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.     

Out‐of‐plane geometrical spreading in anisotropic media

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

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.     

Generalized non‐hyperbolic approximation for qP‐wave relative geometrical spreading in a layered transversely isotropic medium with a vertical symmetry axis

Compensation for geometrical spreading along the ray‐path is important in amplitude variation with offset analysis especially for not strongly attenuative media since it contributes to the seismic amplitude preservation. The P‐wave geometrical spreading factor is described by a non‐hyperbolic moveout approximation using the traveltime parameters that can be estimated from the velocity analysis. We extend the P‐wave relative geometrical spreading approximation from the rational form to the generalized non‐hyperbolic form in a transversely isotropic medium with a vertical symmetry axis. The acoustic approximation is used to reduce the number of parameters. The proposed generalized non‐hyperbolic approximation is developed with parameters defined by two rays: vertical and a reference rays. For numerical examples, we consider two choices for parameter selection by using two specific orientations for reference ray. We observe from the numerical tests that the proposed generalized non‐hyperbolic approximation gives more accurate results in both homogeneous and multi‐layered models than the rational counterpart.     

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.     

弱各向异性介质弹性波的准各向同性近似正演模拟

准各向同性(QI)近似可用于弱各向异性介质的正演模拟.本文通过运用QI方法的零阶和一阶近似,计算了VTI介质模型的地震记录.得出的地震记录与标准各向同性射线理论(IRT)和基于伪谱法的三维地震正演模拟得出的地震记录作了比较,可以认为是精确的合成地震记录.     

Ray tracing and geodesic deviation of the SH and SV reference rays in a heterogeneous generally anisotropic medium which is approximately uniaxial

The coupling ray theory is usually applied to anisotropic common reference rays, but it is more accurate if it is applied to reference rays which are closer to the actual wave paths. If we know that a medium is close to uniaxial (transversely isotropic), it may be advantageous to trace reference rays which resemble the SH–wave and SV–wave rays. This paper is devoted to defining and tracing these SH and SV reference rays of elastic S waves in a heterogeneous generally anisotropic medium which is approximately uniaxial (approximately transversely isotropic), and to the corresponding equations of geodesic deviation (dynamic ray tracing). All presented equations are simultaneously applicable to ordinary and extraordinary reference rays of electromagnetic waves in a generally bianisotropic medium which is approximately uniaxially anisotropic. The improvement of the coupling–ray–theory seismograms calculated along the proposed SH and SV reference rays, compared to the coupling–ray–theory seismograms calculated along the anisotropic common reference rays, has already been numerically demonstrated by the authors in four approximately uniaxial velocity models.     

Rapid directivity detection by azimuthal amplitude spectra inversion

An early detection of the presence of rupture directivity plays a major role in the correct estimation of ground motions and risks associated to the earthquake occurrence. We present here a simple method for a fast detection of rupture directivity, which may be additionally used to discriminate fault and auxiliary planes and have first estimations of important kinematic source parameters, such as rupture length and rupture time. Our method is based on the inversion of amplitude spectra from P-wave seismograms to derive the apparent duration at each station and on the successive modelling of its azimuthal behaviour. Synthetic waveforms are built assuming a spatial point source approximation, and the finite apparent duration of the spatial point source is interpreted in terms of rupture directivity. Since synthetic seismograms for a point source are calculated very quickly, the presence of directivity may be detected within few seconds, once a focal mechanism has been derived. The method is here first tested using synthetic datasets, both for linear and planar sources, and then successfully applied to recent Mw 6.2–6.8 shallow earthquakes in Peloponnese, Greece. The method is suitable for automated application and may be used to improve kinematic waveform modelling approaches.     

SEISMIC REFLECTION AMPLITUDES1

Seismic amplitude variations with offset contain information about the elastic parameters. Prestack amplitude analysis seeks to extract this information by using the variations of the reflection coefficients as functions of angle of incidence. Normally, an approximate formula is used for the reflection coefficients, and variations with offset of the geometrical spreading and the anelastic attenuation are often ignored. Using angle of incidence as the dependent variable is also computationally inefficient since the data are recorded as a function of offset. Improved approximations have been derived for the elastic reflection and transmission coefficients, the geometrical spreading and the complex travel-time (including anelastic attenuation). For a 1 D medium, these approximations are combined to produce seismic reflection amplitudes (P-wave, S-wave or converted wave) as a Taylor series in the offset coordinate. The coefficients of the Taylor series are computed directly from the parameters of the medium, without using the ray parameter. For primary reflected P-waves, dynamic ray tracing has been used to compute the offset variations of the transmission coefficients, the reflection coefficient, the geometrical spreading and the anelastic attenuation. The offset variation of the transmission factor is small, while the variations in the geometrical spreading, absorption and reflection coefficient are all significant. The new approximations have been used for seismic modelling without ray tracing. The amplitude was approximated by a fourth-order polynomial in offset, the traveltime by the normal square-root approximation and the absorption factor by a similar expression. This approximate modelling was compared to dynamic ray tracing, and the results are the same for zero offset and very close for offsets less than the reflector depth.     

