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1.
The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also 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 applied.
The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation.
We derive the equations for estimating the travel-time errors due to the anisotropic-common-ray (and also isotropic-common-ray)
approximation of the coupling ray theory. The errors of the common-ray approximations are calculated along the anisotropic
common rays in smooth velocity models without interfaces. The derivation is based on the general equations for the second-order
perturbations of travel time. 相似文献
2.
The exact analytical solution for the plane S-wave, propagating along the axis of spirality in the simple 1-D anisotropic simplified twisted crystal model, is compared with four different approximate ray-theory solutions. The four different ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. The comparison is carried out numerically, by evaluating both the exact analytical solution and the analytical solutions of the equations of the four ray methods. The comparison simultaneously demonstrates the limits of applicability of the isotropic and anisotropic ray theories, and the superior accuracy of the coupling ray theory over a broad frequency range. The comparison also shows the possible inaccuracy due to the quasi-isotropic perturbation of travel times in the equations of the coupling ray theory. The coupling ray theory thus should definitely be preferred to the isotropic and anisotropic ray theories, but the quasi-isotropic perturbation of travel times should be avoided. Although the simplified twisted crystal model is designed for testing purposes and has no direct relation to geological structures, the wave-propagation phenomena important in the comparison are similar to those in the models of the geological structures.In additional numerical tests, the exact analytical solution is numerically compared with the finite-difference numerical results, and the analytical solutions of the equations of different ray methods are compared with the corresponding numerical results of 3-D ray-tracing programs developed by the authors of the paper. 相似文献
3.
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
4.
Common-ray tracing and dynamic ray tracing for S waves in a smooth elastic anisotropic medium 总被引：1，自引：0，他引：1
L. Klimeš 《Studia Geophysica et Geodaetica》2006,50(3):449-461
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. 相似文献
5.
The coupling ray theory is usually applied to anisotropic common reference rays, but it is more accurate if it is applied to reference rays which are closer to the actual wave paths. If we know that a medium is close to uniaxial (transversely isotropic), it may be advantageous to trace reference rays which resemble the SH–wave and SV–wave rays. This paper is devoted to defining and tracing these SH and SV reference rays of elastic S waves in a heterogeneous generally anisotropic medium which is approximately uniaxial (approximately transversely isotropic), and to the corresponding equations of geodesic deviation (dynamic ray tracing). All presented equations are simultaneously applicable to ordinary and extraordinary reference rays of electromagnetic waves in a generally bianisotropic medium which is approximately uniaxially anisotropic. The improvement of the coupling–ray–theory seismograms calculated along the proposed SH and SV reference rays, compared to the coupling–ray–theory seismograms calculated along the anisotropic common reference rays, has already been numerically demonstrated by the authors in four approximately uniaxial velocity models. 相似文献
6.
L. Klimeš 《Studia Geophysica et Geodaetica》2006,50(3):417-430
Explicit equations for the spatial derivatives and perturbation derivatives of amplitude in both isotropic and anisotropic
media are derived. The spatial and perturbation derivatives of the logarithm of amplitude can be calculated by numerical quadratures
along the rays.
The spatial derivatives of amplitude may be useful in calculating the higher-order terms in the ray series, in calculating
the higher-order amplitude coefficients of Gaussian beams, in estimating the accuracy of zero-order approximations of both
the ray method and Gaussian beams, in estimating the accuracy of the paraxial approximation of individual Gaussian beams,
or in estimating the accuracy of the asymptotic summation of paraxial Gaussian beams. The perturbation derivatives of amplitude
may be useful in perturbation expansions from elastic to viscoelastic media and in estimating the accuracy of the common-ray
approximations of the amplitude in the coupling ray theory. 相似文献
7.
L. Klimeš 《Studia Geophysica et Geodaetica》2006,50(3):431-447
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. 相似文献
8.
The coupling–ray–theory tensor Green function for electromagnetic waves or elastic S waves is frequency dependent, and is usually calculated for many frequencies. This frequency dependence represents no problem in calculating the Green function, but may pose a significant challenge in storing the Green function at the nodes of dense grids, typical for applications such as the Born approximation or non–linear source determination. Storing the Green function at the nodes of dense grids for too many frequencies may be impractical or even unrealistic. We have already proposed the approximation of the coupling–ray–theory tensor Green function, in the vicinity of a given prevailing frequency, by two coupling–ray–theory dyadic Green functions described by their coupling–ray–theory travel times and their coupling–ray–theory amplitudes. The above mentioned prevailing–frequency approximation of the coupling ray theory enables us to interpolate the coupling–ray–theory dyadic Green functions within ray cells, and to calculate them at the nodes of dense grids. For the interpolation within ray cells, we need to separate the pairs of prevailing–frequency coupling–ray–theory dyadic Green functions so that both the first Green function and the second Green function are continuous along rays and within ray cells. We describe the current progress in this field and outline the basic algorithms. The proposed method is equally applicable to both electromagnetic waves and elastic S waves. We demonstrate the preliminary numerical results using the coupling–ray–theory travel times of elastic S waves. 相似文献
9.
