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1.
In viscoelastic media, the slowness vector p of plane waves is complex-valued, p = P + iA. The real-valued vectors P and A are usually called the propagation and the attenuation vector, repectively. For P and A nonparallel, the plane wave is called inhomogeneousThree basic approaches to the determination of the slowness vector of an inhomogeneous plane wave propagating in a homogeneous viscoelastic anisotropic medium are discussed. They differ in the specification of the mathematical form of the slowness vector p. We speak of directional specification, componental specification and mixed specification of the slowness vector. Individual specifications lead to the eigenvalue problems for 3 × 3 or 6 × 6 complex-valued matrices.In the directional specification of the slowness vector, the real-valued unit vectors N and M in the direction of P and A are assumed to be known. This has been the most common specification of the slowness vector used in the seismological literature. In the componental specification, the real-valued unit vectors N and M are not known in advance. Instead, the complex-valued vactorial component p
of slowness vector p into an arbitrary plane with unit normal n is assumed to be known. Finally, the mixed specification is a special case of the componental specification with p
purely imaginary. In the mixed specification, plane represents the plane of constant phase, so that N = ±n. Consequently, unit vector N is known, similarly as in the directional specification. Instead of unit vector M, however, the vectorial component d of the attenuation vector in the plane of constant phase is known.The simplest, most straightforward and transparent algorithms to determine the phase velocities and slowness vectors of inhomogeneous plane waves propagating in viscoelastic anisotropic media are obtained, if the mixed specification of the slowness vector is used. These algorithms are based on the solution of a conventional eigenvalue problem for 6 × 6 complex-valued matrices. The derived equations are quite general and universal. They can be used both for homogeneous and inhomogeneous plane waves, propagating in elastic or viscoelastic, isotropic or anisotropic media. Contrary to the mixed specififcation, the directional specification can hardly be used to determine the slowness vector of inhomogeneous plane waves propagating in viscoelastic anisotropic media. Although the procedure is based on 3 × 3 complex-valued matrices, it yields a cumbersome system of two coupled equations. 相似文献
2.
3.
Propagation in the plane of mirror symmetry of a monoclinic medium, with displacement normal to the plane, is the most general circumstance in anisotropic media for which pure shear-wave propagation can occur at all angles. Because the pure shear mode is uncoupled from the other two modes, its slowness surface in the plane is an ellipse. When the mirror symmetry plane is vertical the pure shear waves in this plane are SH waves and the elliptical SH sheet of the slowness surface is, in general, tilted with respect to the vertical axis. Consider a half-space of such a monoclinic medium, called medium M, overlain by a half-space of isotropic medium I with plane SH waves incident on medium M propagating in the vertical symmetry plane of M. Contrary to the appearance of a lack of symmetry about the vertical axis due to the tilt of the SH-wave slowness ellipse, the reflection and transmission coefficients are symmetrical functions of the angle of incidence, and further, there exists an isotropic medium E with uniquely determined density and shear speed which gives exactly the same reflection and transmission coefficients underlying medium J as does monoclinic medium M. This means that the underlying monoclinic medium M can be replaced by isotropic medium E without changing the reflection and transmission coefficients for all values of the angle of incidence. Thus no set of SH seismic experiments performed in the isotropic medium in the symmetry plane of the underlying half-space can reveal anything about the monoclinic anisotropy of that underlying half-space. Moreover, even when the underlying monoclinic half-space is stratified, there exists a stratified isotropic half-space that gives the identical reflection coefficient as the stratified monoclinic half-space for all angles of incidence and all frequencies. 相似文献
4.
利用微扰理论推导了弱各向异性(WA)介质参数正反演计算的基本公式;给出了在已知慢度矢量的一个分量和偏振矢量情况下确定WA参数的方法.如果这一方法被用于单一的变井源距垂直地震剖面(walkawayVSP)资料,可以获得9个WA参数.这9个WA参数完全决定了qP波和两个qS波在由剖面和钻井所决定的平面内传播的特性.对单条walkawayVSP观测系统数据的产生和WA参数的反演进行了数值模拟计算,对所能确定的WA参数及其可靠性进行了详细的讨论. 相似文献
5.
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. 相似文献
6.
