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.     

Traveltime parameters in a tilted elliptical anisotropic medium‡

In this paper, we derive analytical expressions for one‐way and two‐way kinematical parameters in elliptical tilted transverse isotropy media. We show that the homogeneous elliptical tilted transverse isotropy models result in hyperbolic moveout with a reflection point sideslip x₀, which can be considered as an additional traveltime parameter for one‐way wave propagation. For homogeneous elliptical tilted transverse isotropy models we show that the inversion of one‐way traveltime parameters suffers from the ambiguity for large tilts. It is shown that the accuracy of the inversion is sensitive to the error in x₀. We also derive and invert the traveltime parameters for a vertically heterogeneous elliptical tilted transverse isotropy model with a tilt gradually changing with depth. The a priori knowledge of parameter δ is very important for inversion. The wrong choise of this parameter results in significant errors in inverted model parameters.     

Traveltime inversion and error analysis for layered anisotropy

While tilted transverse isotropy (TTI) is a good approximation of the velocity structure for many dipping and fractured strata, it is still challenging to estimate anisotropic depth models even when the tilted angle is known. With the assumption of weak anisotropy, we present a TTI traveltime inversion approach for models consisting of several thickness-varying layers where the anisotropic parameters are constant for each layer. For each model layer the inversion variables consist of the anisotropic parameters ε and δ, the tilted angle φ of its symmetry axis, layer velocity along the symmetry axis, and thickness variation of the layer. Using this method and synthetic data, we evaluate the effects of errors in some of the model parameters on the inverted values of the other parameters in crosswell and Vertical Seismic Profile (VSP) acquisition geometry. The analyses show that the errors in the layer symmetry axes sensitively affect the inverted values of other parameters, especially δ. However, the impact of errors in δ on the inversion of other parameters is much less than the impact on δ from the errors in other parameters. Hence, a practical strategy is first to invert for the most error-tolerant parameter layer velocity, then progressively invert for ε in crosswell geometry or δ in VSP geometry.     

Seismic anisotropy analysis at the Low-Noise Underground Laboratory (LSBB) of Rustrel (France)

Seismic anisotropy of a fractured karstic limestone massif in sub-parallel underground galleries is studied. As the fractures are mostly vertically oriented, the seismic properties of the massif are approximated by horizontal transverse isotropy (HTI). Several data inversion methods were applied to a seismic dataset of arrival-times of P and S-waves.The applied methods include: isotropic tomography, simple cosine function fit, homogeneous Monte-Carlo anisotropic inversion for the parameters of horizontal transverse isotropy and anisotropic tomography for tilted transversely isotropic bodies. All methods lead to the conclusion that there is indeed an anisotropy present in the rock massif and confirm the direction of maximum velocity parallel to the direction of fracturing. Strong anisotropy of about 15% is found in the studied area. Repeated measurements show variations of the P-wave parameters, but not of the S-wave parameters, which is reflecting a change in water saturation.     

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.     

Accounting for velocity anisotropy in seismic traveltime tomography: a case study from the investigation of the foundations of a Byzantine monumental building

We estimate velocity anisotropy factors from seismic traveltime tomographic data and apply a correction for anisotropy in the inversion procedure to test possible improvements on the traveltime fit and the quality of the resulting tomographic images. We applied the anisotropy correction on a traveltime data set obtained from the investigation of the foundation structure of a monumental building: a Byzantine church from the 11th century AD, in Athens, Greece. Vertical transverse isotropy is represented by one axis of symmetry and one anisotropy magnitude for the entire tomographic inversion grid. We choose the vertical direction for the symmetry axis by analysing the available data set and taking into account information on the character of the foundations of the church from the literature and past excavations. The anisotropy magnitude is determined by testing a series of values of anisotropy and examining their effect on the tomographic inversion results. The best traveltime fit and image quality are obtained with an anisotropy value (V_max/V_min) of 1.6, restricted to the high velocity structures in the subsurface. We believe that this anisotropy value, which is significantly higher than the usual values reported for near‐surface geological material, is related to the fabric of the church foundations, due to the shape of the individual stone blocks and the layout of the stonework. Inversion results obtained with the correction for anisotropy indicate that both the traveltime fit and the image quality are improved, providing an enhanced reconstruction of the velocity field, especially for the high‐velocity features. Based on this enhanced and more reliable reconstruction of velocity distribution, an improved image of the subsurface material character was made possible. In particular, the pattern and state of the church foundations and possible weak ground material areas were revealed more clearly. This improved subsurface knowledge may assist in a better design of restoration measures for monumental buildings such as Byzantine churches.     

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.     

VTI介质P波非双曲时差分析

具有垂直对称轴的横向各向同性介质模型(VTI)是目前各向异性理论研究和多波多分量地震资料叠前成像处理中最常用的一种各向异性模型.VTI介质中反射 P波时距曲线一般不再是双曲线.基于不同的相速度近似公式会得到不同的时距关系式.文中对几种典型的非双曲时距曲线与射线追踪得到的准确时距曲线在不同各向异性强度下进行了对比，结果表明Muir等和Stovas等提出的非双曲时距公式由于过高地考虑了横波垂直速度的影响与精确的时距曲线有很大偏差；Tsvankin等提出的弱各向异性非双曲时距公式在ε-δ<0时误差增大；Alkhalifah等提出的非双曲时距公式在大炮检距任意各向异性强度下都具有较高的精度，适于在实际资料处理中应用.     

