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.     

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.     

二维复杂黏弹性各向异性介质中的地震射线追踪

实际地层中传播的地震波普遍存在速度各向异性和能量衰减现象,因此基于弹性介质假设条件下的地震射线正反演算法具有一定的局限性,而研究黏弹性各向异性介质中的地震波传播规律可为揭示地下结构提供更加可靠的理论依据.射线追踪技术是揭示高频地震波传播特性的有效手段之一,然而绝大多数研究仅限于弹性介质.针对黏弹性各向异性介质,本文首先给出了一种射线速度和振幅衰减的计算方法,然后结合改进型最短路径算法,在实空间内实现了计算复杂介质模型中地震波(qP,qSV,qSH)的射线路径和传播走时以及能量衰减(虚走时).该算法适用于复杂黏弹性各向异性介质.误差分析结果显示,实走时的最大相对误差小于0.13％,虚走时的最大相对误差小于0.55％,表明该算法具有较高的计算精度.     

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.     

多步法快速射线追踪与圆台型高精度走时外插计算

为更好地适应复杂构造的地震偏移成像,本文提出了一套快速射线追踪算法和一种高精度的走时外插计算方法.采用线性多步法的预测-校正公式求解射线追踪方程组,与传统的四阶Runge-Kutta法相比,提高了计算效率.在网格节点上的走时计算中,应用一种基于圆台的外插方法,该方法以射线的方向为轴确定圆台,将轴上的走时外插到圆台内的网格节点上.与传统的矩形体外插方法相比,圆台走时外插方法提高了计算精度,且具有更好的稳定性.另外,该方法利用稀疏分布的射线即可获得高精度的走时表,节省计算量,对复杂构造的偏移成像非常有利,尤其是三维偏移.最后通过逆散射偏移成像算例,验证了算法的有效性和适用性.

     

Modeling,ray tracing,and block nonlinear travel-time inversion in 3D

We describe an integrated forward and inverse three-dimensional modeling system that can deal with complex geological structures. The system has been designed to handle large-scale problems by using a distributed approach. It uses seismic ray tracing for forward simulation, time-to-depth mapping, and nonlinear travel-time inversion.A novel decomposition method is our tool to attack large-scale problems in a parallel approach. The system is fully implemented and we demonstrate its performance with synthetic examples.This research was partially supported by the National Science Foundation under SBIR Grant III-9300992.     

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.     

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     

基于HAFMM的无射线追踪跨孔雷达走时层析成像

本文使用最小二乘线性迭代反演方法对跨孔雷达直达波初至时数据进行反演,每次迭代过程中,用有限差分法求解走时程函方程,并用高精度快速推进方法(HAFMM)进行波前扩展,通过追踪波前避免了进行射线追踪.为了验证该方案,我们对三组合成数据进行了测试,分析了单位矩阵算子、一阶差分算子和拉普拉斯算子等三种不同模型参数加权算子对模型的约束和平滑效果;讨论了FMM和HAFMM对反演精度的影响;测试了LSQR,GMRES和BICGSTAB等三种矩阵反演算法的反演效果.此外,我们还对一组野外实测数据进行了反演,对比了基于本方案以及基于平直射线追踪和弯曲射线追踪的走时层析成像反演效果.对比分析结果表明,使用拉普拉斯算子和HAFMM进行反演能较好地进行目标体重建,而三种矩阵反演方法对反演效果的影响差别不大;并且通过对波前等时线图的分析可以定性地判断异常体的性质和位置;而在对实测数据目标体的重建上,本方案能达到甚至优于弯曲射线算法的重建效果.     

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.     

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

ＩｎｄｉｒｅｃｔａｐｐｒｏａｃｈｍｅｔｈｏｄｆｏｒｓｐｅｃｉｆｉｅｄｅｎｄｐｏｉｎｔｓｓｅｉｓｍｉｃｒａｙｔｒａｃｉｎｇｉｎｔｈｒｅｄｉｍｅｎｓｉｏｎａｌｉｎｈｏｍｏｇｅｎｅｏｕｓｍｅｄｉａＣＨＡＯＦＡＮＸＵ（徐朝繁）ＸＩＡＮＫＡＮＧＺＨ...     

