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.     

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.     

Preserved travel‐time smoothing in orthorhombic media

Certain degree of smoothness of velocity models is required for most ray‐based migration and tomography. Applying conventional smoothing in model parameters results in offset‐dependent travel‐time errors for reflected events, which can be large even for small contrasts in model parameters between the layers. This causes the shift in both the depth and residual moveout of the migrated images. To overcome this problem in transversely isotropic medium with a vertical symmetry axis, the preserved travel‐time smoothing method was proposed earlier. We extend this method for orthorhombic media with and without azimuthal variation between the layers. We illustrate this method for a single interface between two orthorhombic layers and show that the smoothing‐driven errors in travel time are very small for practical application.     

S-wave in 2D acoustic transversely isotropic media with a tilted symmetry axis

In an acoustic transversely isotropic medium, there are two waves that propagate. One is the P-wave and another one is the S-wave (also known as S-wave artefact). This paper is devoted to analyse the S-wave in two-dimensional acoustic transversely isotropic media with a tilted symmetry axis. We derive the S-wave slowness surface and traveltime function in a homogeneous acoustic transversely isotropic medium with a tilted symmetry axis. The S-wave traveltime approximations in acoustic transversely isotropic media with a tilted symmetry axis can be mapped from the counterparts for acoustic transversely isotropic media with a vertical symmetry axis. We consider a layered two-dimensional acoustic transversely isotropic medium with a tilted symmetry axis to analyse the S-wave moveout. We also illustrate the behaviour of the moveout for reflected S-wave and converted waves.     

Scanning anisotropy parameters in horizontal transversely isotropic media

The horizontal transversely isotropic model, with arbitrary symmetry axis orientation, is the simplest effective representative that explains the azimuthal behaviour of seismic data. Estimating the anisotropy parameters of this model is important in reservoir characterisation, specifically in terms of fracture delineation. We propose a travel‐time‐based approach to estimate the anellipticity parameter η and the symmetry axis azimuth ? of a horizontal transversely isotropic medium, given an inhomogeneous elliptic background model (which might be obtained from velocity analysis and well velocities). This is accomplished through a Taylor's series expansion of the travel‐time solution (of the eikonal equation) as a function of parameter η and azimuth angle ?. The accuracy of the travel time expansion is enhanced by the use of Shanks transform. This results in an accurate approximation of the solution of the non‐linear eikonal equation and provides a mechanism to scan simultaneously for the best fitting effective parameters η and ?, without the need for repetitive modelling of travel times. The analysis of the travel time sensitivity to parameters η and ? reveals that travel times are more sensitive to η than to the symmetry axis azimuth ?. Thus, η is better constrained from travel times than the azimuth. Moreover, the two‐parameter scan in the homogeneous case shows that errors in the background model affect the estimation of η and ? differently. While a gradual increase in errors in the background model leads to increasing errors in η, inaccuracies in ?, on the other hand, depend on the background model errors. We also propose a layer‐stripping method valid for a stack of arbitrary oriented symmetry axis horizontal transversely isotropic layers to convert the effective parameters to the interval layer values.     

Angle gathers in wave‐equation imaging for transversely isotropic media

In recent years, wave‐equation imaged data are often presented in common‐image angle‐domain gathers as a decomposition in the scattering angle at the reflector, which provide a natural access to analysing migration velocities and amplitudes. In the case of anisotropic media, the importance of angle gathers is enhanced by the need to properly estimate multiple anisotropic parameters for a proper representation of the medium. We extract angle gathers for each downward‐continuation step from converting offset‐frequency planes into angle‐frequency planes simultaneously with applying the imaging condition in a transversely isotropic with a vertical symmetry axis (VTI) medium. The analytic equations, though cumbersome, are exact within the framework of the acoustic approximation. They are also easily programmable and show that angle gather mapping in the case of anisotropic media differs from its isotropic counterpart, with the difference depending mainly on the strength of anisotropy. Synthetic examples demonstrate the importance of including anisotropy in the angle gather generation as mapping of the energy is negatively altered otherwise. In the case of a titled axis of symmetry (TTI), the same VTI formulation is applicable but requires a rotation of the wavenumbers.     

