各向同性薄层反射理论地震图

本文在各向同性介质假设下,计算了薄层顶、底反射P波和PS波的理论地震图.理论模拟发现薄层的反射P波与PS波是一复合波,包括层内的多次透射和反射转换波型,且具有类似单一界面反射的、脉冲式的波形特征.两类薄层反射整体振幅随着薄层厚度的降低而缩小;在地震子波主频40 Hz条件下,2 m左右厚度的薄层反射相比单一界面反射具有等同的振幅水平.1 m以下极薄层仅有弱反射甚至无反射;薄层反射复合波振幅与炮检距的关系依然成立,但不惟一;单界面反射AVO原理、方法不适用于薄层反射解释与反演.     

Point-to-curve ray tracing

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.     

Theoretical ray seismograms of short periodPKP waves

Summary This paper shows some examples of theoretical seismograms of short period PKP waves calculated by the zero approximation of the ray theory. The influence of the epicentral distance, source time function and the Earth's crust on the form of the seismograms is shown.     

Point-to-curve ray tracing in complicated geological models1

Boundary-value problems (BVPs) for seismic rays generally have multiple solutions. In practical applications the number of solutions can be large. The algorithm presented below solves a one-parameter family of BVPs and makes it easy to obtain all the solutions of a BVP.     

Synthetic seismograms in heterogeneous media by one-return approximation

When reverberations between heterogeneities or resonance scattering can be neglected but accumulated effects of forward scattering are strong, the Born approximation is not valid but the De Wolf approximation can be applied in such cases. In this paper, renormalized MFSB (multiple-forescattering single-backscattering) equations and the dual-domain expression for scalar, acoustic and elastic waves are derived by a unified approach. Two versions of the one-return method (using MFSB approximation) are given: One is the wide-angle dual-domain formulation (thin-slab approximation); the other is the screen approximation. In the screen approximation, which involves a small-angle approximation for the wave-medium interaction, it can be seen clearly that the forward scattered, or transmitted waves are mainly controlled by velocity perturbations; while the backscattered or reflected waves, by impedance perturbations. The validity of the method and the wide-angle capability of the dual-domain implementation are demonstrated by numerical examples. Reflection coefficients of a plane interface derived from numerical simulations by the wide-angle method match the theoretical curves well up to critical angles. For the reflections of a low-velocity slab, the agreement between theory and synthetics only starts to deteriorate for angles greater than 70°. The accuracy of the wide-angle version of the method could be further improved by optimizing the wave-number filtering for the forward propagation and shrinking the step length along the propagation direction.     

Construction of ray synthetic seismograms using interpolation of travel times and ray amplitudes

The seismic wave field, in its high-frequency asymptotic approximation, can be interpolated from a low- to a high-resolution spatial grid of receivers and, possibly, point sources by interpolating the eikonal (travel time) and the amplitude. These quantities can be considered as functions of position only. The travel time and the amplitude are assumed to vary in space only slowly, otherwise the validity conditions of the theory behind would be violated. Relatively coarse spatial sampling is then usually sufficient to obtain their reasonable interpolation. The interpolation is performed in 2-D models of different complexity. The interpolation geometry is either 1-D, 2-D, or 3-D according to the source-receiver distribution. Several interpolation methods are applied: the Fourier interpolation based on the sampling theorem, the linear interpolation, and the interpolation by means of the paraxial approximation. These techniques, based on completely different concepts, are tested by comparing their results with a reference ray-theory solution computed for gathers and grids with fine sampling. The paraxial method holds up as the most efficient and accurate in evaluating travel times from all investigated techniques. However, it is not suitable for approximation of amplitudes, for which the linear interpolation has proved to be universal and accurate enough to provide results acceptable for many seismological applications.     

矩形网格三点Fermat射线追踪技术

矩形网格三点Fermat射线追踪法是基于矩形网格三点扰动法的一种提高计算速度的方法．取矩形网格三个点,在Fermat最小旅行时原则下求取扰动中间点的位置,而不象扰动法那样依次扰动．因此,计算速度比扰动法提高2倍多,同时不受扰动摄动量大小选择的困扰．该方法继承了矩形网格三点扰动法的优点,对任意离散的速度场,总能找到最短时间路径,避免了射线盲区和追踪路径并非时间最短路径问题．     

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.     

