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.     

分块三维速度模型生成及理论地震图的计算

本文提出了在计算机上实现分块三维地壳模型及利用加权最小二乘拟合生成平缓光滑的三维速度函数的方法,给出了适用于分块、块内速度连续变化的三维模型中Cauchy射线追踪的新算法,简介了基于上述方法反射线的基本理论所编制的合成三维理论地震图的程序包RSSGTD.给出的两个盆地状模型的算例表明,所使用的模型生成方法具有模拟复杂地壳结构的能力;与三维样条函数方法比较,最小二乘拟合方法能给出更加适合射线方法合成地震图计算的速度函数,并且内存小、计算速度快;所给出的Cauchy射线追踪算法能够适合块状模型中任何体波射线的追踪.     

Complex rays applied to wave propagation in a viscoelastic medium

In order to trace a ray between known source and receiver locations in a perfectly elastic medium, the take-off angle must be determined, or equialently, the ray parameter. In a viscoelastic medium, the initial value of a second angle, the attenuation angle (the angle between the normal to the plane wavefront and the direction of maximum attenuation), must also be determined. There seems to be no agreement in the literature as to how this should be done. In computing anelastic synthetic seismograms, some authors have simply chosen arbitrary numerical values for the initial attenuation angle, resulting in different raypaths for different choices. There exists, however, a procedure in which the arbitrariness is not present, i.e., in which the raypath is uniquely determined. It consists of computing the value of the anelastic ray parameter for which the phase function is stationary (Fermat's principle). This unique value of the ray parameter gives unique values for the take-off and attenuation angles. The coordinates of points on these stationary raypaths are complex numbers. Such rays are known as complex rays. They have been used to study electromagnetic wave propagation in lossy media. However, ray-synthetic seismograms can be computed by this procedure without concern for the details of complex raypath coordinates. To clarify the nature of complex rays, we study two examples involving a ray passing through a vertically inhomogeneous medium. In the first example, the medium consists of a sequence of discrete homogeneous layers. We find that the coordinates of points on the ray are generally complex (other than the source and receiver points which are usually assumed to lie in real space), except for a ray which is symmetric about an axis down its center, in which case the center point of the ray lies in real space. In the second example, the velocity varies continuously and linearly with depth. We show that, in geneneral, the turning point of the ray lies in complex space (unlike the symmetric ray in the discrete layer case), except if the ratio of the velocity gradient to the complex frequency-dependent velocity at the surface is a real number. We also present a numerical example which demonstrates that the differences between parameters, such as arrival time and raypath angles, for the stationary ray and for rays computed by the above-mentioned arbitrary approaches can be substantial.     

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.     

Application of the edge wave superposition method

Numerical examples of high-frequency synthetic seismograms of body waves in a 2-D layered medium with complex interfaces (faults, wedges, curvilinear, corrugated) are presented. The wave field modeling algorithm combines the possibilities of the ray method and the edge wave superposition method. This approach preserves all advantages of the ray method and eliminates restrictions related to diffraction by boundary edges and to caustic effects in singular regions. The method does not require two-point ray tracing (source-to-receiver), and the position of the source, as well as the type of source, and the position of receivers can be chosen arbitrarily. The memory and the time required for synthetic seismogram computation are similar to ray synthetic seismograms. The computation of the volume of the medium (the Fresnel volume or Fresnel zones), which gives the essential contribution to the wave field, is included in the modeling program package. In the case of complicated irregular interface (or a layered medium with a regular ray field at the last interface), the method displays a high accuracy of wave field computation. Otherwise, the method can be considered a modification of the ray method with regularization by the superposition of edge waves.     

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.     

Modeling multiple elementary waves in layered media using ray field maps

Seismic ray tracing in layered media becomes complicated and demanding when modeling for multiple ray codes (reflection/transmission sequences) and/or dense acquisition geometries. However, we observe some redundancies in current algorithms: (a) the same layers are crossed repeatedly by similar ray segments, and (b) the effort of tracing through a layer is determined by variations in the incoming wavefront rather than the medium. We deal with these redundancies by separating the modeling process in two stages: (Stage 1) compute ray field maps representing all ray segments between each pair of adjacent interfaces, then (Stage 2) for each desired ray code assemble the complete ray field from ray segments by iterative lookup in the ray field maps.     