Moveout analysis for tranversely isotropic media with a tilted symmetry axis

The transversely isotropic (TI) model with a tilted axis of symmetry may be typical, for instance, for sediments near the flanks of salt domes. This work is devoted to an analysis of reflection moveout from horizontal and dipping reflectors in the symmetry plane of TI media that contains the symmetry axis. While for vertical and horizontal transverse isotropy zero-offset reflections exist for the full range of dips up to 90°, this is no longer the case for intermediate axis orientations. For typical homogeneous models with a symmetry axis tilted towards the reflector, wavefront distortions make it impossible to generate specular zero-offset reflected rays from steep interfaces. The ‘missing’ dipping planes can be imaged only in vertically inhomogeneous media by using turning waves. These unusual phenomena may have serious implications in salt imaging. In non-elliptical TI media, the tilt of the symmetry axis may have a drastic influence on normal-moveout (NMO) velocity from horizontal reflectors, as well as on the dependence of NMO velocity on the ray parameter p (the ‘dip-moveout (DMO) signature’). The DMO signature retains the same character as for vertical transverse isotropy only for near-vertical and near-horizontal orientation of the symmetry axis. The behaviour of NMO velocity rapidly changes if the symmetry axis is tilted away from the vertical, with a tilt of ±20° being almost sufficient to eliminate the influence of the anisotropy on the DMO signature. For larger tilt angles and typical positive values of the difference between the anisotropic parameters ε and δ, the NMO velocity increases with p more slowly than in homogeneous isotropic media; a dependence usually caused by a vertical velocity gradient. Dip-moveout processing for a wide range of tilt angles requires application of anisotropic DMO algorithms. The strong influence of the tilt angle on P-wave moveout can be used to constrain the tilt using P-wave NMO velocity in the plane that includes the symmetry axis. However, if the azimuth of the axis is unknown, the inversion for the axis orientation cannot be performed without a 3D analysis of reflection traveltimes on lines with different azimuthal directions.     

怀来、 延庆及其周边地区地壳 介质各向异性研究

提出了利用人工爆破P波走时反演地壳介质方位各向异性参数的方法. 在假定介质是弱各向异性介质的情况下, 使用扰动理论得到了线性化的反演公式, 其中待反演的弱各向异性参数是P波走时的线性函数. 如果在反演公式中参考走时取相同震中距接收点的P波平均走时, 那么所获得的弱各向异性参数与参考介质速度的选取无关. 反演得到的弱各向异性参数可以看作是不同震中距和不同深度范围内介质的等效弱各向异性参数. 等效弱各向异性参数在一定程度上反映了不同深度范围内水平方向相速度随方位的变化. 这种变化可能是不同时期构造应力作用的结果. 2007年中国地震局在首都圈怀来地区实施了一次大吨位人工爆破实验, 以爆破点为中心, 布设了高密度的地震观测台网和台阵. 台站相对于爆破点具有360°的全方位覆盖, 所得到的地震记录数据为研究怀来、 延庆地区地壳介质P波方位各向异性提供了必要条件. 我们通过走时反演获得了与水平方位相关的弱各向异性参数, 并对弱各向异性参数进行坐标变换, 得到了能够直观描述岩石弱各向异性的具有水平对称轴的横向各向同性介质, 给出了对应的3个独立弱各向异性参数及其对称轴方位, 讨论了介质各向异性与构造应力场的关系. 结果表明该地区地壳介质存在明显的方位各向异性, 其最大值约为4.6%.      

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.     

Post‐stack diffraction imaging in vertical transverse isotropy media using non‐hyperbolic moveout approximations

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.     

Interval anisotropic parameter estimation from walkaway vertical seismic profiling data

This paper presents a new explicit method for the estimation of layered vertical transverse isotropic (VTI) anisotropic parameters from walkaway VSP data. This method is based on Dix‐type normal moveout (NMO) inversion. To estimate interval anisotropic parameters above a receiver array, the method uses time arrivals of surface‐related double‐reflected downgoing waves. A three‐term NMO approximation function is used to estimate NMO velocity and a non‐hyperbolic parameter. Assuming the vertical velocity is known from zero‐offset VSP data, Dix‐type inversion is applied to estimate the layered Thomsen anisotropic parameters ?, δ above the receivers array. Model results show reasonable accuracy for estimates through Dix‐type inversion. Results also show that in many cases we can neglect the influence of the velocity gradient on anisotropy estimates. First breaks are used to estimate anisotropic parameters within the walkaway receiver interval. Analytical uncertainty analysis is performed to NMO parameter estimates. Its conclusions are confirmed by modelling.     