Determination of the ray vector (the unit vector specifying the direction of the group velocity vector) corresponding to a given wave normal (the unit vector parallel to the phase velocity vector or slowness vector) in an arbitrary anisotropic medium can be performed using the exact formula following from the ray tracing equations. The determination of the wave normal from the ray vector is, generally, a more complicated task, which is usually solved iteratively. We present a first-order perturbation formula for the approximate determination of the ray vector from a given wave normal and vice versa. The formula is applicable to qP as well as qS waves in directions, in which the waves can be dealt with separately (i.e. outside singular directions of qS waves). Performance of the approximate formulae is illustrated on models of transversely isotropic and orthorhombic symmetry. We show that the formula for the determination of the ray vector from the wave normal yields rather accurate results even for strong anisotropy. The formula for the determination of the wave normal from the ray vector works reasonably well in directions, in which the considered waves have convex slowness surfaces. Otherwise, it can yield, especially for stronger anisotropy, rather distorted results. 相似文献
7.
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. 相似文献
8.
Patrick N.J. Rasolofosaon 《Geophysical Prospecting》2010,58(4):637-655
Seismic anisotropy in geological media is now widely accepted. Parametrizations and explicit approximations for the velocities in such media, considered as purely elastic and moderately anisotropic, are now standards and have even been extended to arbitrary types of anisotropy. In the case of attenuating media, some authors have also recently published different parametrizations and velocity and attenuation approximations in viscoelastic anisotropic media of particular symmetry type (e.g., transversely isotropic or orthorhombic). This paper extends such work to media of arbitrary anisotropy type, that is to say to triclinic media. In the case of homogeneous waves and using the so‐called ‘correspondence principle’, it is shown that the viscoelastic equations (for the phase velocities, phase slownesses, moduli, wavenumbers, etc.) are formally identical to the corresponding purely elastic equations available in the literature provided that all the corresponding quantities are complex (except the unit vector in the propagation direction that remains real). In contrast to previous work, the new parametrization uses complex anisotropy parameters and constitutes a simple extension to viscoelastic media of previous work dealing with non‐attenuating elastic media of arbitrary anisotropy type. We make the link between these new complex anisotropy parameters and measurable parameters, as well as with previously published anisotropy parameters, demonstrating the usefulness of the new parametrization. We compute the explicit complete directional dependence of the exact and of the approximate (first and higher‐order perturbation) complex phase velocities of the three body waves (qP, qS1 and qS2). The exact equations are successfully compared with the ultrasonic phase velocities and phase attenuations of the three body waves measured in a strongly attenuating water‐saturated sample of Vosges sandstone exhibiting moderate velocity anisotropy but very strong attenuation anisotropy. The approximate formulas are checked on experimental data. Compared to the exact solutions, the errors observed on the first‐order approximate velocities are small (<1%) for qP‐waves and moderate (<10%) for qS‐waves. The corresponding errors on the quality factor Q are moderate (<6%) for qP‐waves but critically large (up to 160%) for the qS‐waves. The use of higher‐order approximations substantially improves the accuracy, for instance typical maximum relative errors do not exceed 0.06% on all the velocities and 0.6% on all the quality factors Q, for third‐order approximations. All the results obtained on other rock samples confirm the results obtained on this rock. The simplicity of the derivations and the generality of the results are striking and particularly convenient for practical applications. 相似文献
9.
In this paper, we have considered the reflection and refraction of a plane wave at an interface between two half-spaces. The
lower half-spaces is composed of highly anisotropic triclinic crystalline material and the upper half-space is homogeneous
and isotropic. It has been assumed that due to incidence of a plane quasi-P (qP) wave, three types of waves, namely, quasi-P (qP), quasi-SV (qSV) and quasi-SH (qSH), will be generated in the lower half space whereas P and S waves will be generated in the upper half space. The phase velocities of all the quasi waves have been calculated. It has
been assumed that the direction of particle motion is neither parallel nor perpendicular to the direction of propagation.
Some specific relations have been established between directions of motion and propagation, respectively. The expressions
for reflection coefficients of qP, qSV, qSH and refracted coefficients of P and SV waves are obtained. Results of reflection and refraction coefficients are presented. 相似文献
10.
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. 相似文献
11.