Traveltime computation by wavefront-orientated ray tracing

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.     

Fast computation of ray synthetic seismograms in vertically inhomogeneous media

Summary A procedure of fast computation of body-wave ray synthetic seismograms in vertically inhomogeneous media is suggested. The procedure uses a special approximation of the velocitydepth distribution which guarantees continuity of the first and second derivatives of velocity and does not generate false low-velocity layers (oscillations in the velocity-depth function). The ZESY82 program package, which is based on the suggested procedure, is described. The point source with an arbitrary radiation pattern may be situated at any points of the model, the receivers are situated regularly or irregularly along any profile on the Earth's surface, containing the epicentre. Numerical examples of the synthetic record sections for a model of the Earth's crust and the uppermost mantle are given.

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Synthetic seismograms ofP-waves propagating in solid wedges with free boundaries

A numerical method for the synthesis of seismograms for body wave propagation in solid wedges is presented. The method is based on the superposition of multiple reflections arising from the entrance of a plane primary wave. Therefore the method is restricted to those domains in time-space where no diffracted waves from the wedge axis occur. In spite of this restriction, the dispersion of body waves in wedges can well be studied by this method. Seismograms have been synthesized which show the dispersion of a primary,p-signal propagating in a solid 10°- and a 5°-wedge with free boundaries. For wedge anlges less than 10° the signal from (as distinguished from the wave front) suddently decreases its velocity from that in the infinite medium to about that of the plate wave as the signal approaches the wedge axis. A decrease of the dominant period of the interference signal occurs simultaneously in this transition zone. These observations are concordant with previous seismic model studies [1]. Particle motion diagrams disclose elliptical polarization of the interference signal in the neighboorhood of the wedge axis which changes its sense from prograde to retrograde on passing through the transition zone. This paper has been submitted for publication to Geophysics. It will appear, in Vol. 31 (1966).     

最短路径射线追踪方法及其改进

综述了用网络最短路径算法求解地震射线追踪问题的原理、方法技术以及存在问题和改进措施。特别介绍了作者在最短路径算法基础上，提出的动态网络最短路径地震射线追踪方法。该方法先采集从炮点到整个模型所有节点上的初至旅行时，其中，在一个单元内，对相邻每对已计算出最小旅行时的节点进行线性插值，并利用Fermat原理计算未知节点的最小旅行时；然后，利用同样的方法，从接收点开始，反向追踪炮点到接收点的射线路径、该方法能适于各种复杂的非均匀介质，极大地提高了射线追踪的精度。     

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.

     

合成地震图的广义反射透射系数矩阵方法

简述了在均匀层状介质中，合成地震图的广义反射透射系数矩阵方法，并通过与有人的计算结果对比，检验了经过修改调试后的合成地震图的计算程序。计算与对比表明，经修改调试后的程序对于各种因素，包括震源类型、震源深度、震中、频率和慢度都有较好的普适性，适用于实际的地震波形研究。     

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.     

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.     

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.     

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.     

横向各向同性(VTI)介质中非线性地震波场模拟

数值计算方法是考察非线性弹性波在介质中（如岩石）传播特征的重要手段.非线性弹性波的数值模拟存在陡峭间断面（点）、数值振荡以及误差的指数级增长等现象而破坏数值解的稳定性、收敛性,能否消减上述现象的不利影响成为制约数值方法有效与否的重要因素.文中同时引入了FCT算子和幅值限制器,采用中心差分格式对具有垂直对称轴的横向各向同性（VTI）介质中的二维非线性弹性波进行数值模拟,从而克服了上述困难;介绍了适用于非线性弹性波的吸收边界条件,给出了差分方程的稳定性条件.在验证了方法的有效性后成功地获取了二维VTI介质中非线性弹性波的三分量地震正演记录,表明非线性波在传播过程中会发生波形畸变等现象.