3D description and inversion of reflection moveout of PS‐waves in anisotropic media

Common‐midpoint moveout of converted waves is generally asymmetric with respect to zero offset and cannot be described by the traveltime series t²(x²) conventionally used for pure modes. Here, we present concise parametric expressions for both common‐midpoint (CMP) and common‐conversion‐point (CCP) gathers of PS‐waves for arbitrary anisotropic, horizontally layered media above a plane dipping reflector. This analytic representation can be used to model 3D (multi‐azimuth) CMP gathers without time‐consuming two‐point ray tracing and to compute attributes of PS moveout such as the slope of the traveltime surface at zero offset and the coordinates of the moveout minimum. In addition to providing an efficient tool for forward modelling, our formalism helps to carry out joint inversion of P and PS data for transverse isotropy with a vertical symmetry axis (VTI media). If the medium above the reflector is laterally homogeneous, P‐wave reflection moveout cannot constrain the depth scale of the model needed for depth migration. Extending our previous results for a single VTI layer, we show that the interval vertical velocities of the P‐ and S‐waves (V_P0 and V_S0) and the Thomsen parameters ε and δ can be found from surface data alone by combining P‐wave moveout with the traveltimes of the converted PS(PSV)‐wave. If the data are acquired only on the dip line (i.e. in 2D), stable parameter estimation requires including the moveout of P‐ and PS‐waves from both a horizontal and a dipping interface. At the first stage of the velocity‐analysis procedure, we build an initial anisotropic model by applying a layer‐stripping algorithm to CMP moveout of P‐ and PS‐waves. To overcome the distorting influence of conversion‐point dispersal on CMP gathers, the interval VTI parameters are refined by collecting the PS data into CCP gathers and repeating the inversion. For 3D surveys with a sufficiently wide range of source–receiver azimuths, it is possible to estimate all four relevant parameters (V_P0, V_S0, ε and δ) using reflections from a single mildly dipping interface. In this case, the P‐wave NMO ellipse determined by 3D (azimuthal) velocity analysis is combined with azimuthally dependent traveltimes of the PS‐wave. On the whole, the joint inversion of P and PS data yields a VTI model suitable for depth migration of P‐waves, as well as processing (e.g. transformation to zero offset) of converted waves.     

各向异性TI介质qP反射波走时层析成像

地震走时层析成像是反演地层各向异性参数分布的有效方法,但是关于地震各向异性介质走时层析成像的研究并不多,其技术远远没有达到成熟的阶段.在野外数据采集时,地表反射波观测方式相对井间和垂直地震剖面观测方式的成本更低,利用qP反射波走时反演各向异性参数具有更加广泛的实用价值.本文实现的TI介质地震走时层析成像方法结合了TI介质反射波射线追踪算法、走时扰动方程和非线性共轭梯度算法,它可以对任意强度的TI介质模型进行反演,文中尝试利用qP反射波走时重建TI介质模型的参数图像.利用qP反射波对层状介质模型和块状异常体模型进行走时反演,由于qP波相速度对弹性模量参数和Thomsen参数的偏微分不同,所以可以分别反演弹性模量参数和Thomsen参数.数值模拟结果表明:利用qP反射波可以反演出TI介质模型的弹性模量参数与Thomsen参数,不同模型的走时迭代反演达到了较好的收敛效果,与各向同性介质走时反演结果相比较,各向异性介质走时反演结果具有较好的识别能力.     

正交各向异性介质P波走时分析及Thomsen参数反演

对于包含有垂向裂缝的横向各向同性地层或含有多组正交裂缝的各向同性地层，正交各向异性介质模型是最简单的与实际地层相符的方位各向异性模型.本文对单层水平反射界面正交各向异性模型采用射线追踪法计算了全方位角变化的P波走时，时距曲线表现出强方位各向异性.采用小生境遗传算法，对三条成一定角度的测线的走时信息进行速度和各向异性参数反演.模型算例表明，此方法可以得到高精度的裂缝方位角、P波垂直速度和较高精度的Thomsen各向异性参数.     

Kinematical characteristics of the factorized velocity model

The factorized velocity model that incorporates both vertical heterogeneity and constant anisotropy is one of the complicated analytical models used in seismic data processing and interpretation. In this paper, I derive the analytic equations for offset, traveltime and relative geometrical spreading for the quasi‐compressional (qP‐) waves that can be used for modelling and inversion of the traveltime parameters. I show that the presence of anelliptic anisotropy usually dominates over the vertical heterogeneity with respect to the non‐hyperbolicity of the factorized velocity model.     

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.     

多震相走时联合三参数同时反演成像

采用新近研制的分区多步不规则最短路径多震相地震射线追踪正演技术,结合流行的子空间反演算法,提出了一种联合多震相走时资料进行地震三参数 （速度、反射界面和震源位置） 同时反演的方法技术.数值模拟反演实例、以及与双参数 （速度和反射界面或速度和震源位置） 同时反演的对比分析表明：三参数同时反演成像结果大体接近双参数同时反演成像的结果.另外,噪声敏感性试验表明：所提算法对到时数据中可容许的随机误差并不敏感,结果说明多震相走时的联合三参数同时反演成像方法技术不失为一种提高走时成像空间分辨率、进而降低重建模型参数失真度、行之有效的方法技术.