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1.
The objective is to provide, in one single paper, a complete collection of equations governing kinematic and dynamic ray tracing related to a symmetry plane of an anisotropic medium. Well known systems for kinematic ray tracing and in-plane dynamic ray tracing are reformulated for the purpose of clarity, by taking advantage of a vector representation of the Christoffel matrix elements and related quantities. A generalized formula is derived for the integrand in out-of-plane dynamic ray tracing, pertaining to a monoclinic medium. Integrands corresponding to non-tilted orthorhombic and transversely isotropic media are obtained as special cases. 相似文献
2.
P. M. Bakker 《Pure and Applied Geophysics》1996,148(3-4):583-589
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. 相似文献
3.
Conventional ray tracing for arbitrarily anisotropic and heterogeneous media is expressed in terms of 21 elastic moduli belonging
to a fixed, global, Cartesian coordinate system. Our principle objective is to obtain a new ray-tracing formulation, which
takes advantage of the fact that the number of independent elastic moduli is often less than 21, and that the anisotropy thus
has a simpler nature locally, as is the case for transversely isotropic and orthorhombic media. We have expressed material
properties and ray-tracing quantities (e.g., ray-velocity and slowness vectors) in a local anisotropy coordinate system with
axes changing directions continuously within the model. In this manner, ray tracing is formulated in terms of the minimum
number of required elastic parameters, e.g., four and nine parameters for P-wave propagation in transversely isotropic and
orthorhombic media, plus a number of parameters specifying the rotation matrix connecting local and global coordinates. In
particular, we parameterize this rotation matrix by one, two, or three Euler angles. In the ray-tracing equations, the slowness
vector differentiated with respect to traveltime is related explicitly to the corresponding differentiated slowness vector
for non-varying rotation and the cross product of the ray-velocity and slowness vectors. Our formulation is advantageous with
respect to user-friendliness, efficiency, and memory usage. Another important aspect is that the anisotropic symmetry properties
are conserved when material properties are determined in arbitrary points by linear interpolation, spline function evaluation,
or by other means. 相似文献
4.
Elastic wave propagation in inhomogeneous anisotropic media 总被引：1，自引：0，他引：1
XIU CHENG WEI ) MIN YU DONG ) YUN YAI CHEN ) ) Geoscience Department University of Petroleum Beijing China ) Institute of Geophysics China Seismological Bureau Beijing China 《地震学报(英文版)》1998,11(6):655-667
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5.
A review of the 6 × 6 anisotropic interface ray propagator matrix in Cartesian coordinates and within the framework of the Hamiltonian formalism shows that there is one unique propagator satisfying the symplectic property. This is essential, since the symplecticity furnishes an exact inverse, while an eigenvalue analysis indicates that the propagator may be arbitrarily ill-conditioned. As such, the symplectic interface propagator naturally connects to symplectic ray integration algorithms for smooth media, designed to maintain accuracy. Moreover, several ray invariants for smooth media remain invariant across interfaces. It is straightforward to derive expressions for the interface propagator, both explicit and implicit. Symplecticity is equivalent to the condition that the propagator preserves the eikonal constraint across the interface. The symplectic interface propagator complies with phase matching of the incident and reflected/transmitted ray field, and is therefore in accordance with the earlier derived 4 × 4 matrix in ray-centred coordinates. The symplectic property is related to the symmetry of the second derivative matrix of the reflected/transmitted traveltime field. Thanks to the analytic expression of the symplectic interface propagator, relating interface curvature directly to second derivatives of traveltimes observed at a datum level, numerous applications are available in the area of processing and inversion. 相似文献
6.
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7.
The contravariant components of the wave-propagation metric tensor equal half the second-order partial derivatives of the selected eigenvalue of the Christoffel matrix with respect to the slowness-vector components. The relations of the wave-propagation metric tensor to the curvature matrix and Gaussian curvature of the slowness surface and to the curvature matrix and Gaussian curvature of the ray-velocity surface are demonstrated with the help of ray-centred coordinates. 相似文献
8.