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.     

3D P‐wave traveltime computation in transversely isotropic media using layer‐by‐layer wavefront marching

Subsurface rocks (e.g. shale) may induce seismic anisotropy, such as transverse isotropy. Traveltime computation is an essential component of depth imaging and tomography in transversely isotropic media. It is natural to compute the traveltime using the wavefront marching method. However, tracking the 3D wavefront is expensive, especially in anisotropic media. Besides, the wavefront marching method usually computes the traveltime using the eikonal equation. However, the anisotropic eikonal equation is highly non‐linear and it is challenging to solve. To address these issues, we present a layer‐by‐layer wavefront marching method to compute the P‐wave traveltime in 3D transversely isotropic media. To simplify the wavefront tracking, it uses the traveltime of the previous depth as the boundary condition to compute that of the next depth based on the wavefront marching. A strategy of traveltime computation is designed to guarantee the causality of wave propagation. To avoid solving the non‐linear eikonal equation, it updates traveltime along the expanding wavefront by Fermat's principle. To compute the traveltime using Fermat's principle, an approximate group velocity with high accuracy in transversely isotropic media is adopted to describe the ray propagation. Numerical examples on 3D vertical transverse isotropy and tilted transverse isotropy models show that the proposed method computes the traveltime with high accuracy. It can find applications in modelling and depth migration.     

Out‐of‐plane geometrical spreading in anisotropic media

Two-dimensional seismic processing is successful in media with little structural and velocity variation in the direction perpendicular to the plane defined by the acquisition direction and the vertical axis. If the subsurface is anisotropic, an additional limitation is that this plane is a plane of symmetry. Kinematic ray propagation can be considered as a two-dimensional process in this type of medium. However, two-dimensional processing in a true-amplitude sense requires out-of-plane amplitude corrections in addition to compensation for in-plane amplitude variation. We provide formulae for the out-of-plane geometrical spreading for P- and S-waves in transversely isotropic and orthorhombic media. These are extensions of well-known isotropic formulae.
For isotropic and transversely isotropic media, the ray propagation is independent of the azimuthal angle. The azimuthal direction is defined with respect to a possibly tilted axis of symmetry. The out-of-plane spreading correction can then be calculated by integrating quantities which describe in-plane kinematics along in-plane rays. If, in addition, the medium varies only along the vertical direction and has a vertical axis of symmetry, no ray tracing need be carried out. All quantities affecting the out-of-plane geometrical spreading can be derived from traveltime information available at the observation surface.
Orthorhombic media possess no rotational symmetry and the out-of-plane geometrical spreading includes parameters which, even in principle, are not invertible from in-plane experiments. The exact and approximate formulae derived for P- and S-waves are nevertheless useful for modelling purposes.     

Depth migration of shot records in heterogeneous, transversely isotropic media using optimum explicit operators

A space–frequency domain 2D depth-migration scheme is generalized for imaging in the presence of anisotropy. The anisotropy model used is that of a transversely isotropic (TI) medium with a symmetry axis that can be either vertical or tilted. In the proposed scheme the anisotropy is described in terms of Thomsen parameters; however, the scheme can accommodate a wide range of anisotropy rather than only weak anisotropy. Short spatial convolution operators are used to extrapolate the wavefields recursively in the space–frequency domain for both qP- and qSV-waves. The weighted least-squares method for designing isotropic optimum operators is extended to asymmetric optimum explicit extrapolation operators in the presence of TI media with a tilted symmetry axis. Additionally, an efficient weighted quadratic-programming design method is developed. The short spatial length of the derived operators makes it possible for the proposed scheme to handle lateral inhomogeneities. The performance of the operators, designed by combining the weighted least-squares and weighted quadratic-programming methods, is demonstrated by migration impulse responses of qP and qSV propagation modes for the weak and strong TI models with both vertical and tilted symmetry axes. Finally, a table-driven shot-record depth-migration scheme is proposed, which is illustrated for finite-difference modelled shot records in TI media.     

