Second-Order and Higher-Order Perturbations of Travel Time in Isotropic and Anisotropic Media

The partial derivatives of travel time with respect to model parameters are referred to as perturbations. Explicit equations for the second-order and higher-order perturbations of travel time in both isotropic and anisotropic media are derived. The perturbations of travel time and its spatial derivatives can be calculated by simple numerical quadratures along rays.     

Paraxial ray methods in inhomogeneous anisotropic media: Initial conditions

Paraxial ray methods have found broad applications in the seismic ray method and in numerical modelling and interpretation of high-frequency seismic wave fields propagating in inhomogeneous, isotropic or anisotropic structures. The basic procedure in paraxial ray methods consists in dynamic ray tracing. We derive the initial conditions for dynamic ray equations in Cartesian coordinates, for rays initiated at three types of initial manifolds given in a three-dimensional medium: 1) curved surfaces (surface source), 2) isolated points (point source), and 3) curved, planar and non-planar lines (line source). These initial conditions are very general, valid for homogeneous or inhomogeneous, isotropic or anisotropic media, and for both a constant and a variable initial travel time along the initial manifold. The results presented in the paper considerably extend the possible applications of the paraxial ray method.     

Errors Due to the Common Ray Approximations of the Coupling Ray Theory

The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented.     

Errors due to the anisotropic-common-ray approximation of the coupling ray theory

The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common-ray approximation applied. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. We derive the equations for estimating the travel-time errors due to the anisotropic-common-ray (and also isotropic-common-ray) approximation of the coupling ray theory. The errors of the common-ray approximations are calculated along the anisotropic common rays in smooth velocity models without interfaces. The derivation is based on the general equations for the second-order perturbations of travel time.     

Ray-centred coordinate systems in anisotropic media

Whereas the ray-centred coordinates for isotropic media by Popov and Pšenčík are uniquely defined by the selection of the basis vectors at one point along the ray, there is considerable freedom in selecting the ray-centred coordinates for anisotropic media. We describe the properties common to all ray-centred coordinate systems for anisotropic media and general conditions, which may be imposed on the basis vectors. We then discuss six different particular choices of ray-centred coordinates in an anisotropic medium. This overview may be useful in choosing the ray-centred coordinates best suited for a particular application. The equations are derived for a general homogeneous Hamiltonian of an arbitrary degree and are thus applicable both to the anisotropic-ray-theory rays and anisotropic common S-wave rays.     

各向异性介质中的弹性阻抗及其反演

地震反演已成为油藏描述中的重要组成部分.绝大多数的常规地震反演是叠后地震数据体上进行,很少考虑各向异性存在的情况.随着勘探开发的发展,地震各向异性和叠前地震波阻抗反演引起了人们极大关注.本文在各向同性介质中弹性阻抗研究基础上,推导出了各向异性介质中的弹性阻抗方程,提出了地震各向异性介质中用弹性阻抗进行储层参数描述的技术路线和框架,并对反演过程中存在的问题进行了有益探讨.     

A note on dynamic ray tracing in ray-centered coordinates in anisotropic inhomogeneous media

Dynamic ray tracing plays an important role in paraxial ray methods. In this paper, dynamic ray tracing systems for inhomogeneous anisotropic media, consisting of four linear ordinary differential equations of the first order along the reference ray, are studied. The main attention is devoted to systems expressed in a particularly simple choice of ray-centered coordinates, here referred to as the standard ray-centered coordinates, and in wavefront orthonormal coordinates. These two systems, known from the literature, were derived independently and were given in different forms. In this paper it is proved that both systems are fully equivalent. Consequently, the dynamic ray tracing system, consisting of four equations in wavefront orthonormal coordinates, can also be used if we work in ray-centered coordinates, and vice versa. vcerveny@seis.karlov.mff.cuni.cz     

Incidence, reflection and transmission angles in anisotropic media

IntroductionGenerallyspeaking,theinclusionofanisotropy(exceptdeclaration,anisotropyreferstohomogenousanisotropy)rendersthemathematicalformulationquitecomplicated.Snell'Slawisnotanexceptionandthecalculationofreflectionandtransmissionanglesisnotatrivialtask.ThegraphicalapproachestocalculatingreflectionandtransmissionanglesforanisotropicmediawerepresentedbyAuld(1973)andRokhlin,etal(1986).DaleyandHorn(1977,1979)andSlawinski(1996)deriveSnell'slawintheparticularcasesoftransverselyisotropicandelli…     

Common-ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium

Anisotropic common S-wave rays are traced using the averaged Hamiltonian of both S-wave polarizations. They represent very practical reference rays for calculating S waves by means of the coupling ray theory. They eliminate problems with anisotropic-ray-theory ray tracing through some S-wave slowness-surface singularities and also considerably simplify the numerical algorithm of the coupling ray theory for S waves. The equations required for anisotropic-common-ray tracing for S waves in a smooth elastic anisotropic medium, and for corresponding dynamic ray tracing in Cartesian or ray-centred coordinates, are presented. The equations, for the most part generally known, are summarized in a form which represents a complete algorithm suitable for coding and numerical applications.     