The common ray approximation considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in travel times due to the perturbation from the common reference ray. These travel-time errors can deteriorate the coupling-ray-theory solution at high frequencies. It is thus of principal importance for numerical applications to estimate the errors due to the common ray approximation.We derive the equations for estimating the travel-time errors due to the isotropic and anisotropic common ray approximations of the coupling ray theory. These equations represent the main result of the paper. The derivation is based on the general equations for the second-order perturbations of travel time. The accuracy of the anisotropic common ray approximation can be studied along the isotropic common rays, without tracing the anisotropic common rays.The derived equations are numerically tested in three 1-D models of differing degree of anisotropy. The first-order and second-order perturbation expansions of travel time from the isotropic common rays to anisotropic-ray-theory rays are compared with the anisotropic-ray-theory travel times. The errors due to the isotropic common ray approximation and due to the anisotropic common ray approximation are estimated. In the numerical example, the errors of the anisotropic common ray approximation are considerably smaller than the errors of the isotropic common ray approximation.The effect of the isotropic common ray approximation on the coupling-ray-theory synthetic seismograms is demonstrated graphically. For comparison, the effects of the quasi-isotropic projection of the Green tensor, of the quasi-isotropic approximation of the Christoffel matrix, and of the quasi-isotropic perturbation of travel times on the coupling-ray-theory synthetic seismograms are also shown. The projection of the travel-time errors on the relative errors of the time-harmonic Green tensor is briefly presented. 相似文献
9.
In the computation of paraxial travel times and Gaussian beams, the basic role is played by the second-order derivatives of
the travel-time field at the reference ray. These derivatives can be determined by dynamic ray tracing (DRT) along the ray.
Two basic DRT systems have been broadly used in applications: the DRT system in Cartesian coordinates and the DRT system in
ray-centred coordinates. In this paper, the transformation relations between the second-order derivatives of the travel-time
field in Cartesian and ray-centred coordinates are derived. These transformation relations can be used both in isotropic and
anisotropic media, including computations of complex-valued travel times necessary for the evaluation of Gaussian beams. 相似文献
10.
The coupling ray theory bridges the gap between the isotropic and anisotropic ray theories, and is considerably more accurate than the anisotropic ray theory. The coupling ray theory is often approximated by various quasi-isotropic approximations.Commonly used quasi-isotropic approximations of the coupling ray theory are discussed. The exact analytical solution for the plane S wave, propagating along the axis of spirality in the 1-D anisotropic oblique twisted crystal model, is then numerically compared with the coupling ray theory and its three quasi-isotropic approximations. The three quasi-isotropic approximations of the coupling ray theory are (a) the quasi-isotropic projection of the Green tensor, (b) the quasi-isotropic approximation of the Christoffel matrix, (c) the quasi-isotropic perturbation of travel times. The comparison is carried out numerically in the frequency domain, comparing the exact analytical solution with the results of the 3-D ray tracing and coupling ray theory software. In the oblique twisted crystal model, the three studied quasi-isotropic approximations considerably increase the error of the coupling ray theory. Since these three quasi-isotropic approximations do not noticeably simplify the numerical implementation of the coupling ray theory, they should deffinitely be avoided. The common ray approximations of the coupling ray theory do not affect the plane wave, propagating along the axis of spirality in the 1-D oblique twisted crystal model, and should be studied in more complex models. 相似文献
11.