Reverse time migration with Gaussian beams using optimized ray tracing systems in transversely isotropic media

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.     

A fast and robust two-point ray tracing method in layered media with constant or linearly varying layer velocity

A fast and robust method for two-point ray tracing in one-dimensional layered media is presented. This method is applicable to layered models with constant or linearly varying isotropic layer velocity. For given model properties and source and receiver positions, a ray path can be uniquely determined once its ray parameter (i.e. horizontal slowness) is known. The ray parameter can be obtained by numerically solving the nonlinear offset (i.e. source–receiver horizontal distance) equation using Newton's method, which generally works well at near and mid offsets. However, Newton's method becomes hard to converge at large offsets due to the oversensitivity of offset to ray parameter. Based on the analysis of the characteristic of the offset equation, a modified ray parameter is proposed and used to replace the generic ray parameter in numerical calculation. Numerical experiments show that the iteration process becomes stable and converges rapidly with the modified ray parameter. Moreover, a rational function that asymptotically approximates the shape of the offset equation is introduced for obtaining good initial estimates of the modified ray parameter. Numerical tests show that this method is robust in any situation, and an accurate ray parameter can be obtained within two or three iterations for a wide range of model velocity structure and source–receiver distance. Furthermore, the proposed two-point ray tracing method is easy to implement.     

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.     

Reverse tracing of precursors and earthquake precursors in Taiwan

In this paper, we investigate the precursors of large earthquakes in the eastern region of Taiwan by means of the reverse tracing of precursors. We discuss the parameters which are suitable for the seismic chains and intermedi- ate-term patterns in this region and obtain the threshold of the patterns. Applying the linear discriminate method to the intermediate-term patterns of seismic chains, we present an approach for exploring the precursors of large earthquakes. The results show that this method can reduce the false alarm rate for large earthquakes in this region, and the reverse tracing of precursors can be applied to the eastern region of Taiwan.     

Computation of ray integrals and ray amplitudes in radially symmetric media

Summary A new approximation of the velocity-depth distribution in radially symmetric media is suggested. This approximation guarantees the continuity of velocity and its first and second derivatives, and does not generate false low-velocity layers. It removes false anomalies from the amplitude-distance curve and considerably increases its stability. The evaluation of ray integrals and ray amplitudes using this velocity-depth approximation does not require the computation of any transcendental function and is, therefore, very fast. Numerical examples are presented.     

地震作用下多层砌块房屋墙体倾覆力矩的计算方法

多层混凝土砌块房屋抗震设计时，仅考虑地震引起的剪力作用和整体房屋的抗倾覆性能是不全面的。为此，提出了多层砌块房屋墙体倾覆力矩的计算与分配方法，通过典型砌块房屋的算例，利用本方法得到的墙体倾覆力矩与有限元结构分析软件SATWE的结果相近，证明了该方法的实用性，为多层砌块房屋抗震分析提供了有效手段。     

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.     

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.     

利用射线追踪研究电离层扰动

基于返回散射探测的电离层扰动电离图,本文建立了一个新的对称的准余弦电离层扰动数学模型.基于这种模型,利用HF射线追踪技术合成了HF电离层返回散射电离图,并利用迭代的技术拟合了高频返回散射设备探测的电离层扰动电离图的Pmin-f曲线(Pmin为最小时延,f为工作频率),从而推断了沿探测路径电离层扰动区域的位置及大小,扰动的临界频率波动的幅度.最后基于这种电离层扰动模型,利用射线追踪技术描述了不同电离层扰动参数下的电波传播情况,研究了其天波传播的跳距、覆盖区域的大小及射线“俘获”等.