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.     

Modelling S-wave radiation for SP-waves in isotropic and anisotropic media

Utilizing shear-wave (S-wave) data acquired with compressional waves (P-waves) is becoming more common as joint imaging and inversion techniques improve. Interest in S-waves radiated from vertical sources and buried explosives exploits conversion to P-waves as primary reflections (SP-waves) for reducing acquisition costs and for application to legacy data. However, recent investigations overstate the extent of SP-wave illumination and show isotropic processing results with narrow bandwidth frequency and wavenumber data. I demonstrate that illumination with SP-waves is limited in general to near vertical polar angles up to around 30° or 35° for V_P/V_S of 2 or 3, respectively. At greater angles, S-waves are typically in the P-wave evanescent range and cannot excite SP-wave reflections. Contrary to recent claims, these sources for P-wave do not radiate S_H-waves polarized in horizontal planes in all azimuths. I show these properties for isotropic media with radiation expressions for amplitude derived in vector slowness coordinates. Also, I extend these expressions to transversely isotropic media with a vertical symmetry axis to show agreement with synthetic seismic data that only quasi S_V-waves are radiated and become more narrowly focused towards 45°. Furthermore, in orthorhombic media, synthetic data show that fast S₁- and slow S₂-waves polarized parallel and perpendicular to fractures may appear as S_V- and S_H-waves. For the partially saturated fracture model studied here, S₁-wave radiation has broader azimuthal illumination than slow S₂-waves, which are more narrowly focused in azimuth. These produce SP-wave splitting signatures on vertical component reflection data that are nearly identical to PS-wave signatures on radial horizontal component data. Separating these fast and slow SP-waves is an additional processing challenge.     

Comparison of Quasi-Isotropic Approximations of the Coupling Ray Theory with the Exact Solution in the 1-D Anisotropic “Oblique Twisted Crystal” Model

The coupling ray theory bridges the gap between the isotropic and anisotropic ray theories, and is considerably more accurate than the anisotropic ray theory. The coupling ray theory is often approximated by various quasi-isotropic approximations.Commonly used quasi-isotropic approximations of the coupling ray theory are discussed. The exact analytical solution for the plane S wave, propagating along the axis of spirality in the 1-D anisotropic oblique twisted crystal model, is then numerically compared with the coupling ray theory and its three quasi-isotropic approximations. The three quasi-isotropic approximations of the coupling ray theory are (a) the quasi-isotropic projection of the Green tensor, (b) the quasi-isotropic approximation of the Christoffel matrix, (c) the quasi-isotropic perturbation of travel times. The comparison is carried out numerically in the frequency domain, comparing the exact analytical solution with the results of the 3-D ray tracing and coupling ray theory software. In the oblique twisted crystal model, the three studied quasi-isotropic approximations considerably increase the error of the coupling ray theory. Since these three quasi-isotropic approximations do not noticeably simplify the numerical implementation of the coupling ray theory, they should deffinitely be avoided. The common ray approximations of the coupling ray theory do not affect the plane wave, propagating along the axis of spirality in the 1-D oblique twisted crystal model, and should be studied in more complex models.     

慢度平方三角网格模型VTI介质二维QP波各向异性立体层析反演

基于慢度平方三角网格剖分模型下的各向异性射线扰动理论,建立了具有垂直对称轴的横向各向同性（VTI）介质中二维QP波立体层析所需的FRECHET导数矩阵,实现了慢度平方三角网格模型的VTI介质二维QP波各向异性立体层析.考虑到各向异性参数是影响地震波运动学特征的次一级因素,制定了先反演慢度再反演各向异性参数的反演策略.由于慢度平方三角网格模型的稀疏性,使得数据空间对模型空间的FRECHET导数矩阵规模被大幅压缩,在降低了计算成本的同时也很好的保证了界面位置、背景慢度、各向异性等各个参数的反演精度.理论数据算例证实了FRECHET导数的正确性和反演策略的合理性.

