Paraxial ray methods for anisotropic inhomogeneous media

A new formalism of surface-to-surface paraxial matrices allows a very general and flexible formulation of the paraxial ray theory, equally valid in anisotropic and isotropic inhomogeneous layered media. The formalism is based on conventional dynamic ray tracing in Cartesian coordinates along a reference ray. At any user-selected pair of points of the reference ray, a pair of surfaces may be defined. These surfaces may be arbitrarily curved and oriented, and may represent structural interfaces, data recording surfaces, or merely formal surfaces. A newly obtained factorization of the interface propagator matrix allows to transform the conventional 6 × 6 propagator matrix in Cartesian coordinates into a 6 × 6 surface-to-surface paraxial matrix. This matrix defines the transformation of paraxial ray quantities from one surface to another. The redundant non-eikonal and ray-tangent solutions of the dynamic ray-tracing system in Cartesian coordinates can be easily eliminated from the 6 × 6 surface-to-surface paraxial matrix, and it can be reduced to 4 × 4 form. Both the 6 × 6 and 4 × 4 surface-to-surface paraxial matrices satisfy useful properties, particularly the symplecticity. In their 4 × 4 reduced form, they can be used to solve important boundary-value problems of a four-parametric system of paraxial rays, connecting the two surfaces, similarly as the well-known surface-to-surface matrices in isotropic media in ray-centred coordinates. Applications of such boundary-value problems include the two-point eikonal, relative geometrical spreading, Fresnel zones, the design of migration operators, and more.     

PARABOLIC AND HYPERBOLIC PARAXIAL TWO-POINT TRAVELTIMES IN 3D MEDIA1

The 4 × 4 T -propagator matrix of a 3D central ray determines, among other important seismic quantities, second-order (parabolic or hyperbolic) two-point traveltime approximations of certain paraxial rays in the vicinity of the known central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two-component vector. Here, we provide parabolic and hyperbolic paraxial two-point traveltime approximations using the T -propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three-component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial-ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.     

Eigenrays in 3D heterogeneous anisotropic media,Part II: Dynamics

This paper is the second in a sequel of two papers and dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained in the first paper. We start by formulating the linear, second‐order, Jacobi dynamic ray tracing equation. We then apply a similar finite‐element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction due to caustics (i.e. the amplitude and the phase of the asymptotic form of the Green's function for waves propagating in 3D heterogeneous general anisotropic elastic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point–source and plane‐wave. For the proposed point–source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, similar to the endpoint complexity factor, presented in the first paper, used to define the measure of complexity of the propagated wave/ray phenomena. The new weighted propagation complexity accounts for the normalized relative geometric spreading not only at the receiver point, but along the whole stationary ray path. We propose a criterion based on this parameter as a qualifying factor associated with the given ray solution. To demonstrate the implementation of the proposed method, we use several isotropic and anisotropic benchmark models. For all the examples, we first compute the stationary ray paths, and then compute the geometric spreading and analyse these trajectories for possible caustics. Our primary aim is to emphasize the advantages, transparency and simplicity of the proposed approach.     

Gaussian beams in inhomogeneous media: A review

The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich’s method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich’s method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich’s method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names. The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.     

Point source radiation in inhomogeneous anisotropic structures

The ray formulae for the radiation from point sources in unbounded inhomogeneous isotropic as well as anisotropic media consist of two factors. The first one depends fully on the type and orientation of the source and on the parameters of the medium at the source. We call this factor the directivity function. The second factor depends on the parameters of the medium surrounding the source and this factor is the well-known geometrical spreading. The displacement vector and the radiation pattern defined as a modulus of the amplitude of the displacement vector measured on a unit sphere around the source are both proportional to the ratio of the directivity function and the geometrical spreading.For several reasons it is desirable to separate the two mentioned factors. For example, there are methods in exploration seismics, which separate the effects of the geometrical spreading from the observed wave field (so-called true amplitude concept) and thus require the proposed separation. The separation also has an important impact on computer time savings in modeling seismic wave fields generated by point sources by the ray method. For a given position in a given model, it is sufficient to calculate the geometrical spreading only once. A multitude of various types of point sources with a different orientation can then be calculated at negligible additional cost.In numerical examples we show the effects of anisotropy on the geometrical spreading, the directivity and the radiation pattern. Ray synthetic seismograms due to a point source positioned in an anisotropic medium are also presented and compared with seismograms for an isotropic medium.     

Two-point ray tracing in 3-D

Algorithm for determination of all two-point rays of a given elementary wave by means of the shooting method is presented. The algorithm is designed for general 3-D models composed of inhomogeneous geological blocks separated by curved interfaces. It is independent of the initial conditions for rays and of the initial-value ray tracer. The algorithm described has been coded in Fortran 77, using subroutine packages MODEL and CRT for model specification and for initial-value ray tracing.     