Martin Tygel Bjørn Ursin Einar Iversen Maarten V. de Hoop 《Geophysical Prospecting》2012,60(2):201-216
Starting from a given time‐migrated zero‐offset data volume and time‐migration velocity, recent literature has shown that it is possible to simultaneously trace image rays in depth and reconstruct the depth‐velocity model along them. This, in turn, allows image‐ray migration, namely to map time‐migrated reflections into depth by tracing the image ray until half of the reflection time is consumed. As known since the 1980s, image‐ray migration can be made more complete if, besides reflection time, also estimates of its first and second derivatives with respect to the time‐migration datum coordinates are available. Such information provides, in addition to the location and dip of the reflectors in depth, also an estimation of their curvature. The expressions explicitly relate geological dip and curvature to first and second derivatives of reflection time with respect to time‐migration datum coordinates. Such quantitative relationships can provide useful constraints for improved construction of reflectors at depth in the presence of uncertainty. Furthermore, the results of image‐ray migration can be used to verify and improve time‐migration algorithms and can therefore be considered complementary to those of normal‐ray migration. So far, image‐ray migration algorithms have been restricted to layered models with isotropic smooth velocities within the layers. Using the methodology of surface‐to‐surface paraxial matrices, we obtain a natural extension to smooth or layered anisotropic media. 相似文献
12.
Seismic amplitude variations with offset contain information about the elastic parameters. Prestack amplitude analysis seeks to extract this information by using the variations of the reflection coefficients as functions of angle of incidence. Normally, an approximate formula is used for the reflection coefficients, and variations with offset of the geometrical spreading and the anelastic attenuation are often ignored. Using angle of incidence as the dependent variable is also computationally inefficient since the data are recorded as a function of offset. Improved approximations have been derived for the elastic reflection and transmission coefficients, the geometrical spreading and the complex travel-time (including anelastic attenuation). For a 1 D medium, these approximations are combined to produce seismic reflection amplitudes (P-wave, S-wave or converted wave) as a Taylor series in the offset coordinate. The coefficients of the Taylor series are computed directly from the parameters of the medium, without using the ray parameter. For primary reflected P-waves, dynamic ray tracing has been used to compute the offset variations of the transmission coefficients, the reflection coefficient, the geometrical spreading and the anelastic attenuation. The offset variation of the transmission factor is small, while the variations in the geometrical spreading, absorption and reflection coefficient are all significant. The new approximations have been used for seismic modelling without ray tracing. The amplitude was approximated by a fourth-order polynomial in offset, the traveltime by the normal square-root approximation and the absorption factor by a similar expression. This approximate modelling was compared to dynamic ray tracing, and the results are the same for zero offset and very close for offsets less than the reflector depth. 相似文献
13.
14.
— A P-wave tomographic method for 3-D complex media (3-D distribution of elastic parameters and curved interfaces) with orthorhombic symmetry is presented in this paper. The technique uses an iterative linear approach to the nonlinear travel-time inversion problem. The hypothesis of orthorhombic anisotropy and 3-D inhomogeneity increases the set of parameters describing the model dramatically compared to the isotropic case. Assuming a Factorized Anisotropic Inhomogeneous (FAI) medium and weak anisotropy, we solve the forward problem by a perturbation approach. We use a finite element approach in which the FAI medium is divided into a set of elements with polynomial elastic parameter distributions. Inside each element, analytical expressions for rays and travel times, valid to first-order, are given for P waves in orthorhombic inhomogeneous media. More complex media can be modeled by introducing interfaces separating FAI media with different elastic properties. Simple formulae are given for the Fréchet derivatives of the travel time with respect to the elastic parameters and the interface parameters. In the weak anisotropy hypothesis the P-wave travel times are sensitive only to a subset of the orthorhombic parameters: the six P-wave elastic parameters and the three Euler angles defining the orientation of the mirror planes of symmetry. The P-wave travel times are inverted by minimizing in terms of least-squares the misfit between the observed and calculated travel times. The solution is approached using a Singular Value Decomposition (SVD). The stability of the inversion is ensured by making use of suitable a priori information and/or by applying regularization. The technique is applied to two synthetic data sets, simulating simple Vertical Seismic Profile (VSP) experiments. The examples demonstrate the necessity of good 3-D ray coverage when considering complex anisotropic symmetry. 相似文献
15.
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
16.