     

横向各向同性介质和斜方介质各向异性参数的约束条件

Thomsen提出的横向各向同性(TI)介质各向异性参数(α0、β0、ε、δ、γ)是各向异性理论研究和实际资料处理中的常用参数,Thomsen参数的取值必须符合物理学定律和实际地学情况,随意的取值可能导致无意义乃至错误的结果.本文根据热力学定律和弹性常数的物理意义,结合大量的实测数据,提出常见TI介质的Thomsen参数需要满足以下约束条件:1/4f=1-β02/α02ε>-f/2; 1/2f-1δf-1); -1/2γε)/4(1-f)-1/2.Thomsen参数ε、δ、γ的取值区间主要受P、S波参考速度比β0/α0的约束,其值可正可负,实测参数中ε和γ正多负少;δ、γ、β0/α0的取值范围都有上下界,而ε没有上限;γ与ε正相关且取值上限受到ε的限制;ε、γ与δ之间不存在明显的约束.鉴于椭圆各向异性介质和薄互层等效TI介质在实际应用中的普遍性,专门给出了这两种介质各向异性参数的附加约束条件.一些在实际资料中可观测到的P波特殊偏振方向和SV波三叉现象也能为各向异性参数提供额外的约束.之后,我们将TI介质弹性常数和各向异性参数的约束条件扩展到对正交各向异性介质弹性常数和各向异性参数的约束.本文提出的各向异性参数约束条件简单实用,为各向异性理论研究和数值模拟中参数的选择提供依据,为实际资料的各向异性参数反演提供约束,既能避免出现无物理意义的研究结果,又能加速反演的搜索过程,提高生产效率,具有理论指导意义和实际应用价值.     

Exact traveltime computation in multi-layered transversely isotropic media with vertical symmetry axis

An approach to calculate the accurate ray paths and traveltimes in multi-layered VTI media (transversely isotropic media with a vertical symmetry axis) is proposed. The expressions of phase velocity, group velocity and Snell’s law used for computation are all explicit and exact. The calculation of ray paths and traveltimes for a given ele-mentary wave is equivalent to that of a transmission problem which is much easier to be treated with the formulae proposed. In the section of numerical examples, the proce...     

Physical modeling observations of wave velocity and apparent attenuation in isotropic and anisotropic two-phase media

We study wave propagation through isotropic and anisotropic scatterer distributions in order to observe azimuthal variations in velocity and apparent attenuation. Using thin aluminum plates as physical models, we obtained seismograms for compressional and shear wave propagation through heterogeneous media. Three random distributions of scatterers are studied: circular scatterers in isotropic distributions (modeling circular scatterers), elongated scatterers in isotropic distributions (modeling randomly oriented elliptical scatterers), and elongated scatterers in anisotropic distributions (modeling aligned elliptical scatterers). All scatterers had approximately the same cross-sectional area and were filled with epoxy in order to reduce the impedance contrast. In addition to seismograms recorded for no scatterers, seismograms were recorded for several scatterer volume fractions. Azimuths were measured relative to the long axis of the aligned elongated scatterers. Velocities were calculated using travel times and phase shifts at low frequencies. The velocities measured from the data were compared to simple low-frequency average-velocity theories based on thin lamellae or on distributions of penny-shaped cracks. The apparent attenuation for different scatterer distributions was computed using spectral ratios.Comparisons of the results for circular and randomly oriented elongated scatterers were made to determine the effects of scatterer shape. As expected, the circular and randomly oriented elongated scatterers showed no systematic azimuthal variation in velocity. The velocity anomalies were systematically larger for the randomly oriented elongated scatterers than for the circular scatterers. Both methods of theoretical estimation for the isotropic velocities produced velocities significantly larger than those measured. The spectral ratios showed more apparent attenuation for the randomly oriented elongated scatterers than for the circular scatterers.Comparisons of the results for the randomly oriented and aligned elongated scatterers were made to determine the effects of anisotropy in the scatterer distribution. Compressional waves for the aligned elongated scatterers with wave propagation parallel to the scatterers had larger velocities than for the aligned elongated scatterers with wave propagation perpendicular to the scatterers for all velocity calculations. Shear wave velocities were complicated by an anomalous phase change in the shear wave seismograms for azimuths less than 40° and were not as conclusive. The general trend of the theoretical velocities is similar to the velocities calculated from the data. There are, however, what appear to be significant differences. The spectral ratios showed more apparent attenuation for the randomly oriented elongated scatterers than for the aligned elongated scatterers with wave propagation parallel to the scatterers, and less attenuation than for the aligned elongated scatterers with wave propagation perpendicular to the scatterers.     

