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.     

Common-ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium

Anisotropic common S-wave rays are traced using the averaged Hamiltonian of both S-wave polarizations. They represent very practical reference rays for calculating S waves by means of the coupling ray theory. They eliminate problems with anisotropic-ray-theory ray tracing through some S-wave slowness-surface singularities and also considerably simplify the numerical algorithm of the coupling ray theory for S waves. The equations required for anisotropic-common-ray tracing for S waves in a smooth elastic anisotropic medium, and for corresponding dynamic ray tracing in Cartesian or ray-centred coordinates, are presented. The equations, for the most part generally known, are summarized in a form which represents a complete algorithm suitable for coding and numerical applications.     

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.     

椭球坐标系下多震相地震射线追踪

地震射线追踪方法技术在地震学领域有着较为广泛的应用,然而大多数算法建立在直角坐标系或球坐标系下,实际地球并非完美的球体,而是两极略扁的椭球体,因此,球坐标系下计算结果与真实情况存在一定误差.传统的做法一般是在球坐标系下进行计算,而后进行椭球校正.本文提出了一种直接在椭球体模型中采用分区多步最短路径算法进行多震相地震射线追踪的方法技术,实现了椭球坐标系下多震相地震波射线路径追踪和走时计算.与解析解的对比表明：该算法具有较高的计算精度,适用于任意形状的椭球体,且不需要进行额外的走时校正.数值模拟结果表明,计算所得P波和PcP反射波的走时与AK135走时表的误差小于0.1 s.当震中距较大时,使用球对称模型和椭球体模型计算所得的走时差异显著,说明采用椭球坐标系的必要性.

     

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.     

Indirect-approach method for specified-end points seismic ray tracing in three dimensional inhomogeneous media

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Transient analysis of 1D inhomogeneous media by dynamic inhomogeneous finite element method

The dynamic inhomogeneous finite element method is studied for use in the transient analysis of one dimensional inhomogeneous media. The general formula of the inhomogeneous consistent mass matrix is established based on the shape function. In order to research the advantages of this method, it is compared with the general finite element method. A linear bar element is chosen for the discretization tests of material parameters with two fictitious distributions. And, a numerical example is solved to observe the differences in the results between these two methods. Some characteristics of the dynamic inhomogeneous finite element method that demonstrate its advantages are obtained through comparison with the general finite element method. It is found that the method can be used to solve elastic wave motion problems with a large element scale and a large number of iteration steps.     

On dynamic ray tracing in three dimensional inhomogeneous media

Summary Among the various dynamic ray tracing systems described by ervený and Hron[1] is one particular linear system of second order that readily provides identical parameters (in the ray centred coordinate system) to those that fall out of the system by Popov and Peník[3, 4]. Hence the initial conditions of the latter system for sources and interfaces can easily be used to provide those for the linear system of second order.     

复杂地质层面的网格化模型及两点射线追踪方法

本文针对三维复杂介质的两点射线追踪的适用性和有效性,提出了网格模型和层状模型相结合的建模方式,文中利用规则网格化的层状模型对复杂地质单元进行数值描述,一方面避免了用纯粹复杂地质网格体描述时需要的大量内存,另一方面结合运用规则化的层状模型,必有利于提高计算效率,在计算单元自动生成的射线追踪方法基础上,本文结合波前法提出一种快速、有效的两点射线追踪方法,利用反射波波前在地表形成的射线方位角和倾角坐标体系,能快速、准确地收敛到所有从源点到接收点的射线路径,亦可有效地解决射线路径的多值解难题,从而可适用于复杂地质模型的特点.     

基于波前面三角形网格剖分的波前重建法三维射线追踪

在许多地震反演和偏移成像方法中,都要涉及到射线路径和旅行时的计算.本文将波前面三角形网格剖分和三维波前重建法射线追踪技术结合使用,实现了射线路径和旅行时的准确快速计算.三维波前重建法射线追踪过程中可以保证稳定合理的射线密度,克服了常规射线追踪方法存在阴影区的问题.波前面三角形网格剖分在描述和拆分波前面时更加准确有效,而且不需太多的网格数目,从而提高了射线追踪的精度和效率.该方法在三维复杂构造成像方面有独特的优势,目前在实际的Kirchhoff 偏移中的已经有相关应用.     