— We propose an algorithm for local evaluation of weak anisotropy (WA) parameters from measurements of slowness vector components and/or of particle motions of q P waves at individual receivers in a borehole in a multi-azimuthal multiple-source offset VSP experiment. As a byproduct the algorithm yields approximate angular variation of q P-wave phase velocity. The formulae are derived under assumption of weak but arbitrary anisotropy and lateral inhomogeneity of the medium. The algorithm is thus independent of structural complexities between the source and the receiver. If complete slowness vector is determinable from observed data, then the information about polarization can be used as an independent additional constraint. If only the component of the slowness along the borehole can be determined from observations (which is mostly the case), the inversion without information about polarization is impossible. We present several systems of equations which can be used when different numbers of components of the slowness vector are available. The SVD algorithm is used to solve an overdetermined system of linear equations for WA parameters for two test examples of synthetic multi-azimuthal multiple-source offset VSP data. The system of equations results from approximate first-order perturbation equations for the slowness and polarization vectors of the q P wave. Analysis of singular values and of variances of WA parameters is used for the estimation of chances to recover the sought parameters. Effects of varying number of profiles with sources and of noise added to “observed” data are illustrated. An important observation is that although, due to insufficient data, we often cannot recover all individual WA parameters with sufficient accuracy, angular phase velocity variation can be recovered rather well. 相似文献
12.
Wavefront construction (WFC) methods provide robust tools for computing ray theoretical traveltimes and amplitudes for multivalued wavefields. They simulate a wavefront propagating through a model using a mesh that is refined adaptively to ensure accuracy as rays diverge during propagation. However, an implementation for quasi-shear (qS) waves in anisotropic media can be very difficult, since the two qS slowness surfaces and wavefronts often intersect at shear-wave singularities. This complicates the task of creating the initial wavefront meshes, as a particular wavefront will be the faster qS-wave in some directions, but slower in others. Analogous problems arise during interpolation as the wavefront propagates, when an existing mesh cell that crosses a singularity on the wavefront is subdivided. Particle motion vectors provide the key information for correctly generating and interpolating wavefront meshes, as they will normally change slowly along a wavefront. Our implementation tests particle motion vectors to ensure correct initialization and propagation of the mesh for the chosen wave type and to confirm that the vectors change gradually along the wavefront. With this approach, the method provides a robust and efficient algorithm for modeling shear-wave propagation in a 3-D, anisotropic medium. We have successfully tested the qS-wave WFC in transversely isotropic models that include line singularities and kiss singularities. Results from a VTI model with a strong vertical gradient in velocity also show the accuracy of the implementation. In addition, we demonstrate that the WFC method can model a wavefront with a triplication caused by intrinsic anisotropy and that its multivalued traveltimes are mapped accurately. Finally, qS-wave synthetic seismograms are validated against an independent, full-waveform solution. 相似文献
13.
Conventional ray tracing for arbitrarily anisotropic and heterogeneous media is expressed in terms of 21 elastic moduli belonging
to a fixed, global, Cartesian coordinate system. Our principle objective is to obtain a new ray-tracing formulation, which
takes advantage of the fact that the number of independent elastic moduli is often less than 21, and that the anisotropy thus
has a simpler nature locally, as is the case for transversely isotropic and orthorhombic media. We have expressed material
properties and ray-tracing quantities (e.g., ray-velocity and slowness vectors) in a local anisotropy coordinate system with
axes changing directions continuously within the model. In this manner, ray tracing is formulated in terms of the minimum
number of required elastic parameters, e.g., four and nine parameters for P-wave propagation in transversely isotropic and
orthorhombic media, plus a number of parameters specifying the rotation matrix connecting local and global coordinates. In
particular, we parameterize this rotation matrix by one, two, or three Euler angles. In the ray-tracing equations, the slowness
vector differentiated with respect to traveltime is related explicitly to the corresponding differentiated slowness vector
for non-varying rotation and the cross product of the ray-velocity and slowness vectors. Our formulation is advantageous with
respect to user-friendliness, efficiency, and memory usage. Another important aspect is that the anisotropic symmetry properties
are conserved when material properties are determined in arbitrary points by linear interpolation, spline function evaluation,
or by other means. 相似文献
14.
