at every location in space. Conventional acoustic migration techniques image a scalar wavefield representing the P‐wave mode, in contrast to elastic migration techniques, which image a vector wavefield representing both the P‐ and S‐waves. For elastic imaging, it is desirable that the reconstructed vector fields are decomposed into pure wave modes, such that the imaging condition produces interpretable images, characterizing, for example, PP or PS reflectivity. In anisotropic media, wave mode separation can be achieved by projection of the reconstructed vector fields on the polarization vectors characterizing various wave modes. For heterogeneous media, because polarization directions change with position, wave mode separation needs to be implemented using space‐domain filters. For transversely isotropic media with a tilted symmetry axis, the polarization vectors depend on the elastic material parameters, including the tilt angles. Using these parameters, we separate the wave modes by constructing nine filters corresponding to the nine Cartesian components of the three polarization directions at every grid point. Since the S polarization vectors in transverse isotropic media are not defined in the singular directions, e.g., along the symmetry axes, we construct these vectors by exploiting the orthogonality between the SV and SH polarization vectors, as well as their orthogonality with the P polarization vector. This procedure allows one to separate all three modes, with better preserved P‐wave amplitudes than S‐wave amplitudes. Realistic synthetic examples show that this wave mode separation is effective for both 2D and 3D models with strong heterogeneity and anisotropy.     

Arbitrary-order Taylor series expansion-based viscoacoustic wavefield simulation in 3D vertical transversely isotropic media

Taking the anisotropy of velocity and attenuation into account, we investigate the wavefield simulation of viscoacoustic waves in 3D vertical transversely isotropic attenuating media. The viscoacoustic wave equations with the decoupled amplitude attenuation and phase dispersion are derived from the fractional Laplacian operator and using the acoustic approximation. With respect to the spatially variable fractional Laplacian operator in the formulation, we develop an effective algorithm to realize the viscoacoustic wavefield extrapolation by using the arbitrary-order Taylor series expansion. Based on the approximation, the mixed-domain fractional Laplacian operators are decoupled from the wavenumbers and fractional orders. Thus, the viscoacoustic wave propagation can be conveniently implemented by using a generalized pseudospectral method. In addition, we perform the accuracy and efficiency analyses among first-, second- and third-order Taylor series expansion pseudospectral methods with different quality factors. Considering both the accuracy and computational cost, the second-order Taylor series expansion pseudospectral method can generally satisfy the requirements for most attenuating media. Numerical modelling examples not only illustrate that our decoupled viscoacoustic wave equations can effectively describe the attenuating property of the medium, but also demonstrate the accuracy and the high robustness of our proposed schemes.     

Wavefield dependency on virtual shifts in the source location

The wavefield dependence on a virtual shift in the source location can provide information helpful in velocity estimation and interpolation. However, the second‐order partial differential equation (PDE) that relates changes in the wavefield form (or shape) to lateral perturbations in the source location depends explicitly on lateral derivatives of the velocity field. For velocity models that include lateral velocity discontinuities this is problematic as such derivatives in their classical definition do not exist. As a result, I derive perturbation partial differential wave equations that are independent of direct velocity derivatives and thus, provide possibilities for wavefield shape extrapolation in complex media. These PDEs have the same structure as the wave equation with a source function that depends on the background (original source) wavefield. The solutions of the perturbation equations provide the coefficients of a Taylor's series type expansion for the wavefield. The new formulas introduce changes to the background wavefield only in the presence of lateral velocity variation or in general terms velocity variations in the perturbation direction. The accuracy of the representation, as demonstrated on the Marmousi model, is generally good.     

波动方程深度偏移的频率相关变步长延拓方法

发展了波动方程深度延拓的频率相关变步长深度延拓方法和表驱动的单点波场插值技术.前者通过减少深度延拓的次数减少了波动方程深度偏移的计算量,而后者用很少的计算量实现了等间距、理想采样的深度成像.就同一偏移方法,采用频率相关变步长深度延拓加单点插值,其计算量大约是常规的等间距采样延拓方法的三分之一,但两者的成像效果基本相同.文中以最优分裂Fourier方法为例,用二维理论数据（Marmousi模型）和三维实际地震资料验证了这一方法,但这一方法可适用于各类频率域波动方程深度偏移方法.     

Pseudo‐spectral method using rotated staggered grid for elastic wave propagation in 3D arbitrary anisotropic media

Staggering grid is a very effective way to reduce the Nyquist errors and to suppress the non‐causal ringing artefacts in the pseudo‐spectral solution of first‐order elastic wave equations. However, the straightforward use of a staggered‐grid pseudo‐spectral method is problematic for simulating wave propagation when the anisotropy level is greater than orthorhombic or when the anisotropic symmetries are not aligned with the computational grids. Inspired by the idea of rotated staggered‐grid finite‐difference method, we propose a modified pseudo‐spectral method for wave propagation in arbitrary anisotropic media. Compared with an existing remedy of staggered‐grid pseudo‐spectral method based on stiffness matrix decomposition and a possible alternative using the Lebedev grids, the rotated staggered‐grid‐based pseudo‐spectral method possesses the best balance between the mitigation of artefacts and efficiency. A 2D example on a transversely isotropic model with tilted symmetry axis verifies its effectiveness to suppress the ringing artefacts. Two 3D examples of increasing anisotropy levels demonstrate that the rotated staggered‐grid‐based pseudo‐spectral method can successfully simulate complex wavefields in such anisotropic formations.     

一种三维正交方位各向异性介质岩石 物理建模及弹性波正演模拟方法

通过线性滑动理论和岩石物理等效理论, 将两组正交直立裂隙介质等效为一种正交方位各向异性介质进行三维岩石物理建模, 通过高阶交错网格有限差分求解弹性波动方程模拟地震波在该种介质中的传播过程. 在建模过程中改变物性参数, 分析不同裂隙密度条件下的炮集和波场特征, 以及正交各向异性的方位特征. 研究结果表明, 各向异性强度随裂隙密度等物性增大而增强, 而且这些特征在共炮点道集和波场中均有所体现.      

Weak‐anisotropy approximation for P‐wave reflection coefficient at the boundary between two tilted transversely isotropic media

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.     

Seismic wavefield modeling based on time-domain symplectic and Fourier finite-difference method

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.     

A self-adaptive algorithm for choosing reference velocities in the presence of lateral velocity variations

Based on perturbation theory, the wave equation extrapolation operator with mixed domains has the ability to deal with lateral velocity variations. It is the image method that has undergone much research in seismology. All extrapolation operators face the problem of choosing the reference velocity due to continuation in depth. The wavefield extrapolation operator with a single reference velocity is suitable for media with weak lateral variation. The multi-reference velocity extrapolation operator can cope with severe lateral velocity variations and improve image accuracy. However, the calculation cost is large. We present a self-adaptive approach to automatically determine the number of selected reference velocities according to the complexity of structure and the given velocity threshold value. The approach can be used to construct the SSF, FFD, WXFD, and GSP multi-reference velocity wavefield extrapolation image algorithms. The result of a salt-dome model data test demonstrates that the self-adoptive multi-reference wavefield extrapolation algorithm has the ability to deal with severe lateral velocity variations and can also be used for structure edge detection. The method is flexible and computationally cost-effective.