Application of the Maslov Seismogram Method in Three Dimensions

Asymptotic methods provide an efficient way to compute seismograms in heterogeneous media. However, zeroth-order ray theory, the simplest of the asymptotic methods, often fails because of the presence of caustics. Maslov theory is an extension of zeroth-order ray theory, which gives a uniformly valid expression of the wavefield everywhere, including the caustics. This result is given in terms of an integral of ray data over one or two ray parameters. It is shown in this paper how geometrical arrivals are constructed in the one and two-parameter Maslov integrals.In practice Maslov seismograms have been computed using only one ray parameter. However, in three-dimensional media two parameters are needed to uniquely define a ray. In this paper we present an efficient algorithm to compute two-parameter Maslov integrals. The Maslov integral is evaluated by computing the frequency-to-time Fourier transform prior to integration over the ray parameters. The wavefield is then discretized by smoothing with a boxcar function. The resulting expression, which only requires the results of ordinary kinematic and dynamic ray tracing, cen be computed efficiently and robustly. A numerical example is given that illustrates the use of this algorithm.     

基于频率-慢度域叠前地震记录的合成与并行方法研究

正演模拟是叠前弹性波反演的基础.采用慢度法计算层状介质的叠前地震记录,分别对频率和慢度进行积分变换得到时-空域的地震道集,并对在慢度积分过程中产生的计算噪音提出了解决方案.为得到高精度合成地震记录,需将地层细分,但地层层数很多时,计算量较大;而对地层粗分虽然会大大加快运算速度,但合成记录会丢失很多信息,文中给出了地层的划分原则.该方法能够计算出包括转换波和多次反射在内的全地震响应.但在提高合成记录精度的同时,也导致计算量增大、计算效率降低,因此,本文对基于慢度法全波场模拟进行了并行算法设计,采用计算域分割、工作池并行技术,建立了慢度法全波场正演模拟的并行算法,使得弹性波正演问题求解更加高效,为充分利用叠前地震资料进行叠前反演提供了研究基础.     

区域地震范围内地壳介质的低频响应

本文对于计算P_nl波格林函数的广义地震射线方法进行了改进。包括:1.简化了地震射线的描述方法;2.将射线按其运动学相似性进行分组计算,从而较大地提高了计算速度;3.进一步推广用于计算SH波部分的响应。对于1983年山东菏泽地震和1985年云南禄劝地震,用综合地震图方法模拟了763型长周期地震仪的P_nl波记录,求出两个地震的断层面解答,其结果与有关研究所得该地区构造应力场的图象基本一致。     

Ray–Kirchhoff multicomponent borehole seismic modelling in 3D heterogeneous, anisotropic media

Kirchhoff–Helmholtz (KH) theory is extended to synthesize two-way elastic wave propagation in 3D laterally heterogeneous, anisotropic media. I have developed and tested numerically a specialized algorithm for the generation of three-component synthetic seismograms in multi-layered isotropic and transversely isotropic (TI) media with dipping interfaces and tilted axes of symmetry. This algorithm can be applied to vertical seismic profile (VSP) geometries and works well when the receiver is located near the reflector interface. It is superior to ray methods in predicting elliptical polarization effects observed on radial and transverse components. The algorithm is used to study converted-wave propagation for determining fracture-related shear-wave anisotropy in realistic reservoir models. Results show that all wavefront attributes are strongly affected by the anisotropy. However, it is necessary to resolve a trade-off between the effects of fractures and formation dip prior to converted-wave interpretation. These results provide some assurance that the present scheme is sufficiently versatile to handle shear wave behaviour due to various generalized rays propagating in complex geological models.     

Computation of additional components of the first-order ray approximation in isotropic media

This study shows that the use of the first-order additional components of the ray method in the seismic wave field modeling is easy and that it can bring a substantial improvement of the standard ray results obtained with the zero-order ray approximation only. For the calculation of a first-order additional component, spatial derivatives of the parameters of the medium and spatial derivatives of the zero-order ray amplitude term are necessary. The evaluation of the former derivatives is straightforward; the latter derivatives can be calculated approximately from neighboring rays by substituting the derivatives by finite differences. This allows an effective calculation of the first-order additional terms in arbitrary laterally varying layered media.The importance of the first-order additional terms is demonstrated by the study of individual higher-order terms of the ray series representing elementaryP andS elastodynamic Green functions for a homogeneous isotropic medium. The study shows clearly that the consideration of the first-order additional terms leads to a more substantial decrease of the difference between approximate and exact elementary Green functions than any other higher-order term. With this in mind, effects of the first-order additional terms on the ray synthetic seismograms for aVSP configuration are studied. It is shown that the use of the additional terms leads to such phenomena, unknown in the zero-order approximation of the ray method, like quasi-elliptical and transverse polarization of a singleP wave or longitudinal polarization of a singleS wave.     

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.     

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.     

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.     

