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.     

TI介质局部角度域高斯束叠前深度偏移成像

各向异性射线理论基础上的局部角度域叠前深度偏移方法能够为深度域构造成像与基于角道集的层析反演提供有力支撑,但是对于复杂地质构造而言,高斯度叠前深度偏移在不失高效、灵活等特点的情况下,具有明显的精度优势.为此,本文研究局部角度域理论框架下的高斯束叠前深度偏移方法.为提高算法效率与实用性,文中讨论了一种从经典弹性参数表征的各向异性介质运动学和动力学射线方程演变而来的由相速度表征的简便形式,并提出了一种比较经济的各向异性高斯束近似合成方案.结合地震波局部角度域成像原理,讨论一种适合高斯束偏移的角度参数计算方法.国际上通用的理论模型合成数据试验表明：相比局部角度域Kirchhoff叠前深度偏移成像方法,本文方法具有更高的成像精度与抗噪能力,既适用于复杂构造成像,也可为TI介质深度域偏移速度分析与模型建立提供高效的偏移引擎.     

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.     

快速射线追踪及其在地震层析中的应用

在地震层析研究中,常常涉及地震波在非均匀介质中的传播路径问题,故快速射线追踪是不可缺少的工具。本文讨论了一种以三点法为基础的逐步逼近的射线追踪方法,其优点是计算速度快、精度高,适合地震层析研究的计算要求。     

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

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A note on dynamic ray tracing in ray-centered coordinates in anisotropic inhomogeneous media

Dynamic ray tracing plays an important role in paraxial ray methods. In this paper, dynamic ray tracing systems for inhomogeneous anisotropic media, consisting of four linear ordinary differential equations of the first order along the reference ray, are studied. The main attention is devoted to systems expressed in a particularly simple choice of ray-centered coordinates, here referred to as the standard ray-centered coordinates, and in wavefront orthonormal coordinates. These two systems, known from the literature, were derived independently and were given in different forms. In this paper it is proved that both systems are fully equivalent. Consequently, the dynamic ray tracing system, consisting of four equations in wavefront orthonormal coordinates, can also be used if we work in ray-centered coordinates, and vice versa. vcerveny@seis.karlov.mff.cuni.cz     

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.     

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.     

一种线性走时插值射线追踪改进算法

针对线性走时插值算法(LTI)不能正确追踪逆向传播射线的问题, 目前已提出多种改进算法, 如扩张收缩LTI算法、 循环计算LTI算法、 动态网络最短路径射线追踪算法等, 但这些算法的计算效率普遍偏低. 在分析各种改进LTI算法的优劣后, 本文提出了改进动态网络最短路径射线追踪算法. 该改进算法依据波的传播规律以及LTI算法的基本方程, 排除动态网络最短路径射线追踪算法中大量冗余节点计算, 并采用传统的二叉树堆排序算法对波前阵列节点进行管理. 数值算例表明, 本文提出的改进算法具有较高的计算效率, 其计算效率是动态网络最短路径射线追踪算法的4.5—30倍, 是原始LTI算法的2—6.5倍; 当动态网络最短路径射线追踪算法采用堆排序算法时, 改进算法的计算效率是其3.5—15倍.      

Kirchhoff Helmholtz积分方法计算远震转换波

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。     

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.     

射线法模拟分析井间地震观测的波场特征

按照井间地震的观测系统,用改进的突变点加插值射线追踪方法,追踪每炮每道的射线路径,计算几种主要类型的波沿射线路径的波至时间和射线振幅,制作井间地震多炮多道水平分量和垂直分量的合成记录.并将合成记录选排为井间共炮点道集、共接收点道集、共偏移距道集和共中心深度点道集,系统地分析了不同道集内几种主要类型的地震波的传播特征.对野外观测的实际井间地震记录进行了模拟,从复杂的井间地震记录中,识别出井间地震实际观测到的不同类型的波场,为随后的井间地震资料处理和应用提供了依据.     

起伏地表条件下各向异性地震波最短路径射线追踪

在地震波正反演研究中,考虑起伏地表和地震各向异性具有非常重要的理论意义和实际应用价值.本文在前人研究的基础上,将最短路径追踪算法引入到起伏地表各向异性介质模型的地震波走时计算中.模型剖分时,整体模型划分成正方形单元,起伏边界附近以不规则网格逼近,进而采用非规则节点布置实现非规则网格处的最短路径计算.追踪计算中采用Sena群速度近似公式,得到各向异性地震波的走时,实现了复杂地表情况下各向异性介质模型中地震波的射线追踪.理论模型计算结果显示,本文方法能够可靠地应用于复杂各向异性介质模型,具有较高的计算精度.     

基于射线追踪的微地震多波场正演模拟

对于微地震正演模拟,本文以射线追踪的原理为基础,对两点间的射线追踪问题进行了研究,应用二分算法、改进二分算法和微变网格算法对水平层状匀速模型、弯曲层状匀速模型和复杂地质模型进行射线追踪,使得计算效率和适用范围都得到了很大的改善.文中对每种算法误差范围和计算效率进行了对比验证,对于不同的地质模型,选用合适的算法才能在计算速度和精度上得到双重保证,最后正演模拟了多波三分量记录.在模型建立上引入了超薄层概念,并在前人模拟的直达波、透射波、反射波基础上拟了折射波,使正演模拟的多波场信息更丰富.文中的应用实例及模型结果表明:与二分法相比,改进二分法能够对弯曲界面进行射线追踪,并能保证结果的精度.弯曲层状模型中,改进二分法与微变网格法相比计算速度有显著提高,能够应用到资料的反演中.     

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.     

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.     

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…     

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.     

迭代法在三维网络模型最短路径射线追踪中的应用

目前有许多基于二维网格模型的射线追踪算法,但应用与三维网格模型的射线追踪算法较少,主要是因为三维网格射线追踪算法在算法实现、编程以及图形显示方面都有一定难度.本文提出了一种适用于三维网格射线追踪的迭代算法,并应用于几种理论模型,同时实现了三维绘图显示,通过研究证明迭代算法能在一定程度上提高三维网格射线追踪的精度和速度.     

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.