准各向同性(QI)近似可用于弱各向异性介质的正演模拟.本文通过运用QI方法的零阶和一阶近似,计算了VTI介质模型的地震记录.得出的地震记录与标准各向同性射线理论(IRT)和基于伪谱法的三维地震正演模拟得出的地震记录作了比较,可以认为是精确的合成地震记录. 相似文献
10.
The partial derivatives of travel time with respect to model parameters are referred to as perturbations. Explicit equations for the second-order and higher-order perturbations of travel time in both isotropic and anisotropic media are derived. The perturbations of travel time and its spatial derivatives can be calculated by simple numerical quadratures along rays. 相似文献
11.
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
12.
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. 相似文献
13.
The behaviour of the actual polarization of an electromagnetic wave or elastic S–wave is described by the coupling ray theory, which represents the generalization of both the zero–order isotropic and anisotropic ray theories and provides continuous transition between them. The coupling ray theory is usually applied to anisotropic common reference rays, but it is more accurate if it is applied to reference rays which are closer to the actual wave paths. In a generally anisotropic or bianisotropic medium, the actual wave paths may be approximated by the anisotropic–ray–theory rays if these rays behave reasonably. In an approximately uniaxial (approximately transversely isotropic) anisotropic medium, we can define and trace the SH (ordinary) and SV (extraordinary) reference rays, and use them as reference rays for the prevailing–frequency approximation of the coupling ray theory. In both cases, i.e. for the anisotropic–ray–theory rays or the SH and SV reference rays, we have two sets of reference rays. We thus obtain two arrivals along each reference ray of the first set and have to select the correct one. Analogously, we obtain two arrivals along each reference ray of the second set and have to select the correct one. In this paper, we suggest the way of selecting the correct arrivals. We then demonstrate the accuracy of the resulting prevailing–frequency approximation of the coupling ray theory using elastic S waves along the SH and SV reference rays in four different approximately uniaxial (approximately transversely isotropic) velocity models. 相似文献
14.
We describe the behaviour of the anisotropic–ray–theory S–wave rays in a velocity model with a split intersection singularity. The anisotropic–ray–theory S–wave rays crossing the split intersection singularity are smoothly but very sharply bent. While the initial–value rays can be safely traced by solving Hamilton’s equations of rays, it is often impossible to determine the coefficients of the equations of geodesic deviation (paraxial ray equations, dynamic ray tracing equations) and to solve them numerically. As a result, we often know neither the matrix of geometrical spreading, nor the phase shift due to caustics. We demonstrate the abrupt changes of the geometrical spreading and wavefront curvature of the fast anisotropic–ray–theory S wave. We also demonstrate the formation of caustics and wavefront triplication of the slow anisotropic–ray–theory S wave.Since the actual S waves propagate approximately along the SH and SV reference rays in this velocity model, we compare the anisotropic–ray–theory S–wave rays with the SH and SV reference rays. Since the coupling ray theory is usually calculated along the anisotropic common S–wave rays, we also compare the anisotropic common S–wave rays with the SH and SV reference rays. 相似文献
15.
Analytical expressions for the exact 2 × 2 one-way propagator matrix of a plane S wave, propagating along the axis of spirality in the simple 1-D anisotropic simplified twisted crystal model, are presented. The analytical equations are useful in testing the applicability and accuracy of various approximate wavefield modelling methods, especially of the coupling ray theory and of its various quasi-isotropic approximations and various numerical implementations.In addition to the exact analytical solution of the elastodynamic equation in the simplified twisted crystal model, the analytical solutions of the equations of the four ray methods are given. The ray methods are (a) the coupling ray theory, (b) the coupling ray theory with the quasi-isotropic perturbation of travel times, (c) the anisotropic ray theory, (d) the isotropic ray theory. These four approximate solutions of the elastodynamic equation are roughly compared with the exact solution. Both the exact analytical solution and the analytical ray-theory solutions in the simplified twisted crystal model are also helpful in debugging computer codes for various approximate wavefield modelling methods, especially for the coupling ray theory. 相似文献
16.