Time evolution of the first‐order linear acoustic/elastic wave equation using Lie product formula and Taylor expansion

We propose a new numerical solution to the first‐order linear acoustic/elastic wave equation. This numerical solution is based on the analytic solution of the linear acoustic/elastic wave equation and uses the Lie product formula, where the time evolution operator of the analytic solution is written as a product of exponential matrices where each exponential matrix term is then approximated by Taylor series expansion. Initially, we check the proposed approach numerically and then demonstrate that it is more accurate to apply a Taylor expansion for the exponential function identity rather than the exponential function itself. The numerical solution formulated employs a recursive procedure and also incorporates the split perfectly matched layer boundary condition. Thus, our scheme can be used to extrapolate wavefields in a stable manner with even larger time‐steps than traditional finite‐difference schemes. This new numerical solution is examined through the comparison of the solution of full acoustic wave equation using the Chebyshev expansion approach for the matrix exponential term. Moreover, to demonstrate the efficiency and applicability of our proposed solution, seismic modelling results of three geological models are presented and the processing time for each model is compared with the computing time taking by the Chebyshev expansion method. We also present the result of seismic modelling using the scheme based in Lie product formula and Taylor series expansion for the first‐order linear elastic wave equation in vertical transversely isotropic and tilted transversely isotropic media as well. Finally, a post‐stack migration results are also shown using the proposed method.     

Earthquake location in transversely isotropic media with a tilted symmetry axis

The conventional intersection method for earthquake location in isotropic media is developed in the case of transversely isotropic media with a tilted symmetry axis (TTI media). The hypocenter is determined using its loci, which are calculated through a minimum travel time tree algorithm for ray tracing in TTI media. There are no restrictions on the structural complexity of the model or on the anisotropy strength of the medium. The location method is validated by its application to determine the hypocenter and origin time of an event in a complex TTI structure, in accordance with four hypotheses or study cases: (a) accurate model and arrival times, (b) perturbed model with randomly variable elastic parameter, (c) noisy arrival time data, and (d) incomplete set of observations from the seismic stations. Furthermore, several numerical tests demonstrate that the orientation of the symmetry axis has a significant effect on the hypocenter location when the seismic anisotropy is not very weak. Moreover, if the hypocentral determination is based on an isotropic reference model while the real medium is anisotropic, the resultant location errors can be considerable even though the anisotropy strength does not exceed 6.10%.     

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.     

Perturbation‐based moveout approximations in anisotropic media

The moveout approximations play an important role in seismic data processing. The standard hyperbolic moveout approximation is based on an elliptical background model with two velocities: vertical and normal moveout. We propose a new set of moveout approximations based on a perturbation series in terms of anellipticity parameters using the alternative elliptical background model defined by vertical and horizontal velocities. We start with a transversely isotropic medium with a vertical symmetry axis. Then, we extend this approach to a homogeneous orthorhombic medium. To define the perturbation coefficients for a new background, we solve the eikonal equation with horizontal velocities in transversely isotropic medium with a vertical symmetry axis and orthorhombic media. To stabilise the perturbation series and improve the accuracy, the Shanks transform is applied for all the cases. We select different parameterisations for both velocities and anellipticity parameters for an orthorhombic model. From the comparison in traveltime error, the new moveout approximations result in better accuracy comparing with the standard perturbation‐based methods and other approximations.     

三维VTI介质中波动方程深度偏移的最优分裂Fourier方法

从含Thomsen各向异性参数的qP波相速度表示式出发,建立并求解三维VTI介质中的频散方程,得到三维VTI介质中的相移算子,进而将以相移算子为基础的最优分裂Fourier方法推广到三维VTI介质,发展了一个三维VTI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.文中提出的偏移算法对相移法引入了高阶校正项来补偿介质横向变化的影响,使该方法可应用于横向非均匀VTI介质的陡角度成像,文中给出的偏移脉冲响应很好地证明了这一点.     