利用Lg波谱特征估计地下核爆炸装置的埋深

Lg波同时携带地球介质结构信息和爆炸源引起的二次源信息,是研究地下爆炸震源机制及检测、识别手段的重要震相。基于对某一区域介质结构的了解,将实测Lg波谱特征形态与理论波谱形态进行比对,利用Lg波频谱曲线的低谷点随震源深度"左移"的特征,作为辅助识别手段估计地下核爆炸装置的埋深。研究结果对深入地理解地下核爆炸震源机制有重要的参考意义。     

TheoreticalPKP seismograms near the caustic calculated by sum of reflected waves

Summary This paper gives some examples of theoretical seismograms of PKP waves near the caustic. Seismograms of refracted waves for the original medium are compared with seismograms composed as a sum of the reflected waves, generated at boundaries of a substitute medium. All seismograms are calculated by zero approximation of the ray theory. The influence of some parameters of the source function and of the substitute medium on the results is shown.     

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.     

Part II: Tracing and interpolation1

The numerical tracing of short ray segments and interpolation of new rays between these ray segments are central constituents of the wavefront construction method. In this paper the details of the ray tracing and ray-interpolation procedures are described. The ray-tracing procedure is based on classical ray theory (high-frequency approximation) and it is both accurate and efficient. It is able to compute both kinematic and dynamic parameters at the endpoint of the ray segments, given the same set of parameters at the starting point of the ray. Taylor series are used to approximate the raypath so that the kinematic parameters (new position and new ray tangent) may be found, while a staggered finite-difference approximation gives the dynamic parameters (geometrical spreading). When divergence occurs in some parts of the wavefront, new rays are interpolated. The interpolation procedure uses the kinematic and dynamic parameters of two parent rays to estimate the initial parameters of a new ray on the wavefront between the two rays. Third-order (cubic) interpolation is used for interpolation of position, ray tangent and take-off vector from the source) while linear interpolation is used for the geometrical spreading parameters.     

Complex Rays and Wave Packets for Decaying Signals in Inhomogeneous,Anisotropic and Anelastic Media

Diffraction and anelasticity problems involving decaying, “evanescent” or “inhomogeneous” waves can be studied and modelled using the notion of “complex rays”. The wavefront or “eikonal” equation for such waves is in general complex and leads to rays in complex position-slowness space. Initial conditions must be specified in that domain: for example, even for a wave originating in a perfectly elastic region, the ray to a real receiver in a neighbouring anelastic region generally departs from a complex point on the initial-values surface. Complex ray theory is the formal extension of the usual Hamilton equations to complex domains. Liouville's phase-space-incompressibility theorem and Fermat's stationary-time principle are formally unchanged. However, an infinity of paths exists between two fixed points in complex space all of which give the same final slowness, travel time, amplitude, etc. This does not contradict the fact that for a given receiver position there is a unique point on the initial-values surface from which this infinite complex ray family emanates.In perfectly elastic media complex rays are associated with, for example, evanescent waves in the shadow of a caustic. More generally, caustics in anelastic media may lie just outside the real coordinate subspace and one must trace complex rays around the complex caustic in order to obtain accurate waveforms nearby or the turning waves at greater distances into the lit region. The complex extension of the Maslov method for computing such waveforms is described. It uses the complex extension of the Legendre transformation and the extra freedom of complex rays makes pseudocaustics avoidable. There is no need to introduce a Maslov/KMAH index to account for caustics in the geometrical ray approximation, the complex amplitude being generally continuous. Other singular ray problems, such as the strong coupling around acoustic axes in anisotropic media, may also be addressed using complex rays.Complex rays are insightful and practical for simple models (e.g. homogeneous layers). For more complicated numerical work, though, it would be desirable to confine attention to real position coordinates. Furthermore, anelasticity implies dispersion so that complex rays are generally frequency dependent. The concept of group velocity as the velocity of a spatial or temporal maximum of a narrow-band wave packet does lead to real ray/Hamilton equations. However, envelope-maximum tracking does not itself yield enough information to compute synthetic seismogramsFor anelasticity which is weak in certain precise senses, one can set up a theory of real, dispersive wave-packet tracking suitable for synthetic seismogram calculations in linearly visco-elastic media. The seismologically-accepiable constant-Q rheology of Liu et al. (1976), for example, satisfies the requirements of this wave-packet theory, which is adapted from electromagnetics and presented as a reasonable physical and mathematical basis for ray modelling in inhomogeneous, anisotropic, anelastic media. Dispersion means that one may need to do more work than for elastic media. However, one can envisage perturbation analyses based on the ray theory presented here, as well as extensions like Maslov's which are based on the Hamiltonian properties.