Generation of three dimensional flow fields for statistically anisotropic heterogeneous porous media

A methodology for generating three dimensional (3D) flow fields for statistically anisotropic heterogeneous porous media is presented and demonstrated. The simulated flow fields are shown to exhibit the input spatial correlation structure and observe mass continuity. Sample flow fields are presented in the form of cross sectional slices of the 3D formation. These cross sections demonstrate visually the characteristics of subsurface flow. The method was found to be faster than traditional techniques in terms of its computational requirements. Given this method, it is possible to generate the large number of realizations of a velocity field necessary to compute high order statistics in transport problems.     

Research on propagation properties of elastic waves in two-phase anisotropic media

IntroductionItiswellknownthatanisotropylieswidelyintheundergroundmedia.Anisotropicmediawhicharemetintheseismicengineeringandseismicexplorationofenergyaremainlycausedbytheperiodicthinlayers(PTL)andextensivedilatancyanisotropy(EDA).Insuchmedia,anisotropyleadstomorecomplicatepropagationofseismicwave,thesignificantfeatureinanisotropicmediaisvelocityanisotropy.Infact,undergroundstrataareverycomplicated,whichareusuallycomposedofsolidframeandfluid(suchasoil,gasesorwater)inpores.Inordertostudyseism…     

Weak-contrast approximation of the elastic scattering matrix in anisotropic media

The plane-wave reflection and transmission coefficients at a plane interface between two anisotropic media constitute the elements of the elastic scattering matrix. For a 1-D anisotropic medium the eigenvector decomposition of the system matrix of the transformed elasto-dynamic equations is used to derive a general expression for the scattering matrix. Depending on the normalization of the eigenvectors, the expressions give scattering coefficients for amplitudes or for vertical energy flux.Computing the vertical slownesses and the corresponding polarizations, the eigenvector matrix and its inverse can be found. We give a simple formula for the inverse, regardless of the normalization of the eigenvectors. When the eigenvectors are normalized with respect to amplitudes of displacement (or velocity), the calculation of the scattering matrix for amplitudes is simplified.When the relative changes in all parameters are small, a weak-contrast approximation of the scattering matrix, based on the exactly determined polarization vectors in an average medium, is obtained. The same approximation is also derived directly from the transformed elasto-dynamic equations for a smooth vertically inhomogeneous medium, proving the consistency of the approximation.For monoclinic media, with the mirror symmetry plane parallel to the interface, the approximative scattering matrix is given in terms of analytic expressions for the non-normalized eigenvectors and vertical slownesses. For transversely isotropic media with a vertical axis of symmetry (VTI) and isotropic media, explicit solutions for the weak-contrast approximations of the scattering matrices have been obtained. The scattering matrix for amplitudes for isotropic media is well known. The scattering matrix for vertical energy flux may have applications in AVO analysis and inversion due to the reciprocity of the reflection coefficients for converted waves.Numerical examples for monoclinic and VTI media provide good agreement between the approximative and the exact reflection matrices. It is, however, expected that the approximations cannot be used when the symmetry properties of the two media are very different. This is because the approximation relies on a small relative contrast between the eigenvectors in the two media.Presented at the Workshop Meeting on Seismic Waves in Laterally Inhomogeneous Media, Castle of Trest, Czech Republic, May 22–27, 1995.     

Bayesian Markov chain Monte Carlo inversion for anisotropy of PP- and PS-wave in weakly anisotropic and heterogeneous media

A single set of vertically aligned cracks embedded in a purely isotropic background may be considered as a long-wavelength effective transversely isotropy (HTI) medium with a horizontal symmetry axis. The crack-induced HTI anisotropy can be characterized by the weakly anisotropic parameters introduced by Thomsen. The seismic scattering theory can be utilized for the inversion for the anisotropic parameters in weakly anisotropic and heterogeneous HTI media. Based on the seismic scattering theory, we first derived the linearized PP- and PS-wave reflection coefficients in terms of P- and S-wave impedances, density as well as three anisotropic parameters in HTI media. Then, we proposed a novel Bayesian Markov chain Monte Carlo inversion method of PP- and PS-wave for six elastic and anisotropic parameters directly. Tests on synthetic azimuthal seismic data contaminated by random errors demonstrated that this method appears more accurate, anti-noise and stable owing to the usage of the constrained PS-wave compared with the standards inversion scheme taking only the PP-wave into account.