Migration velocity analysis for tilted transversely isotropic media

Tilted transversely isotropic formations cause serious imaging distortions in active tectonic areas (e.g., fold‐and‐thrust belts) and in subsalt exploration. Here, we introduce a methodology for P‐wave prestack depth imaging in tilted transversely isotropic media that properly accounts for the tilt of the symmetry axis as well as for spatial velocity variations. For purposes of migration velocity analysis, the model is divided into blocks with constant values of the anisotropy parameters ε and δ and linearly varying symmetry‐direction velocity V_P0 controlled by the vertical (k_z) and lateral (k_x) gradients. Since determination of tilt from P‐wave data is generally unstable, the symmetry axis is kept orthogonal to the reflectors in all trial velocity models. It is also assumed that the velocity V_P0 is either known at the top of each block or remains continuous in the vertical direction. The velocity analysis algorithm estimates the velocity gradients k_z and k_x and the anisotropy parameters ε and δ in the layer‐stripping mode using a generalized version of the method introduced by Sarkar and Tsvankin for factorized transverse isotropy with a vertical symmetry axis. Synthetic tests for several models typical in exploration (a syncline, uptilted shale layers near a salt dome and a bending shale layer) confirm that if the symmetry‐axis direction is fixed and V_P0 is known, the parameters k_z, k_x, ε and δ can be resolved from reflection data. It should be emphasized that estimation of ε in tilted transversely isotropic media requires using nonhyperbolic moveout for long offsets reaching at least twice the reflector depth. We also demonstrate that application of processing algorithms designed for a vertical symmetry axis to data from tilted transversely isotropic media may lead to significant misfocusing of reflectors and errors in parameter estimation, even when the tilt is moderate (30°). The ability of our velocity analysis algorithm to separate the anisotropy parameters from the velocity gradients can be also used in lithology discrimination and geologic interpretation of seismic data in complex areas.     

S‐wave kinematics in acoustic transversely isotropic media with a vertical symmetry axis

Acoustic transversely isotropic models are widely used in seismic exploration for P‐wave processing and analysis. In isotropic acoustic media only P‐wave can propagate, while in an acoustic transversely isotropic medium both P and S waves propagate. In this paper, we focus on kinematic properties of S‐wave in acoustic transversely isotropic media. We define new parameters better suited for S‐wave kinematics analysis. We also establish the travel time and relative geometrical spreading equations and analyse their properties. To illustrate the behaviour of the S‐wave in multi‐layered acoustic transversely isotropic media, we define the Dix‐type equations that are different from the ones widely used for the P‐wave propagation.     

任意空间取向TI介质qP和qSV波解耦的波动方程

由于构造运动等作用,TI介质对称轴往往沿空间任意方向分布,具有任意空间取向对称轴的TI（ATI）介质更符合实际地质情况.VTI介质与ATI介质的相速度在形式上具有一致性,VTI介质中地震波的相角对应ATI介质对称轴与地震波传播方向的夹角.本文基于Tsvankin的VTI介质精确相速度公式,利用TI介质对称轴和地震波传播方向上单位向量的数量积和向量积来计算ATI介质的精确相速度.根据弱各向异性假设,导出qP波和qSV波的近似相速度,分析了近似公式的误差,讨论总结了ATI介质qP波和qSV波的相速度特征.本文中的单位向量采用观测坐标系表示,通过相角关系,可以较为方便地由ATI介质近似相速度导出频散关系,然后借助傅里叶逆变换推导出时间-波数域qP波和qSV波解耦的波动方程.数值算例表明本文的波动方程是qP波和qSV波解耦的,波场计算结果稳定,未出现明显的数值频散,验证了本文方法的有效性.