TTI介质是石油地震勘探领域最常用的各向异性介质,快速计算TTI介质射线路径和走时信息有重要的研究意义.TTI介质传统运动学射线追踪方法一般基于任意弹性介质射线方程,利用Bond变换或者四阶张量变换来处理复杂的21个弹性参数,因而非常耗时.实际野外对称轴统一的TTI介质模型,一般可以看成VTI介质模型旋转一定角度获得.为此,本文推导了三维VTI介质射线追踪方程,提出先在本构坐标系中进行VTI介质射线追踪,再通过坐标旋转将射线路径旋转至观测坐标系中,获得TTI介质射线路径.数值模型计算表明该方法高效和精确,较传统方法效率提高了近4倍.在强各向异性等特殊情况下,体波波前面都与理论群速度面一致. 相似文献
17.
电流线追踪电位电阻率层析成像方法初探 总被引：4，自引：3，他引：4
电阻率层析成像技术尽管已有了一些比较好的结果，但从国内外发表的文章可以看出，基本上采用的都是有限元方法，而电阻率层析成像的核心问题也就是雅可比矩阵的求取问题。有限元方法能够很好地实现该问题的求解，但需要的计算机内存及计算时间相当的大，为此我们类比地震学中走时射线追踪技术，开展了电流线追踪电位电阻率层析成像方法研究。 相似文献
18.
Chao-ying Bai Tao Wang Shang-bei Yang Xing-wang Li Guo-jiao Huang 《Journal of Seismology》2016,20(2):475-494
Traditional traveltime inversion for anisotropic medium is, in general, based on a “weak” assumption in the anisotropic property, which simplifies both the forward part (ray tracing is performed once only) and the inversion part (a linear inversion solver is possible). But for some real applications, a general (both “weak” and “strong”) anisotropic medium should be considered. In such cases, one has to develop a ray tracing algorithm to handle with the general (including “strong”) anisotropic medium and also to design a non-linear inversion solver for later tomography. Meanwhile, it is constructive to investigate how much the tomographic resolution can be improved by introducing the later arrivals. For this motivation, we incorporated our newly developed ray tracing algorithm (multistage irregular shortest-path method) for general anisotropic media with a non-linear inversion solver (a damped minimum norm, constrained least squares problem with a conjugate gradient approach) to formulate a non-linear inversion solver for anisotropic medium. This anisotropic traveltime inversion procedure is able to combine the later (reflected) arrival times. Both 2-D/3-D synthetic inversion experiments and comparison tests show that (1) the proposed anisotropic traveltime inversion scheme is able to recover the high contrast anomalies and (2) it is possible to improve the tomographic resolution by introducing the later (reflected) arrivals, but not as expected in the isotropic medium, because the different velocity (qP, qSV and qSH) sensitivities (or derivatives) respective to the different elastic parameters are not the same but are also dependent on the inclination angle. 相似文献
19.
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. 相似文献
20.
Emil Blias 《Geophysical Prospecting》2013,61(3):574-581
I introduce a new explicit form of vertical seismic profile (VSP) traveltime approximation for a 2D model with non‐horizontal boundaries and anisotropic layers. The goal of the new approximation is to dramatically decrease the cost of time calculations by reducing the number of calculated rays in a complex multi‐layered anisotropic model for VSP walkaway data with many sources. This traveltime approximation extends the generalized moveout approximation proposed by Fomel and Stovas. The new equation is designed for borehole seismic geometry where the receivers are placed in a well while the sources are on the surface. For this, the time‐offset function is presented as a sum of odd and even functions. Coefficients in this approximation are determined by calculating the traveltime and its first‐ and second‐order derivatives at five specific rays. Once these coefficients are determined, the traveltimes at other rays are calculated by this approximation. Testing this new approximation on a 2D anisotropic model with dipping boundaries shows its very high accuracy for offsets three times the reflector depths. The new approximation can be used for 2D anisotropic models with tilted symmetry axes for practical VSP geometry calculations. The new explicit approximation eliminates the need of massive ray tracing in a complicated velocity model for multi‐source VSP surveys. This method is designed not for NMO correction but for replacing conventional ray tracing for time calculations. 相似文献