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1.
Andrzej Hanyga 《Pure and Applied Geophysics》1996,148(3-4):387-420
Point-to-curve ray tracing is an attempt at dealing with multiplicity of solutions to a generic boundary-value problem of ray tracing. In a point-to-curve tracing (P2C) the input parameters of the boundary-value problem (BVP), such as the ends of the ray, are allowed to vary along a curve. The solutions of the BVP automatically wander from one solution branch to another generating a nearly complete multi-valued solution of the BVPs.A procedure for transforming an arbitrary iterative algorithm, solving a ray tracing BVP to a corresponding P2C algorithm, is presented. Bifurcations of the solution curve of the P2C problem at caustics are studied and an algorithm for obtaining the bifurcating branches is developed. In particular, transition from real rays to complex rays in a caustic shadow offers an additional link between otherwise disconnected solution curves of the P2C problem. The topological structure of a generic solution curve and its implications for the algorithm are studied. 相似文献
2.
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. 相似文献
3.
Due to the lateral heterogeneity of the upper layers of the Earth, paths of surface waves deviate from arcs of great circles. Because of the sphericity of the Earth, the paths intersect on a hemisphere opposite to the epicenter and form caustics consisting of two branches, with their tangent point being a cusp. For this reason, the field of surface waves cannot be analyzed in terms of the ray theory at distances larger than 90°. The asymptotic approach to the analysis of the field in the vicinity of such caustics is very ill-suited for numerical implementation. The difficulties of such an approach to the field calculation are aggravated by the fact that such caustics are superimposed in some regions. Therefore, it is suggested to use the theorem of representation, according to which the field within a certain contour is expressed as an integral whose integrand contains values of the function itself, its derivative along the normal to the contour, and Green’s function. The field on the contour (the circle bounding a hemisphere centered at the epicenter) is calculated by the ray method because rays do not intersect on this hemisphere. These data are used for the construction of the field on the opposite hemisphere assumed to be homogeneous, which enables the construction of Green’s function for this hemisphere. This limitation is not very stringent because the configuration of rays and caustics on this hemisphere is mainly determined by the field on the circle. The integral in the representation theorem is calculated numerically. Numerical examples are presented for models in which one caustic or two superimposed caustics form. These calculations yield constraints on variations in the amplitude and phase of the wave. Rayleigh wave fields are also calculated for a model of the real Earth. It is shown that, at some points, the Rayleigh wave spectrum can be strongly distorted because caustics corresponding to different periods differ in shape. 相似文献
4.
We describe the behaviour of the anisotropic–ray–theory S–wave rays in a velocity model with a split intersection singularity. The anisotropic–ray–theory S–wave rays crossing the split intersection singularity are smoothly but very sharply bent. While the initial–value rays can be safely traced by solving Hamilton’s equations of rays, it is often impossible to determine the coefficients of the equations of geodesic deviation (paraxial ray equations, dynamic ray tracing equations) and to solve them numerically. As a result, we often know neither the matrix of geometrical spreading, nor the phase shift due to caustics. We demonstrate the abrupt changes of the geometrical spreading and wavefront curvature of the fast anisotropic–ray–theory S wave. We also demonstrate the formation of caustics and wavefront triplication of the slow anisotropic–ray–theory S wave.Since the actual S waves propagate approximately along the SH and SV reference rays in this velocity model, we compare the anisotropic–ray–theory S–wave rays with the SH and SV reference rays. Since the coupling ray theory is usually calculated along the anisotropic common S–wave rays, we also compare the anisotropic common S–wave rays with the SH and SV reference rays. 相似文献
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3D multivalued travel time and amplitude maps 总被引：2，自引：0，他引：2
An algorithm for computing multivalued maps for travel time, amplitude and any other ray related variable in 3D smooth velocity models is presented. It is based on the construction of successive isochrons by tracing a uniformly dense discrete set of rays by fixed travel-time steps. Ray tracing is based on Hamiltonian formulation and includes computation of paraxial matrices. A ray density criterion ensures uniform ray density along isochrons over the entire ray field including caustics. Applications to complex models are shown. 相似文献
7.
