Slowness-domain kinematical characteristics for horizontally layered orthorhombic media. Part II: Pre-critical slowness match

Part II of this paper is a direct continuation of Part I, where we consider the same types of orthorhombic layered media and the same types of pure-mode and converted waves. Like in Part I, the approximations for the slowness-domain kinematical characteristics are obtained by combining power series coefficients in the vicinity of both the normal-incidence ray and an additional wide-angle ray. In Part I, the wide-angle ray was set to be the critical ray (‘critical slowness match’), whereas in Part II we consider a finite long offset associated with a given pre-critical ray (‘pre-critical slowness match’). Unlike the critical slowness match, the approximations in the pre-critical slowness match are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. Moreover, for the pre-critical slowness match, there is no need to distinguish between the high-velocity layer and the other, low-velocity layers. The form of the approximations in both critical and pre-critical slowness matches is the same, where only the wide-angle power series coefficients are different. Comparing the approximated kinematical characteristics with those obtained by exact numerical ray tracing, we demonstrate high accuracy. Furthermore, we show that for all wave types, the accuracy of the pre-critical slowness match is essentially higher than that of the critical slowness match, even for matching slowness values close to the critical slowness. Both approaches can be valuable for implementation, depending on the target offset range and the nature of the subsurface model. The pre-critical slowness match is more accurate for simulating reflection data with conventional offsets. The critical slowness match can be attractive for models with a dominant high-velocity layer, for simulating, for example, refraction events with very long offsets.     

Slowness domain offset and traveltime approximations in layered vertical transversely isotropic media

Considering horizontally layered transversely isotropic media with vertical symmetry axis and all types of pure‐mode and converted waves we present a new wide‐angle series approximation for the kinematical characteristics of reflected waves: horizontal offset, intercept time, and total reflection traveltime as functions of horizontal slowness. The method is based on combining (gluing) both zero‐offset and (large) finite‐offset series coefficients. The horizontal slowness is bounded by the critical value, characterised by nearly horizontal propagation within the layer with the highest horizontal velocity. The suggested approximation uses five parameters to approximate the offset, six parameters to approximate the intercept time or the traveltime, and seven parameters to approximate any two or all three kinematical characteristics. Overall, the method is very accurate for pure‐mode compressional waves and shear waves polarised in the horizontal plane and for converted waves. The application of the method to pure‐mode shear waves polarised in the vertical plane is limited due to cusps and triplications. To demonstrate the high accuracy of the method, we consider a synthetic, multi‐layer model, and we plot the normalised errors with respect to numerical ray tracing.     

Anisotropic inversion of refracted waves in vertical cable data in the presence of dip

Refracted arrivals are analysed to estimate the near‐surface anisotropy of marine sediments using a vertical‐cable (VC) configuration. In the presence of dip, the horizontal and vertical ray‐slownesses are obtained from the observed apparent slownesses in the up‐ and downdip directions using a sum or difference at each azimuth. The multiple azimuths generated by a VC geometry permit the ray‐slowness distribution of the marine sediments to be determined. An inversion procedure is developed to provide dip and anisotropy parameters for refractive layers from the measured refraction traveltimes in multilayered azimuthally isotropic and anisotropic media. Two sets of transversely isotropic models are used to analyse the azimuthal variations of apparent and ray slownesses. In the first set, we fix the anisotropic parameters of the models but vary the dip (0°, 5° and 10°) to test the effects of the presence of dip. In the second set, we vary the P‐wave anisotropy strength (5.2%, 10.3%, 15.8% and 22.0%) to examine the sensitivity and accuracy of ray‐slowness approximations which are independent of dip. We test this inversion procedure on synthetic P‐wave VC data calculated for six different models by a finite‐difference method. The results of applications to real VC data acquired from the North Sea are also presented.     