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.     

Radiation patterns of point sources situated close to structural interfaces and to the earth's surface

The seismic wave field is considerably influenced by local structures close to the source and to the receiver. This applies to sources and receivers situated close to localized inhomogeneities, to structural interfaces, to the earth's surface, etc. In this paper we concentrate our attention mainly to the ray-theoretical radiation patterns of point sources situated close to the structural interfaces and to the earth's surface. In numerical modeling of high-frequency seismic wave fields by the ray method, the interaction of the source with the earth's surface has not usually been taken into account.The proposed procedure of the computation of the radiation patterns of point sources situated directly on structural interfaces and on the earth's surface is based on the zero-order approximation of the ray method, assuming that the length of the ray between the source and the receiver is long. The derived equations are extended to point sources located close to structural interface, to the earth's surface and to thin transition layers using the hybrid ray-reflectivity method, seeervený (1989). The thin layer need not be homogeneous; it may include an arbitrary inner layering (transition layers, laminas, etc.) The only requirement is for the layer to be thin. Roughly speaking, we require its thickness to be less than one quarter of the prevailing wavelength. The hybrid ray-reflectivity method describes well even certain non-ray effects (tunneling.S ^* waves, etc.). Explicit analytical expressions for radiation patterns for all above listed point sources are found. These expression have a local character and may be easily implemented into computer codes designed for the routine computation of ray amplitudes and synthetic ray seismograms in 2-D and 3-D, laterally varying isotropic layered and block structures by the ray method.Numerical examples of radiation patterns ofP andS waves of point sources situated close to the earth's surface and to a thin low-velocity surface layer are presented and discussed. The explosive point source (center of dilatation) and the vertical and horizontal single force point sources are considered. It has been ascertained that the radiation patterns of point sources depend drastically on the depth of the source below the surface even if the depths vary within one quarter of the prevailing wavelength.     

WAVEFRONT DIVERGENCE,MULTIPLES, AND CONVERTED WAVES IN SYNTHETIC SEISMOGRAMS*

Synthetic seismograms can be very useful in aiding understanding of wave propagation through models of real media, verification of geologic models derived from interpretation of field seismic data, and understanding the nature and complexity of wave phenomena. If meaningful results are to be obtained from synthetic seismograms, the method of their computation must, in general, include three-dimensional geometrical spreading of wavefronts associated with highly concentrated (i.e., point) sources. The method should also adequately represent the seismic response of solid-layered media by including enough primaries, multiples, and converted phases to accurately approximate the total wavefield. In addition to these features, it is also very helpful, although not always essential, if the method of seismogram computation provides for explicit identification of wave type and ray path for each arrival. Various seismograms, computed via asymptotic ray theory and an automatic ray generation scheme, are presented for a highly simplified North Sea velocity structure. This is done to illustrate the importance of the above features and to demonstrate the inadequacy of the plane-wave synthesis method of seismogram computation for point sources and the limitations of acoustic models of solid-layered media.     

A NON-GEOMETRICAL SH-ARRIVAL1

The existence of‘*-waves’has, in recent years, prompted a renewed interest in these non-geometrical arrivals, which are generated by point sources located adajcent to plane interfaces. It has led to the re-evaluation of seismic data aquisition techniques and to the question of how to use this real phenomena in enhancing existing seismic interpretation methods. This paper considers a non-geometrical SH-arrival which is generated by a point torque source unrealistically buried within a half-space. The method of solution is essentially the same as presented in an earlier paper, with the modification that the limitation placed on the distance of the source from the interface has been removed in the saddle point method used to obtain a high-frequency approximate solution. In the earlier paper, a preliminary assumption forced the saddle point, which corresponded to the *-wave arrival, to be real when it is generally complex. However, for offsets removed from the distinct ray, the imaginary part of this complex quantity is negligible. A problem which arose when comparing exact synthetic traces with those obtained using zero-order saddle point methods, was the inability to match either the amplitude or phase of the geometrical arrival in the range of offsets when the *-wave and this corresponding geometrical ray were well separated. For this range of offsets the geometrical arrival was approaching grazing incidence and another term in the saddle point expansion of the integral was necessary to rectify this error. This method is also being used to validate the results for higher order terms obtained using asymptotic ray theory. Analytical formulae are given for both the *-wave and the higher order expansion of the geometrical event, together with a comparison of synthetic seismograms using the method developed here and a numerical integration algorithm.     