We present a new ray bending approach, referred to as the Eigenray method, for solving two‐point boundary‐value kinematic and dynamic ray tracing problems in 3D smooth heterogeneous general anisotropic elastic media. The proposed Eigenray method is aimed to provide reliable stationary ray path solutions and their dynamic characteristics, in cases where conventional initial‐value ray shooting methods, followed by numerical convergence techniques, become challenging. The kinematic ray bending solution corresponds to the vanishing first traveltime variation, leading to a stationary path between two fixed endpoints (Fermat's principle), and is governed by the nonlinear second‐order Euler–Lagrange equation. The solution is based on a finite‐element approach, applying the weak formulation that reduces the Euler–Lagrange second‐order ordinary differential equation to the first‐order weighted‐residual nonlinear algebraic equation set. For the kinematic finite‐element problem, the degrees of freedom are discretized nodal locations and directions along the ray trajectory, where the values between the nodes are accurately and naturally defined with the Hermite polynomial interpolation. The target function to be minimized includes two essential penalty (constraint) terms, related to the distribution of the nodes along the path and to the normalization of the ray direction. We distinguish between two target functions triggered by the two possible types of stationary rays: a minimum traveltime and a saddle‐point solution (due to caustics). The minimization process involves the computation of the global (all‐node) traveltime gradient vector and the traveltime Hessian matrix. The traveltime Hessian is used for the minimization process, analysing the type of the stationary ray, and for computing the geometric spreading of the entire resolved stationary ray path. The latter, however, is not a replacement for the dynamic ray tracing solution, since it does not deliver the geometric spreading for intermediate points along the ray, nor the analysis of caustics. Finally, we demonstrate the efficiency and accuracy of the proposed method along three canonical examples. 相似文献
8.
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. 相似文献
9.
Maslov渐近理论与辛几何算法 总被引：26，自引：9，他引：17
为了克服地震层析成像中的焦散问题,本文系统地研究了Ｍａｓｌｏｖ渐近理论与辛几何算法,同时提出了一种基于辛几何算法计算Ｍａｓｌｏｖ波场的数值计算方法,并就其中射线追踪这一重要环节,利用辛算法和非辛Ｒｕｎｇｅ－Ｋｕｔｔａ方法进行了数值模拟．计算结果表明,这两类算法在精度上相差无几,但辛算法的速度要快;在Ｈａｍｉｌｔｏｎ量保持方面,辛算法具有非辛Ｒｕｎｇｅ－Ｋｕｔｔａ方法无可比拟的优越性． 相似文献
10.
Traveltime computation by wavefront-orientated ray tracing 总被引：1，自引：0，他引：1
For multivalued traveltime computation on dense grids, we propose a wavefront‐orientated ray‐tracing (WRT) technique. At the source, we start with a few rays which are propagated stepwise through a smooth two‐dimensional (2D) velocity model. The ray field is examined at wavefronts and a new ray might be inserted between two adjacent rays if one of the following criteria is satisfied: (1) the distance between the two rays is larger than a predefined threshold; (2) the difference in wavefront curvature between the rays is larger than a predefined threshold; (3) the adjacent rays intersect. The last two criteria may lead to oversampling by rays in caustic regions. To avoid this oversampling, we do not insert a ray if the distance between adjacent rays is smaller than a predefined threshold. We insert the new ray by tracing it from the source. This approach leads to an improved accuracy compared with the insertion of a new ray by interpolation, which is the method usually applied in wavefront construction. The traveltimes computed along the rays are used for the estimation of traveltimes on a rectangular grid. This estimation is carried out within a region bounded by adjacent wavefronts and rays. As for the insertion criterion, we consider the wavefront curvature and extrapolate the traveltimes, up to the second order, from the intersection points between rays and wavefronts to a gridpoint. The extrapolated values are weighted with respect to the distances to wavefronts and rays. Because dynamic ray tracing is not applied, we approximate the wavefront curvature at a given point using the slowness vector at this point and an adjacent point on the same wavefront. The efficiency of the WRT technique is strongly dependent on the input parameters which control the wavefront and ray densities. On the basis of traveltimes computed in a smoothed Marmousi model, we analyse these dependences and suggest some rules for a correct choice of input parameters. With suitable input parameters, the WRT technique allows an accurate traveltime computation using a small number of rays and wavefronts. 相似文献
11.
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。 相似文献
12.
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. 相似文献
13.
R. L. Nowack 《Pure and Applied Geophysics》2003,160(3-4):487-507
— In this paper, an overview of the calculation of synthetic seismograms using the Gaussian beam method is presented accompanied by some representative applications and new extensions of the method. Since caustics are a frequent occurrence in seismic wave propagation, modifications to ray theory are often necessary. In the Gaussian beam method, a summation of paraxial Gaussian beams is used to describe the propagation of high-frequency wave fields in smoothly varying inhomogeneous media. Since the beam components are always nonsingular, the method provides stable results over a range of beam parameters. The method has been shown, however, to perform better for some problems when different combinations of beam parameters are used. Nonetheless, with a better understanding of the method as well as new extensions, the summation of Gaussian beams will continue to be a useful tool for the modeling of high-frequency seismic waves in heterogeneous media. 相似文献
14.