Generalized Dix equation and analytic treatment of normal-moveout velocity for anisotropic media*

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well described by the normal-moveout (NMO) velocity defined in the zero-offset limit. In their recent work, Grechka and Tsvankin showed that the azimuthal variation of NMO velocity around a fixed CMP location generally has an elliptical form (i.e. plotting the NMO velocity in each azimuthal direction produces an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for the NMO velocity in anisotropic media of arbitrary symmetry. For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The contribution of anisotropy to NMO velocity is contained in the slowness components of the zero-offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) — quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix-type average of the matrices responsible for the ‘interval’ NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi-axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions. Modelling of normal moveout in general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero-offset) ray. Remarkably, the expressions for geometrical spreading along the zero-offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi-azimuth, multi-offset ray tracing and, therefore, can be used efficiently in traveltime inversion and in devising fast dip-moveout (DMO) processing algorithms for anisotropic media. This technique becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces. The high accuracy of our NMO expressions is illustrated by comparison with ray-traced reflection traveltimes in piecewise-homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P-wave moveout can be used to find the depth-dependent fracture orientation and to evaluate the magnitude of azimuthal anisotropy.     

Normal moveout velocity for pure‐mode and converted waves in layered orthorhombic medium

We study the azimuthally dependent hyperbolic moveout approximation for small angles (or offsets) for quasi‐compressional, quasi‐shear, and converted waves in one‐dimensional multi‐layer orthorhombic media. The vertical orthorhombic axis is the same for all layers, but the azimuthal orientation of the horizontal orthorhombic axes at each layer may be different. By starting with the known equation for normal moveout velocity with respect to the surface‐offset azimuth and applying our derived relationship between the surface‐offset azimuth and phase‐velocity azimuth, we obtain the normal moveout velocity versus the phase‐velocity azimuth. As the surface offset/azimuth moveout dependence is required for analysing azimuthally dependent moveout parameters directly from time‐domain rich azimuth gathers, our phase angle/azimuth formulas are required for analysing azimuthally dependent residual moveout along the migrated local‐angle‐domain common image gathers. The angle and azimuth parameters of the local‐angle‐domain gathers represent the opening angle between the incidence and reflection slowness vectors and the azimuth of the phase velocity ψ_phs at the image points in the specular direction. Our derivation of the effective velocity parameters for a multi‐layer structure is based on the fact that, for a one‐dimensional model assumption, the horizontal slowness and the azimuth of the phase velocity ψ_phs remain constant along the entire ray (wave) path. We introduce a special set of auxiliary parameters that allow us to establish equivalent effective model parameters in a simple summation manner. We then transform this set of parameters into three widely used effective parameters: fast and slow normal moveout velocities and azimuth of the slow one. For completeness, we show that these three effective normal moveout velocity parameters can be equivalently obtained in both surface‐offset azimuth and phase‐velocity azimuth domains.     

Separation of diffracted waves in transversely isotropic media

Imaging diffracted waves can provide useful information about complex subsurface geology and fracture networks. Separation of diffractions from typically more intensive reflected events can be done based on specularity, which measures deviation from Snell’s law. Here, we analyze two formulations of specularity and their applicability to diffraction processing in the presence of anisotropy. We show that the most common definition of specularity, originally introduced for pure modes in isotropic media, remains valid for both pure and converted waves in arbitrarily anisotropic models. The other formulation operates directly with the difference between the slowness projections onto the reflector for the incident and reflected waves. Testing on a VTI (transversely isotropic with a vertical symmetry axis) diffraction ramp model demonstrates that both formulations produce satisfactory results for anisotropic media with appropriate tapering of the specularity gathers. Then separation and imaging of diffractions is performed for the structurally complex VTI Marmousi model. We also analyze the sensitivity of diffractions in the specularity gathers to errors in the symmetry-direction velocity and anellipticity parameter η.     

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.     

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.     