S. S. Kevorkyants 《Izvestiya Physics of the Solid Earth》2013,49(1):19-23
In this paper, the solution of the system of homogeneous Biot equations, which was derived by Biot for the displacement vectors of plane monochrome elastic waves propagating in a homogeneous infinite two-phase medium, is expanded to the case where the propagation area of the elastic waves is limited and the wavefront is a piecewise smooth curved surface. It is shown that the arbitrary system of homogeneous Biot equations for the displacement vectors of the solid and liquid phases can be reduced to three different equations pertaining to the class of Helmholtz equations. From this, irrespective of the geometry of the seismic wavefront and the boundaries of the studied two-phase medium, there is the following. (1) Each displacement vector (of the solid and liquid phase) splits into three independent vectors satisfying three different Helmholtz equations. Two of these vectors correspond to the two types of compressional waves, namely, fast waves (waves of the first kind) and slow waves (waves of the second kind). The third vector describes shear waves. (2) The similar (related to the same wave type) components of the displacement vector in the solid and liquid phases satisfy the same Helmholtz equation and are linked with each other through a corresponding scalar factor that is expressed in terms of the coefficients of the Biot equations. Taking into account the established properties of the displacement vectors in the solid and liquid phases seems to be helpful in the problems dealing with calculation of elastic fields of arbitrary sources in piecewise-homogeneous two-phase media. 相似文献
15.
黏弹各向异性介质中传播不均匀波，其反射、透射模式不仅与介质分界面两侧速度对比有关，还与品质因子Q的对比有关. 用伪谱技术模拟黏弹各向异性介质分界面上波的反射、透射，并与弹性各向异性介质、黏弹各向同性介质和弹性各向同性介质的模拟结果做比较. 计算平面波的反射、透射系数，分析介质的黏弹性和各向异性对反射、透射系数的影响. 数值模拟了一个三层介质模型中的波场，分析两个分界面上产生的反射波的特征. 黏弹各向异性介质中，qS波比qP波衰减程度大. 相似文献
16.
We consider the problem of determining and predicting how the wave speeds in particular directions for a transversely isotropic (TI) medium depend on particular combinations of the density-normalized moduli Aij. The expressions for the qP and qSV velocities are known to depend on four moduli. Normally, we can only determine three independent parameters from qP data, or two from qSZ data, as the others have much lower sensitivity. The resolvable parameters are conveniently described by axial and off-axis parameters: for qP rays, P0°= A11, P90°= A33 and P45°=(A11+ A33)/4 + (A13+2A55)/2; and for qSV rays, S0°= S90°=A55 and S 45°= (A11+ A 33)/4- A13/2. These parameters control the magnitude of the squared-velocities on the axes and at approximately 45°. For an arbitrary TI medium, if the medium is perturbed in a way that preserves a particular parameter, then slowness points in the associated direction and mode witl be approximately preserved in the new medium. we refer to these parameters as ‘push-pins’, i.e. if a parameter is fixed, the associated part of the slowness surface is pinned in place. Because, these five push-pins only contain four independent moduli, we can only fix at most three push-pins. Perturbing one of the other parameters inevitably perturbs the other. Numerical results illustrating the linkage between two push-pins, when three are fixed, are presented. So-called anomalous TI media occur when the roles of the qP and qSV waves are reversed: in some directions the faster ray has transverse polarization. That, in turn, requires anomalous velocities at the push-pins, i.e. S0° > P0°, S45° > P45° and/or S90° > P90° (equivalent to the usual anomalous conditions A11 < A55, < 0 and/or A33 < A55). In the Appendix, we confirm that anomalous sensitivities of the velocities at the five push-pins only occur in such media, although the push-pins still apply if interpreted appropriately. Truly anomalous sensitivities, in which push-pins play no role, only occur in media near the boundary between normal and anomalous. 相似文献
17.