In downhole microseismic monitoring, accurate event location relies on the accuracy of the velocity model. The model can be estimated along with event locations. Anisotropic models are important to get accurate event locations. Taking anisotropy into account makes it possible to use additional data – two S-wave arrivals generated due to shear-wave splitting. However, anisotropic ray tracing requires iterative procedures for computing group velocities, which may become unstable around caustics. As a result, anisotropic kinematic inversion may become time consuming. In this paper, we explore the idea of using simplified ray tracing to locate events and estimate medium parameters. In the simplified ray-tracing algorithm, the group velocity is assumed to be equal to phase velocity in both magnitude and direction. This assumption makes the ray-tracing algorithm five times faster compared to ray tracing based on exact equations. We present a set of tests showing that given perforation-shot data, one can use inversion based on simplified ray-tracing even for moderate-to-strong anisotropic models. When there are no perforation shots, event-location errors may become too large for moderately anisotropic media. 相似文献
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合成三维横向非均匀介质远震体波接收函数的Maslov方法 总被引：5，自引：3，他引：2
三维横向非均匀介质远震体波接收函数的合成算法对于三维介质远震体波接收函数研究至关重要．本文发展了基于Ｍａｓｌｏｖ理论的横向非均匀介质接收函数合成算法,针对远震体波接收函数计算的特点,利用遗传算法完成三维动力学射线追踪,并采用递归算法组织射线的追踪过程,从而在保证追踪精度的同时提高了射线追踪的效率,并可完全避免传统射线追踪过程中的射线编码问题．正演计算结果表明,此法是一种高效、实用的三维横向非均匀介质接收函数计算方法． 相似文献
17.
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. 相似文献
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Prestack depth migration is a key technology for imaging complex reservoirs in media with strong lateral velocity variations. Prestack migrations are broadly separated into ray-based and wave-equation-based methods. Because of its efficiency and flexibility, ray-based Kirchhoff migration is popular in the industry. However, it has difficulties in dealing with the multi-arrivals, caustics and shadow zones. On the other hand, wave-equation-based methods produce images superior to that of the ray-based methods, but they are expensive numerically, especially methods based on two-way propagators in imaging large regions. Therefore, reverse time migration algorithms with Gaussian beams have recently been proposed to reduce the cost, as they combine the high computational efficiency of Gaussian beam migration and the high accuracy of reverse time migration. However, this method was based on the assumption that the subsurface is isotropic. As the acquired azimuth and maximum offsets increase, taking into account the influence of anisotropy on seismic migration is becoming more and more crucial. Using anisotropic ray tracing systems in terms of phase velocity, we proposed an anisotropic reverse time migration using the Gaussian beams method. We consider the influence of anisotropy on the propagation direction and calculate the amplitude of Gaussian beams with optimized correlation coefficients in dynamic ray tracing, which simplifies the calculations and improves the applicability of the proposed method. Numerical tests on anisotropic models demonstrate the efficiency and accuracy of the proposed method, which can be used to image complex structures in the presence of anisotropy in the overburden. 相似文献
20.
对复杂山地介质的非均质性以及介质中地震波运动学特征进行深入研究,对于提高复杂山地区域地震勘探的效果有着重要的理论意义和实际价值.为了研究复杂山地非均质性和该介质中地震波的一些运动特性,提出了一种复杂山地随机介质的建模方法和一种新的射线追踪算法.与常规算法相比,复杂山地随机介质的生成方法采用更贴近实际介质特点的梯度介质作为背景介质,并在模型生成过程中加入地形修正步骤;新提出的GMM-ULTI射线追踪算法,充分融合群推进法、迎风思想、走时插值法的优势,采用先计算走时后追踪射线路径的两步策略完成射线追踪.算法分析与计算实例表明:复杂山地随机介质的生成方法能灵活、精细且更贴近实际地刻画复杂山地介质的非均质特点;新射线追踪算法兼顾精度和效率、能无条件稳定且灵活地适应复杂山地随机介质的特点;同时基于对几个模型试算结果的分析也得出了复杂山地随机介质中的地震波的一些传播规律. 相似文献