Synthetic seismograms in heterogeneous media by one-return approximation

When reverberations between heterogeneities or resonance scattering can be neglected but accumulated effects of forward scattering are strong, the Born approximation is not valid but the De Wolf approximation can be applied in such cases. In this paper, renormalized MFSB (multiple-forescattering single-backscattering) equations and the dual-domain expression for scalar, acoustic and elastic waves are derived by a unified approach. Two versions of the one-return method (using MFSB approximation) are given: One is the wide-angle dual-domain formulation (thin-slab approximation); the other is the screen approximation. In the screen approximation, which involves a small-angle approximation for the wave-medium interaction, it can be seen clearly that the forward scattered, or transmitted waves are mainly controlled by velocity perturbations; while the backscattered or reflected waves, by impedance perturbations. The validity of the method and the wide-angle capability of the dual-domain implementation are demonstrated by numerical examples. Reflection coefficients of a plane interface derived from numerical simulations by the wide-angle method match the theoretical curves well up to critical angles. For the reflections of a low-velocity slab, the agreement between theory and synthetics only starts to deteriorate for angles greater than 70°. The accuracy of the wide-angle version of the method could be further improved by optimizing the wave-number filtering for the forward propagation and shrinking the step length along the propagation direction.     

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.     

方位各向异性粘弹性介质波场数值模拟

当地震信号通过复杂地球介质时，地层除了表现为各向异性，还表现为内在的粘弹性特征.因此，为准确描述地震波在地球介质中的传播特征，理想的地球介质模型应该能够模拟岩石的各向异性特征和衰减特征.本文给出了各向异性粘弹性介质模型的波动方程及其差分格式，并利用有限差分法实现了地震波波场数值模拟.结果表明了该介质模型中地震波场特征与各向异性主轴方位和介质的粘滞性参数之间的关系.     

First-Order Perturbation Theory for Seismic Isochrons

A first-order perturbation theory for seismic isochrons is presented in a model independent form. Two ray concepts are fundamental in this theory, the isochron ray and the velocity ray, for which I obtain first-order approximations to position vectors and slowness vectors. Furthermore, isochron points are connected to a shot and receiver by conventional ray fields. Based on independent perturbation of the shot and receiver ray I obtain first-order approximations to velocity rays. The theory is applicable for 3D inhomogeneous anisotropic media, given that the shot and receiver rays, as well as their perturbations, can be generated with such model generality. The theory has applications in sensitivity analysis of prestack depth migration and in velocity model updating. Numerical examples of isochron and velocity rays are shown for a 2D homogeneous VTI model. The general impression is that the first-order approximation is, with some exceptions, sufficiently accurate for practical applications using an anisotropic velocity model.     

Maslov面波理论地震图

研究了横向非均匀介质中的Ｍａｓｌｏｖ面波渐近理论，在横向弱非均匀介质的假设下，介质的纵向非均匀性反应在局部本征函数中，以局部本征函数近似真本征函数是射线理论的直接推论．由此，三维结构下的面波计算退化为准二维问题．由于本文方法属于慢度法，面波的频散使得在一般情况下得不到与体波ＷＫＢＪ方法相似的褶积结果；在震源函数为高斯波包的假设下，得出了与二维体波Ｍａｓｌｏｖ理论图形式上完全相同的褶积结果．还讨论了吸收介质中的面波波包理论图计算，最后结果与二维体波吸收介质中的结果相似．     

一种弱各向异性介质地震波群速度的近似表示新方法

地震各向异性介质的群速度是关于相角的复杂函数,将其表示成射线角形式较为困难,这给地震各向异性分析以及走时正演模拟等带来诸多不便;另一方面,观测资料表明实际地球介质的地震各向异性通常较弱,这为用射线角近似表示地震波群速度提供了可能.本文基于以射线角近似表示相角的思想,提出了一种弱各向异性条件下,群速度射线角近似表示的新方法.计算表明,在弱地震各向异性条件下,新方法在很宽的射线角范围内,对三种地震波的群速度都能很好地近似,在准SV波计算精度方面显著优于目前通常使用的近似方法.     

Reflection and transmission responses of layered transversely isotropic viscoelastic media