In this paper, ray theoretical amplitudes and travel times are calculated in slightly perturbed velocity models using perturbation analysis. Also, test inversions using travel time and amplitude are computed. The pertubation method is tested using a 3-D velocity model for NORSAR having velocity variations up to 8.0 percent. The perturbed amplitudes are found to be in excellent agreement with the calculated ray amplitudes. Velocity inversions based on travel time and amplitude are next investigated. Perturbation analysis using linearized ray equations is efficiently used to compute amplitude derivatives with respect to model parameters. The trial linearized inversions use smaller velocity variations of 1.7 percent to avoid possible effects due to ray shift, even though the perturbation analysis is valid for larger variations. The trial 2-D inversion results show that linearized amplitude inversions are complementary and not redundant to travel time inversions, even in smoothly varying models. 相似文献
17.
A. B. Druzhinin 《Pure and Applied Geophysics》1996,148(3-4):637-683
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. 相似文献
18.
Luděk Klimeš 《Pure and Applied Geophysics》1996,148(3-4):539-563
A new computational scheme for calculating the first-arrival travel times on a rectangular grid of points is proposed. The new proposed method is of second-order accuracy. This means that the error of the calculated travel time is proportional to the second power of the grid spacing. The method should be sufficiently accurate for all applications in smooth seismic models. On the other hand, the method is not, in its present form, proposed for models with structural interfaces which make the method unstable and generate travel-time errors of the first order. Equations are also presented for the appropriate evaluation of the errors of calculated travel times to check their accuracy, and the proposed method is compared with other numerical methods. The method is developed, described and demonstrated in 2-D, but may also be extended to 3-D models and to general models with structural interfaces. 相似文献
19.
A Fast Sweeping Scheme for Calculating P Wave First-Arrival Travel Times in Transversely Isotropic Media with an Irregular Surface 总被引：2，自引：0，他引：2
Seismic wave propagation shows anisotropic characteristics in many sedimentary rocks. Modern seismic exploration in mountainous areas makes it important to calculate P wave travel times in anisotropic media with irregular surfaces. The challenges in this context are mainly from two aspects. First is how to tackle the irregular surface in a Cartesian coordinate system, and the other lies in solving the anisotropic eikonal equation. Since for anisotropic media the ray (group) velocity direction is not the same as the direction of the travel-time gradient, the travel-time gradient no longer serves as an indicator of the group velocity direction in extrapolating the travel-time field. Recently, a topography-dependent eikonal equation formulated in a curvilinear coordinate system has been established, which is effective for calculating first-arrival travel times in an isotropic model with an irregular surface. Here, we extend the above equation from isotropy to transverse isotropy (TI) by formulating a topography-dependent eikonal equation in TI media in the curvilinear coordinate system, and then use a fast sweeping scheme to solve the topography-dependent anisotropic eikonal equation in the curvilinear coordinate system. Numerical experiments demonstrate the feasibility and accuracy of the scheme in calculating P wave travel times in TI models with an irregular surface. 相似文献
20.
Libor Šachl 《Studia Geophysica et Geodaetica》2013,57(1):84-102
The first-order Born approximation is a weak scattering perturbation method which is a powerful tool. The combination of the Born approximation and the ray theory enables to extend the applicability of the ray theory in terms of the required smoothness of the model and ensures faster computations than with, e.g., the finite difference method. We are motivated to describe and explain the effects of the numerical discretization of the Born integral on the resulting seismograms. We focus on forward modelling and study the cases in which perturbation from the background model contains the interface. We restrict ourselves to isotropic models that contain two homogeneous layers. We compare the 2D and 3D ray-based Bornapproximation seismograms with the ray-theory seismograms. The Born seismograms are computed using a grid of finite extent. We anticipate that the computational grid should contain an appropriate number of gridpoints, otherwise the seismogram would be inaccurate. We also anticipate that the limited size of the computational grid can cause problems. We demonstrate numerically that an incorrect grid can produce significant errors in the amplitude of the wave, or it can shift the seismogram in time. Moreover, the grid boundaries work as interfaces, where spurious waves can be generated. We also attempt to explain these phenomena theoretically. We give and test the options of removing the spurious waves. We show that it is possible to compute the Born approximation in a sparser grid, if we use elastic parameters averaged from some dense grid. 相似文献