B.O. Ruud 《Studia Geophysica et Geodaetica》2006,50(3):479-498
The calculation of reflection and transmission coefficients of plane waves at a plane interface between two homogeneous anelastic
media may become ambiguous because it is not always obvious how to determine the sign of the vertical component of the slowness
vector of the scattered waves. For elastic media, the sign is determined by applying so-called radiation condition when the
slowness vector is complex-valued, but it has long been known that this approach does not work satisfactorily for anelastic
media. Other approaches have been suggested, e.g., by requiring that the reflection and transmission coefficients should vary
continuously with increasing incident angles, or by relating the sign to the direction of the energy flux. In the present
paper, it is shown that these approaches may give different results, and that the results can be inconsistent with the elastic
case even for weak attenuation. Instead, it is demonstrated that the ambiguity in the reflection coefficient can be resolved
by expressing the seismic response of a point source over an interface as a superposition of plane waves and their reflection
coefficients, and solving the resulting integral by the saddle point approximation. Although the saddle point itself (point
of stationary phase) does not provide new insight, the ambiguity is removed by considering the steepest descent path through
the point. Ray synthetic seismograms computed by this method compare well with synthetics computed by the reflectivity method,
which does not suffer from the above-mentioned ambiguity since the integration path is taken along the real axis.
This paper concentrates on the isotropic case, but it is discussed how the result may be extended to layered transversely
isotropic media. The suggested approach, derived for a point source and plane layers, does not directly apply to 2-D or 3-D
laterally inhomogeneous media, or to media of general anisotropy. A generalization of the result found is that the sign of
the vertical slowness components should be chosen according to the energy flux direction for subcritical incidence and according
to the radiation condition for supercritical incidence, even if this creates a discontinuity in the coefficients at the critical
incidence angle. Such a discontinuity is sometimes necessary to get results which are consistent with the elastic case. It
is discussed how the generalized result can be obtained by applying certain continuity criteria for the sub-and supercritical
angle intervals, but the validity of this approach for general models remains to be proved. 相似文献
18.
Ilya Tsvankin 《Geophysical Prospecting》1997,45(3):479-512
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. 相似文献
19.
A new wave equation is derived for modelling viscoacoustic wave propagation in transversely isotropic media under acoustic transverse isotropy approximation. The formulas expressed by fractional Laplacian operators can well model the constant-Q (i.e. frequency-independent quality factor) attenuation, anisotropic attenuation, decoupled amplitude loss and velocity dispersion behaviours. The proposed viscoacoustic anisotropic equation can keep consistent velocity and attenuation anisotropy effects with that of qP-wave in the constant-Q viscoelastic anisotropic theory. For numerical simulations, the staggered-grid pseudo-spectral method is implemented to solve the velocity–stress formulation of wave equation in the time domain. The constant fractional-order Laplacian approximation method is used to cope with spatial variable-order fractional Laplacians for efficient modelling in heterogeneous velocity and Q media. Simulation results for a homogeneous model show the decoupling of velocity dispersion and amplitude loss effects of the constant-Q equation, and illustrate the influence of anisotropic attenuation on seismic wavefields. The modelling example of a layered model illustrates the accuracy of the constant fractional-order Laplacian approximation method. Finally, the Hess vertical transversely isotropic model is used to validate the applicability of the formulation and algorithm for heterogeneous media. 相似文献
20.
Kinematical characteristics of reflected waves in anisotropic elastic media play an important role in the seismic imaging workflow. Considering compressional and converted waves, we derive new, azimuthally dependent, slowness-domain approximations for the kinematical characteristics of reflected waves (radial and transverse offsets, intercept time and traveltime) for layered orthorhombic media with varying azimuth of the vertical symmetry planes. The proposed method can be considered an extension of the well-known ‘generalized moveout approximation’ in the slowness domain, from azimuthally isotropic to azimuthally anisotropic models. For each slowness azimuth, the approximations hold for a wide angle range, combining power series coefficients in the vicinity of both the normal-incidence ray and an additional wide-angle ray. We consider two cases for the wide-angle ray: a ‘critical slowness match’ and a ‘pre-critical slowness match’ studied in Parts I and II of this work, respectively. For the critical slowness match, the approximations are valid within the entire slowness range, up to the critical slowness. For the ‘pre-critical slowness match’, the approximations are valid only within the bounded slowness range; however, the accuracy within the defined range is higher. The critical slowness match is particularly effective when the subsurface model includes a dominant high-velocity layer where, for nearly critical slowness values, the propagation in this layer is almost horizontal. Comparing the approximated kinematical characteristics with those computed by numerical ray tracing, we demonstrate high accuracy. 相似文献