We consider a layered heterogeneous viscoelastic transversely isotropic medium with a vertical symmetry axis (a viscoelastic TIV medium) and parameters that depend on depth only. This takes into account intrinsic attenuation, anisotropy and thin layering. The seismic wavefield is decomposed into up- and downgoing waves scaled by the vertical energy flux. This scaling gives important symmetry relationships for both reflection and transmission (R/T) responses. For a stack of homogeneous layers, the exact reflection response can be computed in a numerically stable way by a simple layer-recursive algorithm. We derive exact plane-wave R/T coefficients and several linear and quadratic approximations between two viscoelastic TIV media, as functions of the real-valued horizontal slowness. The approximations are valid for pre- and post-critical values of horizontal slowness provided that the proper complex square roots are used when computing the vertical slowness. Numerical examples demonstrate that the quadratic approximations can be used for large differences in medium parameters, while the linear approximations can be used for small differences. For weak anisotropy it is sufficient to use an isotropic background medium, while for strong anisotropy it is necessary to use a weak TIV or TIV background medium. We also extend the O'Doherty–Anstey formula to the P- and SV-wave transmission responses of a stack of viscoelastic TIV layers, taking into account intrinsic attenuation, anisotropy and thin layering.     

Quadratic normal moveouts of symmetric reflections in elastic media: A quick tutorial

The design of reflection traveltime approximations for optimal stacking and inversion has always been a subject of much interest in seismic processing. A most prominent role is played by quadratic normal moveouts, namely reflection traveltimes around zero-offset computed as second-order Taylor expansions in midpoint and offset coordinates. Quadratic normal moveouts are best employed to model symmetric reflections, for which the ray code in the downgoing direction coincides with the ray code in the upgoing direction in reverse order. Besides pure (non-converted) primaries, many multiply reflected and converted waves give rise to symmetric reflections. We show that the quadratic normal moveout of a symmetric reflection admits a natural decomposition into a midpoint term and an offset term. These, in turn, can be be formulated as the traveltimes of the one-way normal (N) and normal-incidence-point (NIP) waves, respectively. With the help of this decomposition, which is valid for propagation in isotropic and anisotropic elastic media, we are able to derive, in a simple and didactic way, a unified expression for the quadratic normal moveout of a symmetric reflection in its most general form in 3D. The obtained expression allows for a direct interpretation of its various terms and fully encompasses the effects of velocity gradients and Earth surface topography.     

Slowness and azimuth determination for Bucovina array (BURAR) applying multiple signal techniques

The BURAR seismic array, located in Northern Romania (Bucovina region), is designed to monitor events located in an area poorly covered by other existing seismic stations. In order to use the BURAR array for single-station locations, it is crucial to calibrate the azimuth and slowness parameters, which are currently used in array techniques to locate earthquakes, blasts or nuclear explosions. The goal of this study is to apply “f–k” and plane wave fit techniques in order to constrain the slowness and azimuth parameters at BURAR for teleseismic, regional and local events. The analysis was carried out using P and S waves recorded for events occurred between 2004 and 2008 within a radius of 50° around BURAR. The azimuth values obtained applying both methods strongly deviated from the theoretical values for regions like Central Turkey, Bulgaria, Dodecanese Islands and other parts of Greece, while the ray parameter deviations with respect to a 1-D IASP91 reference model are less significant. For the local events, the anomalies are smaller, except the particular case of Vrancea intermediate-depth earthquakes for which strong azimuth deviations (33.5°), both positive and negative, are observed. We investigate how these systematic deviations in azimuth are explained by the structure lateral heterogeneities which characterize the study region.     

Reflection/transmission laws for slowness vectors in viscoelastic anisotropic media

The reflection/transmission laws (R/T laws) of plane waves at a plane interface between two homogeneous anisotropic viscoelastic (dissipative) halfspaces are discussed. Algorithms for determining the slowness vectors of reflected/transmitted plane waves from the known slowness vector of the incident wave are proposed. In viscoelastic media, the slowness vectors of plane waves are complex-valued, p = P + iA, where P is the propagation vector, and A the attenuation vector. The proposed algorithms may be applied to bulk plane waves (A = 0), homogeneous plane waves (A ≠ 0, P and A parallel), and inhomogeneous plane waves (A ≠ 0, P and A non-parallel). The manner, in which the slowness vector is specified, plays an important role in the algorithms. For unrestricted anisotropy and viscoelasticity, the algorithms require an algebraic equation of the sixth degree to be solved in each halfspace. The degree of the algebraic equation decreases to four or two for simpler cases (isotropic media, plane waves in symmetry planes of anisotropic media). The physical consequences of the proposed algorithms are discussed in detail. vcerveny@seis.karlov.mff.cuni.cz