Spatial derivatives and perturbation derivatives of amplitude in isotropic and anisotropic media

Explicit equations for the spatial derivatives and perturbation derivatives of amplitude in both isotropic and anisotropic media are derived. The spatial and perturbation derivatives of the logarithm of amplitude can be calculated by numerical quadratures along the rays. The spatial derivatives of amplitude may be useful in calculating the higher-order terms in the ray series, in calculating the higher-order amplitude coefficients of Gaussian beams, in estimating the accuracy of zero-order approximations of both the ray method and Gaussian beams, in estimating the accuracy of the paraxial approximation of individual Gaussian beams, or in estimating the accuracy of the asymptotic summation of paraxial Gaussian beams. The perturbation derivatives of amplitude may be useful in perturbation expansions from elastic to viscoelastic media and in estimating the accuracy of the common-ray approximations of the amplitude in the coupling ray theory.     

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     

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.     

Introduction to seismic travel time methods in anisotropic media

Summary Section 1 (and 11) develops the concepts of the front velocity, the front gradient, the travel time in space and on seismometric profiles, the profile velocity and the profile gradient in connection with the propagation of the fronts of elastic waves in solid isotropic and anisotropic media. The sectional velocity and the sectional gradient are defined in terms of the motion of the curve of intersection of a front with a fixed surface. Section 2 (and 12) relates the coefficients of elasticity of the medium, the front types, and their respective rays. In section 12, the theory of fronts of arbitrary shape and of the corresponding rays for any anisotropic, homogeneous or inhomogeneous solid medium is summarized. In section 3 (and 13), the law of reflection and refraction of fronts on surfaces of discontinuity of arbitrary shape is presented. Sections 4 to 6 (and 14 to 16) treat some elementary applications of seismic travel time methods to homogeneous, uniaxially anisotropic media (=transverse isotropy) in greater detail. In section 4 (and 14), the travel time of a direct front generated by a point source is considered and it is shown how the coefficients of elasticity of the medium can be found based on travel time measurements. The seismic prospection of a plane reflector and of a reflecting boundary of arbitrary shape and position are discussed in section 5 (and 15). In section 6 (and 16), the seismic refraction method is used to locate a plane boundary between a homogeneous, uniaxially anisotropic and a homogeneous isotropic medium, where the boundary is perpendicular or at an arbitrary angle to the direction of anisotropy.     

Kirchhoff-Helmholtz积分方法计算远震转换波

Kirchhoff-Helmholtz积分方法推广到远震转换波的合成地震图的计算,其优点是,能够计算复杂界面的地震波。通过与反射率法及动力学射线追踪的对比,表明KH积分方法能很好地模拟远震转换波震相,且精度较高。KH积分方法能够计算任意复杂界面的地震波,是一种研究地壳上地幔结构的有效方法。     

Retrieval of source parameters of an event of the 2000 West Bohemia earthquake swarm assuming an anisotropic crust

We propose an inversion scheme for retrieval of characteristics of seismic point sources, which in contrast to common practice, takes into account anisotropy. If anisotropy is neglected during inversion, the moment tensors retrieved from seismic waves generated by sources situated in anisotropic media may be biased. Instead of the moment tensor, the geometry of the source is retrieved directly in our inversion; if necessary, the moment tensor can be then determined from the source geometry aposteriori. The source geometry is defined by the orientation of the slip vector and the fault normal as well as the strength of the event given by the size of the slip and the area of the fault. This approach allows direct interpretation of the source geometry in terms of shear and tensile faulting. It also makes possible to identify volumetric source changes that occur during rupturing. We apply the described algorithm to one event of the 2000 West Bohemia earthquake swarm episode. For inversion we use information of the direct P waves. The structure is approximated by three different models determined from travel-time observations. The models are inhomogeneous isotropic, inhomogeneous anisotropic, and homogeneous anisotropic. For these models we obtain seismic moments M_T = 3.2 − 3.8 × 10¹⁴ Nm and left-lateral near-vertical oblique normal faulting on a N-S trending rupture surface. The orientation of the rupture surface is consistent with fault-plane solutions of earlier studies and with the spatial distribution of other events during this swarm. The studied event seems to be accompanied by a small amount of crack opening. The amount of crack opening is slightly reduced when the inhomogeneous anisotropic model is assumed, but it persists. These results and additional independent observations seem to indicate that tensile faulting occurs as a result of high fluid pressure.     

Theory of anisotropic dynamic ray tracing in ray-centred coordinates

A 4×4-propagator matrix formalism is presented for anisotropic dynamic ray tracing, including the propagation across curved interfaces. The computations are organised in the same way as in ervený's well-known isotropic propagator matrix formalism. Attention is paid to cases where double eigenvalues of the Christoffel matrix result in unstable expressions in the dynamic ray tracing system, but where geometrical spreading is well-behaved.     