Two-point ray tracing in 3-D

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

最短路径算法下二维层状介质中多次波追踪

在改进后的最短路径算法（MSPM）中引入分区多步计算技术实现了二维层状起伏介质中的多次透射、反射及转换波波前传播的数值模拟,以及相应的走时和射线路径的跟踪计算.其原理是将二维复杂层状模型按速度界面分成若干个独立的计算区域,采用分步计算技术进行多次波的跟踪计算.基于多次波是通过速度界面的简单入射、透射、反射及转换波按一定规律的不同组合,因此可实施分区多步计算技术.通过某一上、下层界面的透射（或透射转换）波实际上是由上层得到的下行波加上由该界面透射的下行波组成,若为转换波则使用不同的速度模型;而经过某一界面的反射（或反射转换）波实际上是由某层内计算得到的下行（或上行）波再加上由该界面反射的上行（或下行）波组成.这样即可得到分区独立计算,并通过速度界面分步连接达到跟踪多次波的目的.计算结果表明MSPM算法下的分区多步计算技术具有单步SPM算法中的诸多优点,即:算法简单、数值计算稳键、计算精度高、速度快及全球解等,因而是解决多次波跟踪计算行之有效的方法.     

Quasi-shear wave ray tracing by wavefront construction in 3-D, anisotropic media

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.     

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.     

最短路径算法下三维层状介质中多次波追踪

本文使用改进后的最短路径算法（MSPM）结合分区多步计算技术实现了三维复杂层状起伏介质中的多次透射、反射及转换波波前传播的数值模拟,以及相应走时和射线路径的跟踪计算.其原理是将三维复杂层状模型按速度界面分成若干个独立的计算区域,采用分步计算技术进行多次波的跟踪计算.基于多次波是通过速度界面简单的入射、透射、反射及转换波按一定规律及原理的不同组合,因此可实施分区多步计算技术.数值模拟实例及误差分析表明分区多步计算技术具有单步最短路径算法中的诸多优点:算法简单、数值计算稳健、计算精度高、速度快及全球解等,因此是解决多次波跟踪计算行之有效的方法.     

Nonlinear ray perturbation theory with its applications to ray tracing and inversion in anisotropic media

A comprehensive approach, based on the general nonlinear ray perturbation theory (Druzhinin, 1991), is proposed for both a fast and accurate uniform asymptotic solution of forward and inverse kinematic problems in anisotropic media. It has been developed to modify the standard ray linearization procedures when they become inconsistent, by providing a predictable truncation error of ray perturbation series. The theoretical background consists in a set of recurrent expressions for the perturbations of all orders for calculating approximately the body wave phase and group velocities, polarization, travel times, ray trajectories, paraxial rays and also the slowness vectors or reflected/transmitted waves in terms of elastic tensor perturbations. We assume that any elastic medium can be used as an unperturbed medium. A total 2-D numerical testing of these expressions has been established within the transverse isotropy to verify the accuracy and convergence of perturbation series when the elastic constants are perturbed. Seismological applications to determine crack-induced anisotropy parameters on VSP travel times for the different wave types in homogeneous and horizontally layered, transversally isotropic and orthorhombic structures are also presented. A number of numerical tests shows that this method is in general stable with respect to the choice of the reference model and the errors in the input data. A proof of uniqueness is provided by an interactive analysis of the sensitivity functions, which are also used for choosing optimum source/receiver locations. Finally, software has been developed for a desktop computer and applied to interpreting specific real VSP observations as well as explaining the results of physical modelling for a 3-D crack model with the estimation of crack parameters.     

Gaussian beams in inhomogeneous media: A review

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

A rapid and accurate two-point ray tracing method in horizontally layered velocity model

Introduction The determination of the ray path between a source and a receiver is fundamental to many seismic problems such as earthquake location, travel time inversion, computation of synthetic seismograms, depth migration, and seismic tomography. The calculation of travel times and their derivatives depend on the determination of ray paths, which is the most time-consuming work in seismic tomography. Therefore, a fast and accurate ray tracing method is particularly important. Many ray-traci…     

一种最短路径射线追踪的快速算法

为提高最短路径射线追踪的精度,需要增加模型的剖分网格和离散节点,并增加子波传播方向,或者采用其他方法改善计算结果,这些处理会带来大量的额外计算.本文的快速算法改进了波前点的管理和子波传播的计算这两项耗时的工作,较大幅度地提高了传统算法的效率.在波前点的管理上,采用按时间步划分区间的方法,实现了波前点的桶排序管理,其效率高于传统方法中常用的堆排序算法. 在子波传播的计算上,利用斯奈尔定律,同时参考来自邻近节点的波的走时,来限定当前子波传播的有效区域,排除大量不需要计算的子波传播方向. 模型实算表明,本文快速算法的计算速度是传统方法的几倍至十多倍.     

弯曲射线追踪中Dijkstra算法的改进与实现

文章针对图论中寻找最短路径的Dijkstra算法内存占用量大,效率低的缺点,对该算法进行了改进,修改后的算法计算效率是原来的四倍,内存使用量和图中节点数呈线性关系.在此基础上,用新算法求出了激发点和接收点的最短走时路径,并由激发接收点的旅行时结合联合迭代法对理论模型和实际场地进行了反演.结果表明：和直射线追踪相比,弯曲射线路径能更好地反演出地质体内部的速度场分布.理论模型和实际探测结果证实改进后的算法是有效的.