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…     

Geometrical spreading factor of head waves in cased and open boreholes

Factors (coefficients) of geometrical spreading of compressional and shear head waves are calculated for an impulse multipole source of elastic oscillations in boreholes. It is shown that the length of the logging tool (i.e., the distance between the source and the nearest receiver) used for sonic measurements and the velocities of elastic waves in the medium both contribute to the factor of geometrical spreading. For a high-velocity formation (the shear wave velocity in the rock is higher than the compressional wave velocity in the fluid that fills the borehole) and a sufficiently long sonic tool with a monopole source, the coefficient of geometrical spreading is approximated by asymptotic formula 1/Z [Roever et al., 1974; Krauklis and Krauklis, 1976], where Z is the length of the tool; i.e., the amplitude of the compressional head wave decreases proportionally to the distance between the source and the receiver. In acoustically soft formations, this approximation is inapplicable even for long tools with length Z > 4 m. Waveforms in cased boreholes have a significant frequency dispersion even in case of good-quality cementing, and the factor of geometrical spreading there depends considerably on the length of the tool and the elastic properties of the rocks.     

On the disturbance due to a spherical distortional pulse in an elastic medium

Summary Theoretical expressions are derived for the displacement, velocity and stress in the time domain induced by an axially symmetric shearing stress applied at the inner surface of a spherical cavity in a hornogeneous, isotropic, elastic medium of infinite extent. Theoretical seismograms are computed for a step source and for three sources with exponential decay in time. A satisfactory time-dependence of the source can be obtained by combining the step source with one or more exponentially decaying sources.     

基于成像射线的偏移剖面和速度时深转换评述

深度域地震波速度模型是地震勘探中最重要的参数之一,是获得精确深度域偏移剖面的基础.本文基于成像射线的概念,阐述了时间偏移域和深度域之间的关系,明确了时间偏移后的像点是通过成像射线与深度域的散射点联系起来的.在有横向变速的情况下,时间偏移剖面需要经过时深转换才能真实反映深度域层位,同样,从时间偏移剖面上提取的速度也需要经过时深转换才能得到深度域的速度.与前者不同的是,在速度的时深转换中,不仅速度的位置需要改变,且其数值也需要进行校正.该校正因子有几种等价的描述形式：度规量、速度扩散因子和几何扩散因子.     

ON THE ROLE OF PARTIAL RAY EXPANSION IN THE COMPUTATION OF RAY SYNTHETIC SEISMOGRAMS FOR LAYERED STRUCTURES1

This paper presents results of testing an efficient ray generation scheme needed whenever ray synthetic seismograms are to be computed for layered models with more than 10‘ thick’layers. Our ray generation algorithm is based on the concept of kinematically equivalent waves (the kinematic analogs) having identical traveltimes along different ray-paths between the source and the receiver, both located on the surface of the model. These waves, existing in any medium composed of laterally homogeneous parallel layers, interfere at any location along the recording surface, thereby producing a composite wavelet whose amplitude and shape depend directly on the number of kinematic analogs (the multiplicity factor). Hence, explicit knowledge of the multiplicity factor is crucial for any analysis based on the amplitude and shape of individual wavelets, such as wavelet shaping, Q estimation, or linearized wavelet inversion. For unconverted waves, such as those discussed in this paper, the multiplicity factor can be computed analytically using formulae given in the Appendix; for converted waves, the multiplicity factor should be computed numerically, using the algorithm employed for the computation of the seismograms presented in a previous paper by one of the authors.     

美国中东部地震台阵波场的三分量波形梯度

波形梯度法是一种全新的台阵数据处理技术,该方法利用子台阵中的波形差异可以得到一些基本的地震波传播参数.本文首次将其应用于美国中东部地震台阵面波的三分量研究当中.首先通过垂向分量的面波进行波形梯度分析,得到垂直分量面波的相速度、传播方向、几何扩散和辐射花样.再通过垂向分量得到的传播方向进行坐标旋转,从而得到了径向分量以及切向分量的面波,再将其应用于波形梯度分析,分别得到径向分量以及切向分量面波的相速度、传播方向、几何扩散和辐射花样.利用2012年8月27日发生在中美洲西海岸地震事件的三分量地震数据的面波波形得到的结果显示,在相同周期,研究区域的三个分量面波的相速度分布横向差异显著.切向分量面波的相速度分布特征差异较大,可能是由于径向各向异性造成的.三个分量的传播方向变化都不大,且切向分量的传播方向变化大于垂直分量与径向分量,说明地震波的切线分量在传播过程中受到的影响更大,同时还可以看出传播方向的变化呈现出条带状的特征.几何扩散和辐射花样都是与地震波的振幅项有关的信息,三个分量的几何扩散特征基本一致.但是由于切向分量传播方向变化相对较大,可能导致了切向分量面波的辐射花样有所差异.     

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.