3D Hybrid Ray-FD and DWN-FD Seismic Modeling for Simple Models Containing Complex Local Structures

Hybrid approaches find broad applications wherever all-in-one modelling of source, path, and site effects is too expensive. Our new 3D hybrid approach allows to compute the seismic wavefield in elastic isotropic models containing a complex local structure embedded in a large, but considerably simpler, regional structure. The hybrid modelling is realized in two successive steps.In the 1^st step, the ray or discrete wave number (DWN) method is used to compute the seismic wavefield due to the source and simple regional structure. The complex local structure is not present. Thus, the excitation contains the source and regional path effects. The time history of this wavefield (excitation), recorded at the points of so called excitation box, is stored on a disk. The excitation box envelopes a small portion of a computational domain.The 2^nd step of the hybrid method, now containing the complex local structure, is computed by finite differences (FD) inside the excitation box and its close vicinity. The excitation from the 1^st step is now used to inject the 1^st step wavefield into the 2^nd step computation. After that, the hybrid combination of the 1^st and 2^nd steps contains the source, regional path, and local structure effects at reasonably lower computational costs than in case of all-in-one modelling.The 3D ray-FD method is tested on models in which the locally complex structure is the well-known Volvi lake basin, embedded in various 1D structures. The wavefield is excited by the point source situated outside the basin. Although the structure outside the excitation box may be less dimensional (2D, 1D, homogeneous), the whole problem is actually 3D due to the 3D features of the structure inside the excitation box, 3D shape of the excitation box, and arbitrary source — excitation-box configuration. Simple (1D) structures outside the excitation box allow for comparison with the alternative hybrid DWN-FD results. However the ray method is suitable for computation of 3D regional structures outside the excitation box. The results from both approaches show a very good agreement for realistic crustal and local structural models.     

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.     

Transformation relations for second-order derivatives of travel time in anisotropic media

In the computation of paraxial travel times and Gaussian beams, the basic role is played by the second-order derivatives of the travel-time field at the reference ray. These derivatives can be determined by dynamic ray tracing (DRT) along the ray. Two basic DRT systems have been broadly used in applications: the DRT system in Cartesian coordinates and the DRT system in ray-centred coordinates. In this paper, the transformation relations between the second-order derivatives of the travel-time field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used both in isotropic and anisotropic media, including computations of complex-valued travel times necessary for the evaluation of Gaussian beams.     

A note on two-point paraxial travel times

Recently, several expressions for the two-point paraxial travel time in laterally varying, isotropic or anisotropic layered media were derived. The two-point paraxial travel time gives the travel time from point S′ to point R′, both these points being situated close to a known reference ray Ω, along which the ray-propagator matrix was calculated by dynamic ray tracing. The reference ray and the position of points S′ and R′ are specified in Cartesian coordinates. Two such expressions for the two-point paraxial travel time play an important role. The first is based on the 4 × 4 ray propagator matrix, computed by dynamic ray tracing along the reference ray in ray-centred coordinates. The second requires the knowledge of the 6 × 6 ray propagator matrix computed by dynamic ray tracing along the reference ray in Cartesian coordinates. Both expressions were derived fully independently, using different methods, and are expressed in quite different forms. In this paper we prove that the two expressions are fully equivalent and can be transformed into each other.     

Space-time ray method for seismic wave fields

Summary The space-time ray method can be applied to the evaluation and continuation (extrapolation) of the complete seismic wave field in laterally inhomogeneous media with curved interfaces. The wave field propagates along certain space-time curves, called space-time rays. Their space projections correspond to standard rays. Examples of possible applications of the space-time ray method, where the standard ray method fails, are as follows: a) The propagation of seismic waves in slightly dissipative media, b) The computation of seismic wave fields generated by seismic sources with direction-dependent source-time variations. c) Downward continuation of the seismic wave field (actual seismograms) measured at the Earth's surface.     

Comparison of Ray Methods with the Exact Solution in the 1-D Anisotropic “Simplified Twisted Crystal” Model

The exact analytical solution for the plane S-wave, propagating along the axis of spirality in the simple 1-D anisotropic simplified twisted crystal model, is compared with four different approximate ray-theory solutions. The four different ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. The comparison is carried out numerically, by evaluating both the exact analytical solution and the analytical solutions of the equations of the four ray methods. The comparison simultaneously demonstrates the limits of applicability of the isotropic and anisotropic ray theories, and the superior accuracy of the coupling ray theory over a broad frequency range. The comparison also shows the possible inaccuracy due to the quasi-isotropic perturbation of travel times in the equations of the coupling ray theory. The coupling ray theory thus should definitely be preferred to the isotropic and anisotropic ray theories, but the quasi-isotropic perturbation of travel times should be avoided. Although the simplified twisted crystal model is designed for testing purposes and has no direct relation to geological structures, the wave-propagation phenomena important in the comparison are similar to those in the models of the geological structures.In additional numerical tests, the exact analytical solution is numerically compared with the finite-difference numerical results, and the analytical solutions of the equations of different ray methods are compared with the corresponding numerical results of 3-D ray-tracing programs developed by the authors of the paper.     

射线法模拟分析井间地震观测的波场特征

按照井间地震的观测系统,用改进的突变点加插值射线追踪方法,追踪每炮每道的射线路径,计算几种主要类型的波沿射线路径的波至时间和射线振幅,制作井间地震多炮多道水平分量和垂直分量的合成记录.并将合成记录选排为井间共炮点道集、共接收点道集、共偏移距道集和共中心深度点道集,系统地分析了不同道集内几种主要类型的地震波的传播特征.对野外观测的实际井间地震记录进行了模拟,从复杂的井间地震记录中,识别出井间地震实际观测到的不同类型的波场,为随后的井间地震资料处理和应用提供了依据.